Research Article The Spiral Coaxial Cabledownloads.hindawi.com/archive/2015/630131.pdf · A spiral...

Research ArticleThe Spiral Coaxial Cable

I M Fabbri

Department of Physics University of Milan Via Celoria 16 20133 Milan Italy

Correspondence should be addressed to I M Fabbri italomariofabbricrfmit

Received 18 September 2014 Revised 10 January 2015 Accepted 12 January 2015

Academic Editor Kamya Yekeh Yazdandoost

Copyright copy 2015 I M Fabbri This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new concept of metal spiral coaxial cable is introduced The solution to Maxwellrsquos equations for the fundamental propagatingTEM eigenmode using a generalization of the Schwarz-Christoffel conformal mapping of the spiral transverse section is providedtogether with the analysis of the impedances and the Poynting vector of the line The new cable may find application as a mediumfor telecommunication and networking or in the sector of the Microwave Photonics A spiral plasmonic coaxial cable could beused to propagate subwavelength surface plasmon polaritons at optical frequencies Furthermore according to the present modelthe myelinated nerves can be considered natural examples of spiral coaxial cables This study suggests that a malformation of thePeters angle which determines the power of the neural signal in the TEM mode causes higherlower power to be transmitted inthe neural networks with respect to the natural level The formulas of the myelin sheaths thickness the diameter of the axon andthe spiral 119892 factor of the lipid bilayers which are mathematically related to the impedances of the spiral coaxial line can make iteasier to analyze the neural line impedance mismatches and the signal disconnections typical of the neurodegenerative diseases

1 Introduction

The coaxial cable invented by Heaviside [1] is a transmissionline composed of an inner conductor surrounded by aninsulating layer and an outer conducting shield The innerand the outer conductors share the same geometrical axis

Nowadays there exist several types of transmission linesthe coaxial cables the hollow waveguides (rectangular cir-cular elliptical and parallel plates [2 3]) the two wires [34] and the channel waveguides (buried strip-loaded ridgerib diffused and graded-dielectric index [5]) The two-wiretransmission line used in conventional circuits is inefficientfor transferring electromagnetic energy at microwave fre-quencies because the fields are not confined in all directionsthe energy escapes by radiation and it has large copper lossesdue to its relatively small surface area

On the other hand the larger surface area of the boundarymakes in general the waveguides more efficient with respectto the coaxials on reducing the copper losses

Dielectric losses in two wires and coaxial lines caused bythe heating of the insulation between the conductors are alsolower in waveguides The insulation behaves as the dielectricof a capacitor and the breakdown of the insulation betweenthe conductors of a transmission line is more frequently aproblem than is the dielectric loss in practical applications [3]

Waveguides are also subject to the dielectric breakdowncaused by the stationary voltage spikes at the nodes of thestanding waves

In spite of the dielectric in waveguides is air it causesarcing which decreases the efficiency of energy transfer andcan severely damage the waveguide

The spiral coaxial cable (SCC) discussed in this paper hasmany advantages with respect to the other types of trans-mission lines first of all the elm energy can be distributedefficiently over a larger area reducing all the undesiredaforementioned effects

The power handling versus frequency is consequentlyhigher for the metal spiral coaxial line (MSCC) with respectto the other lines

The Microwave Photonics (MWP) [6] a new disciplinethat joins together the radio-frequency engineering and opto-electronics represents the future in many civil and defenseapplications like speed of the digital signal processors cabletelevision and optical signal processing

The MSCC could become one of the key objects in thefield of MWP for its characteristics in terms of both powerhandling and energy transfer efficiency

Recently it has also been demonstrated experimentallythat metal coaxial waveguide nanostructures perform at

Hindawi Publishing CorporationInternational Journal of Microwave Science and TechnologyVolume 2015 Article ID 630131 18 pageshttpdxdoiorg1011552015630131

2 International Journal of Microwave Science and Technology

optical frequencies [7] opening up new research pursuitsunexpected in the area of nanoscale waveguiding fieldenhancement imaging with coaxial cavities and negative-index metamaterials [8]

Several different plasmonic waveguiding structureshave been proposed such as metallic nanowires [9 10]metal-dielectric-metal (MDM) structures [11] and metallicnanoparticle arrays [12] for achieving compact integratedphotonic devices

Most of these structures support a highly confined modenear the surface plasmon frequency [11] This study couldbecome the reference point to introduce the MSCC as avalid alternative to guide subwavelength surface plasmonpolaritons (SPPs)

The extraordinary transmission of light through an arrayof subwavelength apertures enhancement which arises fromthe coupling of the incident light with the SPPs through thesurface grating in metal film [13] could result particularlyefficiently on the spiral metal dielectric interface with peri-odic holes

High sensitivity spiral biosensors and spiral photonicintegrated circuits based on nonlinear surface plasmonpolariton optics [14 15] may be implemented

The aimof this pioneeringwork on the SCC is to representan initial landmark in the continuously growing sector of themicrowave research

The popularity of Video on Demand (VoD) and Overthe Top Technology (OTT) services to access high definitionvideos over home interconnected devices of the hybrid fibre-coax (HFC) networks is driving the research toward moreefficient and cost-effective cables Particularly the develop-ment of Converged Cable Access Platforms (CCAP) thatcombine video and data transmission supporting simultane-ous network access of multiple users over a single coaxialcable is flourishing and sustaining the demand for new highspeed transmission media

In a metallic guide the reflection mechanism responsiblefor confining the energy is due to the reflection from theconductors at the boundary [16] whose geometry is strictlyrelated to the propagating modes

Coaxial cables were designed to propagate high frequencyradio signals The principal constraints on performance of acoaxial are attenuation thermal noise and passive intermod-ulation noise (PIM)

In RF applications the wave propagates essentially in thefundamental transverse electric magnetic (TEM) mode thatis the electric and magnetic fields are both perpendicular tothe direction of propagation

In the ideal case the conductors can be considered to haveinfinite conductivity and the TEM eigenmode is the basicpropagating wave (see [17] page 110) along the transmissionline

Practical lines have finite conductivity and this results ina perturbation or change of the TEMmode (see [18] page 119)

Above the cutoff frequency transverse electric (TE) ortransverse magnetic (TM) modes [19] can also propagatewith different velocities within a practical cylindrical coaxinterferingwith each other producing distortion of the signal

The frequency of operation for a specific outer conductorsize is then limited by the highest usable cutoff frequencybefore undesirable modes of propagation occur

In order to prevent higher order modes from beinglaunched the radiuses of the coaxial conductors must bereduced diminishing the amount of power that can betransmitted

On the other hand at high frequencies it is impossible tomake the cylindrical coaxial line in the small size necessaryto propagate the TEMmode alone

The research described in this paper demonstrates thepropagation on the fundamental TEM wave along the idealMSCC

Since the mode of transmission on an ideal line is theTEM wave the relations for input impedance reflectioncoefficient return loss (RL) standing wave ratio (SWR) andso forth given afterward in the next sections are applicablein general to the spiral transmission lines (see [18] Chapter 3)

The metal double spiral coaxial cable or MDSCC result-ing from the superposition of two spiral conductors thatshare the same geometrical axis can be made multi-turnThe amount of heat generated by the losses for heating canbe distributed over a larger area and this would lower thetemperature and raise the reliability of the line

In fact operation at higher temperatures results in areduction in the life expectancy and reliability of the trans-mission line relative to the lower temperature performance

Applications like nanoscale optical components for inte-gration on semiconductor chips could benefit from thesecharacteristics of the MSCC

Where signal integrity is important coaxial cables areneeded to be shielded against radio frequency noise (RFnoise) The multiturn MDSCC is naturally shielded becausethe highest part of the elm energy can be distributed on theinner part of the cable which protects small signals frominterference due to external electric fields

A new class of spiral passive components computer-aided engineering (CAE) tools as well as electromagnetic(EM) simulators is required before new high-frequencyspiral RFmicrowave circuits will be implemented

The spiral geometry occurs widely in nature exampleslike the spiral galaxies are found at the universe level while themyelin bundles are common in the microcosm of the neuroncells

Recently a new spiral optical fibre has been proposedboth in the fundamental mode [20] and in the higher ordermodes [21] operation

Spirals are also of extreme interest to the field of the newmetamaterials and invisible cloaking [22]

Myelinated nerve fibers are micro-spiral coaxial cables(119892 ≪ 1) whose electric behaviour is still today described byneurophysiologists using W Thomsonrsquos (later known as LordKelvin) cable formula [23] of the 1860s which determinesthe velocity of the signal propagating in saltatory conduction[23 24]

Cable theory in neurobiology has a long history havingfirst been applied to neurons in 1863 byCMatteucci [25] whodiscovered that if a constant current flows through a portionof a platinum wire covered with a sheath saturated with fluid

International Journal of Microwave Science and Technology 3

extra-polar current can be led off which corresponds to theelectrotonic current of nerves

Since the 1950sndash60s myelinated nerves have been recog-nized to have a spiral structure and to behave like a high losscoaxial cable [26 27] with negligible inductance

The mathematical model presented in this paper can beused to refine the elm theory of the myelinated nerves bytaking into account their spiral geometry

In a coaxial guide the determination of the electromag-netic fields within any region of the guide is dependent upononersquos ability to explicitly solve the Maxwell field equations inan appropriate coordinate system [28]

Let us considerMaxwellrsquos equations

nabla times = minus119895120596120583

nabla times = 119895120596120598 119864

nabla sdot = 0

nabla sdot = 0

(1)

where the time variation of the fields is assumed to beexp(119895120596119905)

In view of the nature of the boundary surface it isconvenient to separate these field equations into componentsparallel and transverse to the waveguide 119911-axis

This is achieved by scalar and vector multiplication of (1)with 119890

119911 a unit vector in the 119911 direction thus obtaining

nablaperpsdot (119890

119911times

perp) = minus119895120596120583119867

119911


119911times

perp) = 119895120596120598119864

119911

(2)

nablaperp119864

119911minus120597119864

perp

120597119911= minus119895120596120583 (119890

119911times 119867

perp)

nablaperp119867

119911minus120597119867

perp

120597119911= 119895120596120598 (119890

119911times 119864

perp)

(3)

Since the transmission line description of the electromagneticfield within uniform guides is independent of the particularform of the coordinate system employed to describe the crosssection no reference to cross-sectional coordinates is madeon deriving the telegrapherrsquos equation [28 29]

Substituting (2) into (3) we obtain

120597119864perp

120597119911= minus119895119896120585 ( 120598 +

1

1198962nablaperpnablaperp) sdot (

perptimes 119890

119911)

120597119867perp

120597119911= minus119895119896120578 ( 120598 +

1

1198962nablaperpnablaperp) sdot (119890

119911times

perp)

(4)

Vector notation is employed with the following meanings forthe symbols

119864perp

= 119864perp(119909 119910 119911) = the rms electric field intensity

transverse to the 119911-axis119867

perp= 119867

perp(119909 119910 119911) = the rms magnetic field intensity

transverse to the 119911-axis

120578 = intrinsic impedance of the medium 1120578 = radic120583120598119896 = 120596radic120583120598 = 2120587120582 = propagation constant inmediumor the wavenumber (see [28] page 3)nablaperp

= gradient operator transverse to 119911-axis = nabla minus

119890119911(120597120597119911)120598 = unit dyadic defined such that 120598 sdot = sdot 120598 =

Equations (4) and (2) which are fully equivalent to theMaxwell equations make evident the separate dependenceof the field on the cross-sectional coordinates and on thelongitudinal coordinate 119911 The cross-sectional dependencemay be integrated out of (4) by means of a suitable set ofvector orthogonal functions provided they satisfy appropriateconditions on the boundary curve or curves 119904 of the crosssection

Such vector functions are known to be of two types theE-mode functions 1198901015840

119894defined by

1198901015840

119894= minusnabla

perpΦ

119894

ℎ1015840

119894= 119890

119911times 119890

1015840

119894

(5)

where

nabla2

perpΦ

119894+ 119896

10158402

119888119894Φ

119894= 0

Φ119894= 0 on 119904 if 1198961015840

119888119894= 0

120597Φ119894

120597119904= 0 on 119904 if 1198961015840

119888119894= 0

(6)

and the H-mode functions 11989010158401015840119894defined by

11989010158401015840

119894= 119890

119911times nabla

perpΨ

119894

ℎ10158401015840

119894= 119890

119911times 119890

10158401015840

119894

(7)

where

nabla2

perpΨ

119894+ 119896

101584010158402

119888119894Ψ

119894= 0

120597Ψ119894

120597119899= 0 on 119904

(8)

where 119894 denotes a double index and 119899 is the outward normalto 119904 in the cross-section plane

The constants 11989610158401015840

119888119894and 119896

1015840

119888119894are defined as the cutoff wave

numbers or eigenvalues associated with the guide crosssection

The functions 119890119894possess the vector orthogonality proper-

ties

∬1198901015840

119894sdot 119890

1015840

119895119889119878

perp= ∬119890

10158401015840

119894sdot 119890

10158401015840

119895119889119878

perp=

1 for 119894 = 119895

0 for 119894 = 119895

∬1198901015840

119894sdot 119890

10158401015840

119895119889119878

perp= 0

(9)

with the integration extended over the entire guide crosssection with surface 119878

perp


The total average power flow along the guide in the 119911direction is

119875119911=1

2Re(∬119864

perptimes 119867

lowast

perpsdot 119890

119911119889119878

perp) (10)

where all quantities are rms and the asterisk denotes thecomplex conjugate

In TEMmodes both 119864119911and119867

119911vanish and the fields are

fully transverse Their cutoff condition 1198962

119888= 0 or 120596 = 120573119888

(where 119888 is the speed of the light) is equivalent to the followingrelation [28]

perp=1

120578119890119911times

perp (11)

between the electric andmagnetic transverse fields where 120578 =radic120583120598 is the medium impedance so that 120578119888 = 120583 and 120578119888 = 1120598

The electric field perp

is determined from the rest ofMaxwellrsquos equations which read

nablaperptimes

perp= 0

nabla sdot perp= 0

(12)

These are recognized as the field equations of an equivalenttwo-dimensional electrostatic problem

Once the electrostatic solution perpis found the magnetic

field is constructed from (11)Because of the relationship between

perpand

perp the

Poynting vector 119878119911will be

119878119911=1

2Re (

perptimes

lowast

perp) sdot 119890

119911=1

120578

10038161003816100381610038161003816

perp

10038161003816100381610038161003816

2

= 12057810038161003816100381610038161003816

perp

10038161003816100381610038161003816

2

(13)

2 The Spiral Differential Geometry

For the MSCC structures it is difficult to construct solutionsfor Laplacersquos equation with polar or cartesian coordinates

The conformal mapping technique is a powerful methodfor solving two-dimensional potential problems andmappingthe boundaries into a simpler configuration for which solu-tions to Laplacersquos equation are easily found [17 18]

For the specific purposes of the MSCC the followingspiral coordinates based on a generalization of the Schwarz-Christoffelmapping (see appendix) are introduced

119909 = 119890(120575119892minus119892120579) cos (120575 + 120579)

119910 = 119890(120575119892minus119892120579) sin (120575 + 120579)

119911 = 119911

(14)

where 120579 120575 represent the spiral coordinates and 119892 gt 0

is a constant which characterizes the transformation (seeappendix)

As it can be seen in Figure 1 the equation 120575 = constrepresented by a vertical line in the 120575-120579 plane correspondsto a logarithmic spiral into the 119909-119910 plane and a constantcoordinate line of the spiral mapping

Observing (14) it appears clear that for 119892120579 minus 120575lowast

119892 rarr 0where 120575lowast is a constant the curve in the 119909-119910 plane locallyreduces (for |120579| ≪ 1 119892 ≪ 1) to an Archimedean spiral

The region between the two coaxial spirals maps into theregion inside the polygon bounded by the coordinate-lines120579 = 120579

1 120579 = 120579

2and 120575 = 120575

1 120575 = 120575

2 120575 = 120575

1minus 2120587119892

2

(1 + 1198922

) [18](see Figure 1(b))

It is also worth to observe that if 1205752minus120575

1= 2119902120587119892

2

(1+1198922

)119902 isin Z the two spirals 120575 = 120575

1 120575 = 120575

2are identical apart from

a shift of Δ120579119904= 2119902120587(1 + 119892

2

)We then require |120575

2minus120575

1| lt 2120587119892

2

(1+1198922

) in order to avoidcyclic spirals with 120575 gt 120575

2in the middle of the two with 120575 = 120575

1

and 120575 = 1205752

The differential one form of the spiral transformation ordual basis results in

119889119909 =120597119909

120597120575119889120575 +

120597119909

120597120579119889120579 +

120597119909

120597119911119889119911

119889119910 =120597119910

120597120575119889120575 +

120597119910

120597120579119889120579 +

120597119910

120597119911119889119911

119889119911 = 119889119911

(15)

The arc length 119889ℓ is given by

119889ℓ2

= 1198891199092

+ 1198891199102

+ 1198891199112

= 119892120575120575119889120575

2

+ 119892120579120579119889120579

2

+ 119892119911119911119889119911

2

(16)

where

ℎ2

120575= 119892

120575120575= 119890

(2(120575119892)minus2119892120579)

(1 +1

1198922)

ℎ2

120579= 119892

120579120579= 119890

(2(120575119892)minus2119892120579)

(1 + 1198922

)

ℎ2

119911= 119892

119911119911= 1 119892

120575120579= 119892

120575119911= 119892

120579119911= 0

(17)

are the components of themetric tensor and Lame coefficientsThe infinitesimal volume element is given by

119889119881 = 119869 119889120575 119889120579 119889119911 = 119890(2(120575119892)minus2119892120579)

1 + 1198922

119892119889120575 119889120579 119889119911 (18)

where the 119869 is the Jacobian of the spiral transformationLet us now define the spiral natural basis vectors 119890

120575 119890

120579 119890

119911

119890120575=120597

120597120575

= 119890(120575119892minus119892120579)

(1

119892cos (120575 + 120579) minus sin (120575 + 120579)) 119890

119909

+ 119890(120575119892minus119892120579)

(1

119892sin (120575 + 120579) + cos (120575 + 120579)) 119890

119910


0

Conductor 1

Conductor 2

Region I

Region II

or

y

1205791

120579

1205792

1205751

1205752

x

1205751 minus2120587g2

1 + g2

∙

∙SI

SII

(a)

0

Region II

0

1205791

120579

1205792

120575

1205751205752 1205751

Φ

V0

∙

∙

polygonSchwarz-Christoffel

Region I

Con

duct

or 1

Con

duct

or 2

Con

duct

or 1

1205792 minus2120587

1 + g2

1205791 minus2120587g2

1 + g2

SI

SII

(b)

y

z

x

rarrdS120575z

rarrdS120579z

rarrdS120575120579

(c)

Figure 1 (a) The spiral coordinates lines (b) The mapping of the spiral coaxial section and the scalar potential Φ(120575 120579) solution to theequivalent Laplacersquos equation (c) The differential spiral surfaces

119890120579=120597

120597120579

= 119890(120575119892minus119892120579)

(minus119892 cos (120575 + 120579) minus sin (120575 + 120579)) 119890119909

+ 119890(120575119892minus119892120579)

(minus119892 sin (120575 + 120579) + cos (120575 + 120579)) 119890119910

119890119911=120597

120597119911= 119890

119911

(19)

in terms of the cartesian basis vectors 119890119909 119890

119910 119890

119911

The infinitesimal surface elements transverse and longi-tudinal along the 119911-axis (see Figure 1) are given by

10038171003817100381710038171003817119889 119878

120575120579

10038171003817100381710038171003817= 119889119878

perp=

10038171003817100381710038171003817100381710038171003817

120597

120597120575times120597

120597120579

10038171003817100381710038171003817100381710038171003817119889120575 119889120579

= 119890(2120575119892minus2119892120579)

(1

119892+ 119892)119889120575 119889120579

10038171003817100381710038171003817119889 119878

120579119911

10038171003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

120597

120597120579times120597

120597119911

10038171003817100381710038171003817100381710038171003817119889120579 119889119911 = 119890

(120575119892minus119892120579)radic1 + 1198922119889119911 119889120579

10038171003817100381710038171003817119889 119878

120575119911

10038171003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

120597

120597120575times120597

120597119911

10038171003817100381710038171003817100381710038171003817119889120575 119889119911 = 119890

(120575119892minus119892120579)

radic1 + 1198922

119892119889119911 119889120575

(20)

We then define the natural unitary spiral basis vectors

119890120575=119890120575

ℎ120575

119890120579=119890120579

ℎ120579

119890119911=119890119911

ℎ119911

(21)


The usual unitary relations of orthogonality hold that is

119890120575= 119890

120579times 119890

119911 119890

120579= 119890

119911times 119890

120575 119890

119911= 119890

120575times 119890

120579

119890120575sdot 119890

120579= 0 = 119890

120575sdot 119890

119911= 119890

120579sdot 119890

119911= 0

(22)

In Figure 1 a vertical segment in the 120579-120575 plane corre-sponds to a piece of spiral in the 119909-119910 plane the circle is aparticular spiral defined by the relation 120579 = 120575119892

2

minus 119870119892The radius vector in spiral coordinates becomes

119903 =119890(120575119892minus119892120579)

radic1 + 1198922

(119890120575minus 119892119890

120579) + 119911119890

119911 (23)

Logarithmic spirals are analogous to the straight lineTheorthogonal spiral is obtained exactly as for the straight linesby replacing the 119892 factor (which is analogous to the slope forthe straight lines) with 119892

perp= minus1119892

It is also possible to define the orthogonal spiral coordi-nate mapping as follows

119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)

119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)

119911 = 119911

(24)

3 The TEM Mode for the Spiral Waveguide

Let us consider two separate perfectly conducting spiralconductors with uniform cross section infinitely long andoriented parallel to the 119911-axis for such a structure a TEMmode of propagation is possible [18]

Laplacersquos equation of this line transformed by means of aspiral conformalmapping [17 18] which is the generalizationof the polar conformal mapping (see appendix) is

119890minus2(120575119892)+2119892120579

1 + 1198922[119892

21205972

Φ

1205971205752+1205972

Φ

1205971205792] = 0 (25)

where the scalar electric potentialΦ(120575 120579) represents the solu-tion to the equivalent electrostatic problem of the transverseelectromagnetic TEMmode propagating along the MSCC

This equation has to be solved into two separate indepen-dent open regions I II where the solutionmust be continuouswith derivatives

Φ isin C(0)

[[1205751minus

21205871198922

1 + 1198922 120575

2] times (minusinfininfin)]

capC(2)

[[1205751minus

21205871198922

1 + 1198922 120575


Φ isin C(0)

[[1205752 120575


capC(2)

[[1205752 120575


(26)

The derivative of the electric potential represents the electricand the magnetic fields whose values are not continuous at

the two spiral metal boundary walls In Figure 3(a) MDSCCpartially composed of two infinite ideal spiral conductorsfilled with dielectric material having a permittivity 120598 = 120598

0120598119903is

shown The MDSCC has much in common with the parallelplate line [17] the two spiral conductors are consideredinfinitely wide (120579 isin [minusinfininfin]) and separated by Δ120575 =

21205871198922

(1 + 1198922

)The potentialΦ(120575 120579) is subject to the following boundary

conditions in the region I (see Figure 1)

Φ(1205751 120579) = 119881

0

Φ (1205752 120579) = 0 forall120579 isin (minusinfininfin)

(27)

and in the region II

Φ(1205752 120579) = 0

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 119881

0forall120579 isin (minusinfininfin)

(28)

1198810must be the same in both cases of (27) and (28) because

120575 = 1205751and 120575 = 120575

1minus 2120587119892

2

(1 + 1198922

) correspond to the sameconductor (see Figure 1(b) cyclic spiral) and the potentialmust be continuous at the spiral metal walls

By the method of separation of variable let Φ(120575 120579) beexpressed in product form as

Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)

Substituting (29) into (25) and dividing by 119877119875 give

1198922

119877 (120575)

1205972

119877 (120575)

1205971205752+

1

119875 (120579)

1205972

119875 (120579)

1205971205792= 0 (30)

The two terms in (30) must be equal to constants so that

1198922

119877 (120575)

1205972

119877 (120575)

1205971205752= minus119896

2

120575 (31)

1

119875 (120579)

1205972

119875 (120579)

1205971205792= minus119896

2

120579 (32)

1198962

120575+ 119896

2

120579= 0 (33)

The general solution to (32) is

119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896

120579120579) (34)

Now because the boundary conditions (27) (28) do not varywith 120579 the potentialΦ(120575 120579) should not vary with 120579 Thus 119896

120579

must be zero By (33) this implies that 119896120575must also be zero

so that (31) for 119877(120575) reduces to

1205972

119877 (120575)

1205971205752= 0 (35)

and so

Φ (120575 120579) = 119862120575 + 119863 (36)


The equivalent electrostatic problem in the plane (120575 120579) is theproblem of finding the potential distribution between twoplates [18]

Applying the boundary conditions of (27) to (36) givestwo equations for the constants 119862 and119863 in the region I

Φ(1205751 120579) = 0 = 119862I1205751

+ 119863I

Φ (1205752 120579) = 119881

0= 119862I1205752

+ 119863I(37)

At the same time the boundary conditions of (28) into (36)give two equations for the constants 119862 and119863 in the region II

Φ(1205752 120579) = 119881

0= 119862II1205752

+ 119863II

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 0 = 119862II (1205751

minus2120587119892

2

1 + 1198922) + 119863II

(38)

After solving for119862III and119863III we can write the final solutionforΦ(120575 120579)

Φ (120575 120579) =119881

0

1205752minus 120575

1

(120575 minus 1205751)

region I 120579 isin [minusinfininfin] 120575 isin [1205752 120575

1]

Φ (120575 120579) =119881

0

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

(120575 minus 1205751+

21205871198922

1 + 1198922)

region II 120579 isin [minusinfininfin] 120575 isin [1205751minus

21205871198922

1 + 1198922 120575

2]

(39)

The and fields can now be found using (5) and (39)

region I

perp= 119864

120575119890120575= minusnabla

perpΦ = minus

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810

1205752minus 120575

1

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1

119890120579

119867120575= 0

region II

perp= 119864

120575119890120575= minusnabla

perpΦ

= minus119892119890

(minus120575119892+119892120579)

radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120579

119867120575= 0

(40)

While the electric and the magnetic fields together with thesurface charge and current densities vary exponentially withthe spiral coordinates (120575 120579) the potential remains constanton the two conductors

The field distribution for the TEM mode in the MSCCdepicted in Figure 2 is obtained by using (40) and the quiver-MATLAB function

As stated by the Gauss law [30] the whole surface density120590 of charge on each of the two spiral conductors due to thediscontinuity of the electric field is

120590 (120579) = 120598 119864II sdot 119899 minus 120598I sdot 119899 (41)

where 119899 equiv 119890120575is the normal to the spiral surface of the

conductors whilst I and II are the electric fields seen fromthe regions I and II respectively

According to (41) the electric charge distribution followsthe exponential electric field

The two spiral metal conductors are in a parallel configu-ration they have the same potential difference but two differ-ent capacities and two different surface charge distributions

At the same time the total displacement current [30]due to the discontinuity of the magnetic fields at the twoconductors is

119869119878tot

= 119899 times I minus 119899 times II (42)

The time-average stored electric energy per unit length[2 17] in the MDSCC (see Figure 3) is

119882119890=1

2int119878perp

1205981015840

sdot lowast

119889119878perp (43)

while circuit theory gives 119882119890= 119862

1015840

|1198810|2

4 resulting in thefollowing expression for the capacitance per unit length

1198621015840

=1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878

perp sdot

lowast

119889119878perp [Fm] (44)

As in the case of the parallel plate waveguide the MSCC iscomposed of finite strips

The electric field lines at the edge of the finite spiralconductors are not perfect spirals and the field is not entirelycontained between the conductors

The azimuthal length in real multiturn MDSCC isassumed to be much greater than the separation between theconductors (|120579

1minus 120579

2| ≫ Δ120575) with |120579

1| |120579

2| not too high as in

the case of the myelin bundles so that the fringing fields canbe ignored [2]

Furthermore the minimum distance between the twospiral conducting strips is chosen in such a way to avoid thedielectric voltage breakdown

Although the MDSCC line is modeled with two capac-itors it is composed by two and not three conductors as itwould be in the case of the parallel plates


++

+

+

+

+

+

+

++

+

+

+

+

+

Inner conductor

Outer conductor

minusminus minus

minusminusminus

(a)

Outer conductor

Inner conductor+

+

+

+

+

+

+

+

+

+

+

+

+

minus

minus

minus

minusminus

minus

minus

minus

minus

minusminus

minus

(b)

Figure 2 Field distribution for the TEM mode in the (a) MSCC (b) cylindrical coax obtained using the quiver-MATLAB function(simulations on Pentium 4 32 Ghz average CPU time 4min)

The two capacitors are different because their spiraldimensions are different consequently the two capacitancesare determined by

1198621015840

1=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878I

I sdot lowast

I 119889119878perp =119892120598

1015840

(1205792minus 120579

1)

1205751minus 120575

2

1198621015840

2=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878II

II sdot lowast

II119889119878perp

=119892120598

1015840

(1205792minus 120579

1minus 2120587 (1 + 119892

2

))

(1205752minus 120575

1+ 21205871198922 (1 + 1198922))

1205792gt 120579

1

(45)

Thus

1198621015840

tot = 1198621015840

1+ 119862

1015840

2

= 120598119892119882(1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

(46)

This value represents the capacitance 1198621015840

tot = 119862tot119882 (egfaradsmeter) per unit length of the spiral coaxial line withfinite azimuthal dimension 120579

1minus 120579

2for the first greater

capacitor and 1205792minus 120579

1minus 2120587(1 + 119892

2

) for the smaller one (seeFigure 1)

If the number of spiral turns become high enough thedifference in terms of 120579 between the two capacitors will benegligible

In order to determine the inductance 1198711015840 per unit length oftheMDSCC we observe that themagnetic field is orthogonalto the electric field

The magnetic fluxes over the two infinitesimal areas 119889119878I119911120575

and 119889119878II119911120575

are (see Figure 4) 119889ΦIII = IIIperp

sdot 119889 119878III119911120575

while the

total fluxes over the two spiral areas 119878I 119878II according to (40)are

ΦI = ΦII = 1198821198810

120583

120578 (47)

The fluxes per unit length are given by

Φ1015840

III =ΦIII

119882= 119871

1015840

1198680 (48)

Consequently

1198711015840

= 1198850

120583

120578 (49)

where 1198850and 119868

0are the impedances and current of the line

respectivelyAs it can be noted from (48) there is only one current 119868

0

flowing along the spiral coaxial cableThe time-average stored magnetic energy for unit length

(at low frequencies for nondispersive media) of the MDSCCcan be written as [2 17]

119882119898=120583

2int119878perp

sdot lowast

119889119878perp (50)

Circuit theory gives 119882119898

= 1198711198682

04 in terms of the unique

current of the line 1198680and results from the sum of two

contributions119882119898= 119882

1+119882

2

Thus

1198711015840

=120583119885

2

0

1198812

0

(int119878I

sdot lowast

119889119878perp+ int

119878II

sdot lowast

119889119878perp) (51)


Conductor 1

Conductor 2

+

+

++

+ +

+

++

++

++

∙

∙

+

+

+

+

minus

minusminus

minusminus

minusminus

minus

Region I

Region II

120579

1205752

rarrn equiv e120575

1205751

Φ(1205752 120579) = 0

Φ(1205751 120579) = 0

12057511205751

1205752

1205752

SI

SII

(a)

+ + + + + + + + +

_ _ _ _ _ _ ___V0 V0

Q1 Q2

(b)

Inner conductor 2

Inner conductor 1

Outer conductor 1

y z

x

∙

∙

∙

∙

W

Outer conductor 2

JS2 int

JS1 int

JS2 out

JS1 out

(c)

Figure 3 (a) Charge distributions in the electrostaticMDSCC section (b) Parallel capacitors scheme of the electrostaticMDSCC (c) Currentdistributions in the MDSCC

Substituting (40) into (51) by considering the superposi-tion of the two lines and using (49) gives

1198711015840

=120583

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

minus1

1198850=120578

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587(1 + 119892

2

)

1205752minus 120575

1+ 21205871198922(1 + 1198922)

)

minus1

(52)

According to the classical electromagnetism (see eg [16]page 563) a periodic wave incident upon a material bodygives rise to a forced oscillation of free and bound charges

synchronous with the applied field producing a secondaryfield both inside and outside the body the transmittedand reflected waves have the chance to excite propagatingeigenmodes solutions toMaxwellrsquos equations

From the physical point of view the light that passesthrough the entrance of the spiral waveguide is subject tomultiple reflections The historical work of Mie [31 32] forthe case of the spherical topologywill be the reference startingpoint for the analysis of the light that passes through the openMSCC section and it is scattered by the spiral surface

Localized surface plasmon polaritons (LSPP) [15] existingon a good metal surface can be excited propagated andscattered on the spiral lines The enhancement of the elec-tromagnetic field at the metal dielectric spiral interface couldbe responsible for surface-enhanced optical phenomena suchas Raman scattering fluorescence and second harmonicgeneration (SHG) [33]

Nevertheless the continuity of the tangential compo-nents of the magnetic and electric fields on each spiral


Conductor 1

Conductor 2

minusI

WI

rarrBIIperp

rarrBIperp

dSII119911120575

dSI119911120575

∙

∙

(a)

Conductor 1

Conductor 2

W

minusI

∙

I

rarrBperp

dSperp = dSz120593

(b)

WConductor 1

Conductor 2

dSz120575

rarrBperp

minusI

I

(c)

Figure 4 Surfaces for calculation of external inductances of (a) MDSCC (b) cylindrical coaxial line [4] and (c) MSSCC

metal-dielectric interface which is essential in order topropagate the polaritons along the line [15] and includes thespecific frequency-dependent dielectric constant of metals(real and imaginary parts) needs specific simulation meth-ods [11] and dedicated mathematical analysis

All these electromagnetic effects which require advancednumerical techniques validations and comparisons in termsof CPU time involve all the modes that pass through thewaveguide In spite of the interesting results and applicationsthat these analyses could bring to the future of the spiralcoaxial cables their study is beyond the scope of this paper

4 The Spiral Transmission Line

A transmission line consists of two or more conductors [2 417] In this paper we consider two types of spiral transmissionlines their elements of line of infinitesimal length119889119911 depictedin Figure 5 can be modeled as lumped-element circuits

Although the MDSCC line is modeled with two capac-itors it is composed by two conductors with only one realcapacitor The series resistance 1198771015840 per unit length representsthe resistance due to the finite conductivity of the individualconductors and the shunt conductance 1198661015840 per unit length isdue to dielectric loss in the material between the conductors

For lossless lines the three quantities 119885 1198711015840 and 1198621015840 are

related as follows

1198711015840

= 120583119885

120578

1198621015840

= 120598120578

119885

(53)

where 120578 = radic120583120598 is the characteristic impedance of thedielectric medium between the conductors

The equations of the ideal spiral transmission line [4]depicted in Figure 5 are

120597119881

120597119911= minus119871

1015840120597119868

120597119905minus 119877

1015840

119868

120597119868

120597119911= minus119862

1015840120597119881

120597119905minus 119866

1015840

119881

(54)

where1198771015840 is the resistance per unit length of the line expressedin [Ωm] and 119866

1015840 is the conductance per unit length of theline measured in [Sm]

The two equations (54) for 1198771015840

= 0 and 1198661015840

= 0 canbe combined to form DrsquoAlambertrsquos wave equation for either


L998400dz R998400dz

C9984001dz G998400

1dz G9984002dzC998400

2dzV

I I +120597I

120597zdz

V +120597V

120597zdz

dz

(a)

L998400dz R998400dz

V

I

C998400dz G998400dz

I +120597I

120597zdz

V +120597V

120597zdz

dz

(b)

Figure 5 Element 119889119911 (a) MDSCC (b) MSSCC and their lumped-element equivalent circuits obtained using M-file with camlightprogramming tools (run on Pentium 4 32 Ghz average CPU time 8min)

variables [2] whose solutions are waves propagating alongthe ideal line with speed V

1205972

119881

1205971199112=1

V1205972

119881

1205971199052

1205972

119868

1205971199112=1

V1205972

119868

1205971199052 V =

1

radic11987110158401198621015840

(55)

Using the Fourier transform of the signals 119881 119868

119881 (120596) =1

2120587int

infin

minusinfin

119881 (119905) 119890minus119894120596119905

119889119905

119868 (120596) =1

2120587int

infin

minusinfin

119868 (119905) 119890minus119894120596119905

119889119905

(56)

The solution to (55) may be written in terms of exponen-tials

119881 = 119881+119890minus120574119911

+ 119881minus119890120574119911

119868 =1

1198850

(119881+119890minus120574119911

minus 119881minus119890120574119911

)

1205742

= minus1205962

1198711015840

1198621015840

(57)

If a sinusoidal voltage is supplied to MDSCC with loadimpedance 119885

119871at 119911 = 0 the reflection Γ and transmission 120591

coefficients will be

Γ =119881

minus

119881+

=119885

119871minus 119885

0

119885119871+ 119885

0

120591 =119881

119871

119881+

=2119885

119871

119885119871+ 119885

0

(58)

If the terminating impedance is exactly equal to the charac-teristic impedance of the line there is no reflected wave theline is matched with the load According to (49) the reflectedand the transmitted waves of a spiral coaxial line depend onthe number of turns 119899 = Int(Δ1205792120587) on the shift Δ120575 betweenthe spiral walls and on the spiral 119892 factor

5 Waves in a Lossy Spiral CoaxialTransmission Line

Conductors used in transmission lines have finite conductiv-ity and exhibit series resistance 119877 which increases with anincrease in the frequency of operation [17] because of the skineffect Furthermore the two conductors are separated by adielectric medium which have a small amount of dielectricloss due to the polarization consequently a small shuntconductance 119866 is added to the circuit Differentiating thelossy transmission equation (54) we obtain

1205972

119881

1205971199112= 119877

1015840

(1198661015840

119881 + 1198621015840120597119881

120597119905) + 119871

1015840

(1198621015840120597119881

120597119905+ 119862

10158401205972

119881

1205971199052)

1205972

119868

1205971199112= 119877

1015840

(1198661015840

119868 + 1198621015840120597119868

120597119905) + 119871

1015840

(1198621015840120597119868

120597119905+ 119862

10158401205972

119868

1205971199052)

(59)

By using the Fourier transform of the signals 119881 119868 weobtain

120574 = [minus1205962

1198711015840

1198621015840

+ 1198771015840

1198661015840

+ 119894120596 (1198771015840

1198621015840

+ 1198711015840

1198661015840

)]12

1198850= (

1198771015840

+ 1198941205961198711015840

1198661015840 + 1198941205961198711015840)

12

(60)

For most transmission lines the losses are very small that is119877

1015840

≪ 1205961198711015840 and 119866

1015840

≪ 1205961198621015840 a binomial expansion of 120574 then

holds

120574 ≃ 119894120596radic11987110158401198621015840 +1

2

radic11987110158401198621015840 (119877

1015840

1198711015840+119866

1015840

1198621015840) = 120572 + 119894120573 (61)

Thus the phase constant 120573 remains unchanged with respectto the ideal line

The expressions of 1198771015840 reported in Table 2 can be foundfrom the expression of the power loss per unit length due tothe finite conductivity of the two metallic spiral conductors[2] that is

119875119888=119877

119878

2int119878120579119911

119869119878sdot 119869

lowast

119878119889119878

120579119911 (62)


where the argument of the integral is the scalar product of thedisplacement currents [30] flowing along the surfaces of theconductors

In (62) 119877119904= 1(120590120575

119878) is the surface resistance of the

conductors where the skin depth or characteristic depth ofpenetration is defined as 120575

119878= radic2(120596120583120590)

The material filling the space between the conductors isassumed to have a complex permittivity 120598 = 120598

1015840

minus 11989412059810158401015840 a

permeability 120583 = 1205830120583119903 and a loss tangent tan(120575mat) = 120598

10158401015840

1205981015840

The shunt conductance per unit length 1198661015840 reported

in Table 2 can be inferred from the time-average powerdissipated per unit length in a lossy dielectric that is

119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)

The total voltage and current waves on the line can thenbe written as a superposition of an incident and a reflectedwave

119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)

The time-average power flow along the line at the point 119911 is

119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)

When the load is mismatched not all of the available powerfrom the generator is delivered to the load the presence of areflected wave leads to standing waves [2] and themagnitudeof the voltage on the line is not constant

The return loss (RL) is

RL = minus20 log |Γ| [dB] (66)

A measure of the mismatch of a line is the standing waveratio (SWR)

SWR =1 + |Γ|

1 minus |Γ| (67)

At a distance 119911 = minus119897 from the load the input impedance seenlooking toward the load is

119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)

The power delivered to the input of the terminated line at119911 = minus119897 is

119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)

The difference 119875avg minus 119875in corresponds to the power lost in theline [2]

From (58) and (49) it appears clear that |Γ|119875avg RL SWR119885in and the power lost depend critically on the spiral factorsof the line

Particularly it is worth to point out that the 119892 factor actsas a ldquocontrol knobrdquo of the electromagnetic propagation alongthe MDSCC

6 Single Spiral Coaxial Cable andthe Myelinated Nerves

The difficulty of using a single spiral surface to construct acoaxial line is due to the constraint of having the constantpotential on the conductor

The problem can be solved by using two independentstripes of the same single spiral surface with |120579

119891minus120579

119894| le 2120587 and

|1205791| |120579

2|not too high separated by a shiftΔ120575 = 2119899120587119892

2

(1+1198922

)

to form a system of two independent faced conductors withone grounded (as depicted in Figures 5(b) and 6(a))

The metal single spiral coaxial cable (MSSCC) does notdiffer geometrically too much from the cylindrical coaxialdesign especially for 119892 ≪ 1 but the first is an openframework whilst the second is a closed one

Again according to the conformal mapping theory [18]the equivalent electrostatic problem for the MSSCC in theplane (120575 120579) is just the problem of finding the potentialdistribution between two finite coordinate-plates like in thecylindrical case [18]

The potentialΦ(120575 120579) for the TEM wave is now subject tothe following boundary conditions

Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)

Consequently the solution in (36) to Laplacersquos electrostaticequation (25) takes the form

Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)

The electric and magnetic field for the MSSCC is simpli-fied compared to the MDSCC that is

perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)

The total charge 119876 on the innerouter conductors ofMSSCC of length119882 is

119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)


Table 1 Values of capacitance for an average human myelinated nerve obtained with the SSCC and the cylindrical coax models

Fibrediameter[119863]

Axondiameter

[119889]

119892mye 120598myeNumber oflamellae 119899

119897

Core-conductorcapacitance119862mye [34]

Single-coaxcapacitance 119862mye

Colersquosinductance119871mye [36]

Single-coaxinductance 119871mye

≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898

Since the potential difference between the two conductors isΔ119881 = 119881

0 the capacitance per unit length of the MSSCC with

119899 turns between the two spiral conductors takes the followingsimplified form

1198621015840

= 1205981 + 119892

2

119899119892 (74)

The myelin sheath in the ldquocore-conductorrdquo model isan electrically insulating phospholipid multilamellar spiralmembrane surrounding the conducting axons of many neu-rons it consists of units of double bilayers separated by 3 to4 nm thick aqueous layers composed of 75ndash80 lipid and20ndash25 protein The two conductors in myelinated fibrescoincide with the inner conducting axon and the outerconducting extracellular fluid (see Figure 6(b))

The myelin sheath acts as an electrical insulator forminga capacitor surrounding the axon which allows for faster andmore efficient conduction of nerve impulses than unmyeli-nated nerves

In Table 1 a comparison between the SSCC and the coreconductor models [34] of an average human myelinatednerve is proposed

The diameter of the myelinated nerve fibre [35] growsaccording to the formula

119863 = 119889 + 2 times 119899119897times 119896

119897 (75)

where 119899119897is the number of lamellae bilayers 119896

119897is their average

width 119889 is the diameter of the axon and119863 is the diameter ofthe fibre

Now using the formula of the spiral mapping we have

119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912

are the initial and final angles of the myelinsheaths and 120575

119898determine the lipidmembrane spiral contour

For 119892119898≪ 1 as in the case of the myelin the thickness of

the 119899th bilayer is nearly constant and the radius at which itoccurs is 119903

119899= 119890

120575119898119892minus4119899120587119892

By taking as value of the thickness 119896119897≃ 119903

1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)

According to the statistics [35] the nerve fiber diameter119863is linearly related to the axon119889diameter that is119863 = 119862

0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912

(each lipid bilayer consistsof two spiral turns 120579

1198941

≫ 1205791198912

) and using (76) we have thefollowing relation between the number of myelin lamellae 119899

119897

and the diameter 119889 of the axon

119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)

which is confirmed by the statistics [35]In the case of the SCC we have

1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)

where 119899 represents the number of spiral turns between theouter spiral conductor and the inner one

The transmitted power in SCC depends inversely on theimpedance of the line119885

0which is proportional to the 119892 factor

of the spiral and on the number of turnsDuring 1960rsquos Cole [36] presented a circuit model of the

nerves including the inductive effects of the small membranecurrents

In Table 1 a comparison between the Cole and the SCCinductances is proposed

The expressions 1198771015840 and 1198661015840 for the SCC related to the

power loss per unit length due to the finite conductivity ofthe two spiral conductor strips and to the time-average powerdissipated per unit length in the dielectric respectively arereported in Table 2 in a comparison with various types oftransmission lines

The inductance1198711015840

≃ 0 [37] for the core-conductormodelis negligible (59) is then rewritten in the form

119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)

where 120582 and 120591 are called the cable space and time constantsrespectively while119879 is called the time per internodal distanceℓ [37]


Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890

((1205751119892)minus2119892(119901minus119902)120587minus119892120579119894119901)

119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889


7 The Spiral Poynting Vector

On a matched spiral coaxial line the rms voltage 1198810is related

to the total average power flow 119875119911= (12) int

119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)

where the infinitesimal cross section is 119889119878perpequiv 119889 119878

120575120579 of (20)

As the 119892 factor decreases for example in the evolutionof the Schwannrsquos cell around the axon progressively a highernumber of spiral turns are required to yield the same value oftransmitted power Likewise overcoming the power thresh-old in neural networks may provoke nerve inflammation anddisorders or vice versa an amount of power below the naturalrequired level could cause the neural signal to be blocked

In order to change the transmitted power the neuralsystem can modify the number 119899 of turns or the 119892 factor

Peters and Webster [27 38 39] showed that the anglessubtended at the centre of the axon between the internalmesaxon and outer tongue of cytoplasm obey a precisestatistic that is in about 75 of the mature myelin sheathsthey examined the angle that lied within the same quadrantThis work refines the coaxial model for myelinated nervesintroducing the spiral geometry and gives an explanation forthe Peters quadrant mystery [38]The surprising tendency forthe start and finish of themyelin spiral to occur close togetheraccording to this spiral coaxial model comes out from theneed of handling power throughout the nervous system

In fact the Poynting vector of (81) depends linearlyon the Peters angle 120573

119901which represents a finicky control

of the power delivered along the myelinated nerves Amalformation of the Peters angle causes higherlower powerto be transmitted in the neural networks with respect to therequired normal level

8 Conclusions

In this paper two types of metal spiral coaxial cables havebeen proposed the MSCC and the MDSCC

A generalization of the Schwarz-Christoffel [40] confor-mal mapping was used to map the transverse section of

the MSCC into a rectangle and to find the solution to itsequivalent electrostatic Laplacersquos equation

The fundamental TEM wave propagating along theMSCC has been determined together with the impedances ofthe line

Comparisons of the MSCC with the classical cylindricalcoax as well as with the hollow polar waveguide have beendone

The myelinated nerves whose elm model is still basedon the core-conductor theory are analyzed by using thespiral coaxial model and their spiral geometrical factors areprecisely related to the electrical impedances and propagatingelm fields The spiral model could be used to better analyzethe neurodegenerative diseases which are strictly connectedto the geometrical malformations of the myelin bundles

The MDSCC has many advantages compared to thecylindrical coaxial cable because it can be made multiturnthus distributing the energy over a larger area and protectingthe small signals from interference due to external electricfields

The MSCC could have many interesting applications inthe field of video and data transmission as well as for sensinginstrumentationcontrol communication equipment andplasmonic nanostructure at optical wavelength

Appendix

Spiral Generalization ofthe Schwarz-Christoffel Conformal Mapping

We define a spiral conformal coordinate system (119906 V) as oneas specified by a complex analytic function

119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)

where 119892 isin R is a constant [40] and the function 119891(119911) isa generalization of the well-known holomorphic Schwarz-Christoffel [41] formula

119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1

the two formulas of (A2) and (A3) are identicalSince 119891(119911) is holomorphic the derivative 1198911015840

(119911) exists andit is independent of direction

For 119892 = 0 or 1198600isin R the spiral conformal mapping of

(A1)-(A2) coincides with the polar mapping (see [18] page135) the elm propagation along the circular waveguide isthen included in the theoretical treatment of this paper as aparticular case

In terms of cartesian (119909 119910) or polar (119903 120593) coordinates

119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1

Conductor 2 Single spiral

1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)

Conducting outer fluid(extracellular fluid)

Insulating layer

Conducting center(the axon)

(axon cell walls +myelin sheaths)

∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)

Figure 6 SSCC (a) transverse section (b) longitudinal view and (c) the myelin sheaths

Substituting (A2) into (A1) we obtain

119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)

The value of the constant 119870 represents the phase of thetransformation and is related to 119911

0= 119890

minus119870In order to study the spiral coaxial cable a further

normalization of the angles 119906 and V is introduced

119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)

120579 120575 are the two normalized variables Using (A1) (A4)(A6) and

119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)

we obtain the direct complex spiral coordinate transforma-tion that is

119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)

where119870 = 0If 119892 = 0 and 119870 = 0 the two variables 119906 V coincide with

the polar variables ln 119903 120593 (see [18] page 135)The transverse arclength in cartesian or polar coordinates

becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)


or in conformal coordinates

(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)

where the scale factor is the inverse of the modulus of thederivative of the function that is

1198911015840

(119911) =1 minus 119894119892

119911 (A12)

Substituting (A6) into (A11) we have

(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)

Although the scale factors of the variables 120575 and 120579 are notequal their normalized coordinate system is orthogonal andthe potential satisfies the same differential equation that itdoes in the 119909 119910 coordinates [18] By using the variables 119906 andV of the original conformal mapping presented in [40] forwhich the scale factors are identical it is possible to obtainexactly the same results of this paper

The complex variable 119911 = 119909 + 119894119910 here used to describethe spiral conformal mapping is not the same variable ldquo119911rdquothat represents the longitudinal coordinate of the waveguideNevertheless the general treatment of the elm propagationin waveguide [28] and Maxwellrsquos differential operators areseparated into the longitudinal and the transverse parts

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] O Heaviside Electromagnetic Theory vol 1 Dover New YorkNY USA 1950

[2] D M Pozar Microwave Engineering John Wiley amp Sons 4thedition 2011

[3] A S Khan Microwave Engineering Concepts and Fundamen-tals CRC Press New York NY USA 2014

[4] S Ramo J R Whinnery and T Van Duzer Fields and Wavesin Communication Electronics John Wiley amp Sons 3rd edition1993

[5] G Lifante Integrated Photonics Fundamentals John Wiley ampSons Chichester UK 2003

[6] C H Lee Microwave Photonics CRC Press New York NYUSA 2006

[7] R de Waele S P Burgos A Polman and H A AtwaterldquoPlasmon dispersion in coaxial waveguides from single-cavityoptical transmission measurementsrdquo Nano Letters vol 9 no 8pp 2832ndash2837 2009

[8] M S Kushwaha and B D Rouhani ldquoSurface plasmons incoaxial metamaterial cablesrdquo Modern Physics Letters B vol 27no 17 Article ID 1330013 2013

[9] J-C Weeber A Dereux C Girard J R Krenn and J-PGoudonnet ldquoPlasmon polaritons of metallic nanowires forcontrolling submicron propagation of lightrdquo Physical ReviewB Condensed Matter and Materials Physics vol 60 no 12 pp9061ndash9068 1999

[10] H Regneault J M Lourtioz and C Delalande LevensonNanophotonics John Wiley amp Sons New York NY USA 2010

[11] G Veronis Z Yu S Kocaba D A B Miller M L Brongersmaand S Fan ldquoMetal-dielectric-metal plasmonic wave guidedevices for manipulating light at the nanoscalerdquo Chinese OpticsLetters vol 7 no 4 pp 302ndash308 2009

[12] M L Brongersma J W Hartman and H A Atwater ldquoElec-tromagnetic energy transfer and switching in nanoparticlechain arrays below the diffraction limitrdquo Physical Review BmdashCondensed Matter and Materials Physics vol 62 no 24 ppR16356ndashR16359 2000

[13] TW EbbesenH J LezecH F Ghaemi TThio and P AWolffldquoExtraordinary optical transmission through sub-wavelenghthole arraysrdquo Nature vol 391 no 6668 pp 667ndash669 1998

[14] G Boisde and A Harmer Chemical and Biochemical Sensingwith Optical Fibers and Waveguides Arthech House BostonMass USA 1996

[15] A V Zayats I I Smolyaninov and A A Maradudin ldquoNano-optics of surface plasmon polaritonsrdquo Physics Reports vol 408no 3-4 pp 131ndash314 2005

[16] J A Stratton ElectromagneticTheory McGraw-Hill New YorkNY USA 1941

[17] R E Collin Foundations for Microwave Engineering IEEEPress Wiley Interscience New York NY USA 2nd edition2001

[18] R E Collin Field Theory of Guided Waves Mc-Graw Hill NewYork NY USA 1960

[19] L Rayleigh ldquoOn the passage of electric waves through tubesrdquoPhilosophical Magazine vol 43 no 261 pp 125ndash132 1897

[20] I M Fabbri A Lauto and A Lucianetti ldquoA spiral index profilefor high power optical fibersrdquo Journal of Optics A Pure andApplied Optics vol 9 no 11 pp 963ndash971 2007

[21] I M Fabbri A Lucianetti and I Krasikov ldquoOn a Sturm Liou-ville periodic boundary values problemrdquo Integral Transformsand Special Functions vol 20 no 5-6 pp 353ndash364 2009

[22] K Guven E Saenz R Gonzalo E Ozbay and S TretyakovldquoElectromagnetic cloaking with canonical spiral inclusionsrdquoNew Journal of Physics vol 10 Article ID 115037 2008

[23] W T Kelvin ldquoOn the theory of the electric telegraphrdquo Proceed-ings of the Royal Society of London vol 7 pp 382ndash389 1855

[24] W Rall ldquoCore conductor theory and cable properties of neu-ronsrdquo in Handbook of Physiology the Nervous System CellularBiology of Neurons John Wiley amp Sons New York NY USA2011

[25] A H Buck Reference Handbook of the Medical Sciences vol 3of edited by A H Buck Book on Demand New York NY USA1901

[26] A L Hodgkin and A F Huxley ldquoA quantitative descriptionof membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[27] A Peters ldquoFurther observations on the structure of myelinsheaths in the central nervous systemrdquo The Journal of CellBiology vol 20 pp 281ndash296 1964

[28] N Marcuvitz Waveguide Handbook Peter Peregrinus NewYork NY USA 1986


[29] I Boscolo and I M Fabbri ldquoA tunable bragg cavity for anefficient millimeter FEL driven by electrostatic acceleratorsrdquoApplied Physics B Photophysics and Laser Chemistry vol 57 no3 pp 217ndash225 1993

[30] J D Jackson Classical Electrodynamics John Wiley amp SonsNew York NY USA 1962

[31] G Mie ldquoBeitrage zur Optik truber Medien speziell kolloidalerMetallosungenrdquoAnnalen der Physik vol 330 no 3 pp 337ndash4451908 English translated by B Crossland Contributions to theoptics of turbid media particularly of colloidal metal solutionsNasa Royal Aircraft Establishment no 1873 1976

[32] M Born and E Wolf Principles of Optics ElectromagneticTheory of Propagation Cambridge University Press Cam-bridgeUK 1999

[33] V M Agranovich and D L Mills Eds Surface PolaritonsNorth-Holland Amsterdam The Netherlands 1982

[34] YMin K Kristiansen J M Boggs C Husted J A Zasadzinskiand J Israelachvili ldquoInteraction forces and adhesion of sup-portedmyelin lipid bilayersmodulated bymyelin basic proteinrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 9 pp 3154ndash3159 2009

[35] C H Berthold I Nilsson and M Rydmark ldquoAxon diameterandmyelin sheath thickness in nerve fibres of the ventral spinalroot of the seventh lumbar nerve of the adult and developingcatrdquo Journal of Anatomy vol 136 no 3 pp 483ndash508 1983

[36] K Cole Membranes Ions and Impulses A Chapter of ClassicalBiophysics University of California Press Los Angeles CalifUSA 1968

[37] A FHuxley andR Stampfli ldquoEvidence for saltatory conductionin peripheralmyelinated nerve fibresrdquoThe Journal of Physiologyvol 108 no 3 pp 315ndash339 1949

[38] R R Traill Strange Regularities in the Geometry of MyelinNerve-InsulationmdashA Possible Single Cause Ondwelle ShortMonograph no 1 2005

[39] H D Webster ldquoThe geometry of peripheral myelin sheathsduring their formation and growth in rat sciatic nervesrdquo TheJournal of Cell Biology vol 48 no 2 pp 348ndash367 1971

[40] L M B Campos and P J S Gil ldquoOn spiral coordinates withapplication to wave propagationrdquo Journal of Fluid Mechanicsvol 301 pp 153ndash173 1995

[41] Z Nehari Conformal Mapping Dover New York NY USA1975

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



optical frequencies [7] opening up new research pursuitsunexpected in the area of nanoscale waveguiding fieldenhancement imaging with coaxial cavities and negative-index metamaterials [8]

Several different plasmonic waveguiding structureshave been proposed such as metallic nanowires [9 10]metal-dielectric-metal (MDM) structures [11] and metallicnanoparticle arrays [12] for achieving compact integratedphotonic devices

Most of these structures support a highly confined modenear the surface plasmon frequency [11] This study couldbecome the reference point to introduce the MSCC as avalid alternative to guide subwavelength surface plasmonpolaritons (SPPs)

The extraordinary transmission of light through an arrayof subwavelength apertures enhancement which arises fromthe coupling of the incident light with the SPPs through thesurface grating in metal film [13] could result particularlyefficiently on the spiral metal dielectric interface with peri-odic holes

High sensitivity spiral biosensors and spiral photonicintegrated circuits based on nonlinear surface plasmonpolariton optics [14 15] may be implemented

The aimof this pioneeringwork on the SCC is to representan initial landmark in the continuously growing sector of themicrowave research

The popularity of Video on Demand (VoD) and Overthe Top Technology (OTT) services to access high definitionvideos over home interconnected devices of the hybrid fibre-coax (HFC) networks is driving the research toward moreefficient and cost-effective cables Particularly the develop-ment of Converged Cable Access Platforms (CCAP) thatcombine video and data transmission supporting simultane-ous network access of multiple users over a single coaxialcable is flourishing and sustaining the demand for new highspeed transmission media

In a metallic guide the reflection mechanism responsiblefor confining the energy is due to the reflection from theconductors at the boundary [16] whose geometry is strictlyrelated to the propagating modes

Coaxial cables were designed to propagate high frequencyradio signals The principal constraints on performance of acoaxial are attenuation thermal noise and passive intermod-ulation noise (PIM)

In RF applications the wave propagates essentially in thefundamental transverse electric magnetic (TEM) mode thatis the electric and magnetic fields are both perpendicular tothe direction of propagation

In the ideal case the conductors can be considered to haveinfinite conductivity and the TEM eigenmode is the basicpropagating wave (see [17] page 110) along the transmissionline

Practical lines have finite conductivity and this results ina perturbation or change of the TEMmode (see [18] page 119)

Above the cutoff frequency transverse electric (TE) ortransverse magnetic (TM) modes [19] can also propagatewith different velocities within a practical cylindrical coaxinterferingwith each other producing distortion of the signal

The frequency of operation for a specific outer conductorsize is then limited by the highest usable cutoff frequencybefore undesirable modes of propagation occur

In order to prevent higher order modes from beinglaunched the radiuses of the coaxial conductors must bereduced diminishing the amount of power that can betransmitted

On the other hand at high frequencies it is impossible tomake the cylindrical coaxial line in the small size necessaryto propagate the TEMmode alone

The research described in this paper demonstrates thepropagation on the fundamental TEM wave along the idealMSCC

Since the mode of transmission on an ideal line is theTEM wave the relations for input impedance reflectioncoefficient return loss (RL) standing wave ratio (SWR) andso forth given afterward in the next sections are applicablein general to the spiral transmission lines (see [18] Chapter 3)

The metal double spiral coaxial cable or MDSCC result-ing from the superposition of two spiral conductors thatshare the same geometrical axis can be made multi-turnThe amount of heat generated by the losses for heating canbe distributed over a larger area and this would lower thetemperature and raise the reliability of the line

In fact operation at higher temperatures results in areduction in the life expectancy and reliability of the trans-mission line relative to the lower temperature performance

Applications like nanoscale optical components for inte-gration on semiconductor chips could benefit from thesecharacteristics of the MSCC

Where signal integrity is important coaxial cables areneeded to be shielded against radio frequency noise (RFnoise) The multiturn MDSCC is naturally shielded becausethe highest part of the elm energy can be distributed on theinner part of the cable which protects small signals frominterference due to external electric fields

A new class of spiral passive components computer-aided engineering (CAE) tools as well as electromagnetic(EM) simulators is required before new high-frequencyspiral RFmicrowave circuits will be implemented

The spiral geometry occurs widely in nature exampleslike the spiral galaxies are found at the universe level while themyelin bundles are common in the microcosm of the neuroncells

Recently a new spiral optical fibre has been proposedboth in the fundamental mode [20] and in the higher ordermodes [21] operation

Spirals are also of extreme interest to the field of the newmetamaterials and invisible cloaking [22]

Myelinated nerve fibers are micro-spiral coaxial cables(119892 ≪ 1) whose electric behaviour is still today described byneurophysiologists using W Thomsonrsquos (later known as LordKelvin) cable formula [23] of the 1860s which determinesthe velocity of the signal propagating in saltatory conduction[23 24]

Cable theory in neurobiology has a long history havingfirst been applied to neurons in 1863 byCMatteucci [25] whodiscovered that if a constant current flows through a portionof a platinum wire covered with a sheath saturated with fluid








nabla times = 119895120596120598 119864

nabla sdot = 0

nabla sdot = 0

(1)






119911times

perp) = minus119895120596120583119867

119911


119911times

perp) = 119895120596120598119864

119911

(2)

nablaperp119864

119911minus120597119864

perp

120597119911= minus119895120596120583 (119890

119911times 119867

perp)

nablaperp119867

119911minus120597119867

perp

120597119911= 119895120596120598 (119890

119911times 119864

perp)

(3)



120597119864perp

120597119911= minus119895119896120585 ( 120598 +

1


perptimes 119890

119911)

120597119867perp

120597119911= minus119895119896120578 ( 120598 +

1


119911times

perp)

(4)


119864perp



perp= 119867








119894defined by

1198901015840

119894= minusnabla

perpΦ

119894

ℎ1015840

119894= 119890

119911times 119890

1015840

119894

(5)

where

nabla2

perpΦ

119894+ 119896

10158402

119888119894Φ

119894= 0

Φ119894= 0 on 119904 if 1198961015840

119888119894= 0

120597Φ119894

120597119904= 0 on 119904 if 1198961015840

119888119894= 0

(6)


11989010158401015840

119894= 119890

119911times nabla

perpΨ

119894

ℎ10158401015840

119894= 119890

119911times 119890

10158401015840

119894

(7)

where

nabla2

perpΨ

119894+ 119896

101584010158402

119888119894Ψ

119894= 0

120597Ψ119894

120597119899= 0 on 119904

(8)


The constants 11989610158401015840

119888119894and 119896

1015840




ties

∬1198901015840

119894sdot 119890

1015840

119895119889119878

perp= ∬119890

10158401015840

119894sdot 119890

10158401015840

119895119889119878

perp=

1 for 119894 = 119895

0 for 119894 = 119895

∬1198901015840

119894sdot 119890

10158401015840

119895119889119878

perp= 0

(9)


perp



119875119911=1

2Re(∬119864

perptimes 119867

lowast

perpsdot 119890

119911119889119878

perp) (10)





119888= 0 or 120596 = 120573119888


perp=1

120578119890119911times

perp (11)




nablaperptimes

perp= 0

nabla sdot perp= 0

(12)




perpand

perp the


119878119911=1

2Re (

perptimes

lowast

perp) sdot 119890

119911=1

120578

10038161003816100381610038161003816

perp

10038161003816100381610038161003816

2

= 12057810038161003816100381610038161003816

perp

10038161003816100381610038161003816

2

(13)





119909 = 119890(120575119892minus119892120579) cos (120575 + 120579)

119910 = 119890(120575119892minus119892120579) sin (120575 + 120579)

119911 = 119911

(14)







1 120579 = 120579

2and 120575 = 120575

1 120575 = 120575

2 120575 = 120575

1minus 2120587119892

2

(1 + 1198922



1= 2119902120587119892

2

(1+1198922


1 120575 = 120575


a shift of Δ120579119904= 2119902120587(1 + 119892

2


2minus120575

1| lt 2120587119892

2

(1+1198922



1

and 120575 = 1205752


119889119909 =120597119909

120597120575119889120575 +

120597119909

120597120579119889120579 +

120597119909

120597119911119889119911

119889119910 =120597119910

120597120575119889120575 +

120597119910

120597120579119889120579 +

120597119910

120597119911119889119911

119889119911 = 119889119911

(15)


119889ℓ2

= 1198891199092

+ 1198891199102

+ 1198891199112

= 119892120575120575119889120575

2

+ 119892120579120579119889120579

2

+ 119892119911119911119889119911

2

(16)

where

ℎ2

120575= 119892

120575120575= 119890

(2(120575119892)minus2119892120579)

(1 +1

1198922)

ℎ2

120579= 119892

120579120579= 119890

(2(120575119892)minus2119892120579)

(1 + 1198922

)

ℎ2

119911= 119892

119911119911= 1 119892

120575120579= 119892

120575119911= 119892

120579119911= 0

(17)


119889119881 = 119869 119889120575 119889120579 119889119911 = 119890(2(120575119892)minus2119892120579)

1 + 1198922

119892119889120575 119889120579 119889119911 (18)


120575 119890

120579 119890

119911

119890120575=120597

120597120575

= 119890(120575119892minus119892120579)

(1

119892cos (120575 + 120579) minus sin (120575 + 120579)) 119890

119909

+ 119890(120575119892minus119892120579)

(1

119892sin (120575 + 120579) + cos (120575 + 120579)) 119890

119910


0

Conductor 1

Conductor 2

Region I

Region II

or

y

1205791

120579

1205792

1205751

1205752

x

1205751 minus2120587g2

1 + g2

∙

∙SI

SII

(a)

0

Region II

0

1205791

120579

1205792

120575

1205751205752 1205751

Φ

V0

∙

∙


Region I

Con

duct

or 1

Con

duct

or 2

Con

duct

or 1

1205792 minus2120587

1 + g2

1205791 minus2120587g2

1 + g2

SI

SII

(b)

y

z

x

rarrdS120575z

rarrdS120579z

rarrdS120575120579

(c)


119890120579=120597

120597120579

= 119890(120575119892minus119892120579)


+ 119890(120575119892minus119892120579)

(minus119892 sin (120575 + 120579) + cos (120575 + 120579)) 119890119910

119890119911=120597

120597119911= 119890

119911

(19)


119910 119890

119911


10038171003817100381710038171003817119889 119878

120575120579

10038171003817100381710038171003817= 119889119878

perp=

10038171003817100381710038171003817100381710038171003817

120597

120597120575times120597

120597120579

10038171003817100381710038171003817100381710038171003817119889120575 119889120579

= 119890(2120575119892minus2119892120579)

(1

119892+ 119892)119889120575 119889120579

10038171003817100381710038171003817119889 119878

120579119911

10038171003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

120597

120597120579times120597

120597119911

10038171003817100381710038171003817100381710038171003817119889120579 119889119911 = 119890

(120575119892minus119892120579)radic1 + 1198922119889119911 119889120579

10038171003817100381710038171003817119889 119878

120575119911

10038171003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

120597

120597120575times120597

120597119911

10038171003817100381710038171003817100381710038171003817119889120575 119889119911 = 119890

(120575119892minus119892120579)

radic1 + 1198922

119892119889119911 119889120575

(20)


119890120575=119890120575

ℎ120575

119890120579=119890120579

ℎ120579

119890119911=119890119911

ℎ119911

(21)



119890120575= 119890

120579times 119890

119911 119890

120579= 119890

119911times 119890

120575 119890

119911= 119890

120575times 119890

120579

119890120575sdot 119890

120579= 0 = 119890

120575sdot 119890

119911= 119890

120579sdot 119890

119911= 0

(22)


2


119903 =119890(120575119892minus119892120579)

radic1 + 1198922

(119890120575minus 119892119890

120579) + 119911119890

119911 (23)


perp= minus1119892


119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)

119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)

119911 = 119911

(24)




119890minus2(120575119892)+2119892120579

1 + 1198922[119892

21205972

Φ

1205971205752+1205972

Φ

1205971205792] = 0 (25)



Φ isin C(0)

[[1205751minus

21205871198922

1 + 1198922 120575


capC(2)

[[1205751minus

21205871198922

1 + 1198922 120575


Φ isin C(0)

[[1205752 120575


capC(2)

[[1205752 120575


(26)



0120598119903is


21205871198922

(1 + 1198922



Φ(1205751 120579) = 119881

0


(27)


Φ(1205752 120579) = 0

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 119881


(28)


120575 = 1205751and 120575 = 120575

1minus 2120587119892

2

(1 + 1198922



Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)


1198922

119877 (120575)

1205972

119877 (120575)

1205971205752+

1

119875 (120579)

1205972

119875 (120579)

1205971205792= 0 (30)


1198922

119877 (120575)

1205972

119877 (120575)

1205971205752= minus119896

2

120575 (31)

1

119875 (120579)

1205972

119875 (120579)

1205971205792= minus119896

2

120579 (32)

1198962

120575+ 119896

2

120579= 0 (33)


119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896

120579120579) (34)


120579



1205972

119877 (120575)

1205971205752= 0 (35)

and so

Φ (120575 120579) = 119862120575 + 119863 (36)




Φ(1205751 120579) = 0 = 119862I1205751

+ 119863I

Φ (1205752 120579) = 119881

0= 119862I1205752

+ 119863I(37)


Φ(1205752 120579) = 119881

0= 119862II1205752

+ 119863II

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 0 = 119862II (1205751

minus2120587119892

2

1 + 1198922) + 119863II

(38)


Φ (120575 120579) =119881

0

1205752minus 120575

1

(120575 minus 1205751)


1]

Φ (120575 120579) =119881

0

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

(120575 minus 1205751+

21205871198922

1 + 1198922)


21205871198922

1 + 1198922 120575

2]

(39)


region I

perp= 119864

120575119890120575= minusnabla

perpΦ = minus

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810

1205752minus 120575

1

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1

119890120579

119867120575= 0

region II

perp= 119864

120575119890120575= minusnabla

perpΦ

= minus119892119890

(minus120575119892+119892120579)

radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120579

119867120575= 0

(40)










119869119878tot



119882119890=1

2int119878perp

1205981015840

sdot lowast

119889119878perp (43)


1015840

|1198810|2


1198621015840

=1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878

perp sdot

lowast

119889119878perp [Fm] (44)




1minus 120579

2| ≫ Δ120575) with |120579

1| |120579






++

+

+

+

+

+

+

++

+

+

+

+

+

Inner conductor

Outer conductor

minusminus minus

minusminusminus

(a)

Outer conductor

Inner conductor+

+

+

+

+

+

+

+

+

+

+

+

+

minus

minus

minus

minusminus

minus

minus

minus

minus

minusminus

minus

(b)



1198621015840

1=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878I

I sdot lowast

I 119889119878perp =119892120598

1015840

(1205792minus 120579

1)

1205751minus 120575

2

1198621015840

2=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878II

II sdot lowast

II119889119878perp

=119892120598

1015840

(1205792minus 120579

1minus 2120587 (1 + 119892

2

))

(1205752minus 120575

1+ 21205871198922 (1 + 1198922))

1205792gt 120579

1

(45)

Thus

1198621015840

tot = 1198621015840

1+ 119862

1015840

2

= 120598119892119882(1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

(46)



1minus 120579



1minus 2120587(1 + 119892

2





and 119889119878II119911120575


sdot 119889 119878III119911120575

while the


ΦI = ΦII = 1198821198810

120583

120578 (47)


Φ1015840

III =ΦIII

119882= 119871

1015840

1198680 (48)

Consequently

1198711015840

= 1198850

120583

120578 (49)

where 1198850and 119868



0



119882119898=120583

2int119878perp

sdot lowast

119889119878perp (50)


= 1198711198682



contributions119882119898= 119882

1+119882

2

Thus

1198711015840

=120583119885

2

0

1198812

0

(int119878I

sdot lowast

119889119878perp+ int

119878II

sdot lowast

119889119878perp) (51)


Conductor 1

Conductor 2

+

+

++

+ +

+

++

++

++

∙

∙

+

+

+

+

minus

minusminus

minusminus

minusminus

minus

Region I

Region II

120579

1205752

rarrn equiv e120575

1205751

Φ(1205752 120579) = 0

Φ(1205751 120579) = 0

12057511205751

1205752

1205752

SI

SII

(a)

+ + + + + + + + +

_ _ _ _ _ _ ___V0 V0

Q1 Q2

(b)

Inner conductor 2

Inner conductor 1

Outer conductor 1

y z

x

∙

∙

∙

∙

W

Outer conductor 2

JS2 int

JS1 int

JS2 out

JS1 out

(c)



1198711015840

=120583

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

minus1

1198850=120578

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587(1 + 119892

2

)

1205752minus 120575

1+ 21205871198922(1 + 1198922)

)

minus1

(52)







Conductor 1

Conductor 2

minusI

WI

rarrBIIperp

rarrBIperp

dSII119911120575

dSI119911120575

∙

∙

(a)

Conductor 1

Conductor 2

W

minusI

∙

I

rarrBperp

dSperp = dSz120593

(b)

WConductor 1

Conductor 2

dSz120575

rarrBperp

minusI

I

(c)








related as follows

1198711015840

= 120583119885

120578

1198621015840

= 120598120578

119885

(53)



120597119881

120597119911= minus119871

1015840120597119868

120597119905minus 119877

1015840

119868

120597119868

120597119911= minus119862

1015840120597119881

120597119905minus 119866

1015840

119881

(54)




= 0 and 1198661015840



L998400dz R998400dz

C9984001dz G998400

1dz G9984002dzC998400

2dzV

I I +120597I

120597zdz

V +120597V

120597zdz

dz

(a)

L998400dz R998400dz

V

I

C998400dz G998400dz

I +120597I

120597zdz

V +120597V

120597zdz

dz

(b)



1205972

119881

1205971199112=1

V1205972

119881

1205971199052

1205972

119868

1205971199112=1

V1205972

119868

1205971199052 V =

1

radic11987110158401198621015840

(55)


119881 (120596) =1

2120587int

infin

minusinfin

119881 (119905) 119890minus119894120596119905

119889119905

119868 (120596) =1

2120587int

infin

minusinfin

119868 (119905) 119890minus119894120596119905

119889119905

(56)


119881 = 119881+119890minus120574119911

+ 119881minus119890120574119911

119868 =1

1198850

(119881+119890minus120574119911

minus 119881minus119890120574119911

)

1205742

= minus1205962

1198711015840

1198621015840

(57)




Γ =119881

minus

119881+

=119885

119871minus 119885

0

119885119871+ 119885

0

120591 =119881

119871

119881+

=2119885

119871

119885119871+ 119885

0

(58)




1205972

119881

1205971199112= 119877

1015840

(1198661015840

119881 + 1198621015840120597119881

120597119905) + 119871

1015840

(1198621015840120597119881

120597119905+ 119862

10158401205972

119881

1205971199052)

1205972

119868

1205971199112= 119877

1015840

(1198661015840

119868 + 1198621015840120597119868

120597119905) + 119871

1015840

(1198621015840120597119868

120597119905+ 119862

10158401205972

119868

1205971199052)

(59)


120574 = [minus1205962

1198711015840

1198621015840

+ 1198771015840

1198661015840

+ 119894120596 (1198771015840

1198621015840

+ 1198711015840

1198661015840

)]12

1198850= (

1198771015840

+ 1198941205961198711015840

1198661015840 + 1198941205961198711015840)

12

(60)


1015840

≪ 1205961198711015840 and 119866

1015840


holds

120574 ≃ 119894120596radic11987110158401198621015840 +1

2

radic11987110158401198621015840 (119877

1015840

1198711015840+119866

1015840

1198621015840) = 120572 + 119894120573 (61)



119875119888=119877

119878

2int119878120579119911

119869119878sdot 119869

lowast

119878119889119878

120579119911 (62)



In (62) 119877119904= 1(120590120575



119878= radic2(120596120583120590)


1015840

minus 11989412059810158401015840 a


10158401015840

1205981015840



119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)


119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)


119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)





SWR =1 + |Γ|

1 minus |Γ| (67)


119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)


119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)







119891minus120579

119894| le 2120587 and

|1205791| |120579


2

(1+1198922

)





Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)


Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)


perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)


119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)




Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















nabla times = 119895120596120598 119864

nabla sdot = 0

nabla sdot = 0

(1)






119911times

perp) = minus119895120596120583119867

119911


119911times

perp) = 119895120596120598119864

119911

(2)

nablaperp119864

119911minus120597119864

perp

120597119911= minus119895120596120583 (119890

119911times 119867

perp)

nablaperp119867

119911minus120597119867

perp

120597119911= 119895120596120598 (119890

119911times 119864

perp)

(3)



120597119864perp

120597119911= minus119895119896120585 ( 120598 +

1


perptimes 119890

119911)

120597119867perp

120597119911= minus119895119896120578 ( 120598 +

1


119911times

perp)

(4)


119864perp



perp= 119867








119894defined by

1198901015840

119894= minusnabla

perpΦ

119894

ℎ1015840

119894= 119890

119911times 119890

1015840

119894

(5)

where

nabla2

perpΦ

119894+ 119896

10158402

119888119894Φ

119894= 0

Φ119894= 0 on 119904 if 1198961015840

119888119894= 0

120597Φ119894

120597119904= 0 on 119904 if 1198961015840

119888119894= 0

(6)


11989010158401015840

119894= 119890

119911times nabla

perpΨ

119894

ℎ10158401015840

119894= 119890

119911times 119890

10158401015840

119894

(7)

where

nabla2

perpΨ

119894+ 119896

101584010158402

119888119894Ψ

119894= 0

120597Ψ119894

120597119899= 0 on 119904

(8)


The constants 11989610158401015840

119888119894and 119896

1015840




ties

∬1198901015840

119894sdot 119890

1015840

119895119889119878

perp= ∬119890

10158401015840

119894sdot 119890

10158401015840

119895119889119878

perp=

1 for 119894 = 119895

0 for 119894 = 119895

∬1198901015840

119894sdot 119890

10158401015840

119895119889119878

perp= 0

(9)


perp



119875119911=1

2Re(∬119864

perptimes 119867

lowast

perpsdot 119890

119911119889119878

perp) (10)





119888= 0 or 120596 = 120573119888


perp=1

120578119890119911times

perp (11)




nablaperptimes

perp= 0

nabla sdot perp= 0

(12)




perpand

perp the


119878119911=1

2Re (

perptimes

lowast

perp) sdot 119890

119911=1

120578

10038161003816100381610038161003816

perp

10038161003816100381610038161003816

2

= 12057810038161003816100381610038161003816

perp

10038161003816100381610038161003816

2

(13)





119909 = 119890(120575119892minus119892120579) cos (120575 + 120579)

119910 = 119890(120575119892minus119892120579) sin (120575 + 120579)

119911 = 119911

(14)







1 120579 = 120579

2and 120575 = 120575

1 120575 = 120575

2 120575 = 120575

1minus 2120587119892

2

(1 + 1198922



1= 2119902120587119892

2

(1+1198922


1 120575 = 120575


a shift of Δ120579119904= 2119902120587(1 + 119892

2


2minus120575

1| lt 2120587119892

2

(1+1198922



1

and 120575 = 1205752


119889119909 =120597119909

120597120575119889120575 +

120597119909

120597120579119889120579 +

120597119909

120597119911119889119911

119889119910 =120597119910

120597120575119889120575 +

120597119910

120597120579119889120579 +

120597119910

120597119911119889119911

119889119911 = 119889119911

(15)


119889ℓ2

= 1198891199092

+ 1198891199102

+ 1198891199112

= 119892120575120575119889120575

2

+ 119892120579120579119889120579

2

+ 119892119911119911119889119911

2

(16)

where

ℎ2

120575= 119892

120575120575= 119890

(2(120575119892)minus2119892120579)

(1 +1

1198922)

ℎ2

120579= 119892

120579120579= 119890

(2(120575119892)minus2119892120579)

(1 + 1198922

)

ℎ2

119911= 119892

119911119911= 1 119892

120575120579= 119892

120575119911= 119892

120579119911= 0

(17)


119889119881 = 119869 119889120575 119889120579 119889119911 = 119890(2(120575119892)minus2119892120579)

1 + 1198922

119892119889120575 119889120579 119889119911 (18)


120575 119890

120579 119890

119911

119890120575=120597

120597120575

= 119890(120575119892minus119892120579)

(1

119892cos (120575 + 120579) minus sin (120575 + 120579)) 119890

119909

+ 119890(120575119892minus119892120579)

(1

119892sin (120575 + 120579) + cos (120575 + 120579)) 119890

119910


0

Conductor 1

Conductor 2

Region I

Region II

or

y

1205791

120579

1205792

1205751

1205752

x

1205751 minus2120587g2

1 + g2

∙

∙SI

SII

(a)

0

Region II

0

1205791

120579

1205792

120575

1205751205752 1205751

Φ

V0

∙

∙


Region I

Con

duct

or 1

Con

duct

or 2

Con

duct

or 1

1205792 minus2120587

1 + g2

1205791 minus2120587g2

1 + g2

SI

SII

(b)

y

z

x

rarrdS120575z

rarrdS120579z

rarrdS120575120579

(c)


119890120579=120597

120597120579

= 119890(120575119892minus119892120579)


+ 119890(120575119892minus119892120579)

(minus119892 sin (120575 + 120579) + cos (120575 + 120579)) 119890119910

119890119911=120597

120597119911= 119890

119911

(19)


119910 119890

119911


10038171003817100381710038171003817119889 119878

120575120579

10038171003817100381710038171003817= 119889119878

perp=

10038171003817100381710038171003817100381710038171003817

120597

120597120575times120597

120597120579

10038171003817100381710038171003817100381710038171003817119889120575 119889120579

= 119890(2120575119892minus2119892120579)

(1

119892+ 119892)119889120575 119889120579

10038171003817100381710038171003817119889 119878

120579119911

10038171003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

120597

120597120579times120597

120597119911

10038171003817100381710038171003817100381710038171003817119889120579 119889119911 = 119890

(120575119892minus119892120579)radic1 + 1198922119889119911 119889120579

10038171003817100381710038171003817119889 119878

120575119911

10038171003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

120597

120597120575times120597

120597119911

10038171003817100381710038171003817100381710038171003817119889120575 119889119911 = 119890

(120575119892minus119892120579)

radic1 + 1198922

119892119889119911 119889120575

(20)


119890120575=119890120575

ℎ120575

119890120579=119890120579

ℎ120579

119890119911=119890119911

ℎ119911

(21)



119890120575= 119890

120579times 119890

119911 119890

120579= 119890

119911times 119890

120575 119890

119911= 119890

120575times 119890

120579

119890120575sdot 119890

120579= 0 = 119890

120575sdot 119890

119911= 119890

120579sdot 119890

119911= 0

(22)


2


119903 =119890(120575119892minus119892120579)

radic1 + 1198922

(119890120575minus 119892119890

120579) + 119911119890

119911 (23)


perp= minus1119892


119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)

119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)

119911 = 119911

(24)




119890minus2(120575119892)+2119892120579

1 + 1198922[119892

21205972

Φ

1205971205752+1205972

Φ

1205971205792] = 0 (25)



Φ isin C(0)

[[1205751minus

21205871198922

1 + 1198922 120575


capC(2)

[[1205751minus

21205871198922

1 + 1198922 120575


Φ isin C(0)

[[1205752 120575


capC(2)

[[1205752 120575


(26)



0120598119903is


21205871198922

(1 + 1198922



Φ(1205751 120579) = 119881

0


(27)


Φ(1205752 120579) = 0

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 119881


(28)


120575 = 1205751and 120575 = 120575

1minus 2120587119892

2

(1 + 1198922



Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)


1198922

119877 (120575)

1205972

119877 (120575)

1205971205752+

1

119875 (120579)

1205972

119875 (120579)

1205971205792= 0 (30)


1198922

119877 (120575)

1205972

119877 (120575)

1205971205752= minus119896

2

120575 (31)

1

119875 (120579)

1205972

119875 (120579)

1205971205792= minus119896

2

120579 (32)

1198962

120575+ 119896

2

120579= 0 (33)


119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896

120579120579) (34)


120579



1205972

119877 (120575)

1205971205752= 0 (35)

and so

Φ (120575 120579) = 119862120575 + 119863 (36)




Φ(1205751 120579) = 0 = 119862I1205751

+ 119863I

Φ (1205752 120579) = 119881

0= 119862I1205752

+ 119863I(37)


Φ(1205752 120579) = 119881

0= 119862II1205752

+ 119863II

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 0 = 119862II (1205751

minus2120587119892

2

1 + 1198922) + 119863II

(38)


Φ (120575 120579) =119881

0

1205752minus 120575

1

(120575 minus 1205751)


1]

Φ (120575 120579) =119881

0

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

(120575 minus 1205751+

21205871198922

1 + 1198922)


21205871198922

1 + 1198922 120575

2]

(39)


region I

perp= 119864

120575119890120575= minusnabla

perpΦ = minus

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810

1205752minus 120575

1

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1

119890120579

119867120575= 0

region II

perp= 119864

120575119890120575= minusnabla

perpΦ

= minus119892119890

(minus120575119892+119892120579)

radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120579

119867120575= 0

(40)










119869119878tot



119882119890=1

2int119878perp

1205981015840

sdot lowast

119889119878perp (43)


1015840

|1198810|2


1198621015840

=1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878

perp sdot

lowast

119889119878perp [Fm] (44)




1minus 120579

2| ≫ Δ120575) with |120579

1| |120579






++

+

+

+

+

+

+

++

+

+

+

+

+

Inner conductor

Outer conductor

minusminus minus

minusminusminus

(a)

Outer conductor

Inner conductor+

+

+

+

+

+

+

+

+

+

+

+

+

minus

minus

minus

minusminus

minus

minus

minus

minus

minusminus

minus

(b)



1198621015840

1=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878I

I sdot lowast

I 119889119878perp =119892120598

1015840

(1205792minus 120579

1)

1205751minus 120575

2

1198621015840

2=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878II

II sdot lowast

II119889119878perp

=119892120598

1015840

(1205792minus 120579

1minus 2120587 (1 + 119892

2

))

(1205752minus 120575

1+ 21205871198922 (1 + 1198922))

1205792gt 120579

1

(45)

Thus

1198621015840

tot = 1198621015840

1+ 119862

1015840

2

= 120598119892119882(1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

(46)



1minus 120579



1minus 2120587(1 + 119892

2





and 119889119878II119911120575


sdot 119889 119878III119911120575

while the


ΦI = ΦII = 1198821198810

120583

120578 (47)


Φ1015840

III =ΦIII

119882= 119871

1015840

1198680 (48)

Consequently

1198711015840

= 1198850

120583

120578 (49)

where 1198850and 119868



0



119882119898=120583

2int119878perp

sdot lowast

119889119878perp (50)


= 1198711198682



contributions119882119898= 119882

1+119882

2

Thus

1198711015840

=120583119885

2

0

1198812

0

(int119878I

sdot lowast

119889119878perp+ int

119878II

sdot lowast

119889119878perp) (51)


Conductor 1

Conductor 2

+

+

++

+ +

+

++

++

++

∙

∙

+

+

+

+

minus

minusminus

minusminus

minusminus

minus

Region I

Region II

120579

1205752

rarrn equiv e120575

1205751

Φ(1205752 120579) = 0

Φ(1205751 120579) = 0

12057511205751

1205752

1205752

SI

SII

(a)

+ + + + + + + + +

_ _ _ _ _ _ ___V0 V0

Q1 Q2

(b)

Inner conductor 2

Inner conductor 1

Outer conductor 1

y z

x

∙

∙

∙

∙

W

Outer conductor 2

JS2 int

JS1 int

JS2 out

JS1 out

(c)



1198711015840

=120583

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

minus1

1198850=120578

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587(1 + 119892

2

)

1205752minus 120575

1+ 21205871198922(1 + 1198922)

)

minus1

(52)







Conductor 1

Conductor 2

minusI

WI

rarrBIIperp

rarrBIperp

dSII119911120575

dSI119911120575

∙

∙

(a)

Conductor 1

Conductor 2

W

minusI

∙

I

rarrBperp

dSperp = dSz120593

(b)

WConductor 1

Conductor 2

dSz120575

rarrBperp

minusI

I

(c)








related as follows

1198711015840

= 120583119885

120578

1198621015840

= 120598120578

119885

(53)



120597119881

120597119911= minus119871

1015840120597119868

120597119905minus 119877

1015840

119868

120597119868

120597119911= minus119862

1015840120597119881

120597119905minus 119866

1015840

119881

(54)




= 0 and 1198661015840



L998400dz R998400dz

C9984001dz G998400

1dz G9984002dzC998400

2dzV

I I +120597I

120597zdz

V +120597V

120597zdz

dz

(a)

L998400dz R998400dz

V

I

C998400dz G998400dz

I +120597I

120597zdz

V +120597V

120597zdz

dz

(b)



1205972

119881

1205971199112=1

V1205972

119881

1205971199052

1205972

119868

1205971199112=1

V1205972

119868

1205971199052 V =

1

radic11987110158401198621015840

(55)


119881 (120596) =1

2120587int

infin

minusinfin

119881 (119905) 119890minus119894120596119905

119889119905

119868 (120596) =1

2120587int

infin

minusinfin

119868 (119905) 119890minus119894120596119905

119889119905

(56)


119881 = 119881+119890minus120574119911

+ 119881minus119890120574119911

119868 =1

1198850

(119881+119890minus120574119911

minus 119881minus119890120574119911

)

1205742

= minus1205962

1198711015840

1198621015840

(57)




Γ =119881

minus

119881+

=119885

119871minus 119885

0

119885119871+ 119885

0

120591 =119881

119871

119881+

=2119885

119871

119885119871+ 119885

0

(58)




1205972

119881

1205971199112= 119877

1015840

(1198661015840

119881 + 1198621015840120597119881

120597119905) + 119871

1015840

(1198621015840120597119881

120597119905+ 119862

10158401205972

119881

1205971199052)

1205972

119868

1205971199112= 119877

1015840

(1198661015840

119868 + 1198621015840120597119868

120597119905) + 119871

1015840

(1198621015840120597119868

120597119905+ 119862

10158401205972

119868

1205971199052)

(59)


120574 = [minus1205962

1198711015840

1198621015840

+ 1198771015840

1198661015840

+ 119894120596 (1198771015840

1198621015840

+ 1198711015840

1198661015840

)]12

1198850= (

1198771015840

+ 1198941205961198711015840

1198661015840 + 1198941205961198711015840)

12

(60)


1015840

≪ 1205961198711015840 and 119866

1015840


holds

120574 ≃ 119894120596radic11987110158401198621015840 +1

2

radic11987110158401198621015840 (119877

1015840

1198711015840+119866

1015840

1198621015840) = 120572 + 119894120573 (61)



119875119888=119877

119878

2int119878120579119911

119869119878sdot 119869

lowast

119878119889119878

120579119911 (62)



In (62) 119877119904= 1(120590120575



119878= radic2(120596120583120590)


1015840

minus 11989412059810158401015840 a


10158401015840

1205981015840



119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)


119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)


119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)





SWR =1 + |Γ|

1 minus |Γ| (67)


119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)


119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)







119891minus120579

119894| le 2120587 and

|1205791| |120579


2

(1+1198922

)





Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)


Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)


perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)


119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)




Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119875119911=1

2Re(∬119864

perptimes 119867

lowast

perpsdot 119890

119911119889119878

perp) (10)





119888= 0 or 120596 = 120573119888


perp=1

120578119890119911times

perp (11)




nablaperptimes

perp= 0

nabla sdot perp= 0

(12)




perpand

perp the


119878119911=1

2Re (

perptimes

lowast

perp) sdot 119890

119911=1

120578

10038161003816100381610038161003816

perp

10038161003816100381610038161003816

2

= 12057810038161003816100381610038161003816

perp

10038161003816100381610038161003816

2

(13)





119909 = 119890(120575119892minus119892120579) cos (120575 + 120579)

119910 = 119890(120575119892minus119892120579) sin (120575 + 120579)

119911 = 119911

(14)







1 120579 = 120579

2and 120575 = 120575

1 120575 = 120575

2 120575 = 120575

1minus 2120587119892

2

(1 + 1198922



1= 2119902120587119892

2

(1+1198922


1 120575 = 120575


a shift of Δ120579119904= 2119902120587(1 + 119892

2


2minus120575

1| lt 2120587119892

2

(1+1198922



1

and 120575 = 1205752


119889119909 =120597119909

120597120575119889120575 +

120597119909

120597120579119889120579 +

120597119909

120597119911119889119911

119889119910 =120597119910

120597120575119889120575 +

120597119910

120597120579119889120579 +

120597119910

120597119911119889119911

119889119911 = 119889119911

(15)


119889ℓ2

= 1198891199092

+ 1198891199102

+ 1198891199112

= 119892120575120575119889120575

2

+ 119892120579120579119889120579

2

+ 119892119911119911119889119911

2

(16)

where

ℎ2

120575= 119892

120575120575= 119890

(2(120575119892)minus2119892120579)

(1 +1

1198922)

ℎ2

120579= 119892

120579120579= 119890

(2(120575119892)minus2119892120579)

(1 + 1198922

)

ℎ2

119911= 119892

119911119911= 1 119892

120575120579= 119892

120575119911= 119892

120579119911= 0

(17)


119889119881 = 119869 119889120575 119889120579 119889119911 = 119890(2(120575119892)minus2119892120579)

1 + 1198922

119892119889120575 119889120579 119889119911 (18)


120575 119890

120579 119890

119911

119890120575=120597

120597120575

= 119890(120575119892minus119892120579)

(1

119892cos (120575 + 120579) minus sin (120575 + 120579)) 119890

119909

+ 119890(120575119892minus119892120579)

(1

119892sin (120575 + 120579) + cos (120575 + 120579)) 119890

119910


0

Conductor 1

Conductor 2

Region I

Region II

or

y

1205791

120579

1205792

1205751

1205752

x

1205751 minus2120587g2

1 + g2

∙

∙SI

SII

(a)

0

Region II

0

1205791

120579

1205792

120575

1205751205752 1205751

Φ

V0

∙

∙


Region I

Con

duct

or 1

Con

duct

or 2

Con

duct

or 1

1205792 minus2120587

1 + g2

1205791 minus2120587g2

1 + g2

SI

SII

(b)

y

z

x

rarrdS120575z

rarrdS120579z

rarrdS120575120579

(c)


119890120579=120597

120597120579

= 119890(120575119892minus119892120579)


+ 119890(120575119892minus119892120579)

(minus119892 sin (120575 + 120579) + cos (120575 + 120579)) 119890119910

119890119911=120597

120597119911= 119890

119911

(19)


119910 119890

119911


10038171003817100381710038171003817119889 119878

120575120579

10038171003817100381710038171003817= 119889119878

perp=

10038171003817100381710038171003817100381710038171003817

120597

120597120575times120597

120597120579

10038171003817100381710038171003817100381710038171003817119889120575 119889120579

= 119890(2120575119892minus2119892120579)

(1

119892+ 119892)119889120575 119889120579

10038171003817100381710038171003817119889 119878

120579119911

10038171003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

120597

120597120579times120597

120597119911

10038171003817100381710038171003817100381710038171003817119889120579 119889119911 = 119890

(120575119892minus119892120579)radic1 + 1198922119889119911 119889120579

10038171003817100381710038171003817119889 119878

120575119911

10038171003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

120597

120597120575times120597

120597119911

10038171003817100381710038171003817100381710038171003817119889120575 119889119911 = 119890

(120575119892minus119892120579)

radic1 + 1198922

119892119889119911 119889120575

(20)


119890120575=119890120575

ℎ120575

119890120579=119890120579

ℎ120579

119890119911=119890119911

ℎ119911

(21)



119890120575= 119890

120579times 119890

119911 119890

120579= 119890

119911times 119890

120575 119890

119911= 119890

120575times 119890

120579

119890120575sdot 119890

120579= 0 = 119890

120575sdot 119890

119911= 119890

120579sdot 119890

119911= 0

(22)


2


119903 =119890(120575119892minus119892120579)

radic1 + 1198922

(119890120575minus 119892119890

120579) + 119911119890

119911 (23)


perp= minus1119892


119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)

119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)

119911 = 119911

(24)




119890minus2(120575119892)+2119892120579

1 + 1198922[119892

21205972

Φ

1205971205752+1205972

Φ

1205971205792] = 0 (25)



Φ isin C(0)

[[1205751minus

21205871198922

1 + 1198922 120575


capC(2)

[[1205751minus

21205871198922

1 + 1198922 120575


Φ isin C(0)

[[1205752 120575


capC(2)

[[1205752 120575


(26)



0120598119903is


21205871198922

(1 + 1198922



Φ(1205751 120579) = 119881

0


(27)


Φ(1205752 120579) = 0

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 119881


(28)


120575 = 1205751and 120575 = 120575

1minus 2120587119892

2

(1 + 1198922



Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)


1198922

119877 (120575)

1205972

119877 (120575)

1205971205752+

1

119875 (120579)

1205972

119875 (120579)

1205971205792= 0 (30)


1198922

119877 (120575)

1205972

119877 (120575)

1205971205752= minus119896

2

120575 (31)

1

119875 (120579)

1205972

119875 (120579)

1205971205792= minus119896

2

120579 (32)

1198962

120575+ 119896

2

120579= 0 (33)


119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896

120579120579) (34)


120579



1205972

119877 (120575)

1205971205752= 0 (35)

and so

Φ (120575 120579) = 119862120575 + 119863 (36)




Φ(1205751 120579) = 0 = 119862I1205751

+ 119863I

Φ (1205752 120579) = 119881

0= 119862I1205752

+ 119863I(37)


Φ(1205752 120579) = 119881

0= 119862II1205752

+ 119863II

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 0 = 119862II (1205751

minus2120587119892

2

1 + 1198922) + 119863II

(38)


Φ (120575 120579) =119881

0

1205752minus 120575

1

(120575 minus 1205751)


1]

Φ (120575 120579) =119881

0

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

(120575 minus 1205751+

21205871198922

1 + 1198922)


21205871198922

1 + 1198922 120575

2]

(39)


region I

perp= 119864

120575119890120575= minusnabla

perpΦ = minus

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810

1205752minus 120575

1

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1

119890120579

119867120575= 0

region II

perp= 119864

120575119890120575= minusnabla

perpΦ

= minus119892119890

(minus120575119892+119892120579)

radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120579

119867120575= 0

(40)










119869119878tot



119882119890=1

2int119878perp

1205981015840

sdot lowast

119889119878perp (43)


1015840

|1198810|2


1198621015840

=1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878

perp sdot

lowast

119889119878perp [Fm] (44)




1minus 120579

2| ≫ Δ120575) with |120579

1| |120579






++

+

+

+

+

+

+

++

+

+

+

+

+

Inner conductor

Outer conductor

minusminus minus

minusminusminus

(a)

Outer conductor

Inner conductor+

+

+

+

+

+

+

+

+

+

+

+

+

minus

minus

minus

minusminus

minus

minus

minus

minus

minusminus

minus

(b)



1198621015840

1=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878I

I sdot lowast

I 119889119878perp =119892120598

1015840

(1205792minus 120579

1)

1205751minus 120575

2

1198621015840

2=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878II

II sdot lowast

II119889119878perp

=119892120598

1015840

(1205792minus 120579

1minus 2120587 (1 + 119892

2

))

(1205752minus 120575

1+ 21205871198922 (1 + 1198922))

1205792gt 120579

1

(45)

Thus

1198621015840

tot = 1198621015840

1+ 119862

1015840

2

= 120598119892119882(1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

(46)



1minus 120579



1minus 2120587(1 + 119892

2





and 119889119878II119911120575


sdot 119889 119878III119911120575

while the


ΦI = ΦII = 1198821198810

120583

120578 (47)


Φ1015840

III =ΦIII

119882= 119871

1015840

1198680 (48)

Consequently

1198711015840

= 1198850

120583

120578 (49)

where 1198850and 119868



0



119882119898=120583

2int119878perp

sdot lowast

119889119878perp (50)


= 1198711198682



contributions119882119898= 119882

1+119882

2

Thus

1198711015840

=120583119885

2

0

1198812

0

(int119878I

sdot lowast

119889119878perp+ int

119878II

sdot lowast

119889119878perp) (51)


Conductor 1

Conductor 2

+

+

++

+ +

+

++

++

++

∙

∙

+

+

+

+

minus

minusminus

minusminus

minusminus

minus

Region I

Region II

120579

1205752

rarrn equiv e120575

1205751

Φ(1205752 120579) = 0

Φ(1205751 120579) = 0

12057511205751

1205752

1205752

SI

SII

(a)

+ + + + + + + + +

_ _ _ _ _ _ ___V0 V0

Q1 Q2

(b)

Inner conductor 2

Inner conductor 1

Outer conductor 1

y z

x

∙

∙

∙

∙

W

Outer conductor 2

JS2 int

JS1 int

JS2 out

JS1 out

(c)



1198711015840

=120583

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

minus1

1198850=120578

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587(1 + 119892

2

)

1205752minus 120575

1+ 21205871198922(1 + 1198922)

)

minus1

(52)







Conductor 1

Conductor 2

minusI

WI

rarrBIIperp

rarrBIperp

dSII119911120575

dSI119911120575

∙

∙

(a)

Conductor 1

Conductor 2

W

minusI

∙

I

rarrBperp

dSperp = dSz120593

(b)

WConductor 1

Conductor 2

dSz120575

rarrBperp

minusI

I

(c)








related as follows

1198711015840

= 120583119885

120578

1198621015840

= 120598120578

119885

(53)



120597119881

120597119911= minus119871

1015840120597119868

120597119905minus 119877

1015840

119868

120597119868

120597119911= minus119862

1015840120597119881

120597119905minus 119866

1015840

119881

(54)




= 0 and 1198661015840



L998400dz R998400dz

C9984001dz G998400

1dz G9984002dzC998400

2dzV

I I +120597I

120597zdz

V +120597V

120597zdz

dz

(a)

L998400dz R998400dz

V

I

C998400dz G998400dz

I +120597I

120597zdz

V +120597V

120597zdz

dz

(b)



1205972

119881

1205971199112=1

V1205972

119881

1205971199052

1205972

119868

1205971199112=1

V1205972

119868

1205971199052 V =

1

radic11987110158401198621015840

(55)


119881 (120596) =1

2120587int

infin

minusinfin

119881 (119905) 119890minus119894120596119905

119889119905

119868 (120596) =1

2120587int

infin

minusinfin

119868 (119905) 119890minus119894120596119905

119889119905

(56)


119881 = 119881+119890minus120574119911

+ 119881minus119890120574119911

119868 =1

1198850

(119881+119890minus120574119911

minus 119881minus119890120574119911

)

1205742

= minus1205962

1198711015840

1198621015840

(57)




Γ =119881

minus

119881+

=119885

119871minus 119885

0

119885119871+ 119885

0

120591 =119881

119871

119881+

=2119885

119871

119885119871+ 119885

0

(58)




1205972

119881

1205971199112= 119877

1015840

(1198661015840

119881 + 1198621015840120597119881

120597119905) + 119871

1015840

(1198621015840120597119881

120597119905+ 119862

10158401205972

119881

1205971199052)

1205972

119868

1205971199112= 119877

1015840

(1198661015840

119868 + 1198621015840120597119868

120597119905) + 119871

1015840

(1198621015840120597119868

120597119905+ 119862

10158401205972

119868

1205971199052)

(59)


120574 = [minus1205962

1198711015840

1198621015840

+ 1198771015840

1198661015840

+ 119894120596 (1198771015840

1198621015840

+ 1198711015840

1198661015840

)]12

1198850= (

1198771015840

+ 1198941205961198711015840

1198661015840 + 1198941205961198711015840)

12

(60)


1015840

≪ 1205961198711015840 and 119866

1015840


holds

120574 ≃ 119894120596radic11987110158401198621015840 +1

2

radic11987110158401198621015840 (119877

1015840

1198711015840+119866

1015840

1198621015840) = 120572 + 119894120573 (61)



119875119888=119877

119878

2int119878120579119911

119869119878sdot 119869

lowast

119878119889119878

120579119911 (62)



In (62) 119877119904= 1(120590120575



119878= radic2(120596120583120590)


1015840

minus 11989412059810158401015840 a


10158401015840

1205981015840



119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)


119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)


119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)





SWR =1 + |Γ|

1 minus |Γ| (67)


119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)


119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)







119891minus120579

119894| le 2120587 and

|1205791| |120579


2

(1+1198922

)





Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)


Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)


perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)


119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)




Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

Conductor 1

Conductor 2

Region I

Region II

or

y

1205791

120579

1205792

1205751

1205752

x

1205751 minus2120587g2

1 + g2

∙

∙SI

SII

(a)

0

Region II

0

1205791

120579

1205792

120575

1205751205752 1205751

Φ

V0

∙

∙


Region I

Con

duct

or 1

Con

duct

or 2

Con

duct

or 1

1205792 minus2120587

1 + g2

1205791 minus2120587g2

1 + g2

SI

SII

(b)

y

z

x

rarrdS120575z

rarrdS120579z

rarrdS120575120579

(c)


119890120579=120597

120597120579

= 119890(120575119892minus119892120579)


+ 119890(120575119892minus119892120579)

(minus119892 sin (120575 + 120579) + cos (120575 + 120579)) 119890119910

119890119911=120597

120597119911= 119890

119911

(19)


119910 119890

119911


10038171003817100381710038171003817119889 119878

120575120579

10038171003817100381710038171003817= 119889119878

perp=

10038171003817100381710038171003817100381710038171003817

120597

120597120575times120597

120597120579

10038171003817100381710038171003817100381710038171003817119889120575 119889120579

= 119890(2120575119892minus2119892120579)

(1

119892+ 119892)119889120575 119889120579

10038171003817100381710038171003817119889 119878

120579119911

10038171003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

120597

120597120579times120597

120597119911

10038171003817100381710038171003817100381710038171003817119889120579 119889119911 = 119890

(120575119892minus119892120579)radic1 + 1198922119889119911 119889120579

10038171003817100381710038171003817119889 119878

120575119911

10038171003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

120597

120597120575times120597

120597119911

10038171003817100381710038171003817100381710038171003817119889120575 119889119911 = 119890

(120575119892minus119892120579)

radic1 + 1198922

119892119889119911 119889120575

(20)


119890120575=119890120575

ℎ120575

119890120579=119890120579

ℎ120579

119890119911=119890119911

ℎ119911

(21)



119890120575= 119890

120579times 119890

119911 119890

120579= 119890

119911times 119890

120575 119890

119911= 119890

120575times 119890

120579

119890120575sdot 119890

120579= 0 = 119890

120575sdot 119890

119911= 119890

120579sdot 119890

119911= 0

(22)


2


119903 =119890(120575119892minus119892120579)

radic1 + 1198922

(119890120575minus 119892119890

120579) + 119911119890

119911 (23)


perp= minus1119892


119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)

119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)

119911 = 119911

(24)




119890minus2(120575119892)+2119892120579

1 + 1198922[119892

21205972

Φ

1205971205752+1205972

Φ

1205971205792] = 0 (25)



Φ isin C(0)

[[1205751minus

21205871198922

1 + 1198922 120575


capC(2)

[[1205751minus

21205871198922

1 + 1198922 120575


Φ isin C(0)

[[1205752 120575


capC(2)

[[1205752 120575


(26)



0120598119903is


21205871198922

(1 + 1198922



Φ(1205751 120579) = 119881

0


(27)


Φ(1205752 120579) = 0

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 119881


(28)


120575 = 1205751and 120575 = 120575

1minus 2120587119892

2

(1 + 1198922



Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)


1198922

119877 (120575)

1205972

119877 (120575)

1205971205752+

1

119875 (120579)

1205972

119875 (120579)

1205971205792= 0 (30)


1198922

119877 (120575)

1205972

119877 (120575)

1205971205752= minus119896

2

120575 (31)

1

119875 (120579)

1205972

119875 (120579)

1205971205792= minus119896

2

120579 (32)

1198962

120575+ 119896

2

120579= 0 (33)


119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896

120579120579) (34)


120579



1205972

119877 (120575)

1205971205752= 0 (35)

and so

Φ (120575 120579) = 119862120575 + 119863 (36)




Φ(1205751 120579) = 0 = 119862I1205751

+ 119863I

Φ (1205752 120579) = 119881

0= 119862I1205752

+ 119863I(37)


Φ(1205752 120579) = 119881

0= 119862II1205752

+ 119863II

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 0 = 119862II (1205751

minus2120587119892

2

1 + 1198922) + 119863II

(38)


Φ (120575 120579) =119881

0

1205752minus 120575

1

(120575 minus 1205751)


1]

Φ (120575 120579) =119881

0

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

(120575 minus 1205751+

21205871198922

1 + 1198922)


21205871198922

1 + 1198922 120575

2]

(39)


region I

perp= 119864

120575119890120575= minusnabla

perpΦ = minus

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810

1205752minus 120575

1

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1

119890120579

119867120575= 0

region II

perp= 119864

120575119890120575= minusnabla

perpΦ

= minus119892119890

(minus120575119892+119892120579)

radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120579

119867120575= 0

(40)










119869119878tot



119882119890=1

2int119878perp

1205981015840

sdot lowast

119889119878perp (43)


1015840

|1198810|2


1198621015840

=1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878

perp sdot

lowast

119889119878perp [Fm] (44)




1minus 120579

2| ≫ Δ120575) with |120579

1| |120579






++

+

+

+

+

+

+

++

+

+

+

+

+

Inner conductor

Outer conductor

minusminus minus

minusminusminus

(a)

Outer conductor

Inner conductor+

+

+

+

+

+

+

+

+

+

+

+

+

minus

minus

minus

minusminus

minus

minus

minus

minus

minusminus

minus

(b)



1198621015840

1=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878I

I sdot lowast

I 119889119878perp =119892120598

1015840

(1205792minus 120579

1)

1205751minus 120575

2

1198621015840

2=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878II

II sdot lowast

II119889119878perp

=119892120598

1015840

(1205792minus 120579

1minus 2120587 (1 + 119892

2

))

(1205752minus 120575

1+ 21205871198922 (1 + 1198922))

1205792gt 120579

1

(45)

Thus

1198621015840

tot = 1198621015840

1+ 119862

1015840

2

= 120598119892119882(1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

(46)



1minus 120579



1minus 2120587(1 + 119892

2





and 119889119878II119911120575


sdot 119889 119878III119911120575

while the


ΦI = ΦII = 1198821198810

120583

120578 (47)


Φ1015840

III =ΦIII

119882= 119871

1015840

1198680 (48)

Consequently

1198711015840

= 1198850

120583

120578 (49)

where 1198850and 119868



0



119882119898=120583

2int119878perp

sdot lowast

119889119878perp (50)


= 1198711198682



contributions119882119898= 119882

1+119882

2

Thus

1198711015840

=120583119885

2

0

1198812

0

(int119878I

sdot lowast

119889119878perp+ int

119878II

sdot lowast

119889119878perp) (51)


Conductor 1

Conductor 2

+

+

++

+ +

+

++

++

++

∙

∙

+

+

+

+

minus

minusminus

minusminus

minusminus

minus

Region I

Region II

120579

1205752

rarrn equiv e120575

1205751

Φ(1205752 120579) = 0

Φ(1205751 120579) = 0

12057511205751

1205752

1205752

SI

SII

(a)

+ + + + + + + + +

_ _ _ _ _ _ ___V0 V0

Q1 Q2

(b)

Inner conductor 2

Inner conductor 1

Outer conductor 1

y z

x

∙

∙

∙

∙

W

Outer conductor 2

JS2 int

JS1 int

JS2 out

JS1 out

(c)



1198711015840

=120583

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

minus1

1198850=120578

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587(1 + 119892

2

)

1205752minus 120575

1+ 21205871198922(1 + 1198922)

)

minus1

(52)







Conductor 1

Conductor 2

minusI

WI

rarrBIIperp

rarrBIperp

dSII119911120575

dSI119911120575

∙

∙

(a)

Conductor 1

Conductor 2

W

minusI

∙

I

rarrBperp

dSperp = dSz120593

(b)

WConductor 1

Conductor 2

dSz120575

rarrBperp

minusI

I

(c)








related as follows

1198711015840

= 120583119885

120578

1198621015840

= 120598120578

119885

(53)



120597119881

120597119911= minus119871

1015840120597119868

120597119905minus 119877

1015840

119868

120597119868

120597119911= minus119862

1015840120597119881

120597119905minus 119866

1015840

119881

(54)




= 0 and 1198661015840



L998400dz R998400dz

C9984001dz G998400

1dz G9984002dzC998400

2dzV

I I +120597I

120597zdz

V +120597V

120597zdz

dz

(a)

L998400dz R998400dz

V

I

C998400dz G998400dz

I +120597I

120597zdz

V +120597V

120597zdz

dz

(b)



1205972

119881

1205971199112=1

V1205972

119881

1205971199052

1205972

119868

1205971199112=1

V1205972

119868

1205971199052 V =

1

radic11987110158401198621015840

(55)


119881 (120596) =1

2120587int

infin

minusinfin

119881 (119905) 119890minus119894120596119905

119889119905

119868 (120596) =1

2120587int

infin

minusinfin

119868 (119905) 119890minus119894120596119905

119889119905

(56)


119881 = 119881+119890minus120574119911

+ 119881minus119890120574119911

119868 =1

1198850

(119881+119890minus120574119911

minus 119881minus119890120574119911

)

1205742

= minus1205962

1198711015840

1198621015840

(57)




Γ =119881

minus

119881+

=119885

119871minus 119885

0

119885119871+ 119885

0

120591 =119881

119871

119881+

=2119885

119871

119885119871+ 119885

0

(58)




1205972

119881

1205971199112= 119877

1015840

(1198661015840

119881 + 1198621015840120597119881

120597119905) + 119871

1015840

(1198621015840120597119881

120597119905+ 119862

10158401205972

119881

1205971199052)

1205972

119868

1205971199112= 119877

1015840

(1198661015840

119868 + 1198621015840120597119868

120597119905) + 119871

1015840

(1198621015840120597119868

120597119905+ 119862

10158401205972

119868

1205971199052)

(59)


120574 = [minus1205962

1198711015840

1198621015840

+ 1198771015840

1198661015840

+ 119894120596 (1198771015840

1198621015840

+ 1198711015840

1198661015840

)]12

1198850= (

1198771015840

+ 1198941205961198711015840

1198661015840 + 1198941205961198711015840)

12

(60)


1015840

≪ 1205961198711015840 and 119866

1015840


holds

120574 ≃ 119894120596radic11987110158401198621015840 +1

2

radic11987110158401198621015840 (119877

1015840

1198711015840+119866

1015840

1198621015840) = 120572 + 119894120573 (61)



119875119888=119877

119878

2int119878120579119911

119869119878sdot 119869

lowast

119878119889119878

120579119911 (62)



In (62) 119877119904= 1(120590120575



119878= radic2(120596120583120590)


1015840

minus 11989412059810158401015840 a


10158401015840

1205981015840



119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)


119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)


119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)





SWR =1 + |Γ|

1 minus |Γ| (67)


119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)


119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)







119891minus120579

119894| le 2120587 and

|1205791| |120579


2

(1+1198922

)





Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)


Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)


perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)


119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)




Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119890120575= 119890

120579times 119890

119911 119890

120579= 119890

119911times 119890

120575 119890

119911= 119890

120575times 119890

120579

119890120575sdot 119890

120579= 0 = 119890

120575sdot 119890

119911= 119890

120579sdot 119890

119911= 0

(22)


2


119903 =119890(120575119892minus119892120579)

radic1 + 1198922

(119890120575minus 119892119890

120579) + 119911119890

119911 (23)


perp= minus1119892


119909 = 119890(minus119892120575+120579119892) cos (120575 + 120579)

119910 = 119890(minus119892120575+120579119892) sin (120575 + 120579)

119911 = 119911

(24)




119890minus2(120575119892)+2119892120579

1 + 1198922[119892

21205972

Φ

1205971205752+1205972

Φ

1205971205792] = 0 (25)



Φ isin C(0)

[[1205751minus

21205871198922

1 + 1198922 120575


capC(2)

[[1205751minus

21205871198922

1 + 1198922 120575


Φ isin C(0)

[[1205752 120575


capC(2)

[[1205752 120575


(26)



0120598119903is


21205871198922

(1 + 1198922



Φ(1205751 120579) = 119881

0


(27)


Φ(1205752 120579) = 0

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 119881


(28)


120575 = 1205751and 120575 = 120575

1minus 2120587119892

2

(1 + 1198922



Φ (120575 120579) = 119877 (120575) 119875 (120579) (29)


1198922

119877 (120575)

1205972

119877 (120575)

1205971205752+

1

119875 (120579)

1205972

119875 (120579)

1205971205792= 0 (30)


1198922

119877 (120575)

1205972

119877 (120575)

1205971205752= minus119896

2

120575 (31)

1

119875 (120579)

1205972

119875 (120579)

1205971205792= minus119896

2

120579 (32)

1198962

120575+ 119896

2

120579= 0 (33)


119875 (120579) = 119860 cos (119896120579120579) + 119861 sin (119896

120579120579) (34)


120579



1205972

119877 (120575)

1205971205752= 0 (35)

and so

Φ (120575 120579) = 119862120575 + 119863 (36)




Φ(1205751 120579) = 0 = 119862I1205751

+ 119863I

Φ (1205752 120579) = 119881

0= 119862I1205752

+ 119863I(37)


Φ(1205752 120579) = 119881

0= 119862II1205752

+ 119863II

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 0 = 119862II (1205751

minus2120587119892

2

1 + 1198922) + 119863II

(38)


Φ (120575 120579) =119881

0

1205752minus 120575

1

(120575 minus 1205751)


1]

Φ (120575 120579) =119881

0

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

(120575 minus 1205751+

21205871198922

1 + 1198922)


21205871198922

1 + 1198922 120575

2]

(39)


region I

perp= 119864

120575119890120575= minusnabla

perpΦ = minus

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810

1205752minus 120575

1

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1

119890120579

119867120575= 0

region II

perp= 119864

120575119890120575= minusnabla

perpΦ

= minus119892119890

(minus120575119892+119892120579)

radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120579

119867120575= 0

(40)










119869119878tot



119882119890=1

2int119878perp

1205981015840

sdot lowast

119889119878perp (43)


1015840

|1198810|2


1198621015840

=1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878

perp sdot

lowast

119889119878perp [Fm] (44)




1minus 120579

2| ≫ Δ120575) with |120579

1| |120579






++

+

+

+

+

+

+

++

+

+

+

+

+

Inner conductor

Outer conductor

minusminus minus

minusminusminus

(a)

Outer conductor

Inner conductor+

+

+

+

+

+

+

+

+

+

+

+

+

minus

minus

minus

minusminus

minus

minus

minus

minus

minusminus

minus

(b)



1198621015840

1=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878I

I sdot lowast

I 119889119878perp =119892120598

1015840

(1205792minus 120579

1)

1205751minus 120575

2

1198621015840

2=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878II

II sdot lowast

II119889119878perp

=119892120598

1015840

(1205792minus 120579

1minus 2120587 (1 + 119892

2

))

(1205752minus 120575

1+ 21205871198922 (1 + 1198922))

1205792gt 120579

1

(45)

Thus

1198621015840

tot = 1198621015840

1+ 119862

1015840

2

= 120598119892119882(1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

(46)



1minus 120579



1minus 2120587(1 + 119892

2





and 119889119878II119911120575


sdot 119889 119878III119911120575

while the


ΦI = ΦII = 1198821198810

120583

120578 (47)


Φ1015840

III =ΦIII

119882= 119871

1015840

1198680 (48)

Consequently

1198711015840

= 1198850

120583

120578 (49)

where 1198850and 119868



0



119882119898=120583

2int119878perp

sdot lowast

119889119878perp (50)


= 1198711198682



contributions119882119898= 119882

1+119882

2

Thus

1198711015840

=120583119885

2

0

1198812

0

(int119878I

sdot lowast

119889119878perp+ int

119878II

sdot lowast

119889119878perp) (51)


Conductor 1

Conductor 2

+

+

++

+ +

+

++

++

++

∙

∙

+

+

+

+

minus

minusminus

minusminus

minusminus

minus

Region I

Region II

120579

1205752

rarrn equiv e120575

1205751

Φ(1205752 120579) = 0

Φ(1205751 120579) = 0

12057511205751

1205752

1205752

SI

SII

(a)

+ + + + + + + + +

_ _ _ _ _ _ ___V0 V0

Q1 Q2

(b)

Inner conductor 2

Inner conductor 1

Outer conductor 1

y z

x

∙

∙

∙

∙

W

Outer conductor 2

JS2 int

JS1 int

JS2 out

JS1 out

(c)



1198711015840

=120583

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

minus1

1198850=120578

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587(1 + 119892

2

)

1205752minus 120575

1+ 21205871198922(1 + 1198922)

)

minus1

(52)







Conductor 1

Conductor 2

minusI

WI

rarrBIIperp

rarrBIperp

dSII119911120575

dSI119911120575

∙

∙

(a)

Conductor 1

Conductor 2

W

minusI

∙

I

rarrBperp

dSperp = dSz120593

(b)

WConductor 1

Conductor 2

dSz120575

rarrBperp

minusI

I

(c)








related as follows

1198711015840

= 120583119885

120578

1198621015840

= 120598120578

119885

(53)



120597119881

120597119911= minus119871

1015840120597119868

120597119905minus 119877

1015840

119868

120597119868

120597119911= minus119862

1015840120597119881

120597119905minus 119866

1015840

119881

(54)




= 0 and 1198661015840



L998400dz R998400dz

C9984001dz G998400

1dz G9984002dzC998400

2dzV

I I +120597I

120597zdz

V +120597V

120597zdz

dz

(a)

L998400dz R998400dz

V

I

C998400dz G998400dz

I +120597I

120597zdz

V +120597V

120597zdz

dz

(b)



1205972

119881

1205971199112=1

V1205972

119881

1205971199052

1205972

119868

1205971199112=1

V1205972

119868

1205971199052 V =

1

radic11987110158401198621015840

(55)


119881 (120596) =1

2120587int

infin

minusinfin

119881 (119905) 119890minus119894120596119905

119889119905

119868 (120596) =1

2120587int

infin

minusinfin

119868 (119905) 119890minus119894120596119905

119889119905

(56)


119881 = 119881+119890minus120574119911

+ 119881minus119890120574119911

119868 =1

1198850

(119881+119890minus120574119911

minus 119881minus119890120574119911

)

1205742

= minus1205962

1198711015840

1198621015840

(57)




Γ =119881

minus

119881+

=119885

119871minus 119885

0

119885119871+ 119885

0

120591 =119881

119871

119881+

=2119885

119871

119885119871+ 119885

0

(58)




1205972

119881

1205971199112= 119877

1015840

(1198661015840

119881 + 1198621015840120597119881

120597119905) + 119871

1015840

(1198621015840120597119881

120597119905+ 119862

10158401205972

119881

1205971199052)

1205972

119868

1205971199112= 119877

1015840

(1198661015840

119868 + 1198621015840120597119868

120597119905) + 119871

1015840

(1198621015840120597119868

120597119905+ 119862

10158401205972

119868

1205971199052)

(59)


120574 = [minus1205962

1198711015840

1198621015840

+ 1198771015840

1198661015840

+ 119894120596 (1198771015840

1198621015840

+ 1198711015840

1198661015840

)]12

1198850= (

1198771015840

+ 1198941205961198711015840

1198661015840 + 1198941205961198711015840)

12

(60)


1015840

≪ 1205961198711015840 and 119866

1015840


holds

120574 ≃ 119894120596radic11987110158401198621015840 +1

2

radic11987110158401198621015840 (119877

1015840

1198711015840+119866

1015840

1198621015840) = 120572 + 119894120573 (61)



119875119888=119877

119878

2int119878120579119911

119869119878sdot 119869

lowast

119878119889119878

120579119911 (62)



In (62) 119877119904= 1(120590120575



119878= radic2(120596120583120590)


1015840

minus 11989412059810158401015840 a


10158401015840

1205981015840



119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)


119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)


119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)





SWR =1 + |Γ|

1 minus |Γ| (67)


119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)


119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)







119891minus120579

119894| le 2120587 and

|1205791| |120579


2

(1+1198922

)





Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)


Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)


perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)


119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)




Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Φ(1205751 120579) = 0 = 119862I1205751

+ 119863I

Φ (1205752 120579) = 119881

0= 119862I1205752

+ 119863I(37)


Φ(1205752 120579) = 119881

0= 119862II1205752

+ 119863II

Φ(1205751minus

21205871198922

1 + 1198922 120579) = 0 = 119862II (1205751

minus2120587119892

2

1 + 1198922) + 119863II

(38)


Φ (120575 120579) =119881

0

1205752minus 120575

1

(120575 minus 1205751)


1]

Φ (120575 120579) =119881

0

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

(120575 minus 1205751+

21205871198922

1 + 1198922)


21205871198922

1 + 1198922 120575

2]

(39)


region I

perp= 119864

120575119890120575= minusnabla

perpΦ = minus

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810

1205752minus 120575

1

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1

119890120579

119867120575= 0

region II

perp= 119864

120575119890120575= minusnabla

perpΦ

= minus119892119890

(minus120575119892+119892120579)

radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575

=119892119890

(minus120575119892+119892120579)

120578radic1 + 1198922

1198810

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

119890120579

119867120575= 0

(40)










119869119878tot



119882119890=1

2int119878perp

1205981015840

sdot lowast

119889119878perp (43)


1015840

|1198810|2


1198621015840

=1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878

perp sdot

lowast

119889119878perp [Fm] (44)




1minus 120579

2| ≫ Δ120575) with |120579

1| |120579






++

+

+

+

+

+

+

++

+

+

+

+

+

Inner conductor

Outer conductor

minusminus minus

minusminusminus

(a)

Outer conductor

Inner conductor+

+

+

+

+

+

+

+

+

+

+

+

+

minus

minus

minus

minusminus

minus

minus

minus

minus

minusminus

minus

(b)



1198621015840

1=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878I

I sdot lowast

I 119889119878perp =119892120598

1015840

(1205792minus 120579

1)

1205751minus 120575

2

1198621015840

2=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878II

II sdot lowast

II119889119878perp

=119892120598

1015840

(1205792minus 120579

1minus 2120587 (1 + 119892

2

))

(1205752minus 120575

1+ 21205871198922 (1 + 1198922))

1205792gt 120579

1

(45)

Thus

1198621015840

tot = 1198621015840

1+ 119862

1015840

2

= 120598119892119882(1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

(46)



1minus 120579



1minus 2120587(1 + 119892

2





and 119889119878II119911120575


sdot 119889 119878III119911120575

while the


ΦI = ΦII = 1198821198810

120583

120578 (47)


Φ1015840

III =ΦIII

119882= 119871

1015840

1198680 (48)

Consequently

1198711015840

= 1198850

120583

120578 (49)

where 1198850and 119868



0



119882119898=120583

2int119878perp

sdot lowast

119889119878perp (50)


= 1198711198682



contributions119882119898= 119882

1+119882

2

Thus

1198711015840

=120583119885

2

0

1198812

0

(int119878I

sdot lowast

119889119878perp+ int

119878II

sdot lowast

119889119878perp) (51)


Conductor 1

Conductor 2

+

+

++

+ +

+

++

++

++

∙

∙

+

+

+

+

minus

minusminus

minusminus

minusminus

minus

Region I

Region II

120579

1205752

rarrn equiv e120575

1205751

Φ(1205752 120579) = 0

Φ(1205751 120579) = 0

12057511205751

1205752

1205752

SI

SII

(a)

+ + + + + + + + +

_ _ _ _ _ _ ___V0 V0

Q1 Q2

(b)

Inner conductor 2

Inner conductor 1

Outer conductor 1

y z

x

∙

∙

∙

∙

W

Outer conductor 2

JS2 int

JS1 int

JS2 out

JS1 out

(c)



1198711015840

=120583

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

minus1

1198850=120578

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587(1 + 119892

2

)

1205752minus 120575

1+ 21205871198922(1 + 1198922)

)

minus1

(52)







Conductor 1

Conductor 2

minusI

WI

rarrBIIperp

rarrBIperp

dSII119911120575

dSI119911120575

∙

∙

(a)

Conductor 1

Conductor 2

W

minusI

∙

I

rarrBperp

dSperp = dSz120593

(b)

WConductor 1

Conductor 2

dSz120575

rarrBperp

minusI

I

(c)








related as follows

1198711015840

= 120583119885

120578

1198621015840

= 120598120578

119885

(53)



120597119881

120597119911= minus119871

1015840120597119868

120597119905minus 119877

1015840

119868

120597119868

120597119911= minus119862

1015840120597119881

120597119905minus 119866

1015840

119881

(54)




= 0 and 1198661015840



L998400dz R998400dz

C9984001dz G998400

1dz G9984002dzC998400

2dzV

I I +120597I

120597zdz

V +120597V

120597zdz

dz

(a)

L998400dz R998400dz

V

I

C998400dz G998400dz

I +120597I

120597zdz

V +120597V

120597zdz

dz

(b)



1205972

119881

1205971199112=1

V1205972

119881

1205971199052

1205972

119868

1205971199112=1

V1205972

119868

1205971199052 V =

1

radic11987110158401198621015840

(55)


119881 (120596) =1

2120587int

infin

minusinfin

119881 (119905) 119890minus119894120596119905

119889119905

119868 (120596) =1

2120587int

infin

minusinfin

119868 (119905) 119890minus119894120596119905

119889119905

(56)


119881 = 119881+119890minus120574119911

+ 119881minus119890120574119911

119868 =1

1198850

(119881+119890minus120574119911

minus 119881minus119890120574119911

)

1205742

= minus1205962

1198711015840

1198621015840

(57)




Γ =119881

minus

119881+

=119885

119871minus 119885

0

119885119871+ 119885

0

120591 =119881

119871

119881+

=2119885

119871

119885119871+ 119885

0

(58)




1205972

119881

1205971199112= 119877

1015840

(1198661015840

119881 + 1198621015840120597119881

120597119905) + 119871

1015840

(1198621015840120597119881

120597119905+ 119862

10158401205972

119881

1205971199052)

1205972

119868

1205971199112= 119877

1015840

(1198661015840

119868 + 1198621015840120597119868

120597119905) + 119871

1015840

(1198621015840120597119868

120597119905+ 119862

10158401205972

119868

1205971199052)

(59)


120574 = [minus1205962

1198711015840

1198621015840

+ 1198771015840

1198661015840

+ 119894120596 (1198771015840

1198621015840

+ 1198711015840

1198661015840

)]12

1198850= (

1198771015840

+ 1198941205961198711015840

1198661015840 + 1198941205961198711015840)

12

(60)


1015840

≪ 1205961198711015840 and 119866

1015840


holds

120574 ≃ 119894120596radic11987110158401198621015840 +1

2

radic11987110158401198621015840 (119877

1015840

1198711015840+119866

1015840

1198621015840) = 120572 + 119894120573 (61)



119875119888=119877

119878

2int119878120579119911

119869119878sdot 119869

lowast

119878119889119878

120579119911 (62)



In (62) 119877119904= 1(120590120575



119878= radic2(120596120583120590)


1015840

minus 11989412059810158401015840 a


10158401015840

1205981015840



119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)


119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)


119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)





SWR =1 + |Γ|

1 minus |Γ| (67)


119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)


119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)







119891minus120579

119894| le 2120587 and

|1205791| |120579


2

(1+1198922

)





Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)


Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)


perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)


119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)




Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










++

+

+

+

+

+

+

++

+

+

+

+

+

Inner conductor

Outer conductor

minusminus minus

minusminusminus

(a)

Outer conductor

Inner conductor+

+

+

+

+

+

+

+

+

+

+

+

+

minus

minus

minus

minusminus

minus

minus

minus

minus

minusminus

minus

(b)



1198621015840

1=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878I

I sdot lowast

I 119889119878perp =119892120598

1015840

(1205792minus 120579

1)

1205751minus 120575

2

1198621015840

2=

1205981015840

10038161003816100381610038161198810

10038161003816100381610038162int119878II

II sdot lowast

II119889119878perp

=119892120598

1015840

(1205792minus 120579

1minus 2120587 (1 + 119892

2

))

(1205752minus 120575

1+ 21205871198922 (1 + 1198922))

1205792gt 120579

1

(45)

Thus

1198621015840

tot = 1198621015840

1+ 119862

1015840

2

= 120598119892119882(1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

(46)



1minus 120579



1minus 2120587(1 + 119892

2





and 119889119878II119911120575


sdot 119889 119878III119911120575

while the


ΦI = ΦII = 1198821198810

120583

120578 (47)


Φ1015840

III =ΦIII

119882= 119871

1015840

1198680 (48)

Consequently

1198711015840

= 1198850

120583

120578 (49)

where 1198850and 119868



0



119882119898=120583

2int119878perp

sdot lowast

119889119878perp (50)


= 1198711198682



contributions119882119898= 119882

1+119882

2

Thus

1198711015840

=120583119885

2

0

1198812

0

(int119878I

sdot lowast

119889119878perp+ int

119878II

sdot lowast

119889119878perp) (51)


Conductor 1

Conductor 2

+

+

++

+ +

+

++

++

++

∙

∙

+

+

+

+

minus

minusminus

minusminus

minusminus

minus

Region I

Region II

120579

1205752

rarrn equiv e120575

1205751

Φ(1205752 120579) = 0

Φ(1205751 120579) = 0

12057511205751

1205752

1205752

SI

SII

(a)

+ + + + + + + + +

_ _ _ _ _ _ ___V0 V0

Q1 Q2

(b)

Inner conductor 2

Inner conductor 1

Outer conductor 1

y z

x

∙

∙

∙

∙

W

Outer conductor 2

JS2 int

JS1 int

JS2 out

JS1 out

(c)



1198711015840

=120583

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

minus1

1198850=120578

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587(1 + 119892

2

)

1205752minus 120575

1+ 21205871198922(1 + 1198922)

)

minus1

(52)







Conductor 1

Conductor 2

minusI

WI

rarrBIIperp

rarrBIperp

dSII119911120575

dSI119911120575

∙

∙

(a)

Conductor 1

Conductor 2

W

minusI

∙

I

rarrBperp

dSperp = dSz120593

(b)

WConductor 1

Conductor 2

dSz120575

rarrBperp

minusI

I

(c)








related as follows

1198711015840

= 120583119885

120578

1198621015840

= 120598120578

119885

(53)



120597119881

120597119911= minus119871

1015840120597119868

120597119905minus 119877

1015840

119868

120597119868

120597119911= minus119862

1015840120597119881

120597119905minus 119866

1015840

119881

(54)




= 0 and 1198661015840



L998400dz R998400dz

C9984001dz G998400

1dz G9984002dzC998400

2dzV

I I +120597I

120597zdz

V +120597V

120597zdz

dz

(a)

L998400dz R998400dz

V

I

C998400dz G998400dz

I +120597I

120597zdz

V +120597V

120597zdz

dz

(b)



1205972

119881

1205971199112=1

V1205972

119881

1205971199052

1205972

119868

1205971199112=1

V1205972

119868

1205971199052 V =

1

radic11987110158401198621015840

(55)


119881 (120596) =1

2120587int

infin

minusinfin

119881 (119905) 119890minus119894120596119905

119889119905

119868 (120596) =1

2120587int

infin

minusinfin

119868 (119905) 119890minus119894120596119905

119889119905

(56)


119881 = 119881+119890minus120574119911

+ 119881minus119890120574119911

119868 =1

1198850

(119881+119890minus120574119911

minus 119881minus119890120574119911

)

1205742

= minus1205962

1198711015840

1198621015840

(57)




Γ =119881

minus

119881+

=119885

119871minus 119885

0

119885119871+ 119885

0

120591 =119881

119871

119881+

=2119885

119871

119885119871+ 119885

0

(58)




1205972

119881

1205971199112= 119877

1015840

(1198661015840

119881 + 1198621015840120597119881

120597119905) + 119871

1015840

(1198621015840120597119881

120597119905+ 119862

10158401205972

119881

1205971199052)

1205972

119868

1205971199112= 119877

1015840

(1198661015840

119868 + 1198621015840120597119868

120597119905) + 119871

1015840

(1198621015840120597119868

120597119905+ 119862

10158401205972

119868

1205971199052)

(59)


120574 = [minus1205962

1198711015840

1198621015840

+ 1198771015840

1198661015840

+ 119894120596 (1198771015840

1198621015840

+ 1198711015840

1198661015840

)]12

1198850= (

1198771015840

+ 1198941205961198711015840

1198661015840 + 1198941205961198711015840)

12

(60)


1015840

≪ 1205961198711015840 and 119866

1015840


holds

120574 ≃ 119894120596radic11987110158401198621015840 +1

2

radic11987110158401198621015840 (119877

1015840

1198711015840+119866

1015840

1198621015840) = 120572 + 119894120573 (61)



119875119888=119877

119878

2int119878120579119911

119869119878sdot 119869

lowast

119878119889119878

120579119911 (62)



In (62) 119877119904= 1(120590120575



119878= radic2(120596120583120590)


1015840

minus 11989412059810158401015840 a


10158401015840

1205981015840



119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)


119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)


119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)





SWR =1 + |Γ|

1 minus |Γ| (67)


119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)


119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)







119891minus120579

119894| le 2120587 and

|1205791| |120579


2

(1+1198922

)





Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)


Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)


perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)


119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)




Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Conductor 1

Conductor 2

+

+

++

+ +

+

++

++

++

∙

∙

+

+

+

+

minus

minusminus

minusminus

minusminus

minus

Region I

Region II

120579

1205752

rarrn equiv e120575

1205751

Φ(1205752 120579) = 0

Φ(1205751 120579) = 0

12057511205751

1205752

1205752

SI

SII

(a)

+ + + + + + + + +

_ _ _ _ _ _ ___V0 V0

Q1 Q2

(b)

Inner conductor 2

Inner conductor 1

Outer conductor 1

y z

x

∙

∙

∙

∙

W

Outer conductor 2

JS2 int

JS1 int

JS2 out

JS1 out

(c)



1198711015840

=120583

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205752minus 120575

1+ 21205871198922 (1 + 1198922)

)

minus1

1198850=120578

119892sdot (

1205792minus 120579

1

1205751minus 120575

2

+1205792minus 120579

1minus 2120587(1 + 119892

2

)

1205752minus 120575

1+ 21205871198922(1 + 1198922)

)

minus1

(52)







Conductor 1

Conductor 2

minusI

WI

rarrBIIperp

rarrBIperp

dSII119911120575

dSI119911120575

∙

∙

(a)

Conductor 1

Conductor 2

W

minusI

∙

I

rarrBperp

dSperp = dSz120593

(b)

WConductor 1

Conductor 2

dSz120575

rarrBperp

minusI

I

(c)








related as follows

1198711015840

= 120583119885

120578

1198621015840

= 120598120578

119885

(53)



120597119881

120597119911= minus119871

1015840120597119868

120597119905minus 119877

1015840

119868

120597119868

120597119911= minus119862

1015840120597119881

120597119905minus 119866

1015840

119881

(54)




= 0 and 1198661015840



L998400dz R998400dz

C9984001dz G998400

1dz G9984002dzC998400

2dzV

I I +120597I

120597zdz

V +120597V

120597zdz

dz

(a)

L998400dz R998400dz

V

I

C998400dz G998400dz

I +120597I

120597zdz

V +120597V

120597zdz

dz

(b)



1205972

119881

1205971199112=1

V1205972

119881

1205971199052

1205972

119868

1205971199112=1

V1205972

119868

1205971199052 V =

1

radic11987110158401198621015840

(55)


119881 (120596) =1

2120587int

infin

minusinfin

119881 (119905) 119890minus119894120596119905

119889119905

119868 (120596) =1

2120587int

infin

minusinfin

119868 (119905) 119890minus119894120596119905

119889119905

(56)


119881 = 119881+119890minus120574119911

+ 119881minus119890120574119911

119868 =1

1198850

(119881+119890minus120574119911

minus 119881minus119890120574119911

)

1205742

= minus1205962

1198711015840

1198621015840

(57)




Γ =119881

minus

119881+

=119885

119871minus 119885

0

119885119871+ 119885

0

120591 =119881

119871

119881+

=2119885

119871

119885119871+ 119885

0

(58)




1205972

119881

1205971199112= 119877

1015840

(1198661015840

119881 + 1198621015840120597119881

120597119905) + 119871

1015840

(1198621015840120597119881

120597119905+ 119862

10158401205972

119881

1205971199052)

1205972

119868

1205971199112= 119877

1015840

(1198661015840

119868 + 1198621015840120597119868

120597119905) + 119871

1015840

(1198621015840120597119868

120597119905+ 119862

10158401205972

119868

1205971199052)

(59)


120574 = [minus1205962

1198711015840

1198621015840

+ 1198771015840

1198661015840

+ 119894120596 (1198771015840

1198621015840

+ 1198711015840

1198661015840

)]12

1198850= (

1198771015840

+ 1198941205961198711015840

1198661015840 + 1198941205961198711015840)

12

(60)


1015840

≪ 1205961198711015840 and 119866

1015840


holds

120574 ≃ 119894120596radic11987110158401198621015840 +1

2

radic11987110158401198621015840 (119877

1015840

1198711015840+119866

1015840

1198621015840) = 120572 + 119894120573 (61)



119875119888=119877

119878

2int119878120579119911

119869119878sdot 119869

lowast

119878119889119878

120579119911 (62)



In (62) 119877119904= 1(120590120575



119878= radic2(120596120583120590)


1015840

minus 11989412059810158401015840 a


10158401015840

1205981015840



119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)


119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)


119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)





SWR =1 + |Γ|

1 minus |Γ| (67)


119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)


119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)







119891minus120579

119894| le 2120587 and

|1205791| |120579


2

(1+1198922

)





Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)


Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)


perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)


119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)




Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Conductor 1

Conductor 2

minusI

WI

rarrBIIperp

rarrBIperp

dSII119911120575

dSI119911120575

∙

∙

(a)

Conductor 1

Conductor 2

W

minusI

∙

I

rarrBperp

dSperp = dSz120593

(b)

WConductor 1

Conductor 2

dSz120575

rarrBperp

minusI

I

(c)








related as follows

1198711015840

= 120583119885

120578

1198621015840

= 120598120578

119885

(53)



120597119881

120597119911= minus119871

1015840120597119868

120597119905minus 119877

1015840

119868

120597119868

120597119911= minus119862

1015840120597119881

120597119905minus 119866

1015840

119881

(54)




= 0 and 1198661015840



L998400dz R998400dz

C9984001dz G998400

1dz G9984002dzC998400

2dzV

I I +120597I

120597zdz

V +120597V

120597zdz

dz

(a)

L998400dz R998400dz

V

I

C998400dz G998400dz

I +120597I

120597zdz

V +120597V

120597zdz

dz

(b)



1205972

119881

1205971199112=1

V1205972

119881

1205971199052

1205972

119868

1205971199112=1

V1205972

119868

1205971199052 V =

1

radic11987110158401198621015840

(55)


119881 (120596) =1

2120587int

infin

minusinfin

119881 (119905) 119890minus119894120596119905

119889119905

119868 (120596) =1

2120587int

infin

minusinfin

119868 (119905) 119890minus119894120596119905

119889119905

(56)


119881 = 119881+119890minus120574119911

+ 119881minus119890120574119911

119868 =1

1198850

(119881+119890minus120574119911

minus 119881minus119890120574119911

)

1205742

= minus1205962

1198711015840

1198621015840

(57)




Γ =119881

minus

119881+

=119885

119871minus 119885

0

119885119871+ 119885

0

120591 =119881

119871

119881+

=2119885

119871

119885119871+ 119885

0

(58)




1205972

119881

1205971199112= 119877

1015840

(1198661015840

119881 + 1198621015840120597119881

120597119905) + 119871

1015840

(1198621015840120597119881

120597119905+ 119862

10158401205972

119881

1205971199052)

1205972

119868

1205971199112= 119877

1015840

(1198661015840

119868 + 1198621015840120597119868

120597119905) + 119871

1015840

(1198621015840120597119868

120597119905+ 119862

10158401205972

119868

1205971199052)

(59)


120574 = [minus1205962

1198711015840

1198621015840

+ 1198771015840

1198661015840

+ 119894120596 (1198771015840

1198621015840

+ 1198711015840

1198661015840

)]12

1198850= (

1198771015840

+ 1198941205961198711015840

1198661015840 + 1198941205961198711015840)

12

(60)


1015840

≪ 1205961198711015840 and 119866

1015840


holds

120574 ≃ 119894120596radic11987110158401198621015840 +1

2

radic11987110158401198621015840 (119877

1015840

1198711015840+119866

1015840

1198621015840) = 120572 + 119894120573 (61)



119875119888=119877

119878

2int119878120579119911

119869119878sdot 119869

lowast

119878119889119878

120579119911 (62)



In (62) 119877119904= 1(120590120575



119878= radic2(120596120583120590)


1015840

minus 11989412059810158401015840 a


10158401015840

1205981015840



119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)


119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)


119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)





SWR =1 + |Γ|

1 minus |Γ| (67)


119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)


119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)







119891minus120579

119894| le 2120587 and

|1205791| |120579


2

(1+1198922

)





Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)


Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)


perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)


119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)




Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










L998400dz R998400dz

C9984001dz G998400

1dz G9984002dzC998400

2dzV

I I +120597I

120597zdz

V +120597V

120597zdz

dz

(a)

L998400dz R998400dz

V

I

C998400dz G998400dz

I +120597I

120597zdz

V +120597V

120597zdz

dz

(b)



1205972

119881

1205971199112=1

V1205972

119881

1205971199052

1205972

119868

1205971199112=1

V1205972

119868

1205971199052 V =

1

radic11987110158401198621015840

(55)


119881 (120596) =1

2120587int

infin

minusinfin

119881 (119905) 119890minus119894120596119905

119889119905

119868 (120596) =1

2120587int

infin

minusinfin

119868 (119905) 119890minus119894120596119905

119889119905

(56)


119881 = 119881+119890minus120574119911

+ 119881minus119890120574119911

119868 =1

1198850

(119881+119890minus120574119911

minus 119881minus119890120574119911

)

1205742

= minus1205962

1198711015840

1198621015840

(57)




Γ =119881

minus

119881+

=119885

119871minus 119885

0

119885119871+ 119885

0

120591 =119881

119871

119881+

=2119885

119871

119885119871+ 119885

0

(58)




1205972

119881

1205971199112= 119877

1015840

(1198661015840

119881 + 1198621015840120597119881

120597119905) + 119871

1015840

(1198621015840120597119881

120597119905+ 119862

10158401205972

119881

1205971199052)

1205972

119868

1205971199112= 119877

1015840

(1198661015840

119868 + 1198621015840120597119868

120597119905) + 119871

1015840

(1198621015840120597119868

120597119905+ 119862

10158401205972

119868

1205971199052)

(59)


120574 = [minus1205962

1198711015840

1198621015840

+ 1198771015840

1198661015840

+ 119894120596 (1198771015840

1198621015840

+ 1198711015840

1198661015840

)]12

1198850= (

1198771015840

+ 1198941205961198711015840

1198661015840 + 1198941205961198711015840)

12

(60)


1015840

≪ 1205961198711015840 and 119866

1015840


holds

120574 ≃ 119894120596radic11987110158401198621015840 +1

2

radic11987110158401198621015840 (119877

1015840

1198711015840+119866

1015840

1198621015840) = 120572 + 119894120573 (61)



119875119888=119877

119878

2int119878120579119911

119869119878sdot 119869

lowast

119878119889119878

120579119911 (62)



In (62) 119877119904= 1(120590120575



119878= radic2(120596120583120590)


1015840

minus 11989412059810158401015840 a


10158401015840

1205981015840



119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)


119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)


119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)





SWR =1 + |Γ|

1 minus |Γ| (67)


119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)


119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)







119891minus120579

119894| le 2120587 and

|1205791| |120579


2

(1+1198922

)





Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)


Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)


perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)


119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)




Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











In (62) 119877119904= 1(120590120575



119878= radic2(120596120583120590)


1015840

minus 11989412059810158401015840 a


10158401015840

1205981015840



119875119889=120596120598

10158401015840

2int119878Iperp

sdot lowast

119889119878perp+120596120598

10158401015840

2int119878IIperp

sdot lowast

119889119878perp (63)


119881 = 119881+(119890

minus120574119911

+ Γ119890120574119911

)

119868 =119881

+

1198850

(119890minus120574119911

minus Γ119890120574119911

)

(64)


119875avg =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1 minus |Γ|2

) (65)





SWR =1 + |Γ|

1 minus |Γ| (67)


119885in = 1198850

119885119871+ 119894119885

0tan 120574119897

119885119871minus 119894119885

0tan 120574119897

(68)


119875in =1

2

10038161003816100381610038161003816119881

0+

10038161003816100381610038161003816

2

1198850

(1198902120572119897

minus |Γ|2

1198902120572119897

) (69)







119891minus120579

119894| le 2120587 and

|1205791| |120579


2

(1+1198922

)





Φ(1205751 120579) = 0 = 119862

1198981205751+ 119863

119898

Φ(1205751+2119899120587119892

2

1 + 1198922 120579) = 119881

0= 119862

119898(120575

1+2119899120587119892

2

1 + 1198922) + 119863

119898

forall120579 isin [120579119894 120579

119891]

10038161003816100381610038161003816120579119894minus 120579

119891

10038161003816100381610038161003816le 2120587

(70)


Φ (120575 120579) = 1198810

1 + 1198922

21198991205871198922(120575 minus 120575

1) (71)


perp= 119864

120575119890120575= minusnabla

perpΦ =

119890(minus120575119892+119892120579)

radic1 + 1198922

1198921198810(1 + 119892

2

)

21198991205871198922119890120575

119864120579= 0

perp= 119867

120579119890120579=1

120578119890119911times 119864

120575119890120575= minus

119890(minus120575119892+119892120579)

120578radic1 + 1198922

1198810(1 + 119892

2

)

2119899120587119892119890120579

119867120575= 0

forall120579 isin [1205791198941

1205791198912

] 120575 isin [1205751 120575

1+2119899120587119892

2

1 + 1198922]

(72)


119876 = int119878119898

120590119889119878120579119911= 119882120598

1198810(1 + 119892

2

)

119899119892 (73)




Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Axondiameter

[119889]


119897





≃2 120583m ≃14120583m ≃00009 ≃13 ≃161205980120598mye

2120587

log(119863119889) 1205980120598mye

1 + 1198922

mye

2119899119897119892mye

120583mye

2120587log(119863119889) 120583mye119899119897

119892mye

1 + 1198922

mye

≃46119899119865119898

≃4119899119865119898

≃30119899119867119890119899119903119910

119898≃20

119899119867119890119899119903119910

119898




1198621015840

= 1205981 + 119892

2

119899119892 (74)





119863 = 119889 + 2 times 119899119897times 119896

119897 (75)





119889 = 2119890120575119898119892119898minus1198921198981205791198941

119863 = 2119890120575119898119892119898minus1198921198981205791198912

(76)

where 12057911989411198912





119899= 119890

120575119898119892minus4119899120587119892


1minus 119903

0= 119903

0(1 minus

119890minus4119892119898120587

) ≃ 0018 120583m [35] we have

119892mye ≃1

4120587ln( 119889

119889 minus 2119896119897

) (77)


0+119862

1119889

By taking 4120587119899119897= 120579

1198941

minus 1205791198912


1198941

≫ 1205791198912


119897


119899119897(119889) = Int 1

4120587119892119898

log [119862

0+ 119862

1119889

119889] (78)


1198711015840

= 120583119899119892

1 + 1198922

1198850= 120578119899

119892

1 + 1198922

(79)











119881 = 12058221205972

119881

1205971199112minus 120591

120597119881

120597119905

120582 =1

radic11987710158401198661015840

120591 =119862

1015840

1198661015840

119879 =120591ℓ

2

1205822= 119877

1015840

1198621015840

ℓ2

(80)



Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table2Transm

issionparametersfor

theM

DSC

CMSSCC

the

cylin

dricalcoaxand

thep

arallelplatelines

Dou

bles

piralcoax

Sing

lespira

lcoax

Cylin

dricalcoax

Parallelplate

1205751

1205752

a 21

a 22

a 11

a 12

1205791 1205792

a 21

a 22

a 11

a 12

a

b

d

D

1198711015840

120583 119892

1

(((1205792minus

1205791)(1205751minus

1205752))+

((1205792minus

1205791minus

(2120587(1+

1198922)))(1205752minus

1205751+

(21205871198922(1+

1198922)))))

120583

119899119892

1+

1198922

120583

2120587

ln119887 119886

120583

119889 119863

1198621015840

1205981015840119892119882

(

1205792minus

1205791

1205751minus

1205752

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205752minus

1205751+

(21205871198922(1+

1198922))

)1205981015840(1+

1198922)

119899119892

1205981015840

2120587

ln119887119886

1205981015840119863 119889

1198771015840

119877119878

16119892radic1+

1198922

1

(1205792minus

1205791minus

(120587(1+

1198922)))2

((1(1205751minus

1205752))+

1(1205752minus

1205751)+

(21205871198922(1+

1198922)))2

times

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1

11988622

(

1

(1205752minus

1205751)2

+

119890(minus(2119892120587(1+1198922)))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988621

(

1

(1205752minus

1205751)2

+

1

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

+

1

11988612

(

1

(1205752minus

1205751)2

+

119890minus(4119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

minus

1

11988611

(

1

(1205752minus

1205751)2

+

119890minus(2119892120587(1+1198922))

(1205751minus

1205752+

(21205871198922(1+

1198922)))2)

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

119886119901119902=

119890(120575119901119892)minus119892120579119902

119901119902=

12

119877119878

81205872radic1+

1198922

times

[ [ [ [ [

1

11988611

minus

1

11988612

+

119890minus(21198991205871198922(1+1198922))

11988621

minus

119890minus(21198991205871198922(1+1198922))

11988622

] ] ] ] ]

119886119901119902=

119890


119901119902=

12

119877119878

2120587

(

1 119886

+

1 119887

)

2119877119878

119863

1198661015840

12059612059810158401015840119892(

1205792minus

1205791

(1205752minus

1205751)

+

1205792minus

1205791minus

(2120587(1+

1198922))

1205751minus

1205752+

(21205871198922(1+

1198922))

)

12059612059810158401015840

119892

(1+

1198922)

212058712059612059810158401015840

ln119887119886

12059612059810158401015840119863

119889





119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119878perp

times lowast

sdot 119890119911119889119878

perp

by

119875119911

=

1

2int

1205752

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

+int

1205751+21205871198922(1+119892

2)

1205752

int

1205792

1205791+2120587(1+119892

2)

times lowast

sdot 119890119911119889119878

perp

=1

2radic120598

120583119892119881

2

0(1205792minus 120579

1

1205752minus 120575

1

+1205792minus 120579

1minus 2120587 (1 + 119892

2

)

1205751minus 120575

2+ 21205871198922 (1 + 1198922)

)

double coax

1

2int

1205751+2119899120587119892

2(1+119892

2)

1205751

int

1205792

1205791

times lowast

sdot 119890119911119889119878

perp

=1

120578

1 + 1198922

2119892119899119881

2

0 single coax

(81)


120575120579 of (20)







8 Conclusions









Appendix



119908 = 119906 + 119894V 119908 = 119891 (119911) (A1)

119891 (119911) = 1198600int

119911

1199110

1

120577119889120577 119860

0= 1 minus 119894119892 119911

0= 0 (A2)


119882(119911) = 1198600int

119911

1199110

119899

prod

119896=1

(120577 minus 120577119896)minus120572119896120587

119889120577 + 1198610

1198600 119861

0isin C

(A3)

because for 1205721= 120587 120577

1= 0 and 120572

119896= 0 forall119896 gt 1 120577

119896= 0 forall119896 ge 1






119911 = 119909 + 119894119910 = 119903119890119894120593

(A4)


Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Conductor 1


1205751 +2120587g2

1 + g

1205751120579i

120579f

120579f

+

++

+

+

+

+

+

+

minus

minusminus

minus

minus

minus

minus

120579i

(a)


Insulating layer



∙

∙

∙

∙

∙

∙∙

D

d

(b)

D2 d2

kl

minus

(c)



119906 + 119894V = (1 minus 119894119892) log 119911 + 119870 = 119891 (119911) (A5)


0= 119890



119906 + 119894V =1 + 119892

2

119892120575 + 119894 (1 + 119892

2

) 120579 (A6)


119908 = (1 minus 119894119892) (log 119903 + 119894120593) + 119870 (A7)


119911 = 119890120575119892minus119892120579+119894(120575+120579)

(A8)



becomes

(119889ℓ)2

= |119889119911|2

= (119889119909)2

+ (119889119910)2

= (119889119903)2

+ (119903119889120593)2

(A9)

where

|119889119911|2

=10038161003816100381610038161003816119891

1015840

(119911)10038161003816100381610038161003816

minus2

|119889119908|2

(A10)



(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(119889ℓ)2

= |119904|2

((119889119906)2

+ (119889V)2) |119904| equiv1

10038161003816100381610038161198911015840 (119911)

1003816100381610038161003816

(A11)


1198911015840

(119911) =1 minus 119894119892

119911 (A12)


(119889ℓ)2

= |119878|2

((119889120575

119892)

2

+ (119889120579)2

) (A13)

where

|119878| = (1 + 1198922

) |119904| (A14)





References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article The Spiral Coaxial Cabledownloads.hindawi.com/archive/2015/630131.pdf · A spiral...

Documents

Transcript of Research Article The Spiral Coaxial Cabledownloads.hindawi.com/archive/2015/630131.pdf · A spiral...