Electromagnetics (ENGR 367) Types of T-lines and Their Applications.

ElectromagneticsElectromagnetics(ENGR 367)(ENGR 367)

Types of T-lines Types of T-lines

and Their Applicationsand Their Applications

Outline of LectureOutline of Lecture

• Identify Types of T-lines versus WaveguidesIdentify Types of T-lines versus Waveguides

• Detail construction and applications of each Detail construction and applications of each typetype

• Provide formulas for T-line parameters of each Provide formulas for T-line parameters of each type that depend ontype that depend on– LF or HF operationLF or HF operation– Lossy or Lossless conditionsLossy or Lossless conditions

• Show examples of T-line parameter calculationShow examples of T-line parameter calculation

• Draw some conclusionsDraw some conclusions

Types of T-linesTypes of T-lines

• Coaxial cableCoaxial cable

• Two-wire lineTwo-wire line

• Stripline and Microstrip LineStripline and Microstrip Line

• Other specialized variationsOther specialized variations

T-lines versus WaveguidesT-lines versus Waveguides

• T-lines operate in the TEM mode only T-lines operate in the TEM mode only ((TTransverse ransverse EElectrolectroMMagnetic waves)agnetic waves)

• Waveguides carry EM waves that Waveguides carry EM waves that – May propagate in the TM or TE modes at May propagate in the TM or TE modes at

higher frequencies and may perform more higher frequencies and may perform more efficiently than T-lines in the microwave efficiently than T-lines in the microwave rangerange

– Take a zig-zag path as they reflect off the Take a zig-zag path as they reflect off the conducting boundaries and propagate conducting boundaries and propagate along the main axis of the waveguidealong the main axis of the waveguide

Detailed Description of Detailed Description of IndividualIndividual

T-line Types and Their T-line Types and Their ApplicationsApplications• Coaxial Cable: Basic ConstructionCoaxial Cable: Basic Construction

Radio-grade flexible RG-RG-59 (Z59 (Z00 = 75 = 75 )) coaxial cable.

A: outer plastic sheathB: copper screenC: inner dielectric insulatorD: copper core

Other standard types have similar construction.

from http://en.wikipedia.org/wiki/Coax

Coaxial CableCoaxial Cable

• Other aspects of basic constructionOther aspects of basic construction– Stiffness options available on the marketStiffness options available on the market

•Flexible type: has braided sheathFlexible type: has braided sheath

•Rigid type: has a solid sheathRigid type: has a solid sheath

(Sheath in either case typically made of Cu)(Sheath in either case typically made of Cu)

– Dielectric insulating layerDielectric insulating layer•Thickness and permittivity determine Thickness and permittivity determine

– Characteristic Impedance (ZCharacteristic Impedance (Z00))

– Attenuation (Attenuation ())

•May be either solid or perforatedMay be either solid or perforated

Applications of Coaxial Applications of Coaxial CableCable

• Short runs to connectShort runs to connect– Home video equipmentHome video equipment– Ham radio setups (transciever Ham radio setups (transciever

antenna)antenna)– Satellite TV (dish Rx Satellite TV (dish Rx set) set)– Cable modem & VSAT for Internet AccessCable modem & VSAT for Internet Access– Broadcast radio communication (Tx Broadcast radio communication (Tx

ant.)ant.)

Applications of Coaxial Applications of Coaxial CableCable

• Long runs to connectLong runs to connect

– Formerly radio and TV networksFormerly radio and TV networks

(Now replaced by optical and satellite (Now replaced by optical and satellite networks)networks)

– Presently cable TV signalsPresently cable TV signals

Specialized Variations of Specialized Variations of CoaxCoax

• Triaxial Cable (triax): includes a 3Triaxial Cable (triax): includes a 3rdrd layer layer of shielding, insulation and of shielding, insulation and sheathing, sheathing, the latter is grounded to the latter is grounded to further reduce further reduce outside interferenceoutside interference

• Twin-axial cable (twinax): balanced Twin-axial cable (twinax): balanced twisted pair within a cylindrical twisted pair within a cylindrical

shield shield for shielded and balanced for shielded and balanced differential differential signalssignals

• Multi-conductor coaxMulti-conductor coax

Review Skin DepthReview Skin Depth

• A measure of the depth electromagnetic A measure of the depth electromagnetic waves penetrate into a conductor where the waves penetrate into a conductor where the amplitude has decayed by eamplitude has decayed by e-1-1 = 0.368 = 0.368

• Note that as frequency (f) Note that as frequency (f) increasesincreases, the skin , the skin depth becomes depth becomes smallersmaller and more significant! and more significant!

1 [m]

f

Coax T-line GeometryCoax T-line Geometry

T-line Parameters for CoaxT-line Parameters for Coax(from Hayt & Buck, 7/e, pp. 483-(from Hayt & Buck, 7/e, pp. 483-

484)484) • Assuming HF operation such that the skin Assuming HF operation such that the skin

depth depth << a = radius of inner << a = radius of inner conductorconductor– Lossless approx. (Lossless approx. (RR << << LL and and GG << << CC):):

– Modifications for the lossy case:Modifications for the lossy case:

0

2ln (magnetostatics) (electrostatics)

2 ln( / )

1 60ln ln

2

ext

ext

r

b

a b a

b bZ

a a

L C

LC

ext0

2 1 1 1 (dielectric); (conductors)

ln( / ) 2

( )

( )

diel

cb a πδσ a b

jZ

j

G R

R LG C

T-line Parameters for CoaxT-line Parameters for Coax (from Hayt & Buck, 7/e, p. 485)(from Hayt & Buck, 7/e, p. 485)

• Modifications for LF operation where the Modifications for LF operation where the skin effect is negligible (skin effect is negligible ( >> a): current >> a): current distributes uniformly throughout conductor distributes uniformly throughout conductor cross-sections cross-sections – Resistance of conductors increasesResistance of conductors increases

2 2 2

1 1 1

( )

takes into account the entire cross-sections

of the inner and outer conductors

c a c b

R

T-line Parameters for CoaxT-line Parameters for Coax (from Hayt & Buck, 7/e, p. 485)(from Hayt & Buck, 7/e, p. 485)

• Modifications for LF operation (Modifications for LF operation ( >> a): >> a):– Internal inductance of conductors becomes significantInternal inductance of conductors becomes significant

• For intermediate frequencies (For intermediate frequencies ( a): a):– Expressions for parameters become more complicatedExpressions for parameters become more complicated– One can refer to handbook values as neededOne can refer to handbook values as needed

,int ,int

,int ,int 0

(as prev. derived); ( , ) (more complicated)8 8

(with , as modified for LF)

a b

a b

f b c

jZ

j

L L

R LL L L R L

G C

Example of Example of Parameter Calculations for Parameter Calculations for

CoaxCoax• Exercise 1 (D14.2a, H&B, 7/e, p. 486)Exercise 1 (D14.2a, H&B, 7/e, p. 486)

GivenGiven: a coax T-line with : a coax T-line with aa = 4 mm, = 4 mm, bb = 17.5 mm, = 17.5 mm, and and cc = 20 mm. Each conductor has = 20 mm. Each conductor has = 2 x 10 = 2 x 1077 S/m, and the dielectric has S/m, and the dielectric has rr = 1, = 1, r r = 3, and = 3, and // = 0.025. = 0.025.

FindFind: : LL, , CC, , RR, , GG, and Z, and Z00 at 150 MHz. at 150 MHz.

SolutionSolution: 1: 1stst find the skin depth; compare to find the skin depth; compare to aa6 7 7

6

1 1

(150 10 ) (4 10 )(2 10 )

9.2 10 m 9.2 m 4 mm

use HF model

f

a


CoaxCoax• Exercise 1 (continued)Exercise 1 (continued)

– SolutionSolution: next calculate coax T-line parameters: next calculate coax T-line parameters

7

ext

12

6 7 3 3c

d

4 10 17.5ln ln 295 nH/m

2 2

2 2 (3)(8.85 10 )113 pF/m

ln( / ) ln(17.5 / 4)

1 1 1 1 1 1266m /m

2 2 (9.2 10 )(2 10 ) 4 10 17.5 10

2 2 (0.025 )

ln( / ) ln(

b

a a

b a

a b

b a b

L

C

R

G6 122 (0.025)(2 )(150 10 )(3)(8.85 10 )

2.66 mS/m/ ) ln(17.5 / 4)a


CoaxCoax• Exercise 1 (continued)Exercise 1 (continued)

– SolutionSolution: check validity of lossless approx. for : check validity of lossless approx. for ZZ00 6 9

ext

ext

6 12

9ext

0 12

2 (150 10 )(295 10 ) 278 /m

266 m /m

2 (150 10 )(113 10 ) 106 mS/m

2.66 mS/m

295 1051 (lossless approx.)

113 10Z

L

L R

C

C G

LC

Two-wire LineTwo-wire Line

• Basic ConstructionBasic Construction– Two parallel circular conductors of equal radius Two parallel circular conductors of equal radius

and conductivity enclosed in a plastic and conductivity enclosed in a plastic insulating materialinsulating material

– Dielectric insulatorDielectric insulator• Provides mechanical spacing and some rigidityProvides mechanical spacing and some rigidity

• Affects ZAffects Z00 and and

Applications of Two-wire Applications of Two-wire LineLine

• As a lead-in to carry low level signals from As a lead-in to carry low level signals from antenna over a short run to a TV or FM Rxantenna over a short run to a TV or FM Rx

• Connections in regular telephone networksConnections in regular telephone networks

• In the conceptual development of a more In the conceptual development of a more sophisticated waveguide systems sophisticated waveguide systems (Fast Neutron Research Facility of CMU (Fast Neutron Research Facility of CMU

www.fnrf.science.cmu.ac.th/theory/ wavegu www.fnrf.science.cmu.ac.th/theory/ waveguide/Waveguide%2 0 theory%2 0 2 .htmlide/Waveguide%2 0 theory%2 0 2 .html) )

Parameters of Two-wire LineParameters of Two-wire Line(from Hayt & Buck, 7/e, pp. 486-(from Hayt & Buck, 7/e, pp. 486-

487)487)• For HF operation (For HF operation ( << a): << a):

– Lossless approximationLossless approximation

1ext 1

1ext0

1

cosh ( / 2 );cosh ( / 2 )

1cosh ( / 2 )

120cosh ( / 2 )

r

d ad a

Z d a

d a

L C

LC

Parameters of Two-wire LineParameters of Two-wire Line (from Hayt & Buck, 7/e, pp. 486-487)(from Hayt & Buck, 7/e, pp. 486-487)

• For HF operation (For HF operation ( << a): << a):– Modifications for Lossy conditionsModifications for Lossy conditions

• In LF operation: In LF operation: must modify above by including must modify above by including R R and and LL over entire cross-section of conductors as over entire cross-section of conductors as for coax for coax

.1

.

0

1 and

cosh ( / 2 )

(with and as above)

diel

conda d a

jZ

j

R G

R LR G

G C

Parameters of Two-wire LineParameters of Two-wire Line (from Hayt & Buck, 7/e, p. 487)(from Hayt & Buck, 7/e, p. 487)

• For LF operation (For LF operation ( >> a): >> a):– Inductance per unit length increases by twice Inductance per unit length increases by twice

the internal inductance of a straight round wirethe internal inductance of a straight round wire

– Resistance per unit length becomes twice the dc Resistance per unit length becomes twice the dc resistance of a wire of radius resistance of a wire of radius aa and conductivity and conductivity cc

11cosh

4 2

d

a

L

2c

2

a R

Example of Parameter Example of Parameter Calculation for Two-wire LineCalculation for Two-wire Line

• Exercise 2 (D14.3, H&B, 7/e, p. 487)Exercise 2 (D14.3, H&B, 7/e, p. 487)GivenGiven: a two-wire T-line with conductors each of : a two-wire T-line with conductors each of

radius 0.8 mm and conductivity 3 x 10radius 0.8 mm and conductivity 3 x 1077 S/m, S/m, separated by 0.8 cm in a dielectric where separated by 0.8 cm in a dielectric where rr = = 2.5, 2.5, rr = 1, and = 1, and dd = 4x10 = 4x10-9-9 S/m S/m

FindFind: : , , CC, , GG, , LL & & R R at 60 Hzat 60 Hz

SolutionSolution: validate LF model by comparing : validate LF model by comparing to to aa

7 7c

1 111.9 mm

(60) (4 10 )(3 10 )

0.8 mm so LF model applies

f

a

Example of Parameter Example of Parameter Calculation for Two-wire LineCalculation for Two-wire Line

• Exercise 2 (D14.3, H&B, 7/e, p. 487)Exercise 2 (D14.3, H&B, 7/e, p. 487)SolutionSolution: calculate the LF two-wire T-line circuit : calculate the LF two-wire T-line circuit

param’sparam’s

1 1

d1 1

1 1

2 4 2 7c

(2.5)(8.85 pF/m)30 pF/m

cosh ( / 2 ) cosh (8 /1.6)

(4 nS/m)5.5 nS/m

cosh ( / 2 ) cosh (8 /1.6)

1 (400 nH/m) 1= cosh ( ) cosh ( ) 1.0 H/m

4 2 4 2

2 233

(8 10 ) (3 10 )

d a

d a

d d

a a

a

C

G

L

R m /m

Striplines and Microstrip Striplines and Microstrip LinesLines

• Basic Construction: StriplineBasic Construction: Stripline– Single or double track strip of Cu Single or double track strip of Cu

imbedded in dielectric material imbedded in dielectric material sandwiched between conducting ground sandwiched between conducting ground planes on top planes on top andand bottom bottom


• Basic Construction: Microstrip Basic Construction: Microstrip – Single or double track strip of Cu on top Single or double track strip of Cu on top

of a dielectric substrate material above of a dielectric substrate material above a single conducting ground planea single conducting ground plane

• In practice, superstrate material may In practice, superstrate material may be be other than airother than air


• Basic Construction: dielectric materialBasic Construction: dielectric material

For HF applications, special substrate For HF applications, special substrate and superstrate (if present) materials and superstrate (if present) materials must be used with high uniformity and must be used with high uniformity and low loss tangent (tanlow loss tangent (tan = = ’/’/’) such as ’) such as Rogers RT Duroid or FR4 as Rogers RT Duroid or FR4 as manufactured for this application manufactured for this application (common PCB will not work!)(common PCB will not work!)

Applications of Applications of Stripline & Microstrip LineStripline & Microstrip Line

• Trace connections between devices in PCB Trace connections between devices in PCB microelectronic circuitsmicroelectronic circuits

• T-line connections between HF devices easily T-line connections between HF devices easily integrated with surface mount, distributed integrated with surface mount, distributed elements and microstrip antennas.elements and microstrip antennas.

• HF communication systems and devices with HF communication systems and devices with compact, flat, lightweight, constraints and compact, flat, lightweight, constraints and short run requirements such as cell phones, short run requirements such as cell phones, portable PCs and and other wireless mobile portable PCs and and other wireless mobile devicesdevices

Parameters for Microstrip Line Parameters for Microstrip Line (Single Track)(Single Track)

• If the strip width is large (w >> d), If the strip width is large (w >> d), then the structure acts like a parallel then the structure acts like a parallel plate line where for the low loss caseplate line where for the low loss case

0

377

r

d dZ

w w


• If w If w d or w < d (as typical for microstrip) then a d or w < d (as typical for microstrip) then a quasi TEM mode may be assumed to account for quasi TEM mode may be assumed to account for the propagation of waves through the two different the propagation of waves through the two different materials (e.g., air or superstrate and substrate materials (e.g., air or superstrate and substrate dielectrics)dielectrics)

• At low frequencies (f < 1.5 GHz) assuming At low frequencies (f < 1.5 GHz) assuming negligible losses over a short run the propagation negligible losses over a short run the propagation velocity isvelocity is p0

ext 0 0 0

pd

0

0p

ext r,eff

1 1(in air only)

1(in dielectric only)

1 (in dielectric-air medium)

r

v c

cv

cv c

L C

CCL C


• Definition of Effective Dielectric Constant Definition of Effective Dielectric Constant ((r,effr,eff):):

acts as a weighted average of the air (or acts as a weighted average of the air (or superstrate) superstrate)

and substrate dielectric constants with a and substrate dielectric constants with a proportionproportion

determined by the field filling factor (q)determined by the field filling factor (q)

2

r,eff r,eff0 p

r,eff

1 where ( 1)

2

or 1 ( 1) where 0.5 1

and as 1; 0.5

(parallel plate case; simple average case)

r r

r

c

v

q q

w wq q

d d

CC


• Empirical Formulas for Empirical Formulas for r,effr,eff assuming w/d > assuming w/d > 1.31.3– in terms of w/d for application to the dimensions in terms of w/d for application to the dimensions

of a pre-fabricated lineof a pre-fabricated line

– In terms of In terms of rr and Z and Z00 for finding the necessary for finding the necessary r,eff r,eff based on a desired Zbased on a desired Z00

.555

r rr,eff

1 11 10

2 2

d

w

1r,eff r r r 10 0[0.96 (0.109 0.004 )(log (10 ) 1)]Z


• Characteristic Impedance (ZCharacteristic Impedance (Z00):):– Based on the air-filled equivalent microstrip lineBased on the air-filled equivalent microstrip line

– For w/d < 3.3For w/d < 3.3

airair0

0 0

r,eff

where may be found

by curvilinear or other numerical methods

ZZ Z

air 20 60ln 4( / ) 16( / ) 2Z d w d w

Example of Parameter Example of Parameter Calculation for Microstrip LineCalculation for Microstrip Line

• Exercise 3 (D14.4, H&B, 7/e, p. )Exercise 3 (D14.4, H&B, 7/e, p. )GivenGiven: a 2 mm wide microstrip line is fabricated : a 2 mm wide microstrip line is fabricated

on a 1 mm thick substrate of lithium niobate on a 1 mm thick substrate of lithium niobate ((rr = 4.8). = 4.8).

FindFind: : r,effr,eff, Z, Z00 and v and vpp

SolutionSolution: : since w/d = 2 > 1.3:since w/d = 2 > 1.3:

.555.555

r rr,eff

air 20

2

air0

0

r,eff

1 1 5.8 3.8 11 10 1 10 3.6

2 2 2 2 2

60ln 4( / ) 16( / ) 2

60ln 4(1/ 2) 16(1/ 2) 2 90

90 Z = 47

3.6

d

w

Z d w d w

Z

ConclusionsConclusions

• Types of T-lines such as coax, two-wire, Types of T-lines such as coax, two-wire, microstrip lines and other variations have microstrip lines and other variations have – relative advantages in certain applications relative advantages in certain applications – a unique construction that determines their a unique construction that determines their

circuit parameterscircuit parameters

• Circuit parameters of T-lines depend on Circuit parameters of T-lines depend on LF or HF operation as indicated by skin LF or HF operation as indicated by skin depth versus conductor size; parameters depth versus conductor size; parameters may be found by calculation or in a may be found by calculation or in a handbookhandbook


• While coax has shielding to minimize While coax has shielding to minimize interference in low signal applications, it interference in low signal applications, it may be too large and bulky for PCB and may be too large and bulky for PCB and microelectronic circuit applicationsmicroelectronic circuit applications

• While microstrip lines are easy to While microstrip lines are easy to fabricate and compatible with fabricate and compatible with microelectronic systems, they are microelectronic systems, they are analytically complex requiring numerical analytically complex requiring numerical methods for accuracy methods for accuracy


• For higher frequency applications, For higher frequency applications, waveguides (including rectangular or waveguides (including rectangular or cylindrical “pipes,” parallel plates, cylindrical “pipes,” parallel plates, dielectric slabs and optical fiber types) dielectric slabs and optical fiber types) become more efficient than T-linesbecome more efficient than T-lines

• The analysis of waveguides becomes The analysis of waveguides becomes a more advanced task with fully EM a more advanced task with fully EM field wave equations field wave equations

References and Other References and Other ResourcesResources

• Hayt & Buck, Hayt & Buck, Engineering Engineering ElectromagneticsElectromagnetics, 7/e, McGraw Hill: , 7/e, McGraw Hill: New York, 2006.New York, 2006.

• Kraus & Fleisch, Kraus & Fleisch, Electromagnetics with Electromagnetics with ApplicationsApplications, 5/e, McGraw Hill: New , 5/e, McGraw Hill: New York, 1999.York, 1999.

Electromagnetics (ENGR 367) Types of T-lines and Their Applications.

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Transcript of Electromagnetics (ENGR 367) Types of T-lines and Their Applications.