· are traveling at velocity (1 ) sn f relative to a station-ary reference frame where λ is the...

J. Electromagnetic Analysis & Applications, 2010, 2, 403-456 Published Online July 2010 in SciRes (http://www.SciRP.org/journal/jemaa/)

Copyright © 2010 SciRes. JEMAA

TABLE OF CONTENTS

Volume 2 Number 7 July 2010

Analysis of Double and Single Sided Induction Heating Systems by Layer Theory Approach

L. J. B. Qaseer……………………………………………………………………………………………………………………403

Wave Propagation in Nanocomposite Materials

P. Hillion………………………………………………………………………………………………………………………………411

Simulation on DC Current Distribution in AC Power Grid under HVDC Ground-Return-Mode

G. H. Mei, Y. Z. Sun, Y. C. Liu…………………………………………………………………………………………………418

Polynomial-Based Evaluation of the Impact of Aperture Phase Taper on the Gain of Rectangular Horns

K. B. Baltzis……………………………………………………………………………………………………………………424

Influence of Magnetic Field Intensity on the Temperature Dependence of Magnetization of

Ni2.08Mn0.96Ga0.96 Alloy

K. Y. Mulyukov, I. I. Musabirov…………………………………………………………………………………………………431

H0i-Eigenwave Characteristics of a Periodic Iris-Loaded Circular Waveguide

S. K. Katenev, H. Shi………………………………………………………………………………………………………………436

The Design of Circular Microstrip Patch Antenna by Using Quasi-Newton Algorithm of ANN

A. Mishra, G. B. Janvale, B. V. Pawar, P. M. Patil………………………………………………………………………………444

Return Signal Intensity Ratio Modulates the Impact of Background Signal on Ozone DIAL Night Time

Measurement in the Troposphere

N. W. Cao, T. Fuckuchi, T. Fujii, Z. R. Chen, J. S. Huang………………………………………………………………………450

Journal of Electromagnetic Analysis and Applications (JEMAA)

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J. Electromagnetic Analysis & Applications, 2010, 2, 403-410 doi:10.4236/jemaa.2010.27052 Published Online July 2010 (http://www.SciRP.org/journal/jemaa)


403


Layth Jameel Buni Qaseer

General & Theoretical Electrical Engineering Department, University of Duisburg-Essen, Duisburg, Germany. Email: [email protected] Received February 11th, 2010; revised April 8th, 2010; accepted April 15th, 2010.

ABSTRACT

The iterative layer theory approach is applied to the analysis of double sided and single sided induction heating systems for continuous heating of thin metal strips. The excitation is transverse to the direction of strip motion and can be three phase or single phase. Nonmagnetic as well as ferromagnetic strips are employed. The important system parameters, namely, strip resistance, reactance, induced power and electromagnetic force are introduced. Accuracy of the method is verified with measurement of practical induction heating system together with comparison to numerical and analytical methods. Keywords: Eddy Currents, Electromagnetic Analysis, Energy Conversion, Induction Heating

1. Introduction

THREE phase induction heating systems such as transverse flux induction heating (TFIH) systems and traveling wave induction heating (TWIH) systems have been ex-tensively studied in recent years. While numerical tech-niques are more popular and particularly useful for in-vestigating the induced current and power distributions taking into account longitudinal and transverse edge ef-fects, analytical methods are more convenient for the integral parameters determination and analysis.

3-D finite element method (FEM) has been employed in the analysis of TFIH systems [1-8] while 2-D and 3-D FEM have been employed in the analysis of TWIH systems [9-13]. Few papers relating to analytical methods for the analysis of single phase and traveling wave but cylindrical induction heating systems have been published [14-17].

Only a few researchers pay attention to this area in the world.

The TWIH is not fully appreciated with respect to their main advantages and possible industrial applica-tions [18].

A. Ali, V. Bukanin from St. Petersburg Electrotechn- ical University in Russia and F. Dughiero, M. Frozen, S. Lupi, P. Siega, V. Nemkov from University of Padua in Italy have obtained significant achievements in this area. In the very recent years, Takamitsu Sekine, Hideo Tomita, Shuji Obata and Yokio Saito from Tokyo Den-

ki University in Japan have designed an excellent trav-eling wave induction heating system and carried out experiment [19].

An analytical method based on the decomposition of the main magnetic flux imposed by means of an excitation coil into partial magnetic fluxes along different regions that comprise the assembly. The basic circuit parameters that feature the electric performance in induction heating devices having an excitation axial winding as found in induction motors for generating rotary magnetic fields are mathematically modeled [20].

Modern analytical approaches using transmission line terminology [21-23] are confined to lossless or low conductivity (dielectric) media where displacement currents are prominent at microwave frequencies in the order of hundreds of gigahertz, which is not the case as in this approach where induced power is the major objective of induction heating, moreover these methods are primarily applied to isotropic media while the layer theory is applied to both isotropic and anisotropic media, also it is not mentioned in these references whether these approaches may be used in the case of three phase (traveling wave) excitation.

The layer theory approach has been mainly used for the analysis of linear, tubular linear and helical motion induction motors as given in [24-26].

TWIH systems widely common in literature are of the double sided induction heating (DSIH) system type as it employs upper and lower excitation inductors with res-



404

pect to the long and thin continuously moving strip. Sin-gle sided induction heating (SSIH) systems are TWIH systems that employ one inductor for exciting the metal strip while single phase induction heating systems are commonly known as longitudinal flux induction heating (LFIH) systems.

The primary object of this paper is to propose a general mathematical model for the induction heating system using the actual topology for single phase and three phase excitations with any number of poles for SSIH and DSIH systems. As a second object, the paper employs the multi-layer approach with the appropriate current sheet to calculate the flux density components, induced power in the strip, terminal impedance and the magnetic force acting on the strip in the direction of field travel in the case of TWIH systems.

2. Mathematical Model

2.1 Three – Phase Excitation

A general multi-region problem is analyzed. Figure 1 shows a cross-section of the N-region model used in the theory. The model is taken to be a set of planar regions. The current sheet lies between regions r and r + 1.

.

.

.

1N 1N 1N

NN N

1NH

2NH

1rH

rHrH J

1NB

2NB

1rB

rB

x

y

z

.

.

.

rz g

1rz g

1r

1 11

2 22

3 33

1z g

2H

1H

2B

1B2z g

1rB

1rH

r1rz g

2Nz g

1Nz g

σ1 μ1

σ3 μ3

σ2 μ2

σN-1 μN-1

σN μN

Figure 1. General model with current sheet at boundary rz = g

The current sheet varies sinusoidally in the y-direction and with time. It is of infinite extent in the x-direction and infinitesimally thin in the z-direction.

Regions 1-N are layers of materials where the general region n has a conductivity σn and anisotropic relative permeability µn. The anisotropy is an approximation made in order to deal with slotted regions. The regions are traveling at velocity (1 )ns f relative to a station-

ary reference frame where λ is the wavelength of applied field, f is the frequency and sn is the slip in region n de-fined as

nn

fs

f

where fn is frequency of the field experienced by region n.

In this frame the traveling field has a velocity f .

It is assumed that displacement current is negligible and magnetic saturation is neglected. Maxwell’s equations for any region in the model are

H J (1)

BE

t

(2)

0B (3)

0E (4)

J E (5)

B H (6)

The boundary conditions may be summarized as follows

1) The normal component of the magnetic flux density Bz is continuous across a boundary.

2) All field components vanish at z . 3) The tangential component of magnetic field stre-

ngth Hy is continuous across a boundary, but allowance must be made for the current sheet in the manner ex-plained in Section 3.

2.1.1 Excitation Current Density It is assumed that the winding produces perfect sinusoi-dal traveling wave. The line current density may be rep-resented as

Re exp[ ( )]J J j t ky (7)

where ,J and k are the line current density, angular frequency and wave length factor respectively. The line current density is given by

6 2 effN IJ

p

where I, p, τ and Neff are the r.m.s. value of the phase current, number of poles, pole pitch and effective number



405

of series turns per phase respectively. The wave length factor is defined as

k

(8)

2.1.2 Field Equation of a General Region As a first step in the analysis the field components of a general region are derived, assuming that all fields vary as exp[ j(ωt – ky)], and omitting this factor for simplicity reasons from all the field expressions that follow. Taking only the x- component from both sides of (2) yields

0xB

From (3) we have

zy

BjkB

z

which leads to 2

2

yzBB

jkzz

(9)

Taking the z- component from both sides of (2) yields

x zE Bk

(10)

Taking only the x-component from both sides of (1) yields

yzx

BBE

y z

(11)

Therefore, using (9) and (10) into (11) yields 2

22

zZ

Bα B

z

(12)

where 2 2

0 rα k jωμ μ σ

The solution is given by

cosh( ) sinh( )zB A αz C αz (13)

where A and C are arbitrary constants to be determined from the boundary conditions.

From (3) we get

0

sinh( ) cosh( )yr

αH A αz C αz

jkμ μ (14)

2.1.3 Field Calculations at the Region Boundaries Figure 2 shows a general region n of thickness Sn the normal component of magnetic flux density on the lower boundary is Bn-1 and the tangential component of mag-netic field strength is Hn-1. The corresponding values on the upper boundary are nB and nH . From (13) and (14)

, cosh( ) sinh( )z n n n n nB A α z C α z (15)

1nz g

nH

1nH

nB

1nB

nz g

nSregion nnf

Figure 2. General region n

,0

sinh( ) cosh( )ny n n n n n

n

αH A α z C α z

jkμ μ (16)

Equivalent expressions for Bz,n-1 and Hz,n-1 can be found by replacing zn by zn-1. Now for the regions where

1n or N

, , 1

, , 1

cosh( ) sinh( )

cosh( )sinh( )

z n z nn n n n

n

n ny n n n n y n

B Bα S α S

β

α SH β α S H

(17)

or

, , 1

, , 1

z n z n

ny n y n

B BT

H H

(18)

where

0

nn

n

αβ

jμ μ k (19)

Hence given the values of Bz and Hy at the lower boundary of a region, the values of Bz and Hy at the upper boundary are immediately obtainable from this simple transfer matrix relation. At the boundaries where no ex-citation current sheet exists, Bz and Hy are continuous; thus for example, if two regions are considered with no current sheet at the common boundary, knowing Bz and Hy at the lower boundary of the first region, Bz and Hy at the upper boundary of the second region can be calcu-lated by successive use of the underlying two transfer matrices. Considering the current sheet to be at rz g ,

then

, ,y n y nH H , n r (20)

and

, ,y n y nH H J , n r (21)

where ,y n

H is the tangential magnetic field strength in

close lower proximity to the boundary and ,z nH is the

tangential magnetic field strength in close upper prox-imity to the boundary.

Given the current sheet excitation at rz g the over-

all structure divides into an upper part, which is modeled according to



406

, 11 2

, 1

z nn n

y n

BT T

H

… ,

1,

z rr

y r

BT

H J

(22)

and an inner part which supports the following relation

,1

,

z rr r

y r

BT T

H

… ,1

2,1

z

y

BT

H

(23)

If the top region is now considered, then as z , tanh( ) 1αz and all field quantities tend to zero, hence

on the boundary 1Ng the field quantities are related by

1 1N N NH β B (24)

Therefore at any z within region N the field quantities become

1 1expz N N NB B α g z

1 1exp ( )y N N N NH β B α g z

Considering the bottom region where n = 1, the field quantities are related by

1 1 1H β B (25)

and at any z within region 1

1 1 1expzB B α z g

1 1 1 1exp ( )yH β B α z g

2.1.4 Surface Impedance Calculations The surface impedance looking outwards at a boundary of nz g is defined as

, ,1

, ,

x n z nn

y n y n

E BZ

H k H

(26)

and the surface impedance looking inwards is defined as

, ,

, ,

x n z nn

y n y n

E BZ

H k H

(27)

Using the method obtained in [17] with the values of Bz,N-1, Hy,N-1, Bz,1, Hy,1 and [Tn] as derived in the previous section then

1

1

r rin

r r

Z ZZ

Z Z

(28)

where Zin is the input surface impedance at the current sheet and Zr+1 and Zr are the surface impedances looking outwards and inwards at the current sheet. Substituting for Zr and Zr+1 using (26) and (27) respectively, and re-arranging the terms yields

,

, ,

x rin

y r y r

EZ

H H

(29)

Substituting (21) into (29) yields

,x rin

EZ

J

(30)

Thus the input surface impedance at the current sheet has been determined. This means that all field compo-nents can be found by making use of this and (27), (22), (23).

2.1.5 Terminal Impedance, Power and Tangential Force

The terminal impedance per phase per metre of axial length can be derived [17] in terms of Zin as

224 efft in

NZ Z

λp Ω/m (31)

Having found Ex, Bz and Hy at all boundaries, it is then a simple matter to calculate the power entering a region through the concept of Poynting vector. The time average power density passing through a surface is given by

*1Re

2P E H W/m2

Hence the time average power density flowing up-wards from the current sheet at rz g is given by

*, , ,

1Re

2in r x r y rP E H

and the time average power density flowing downwards from the current sheet at rz g is given by

*, , ,

1Re

2in r x r y rP E H

The net power density in a region is the difference between the power in and power out

* *, , , ,Re

2in z r y r z r y r

ωP B H B H

k (32)

It follows that the tangential force density Fy acting on the strip is the net power density induced divided by traveling wave velocity λf

iny

PF

λf N/m2 (33)

2.1.5.1 Single Phase Excitation This is a simpler problem than the three phase one and only regions below or above the current sheet (depending where the strip is located) need be considered. Other re-gions do not in any way affect the field distribution. The excitation is provided by a single coil of N1 turns per metre of axial length carrying alternating current in the transverse direction. The line current density takes the form



407

Re exp( )J J jωt (34)

with

12J N I

In this case 0k , all field relations, Maxwell’s equa-tions and boundary conditions hold. The solution is given by

cosh( ) sinh( )zB A αz C αz (35)

where 2

0 rα jωμ μ σ (36)

If the plane 0z passes through the central axis of the strip then

/2 cosh( / 2) sinh( / 2)bB A αb C αb

where b is the thickness of the strip and /2bB is the axial

(tangential) component of magnetic flux density at the upper surface of the strip. The axial component of mag-netic flux density at the lower surface of the strip is given by

/2 cosh( / 2) sinh( / 2)bB A αb C αb

Hence we can write

/2 /2 0 cosh( / 2)b bB B B αb (37)

where B0 is the axial component of magnetic flux density at the centre of the strip.

The input surface impedance at the current sheet be-comes

in rZ Z (38)

where Zr is obtained using (27). The net power density induced in the strip is obtained

using the concept of Poynting vector and therefore the net power density in the strip is

* *, 1 , 1 , ,

2

2

1Re

2

Re tanh( / 2)

in x n y n x n y nP E H E H

Jα αb w m

σ

(39)

The terminal impedance is given by the relationship

tanh( / 2)t

αZ αb

σ (40)

3. Numerical Results

The solution procedure that has been described in the previous sections is used to analyze two examples to check validity and accuracy. One example is a single phase practical induction heating system with ferromagnetic strip [27]. The importance of this example is that meas-urement is available in addition to calculation. The other

example is based on FEM solution for DSIH system [9]. For comparison reasons, FEM computation is adopted

in our analysis which is widely used as a numerical tech-nique for this kind of applications.

In our implementation, the field domain is divided into a number of regions, each being defined by its coordi-nates, permeability and conductivity. Each region is de-scritized using first order triangular elements [28]. The induced power in the strip is obtained through the solu-tion of governing differential equation for each nodal magnetic vector potential. Three values of power are computed: the power integrated over the coil, the air gap power and the power integrated over the strip.

The solution is assumed to be convergent when these three values do not differ by more than 1% which is termed as the power mismatch or power imbalance.

3.1 Practical Single Phase Induction Heating System

Problem data are given in Table 1. Results obtained us-ing the layer theory approach and FEM are in good agreement with measurement as shown in Table 2. This agreement is attributed to the fact that strip thickness is very small compared to strip length and width which coincides with the assumptions made in the mathematical model.

Table 1. Problem data for practical induction heating sys-tem (based on [27], Ex. 13)

Strip thickness, (mm) 1.56 Strip width, (mm) 1220 Strip length, (mm) 1270 Relative permeability of strip 50 Mean strip conductivity, (S/m) 1.333 × 106 Production rate, (ton/hr) 9.072 Heat cycle, (sec) 7.5 Speed of strip, (m/s) 0.169 Frequency, (Hz) 9600 Coil axial length, (mm) 1270 Coil width, (mm) 1270 Air gap length, (mm) 73.32 Amplitude of line current density, (kA/m) 31.831

Table 2. Computed parameters of the practical single phase induction heating system

Parameter Measuredvalue [18]

Value calculated by empirical formula [18]

FEMLayer theory value

Strip power (kW)

1490 1856.3 1824.9 1903.9

Strip resistance(Ω)

— 1.86 1.82 1.9

Reactance (Ω)

— 2.32 2.35 2.37



408

Figure 3 shows the variation of the axial component of magnetic flux density along strip depth at mid coil axial length for single phase model using FEM analysis and the layer theory approach. The agreement between the results of both methods may be considered good with a maximum relative deviation of 4.9%. It is shown in this figure that the axial component of magnetic flux density decreases rapidly (exponentially) from the surface of the strip for both sides due to skin effect. Obviously there is no normal component for single phase induction heating system and this can be derived directly from Maxwell’s equations.

3.2 Single Sided and Double Sided Traveling Wave Induction Heating Systems

Reference [9] employed a double sided induction heating system whose data are given in Table 3.

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.81.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Strip Depth (mm)

Axi

al F

lux

De

nsi

ty (

T)

layer theory finite elementlayer theory finite elementlayer theory finite elementlayer theory finite element

Central axialplane of strip

Figure 3. Variation of axial flux density component with strip depth for single phase induction heating model

Table 3. Problem data for traveling wave DSIH and SSIH systems (based on Reference [9])

Strip thickness, (mm) 2 Strip width, (mm) 1000 Strip length, (mm) 960 Relative permeability of strip 1 Mean strip conductivity, (S/m) 3.03 × 107 Axial pole pitch, (mm) 480 Slot pitch, (mm) 160 Slot width, (mm) 80 Slot depth, (mm) 40 Slots per pole per phase 1 Number of axial poles 2 Number of conductors per slot 8 Frequency, (Hz) 50 Inductor axial length, (mm) 960 Inductor width, (mm) 1000 Magnetic yoke depth, (mm) 80 Air gap length between yoke & strip, (mm) 15 Amplitude of line current density, (kA/m) 200 Input phase voltage, (V) 220

For the sake of comparison, the same model is adopted as a single sided induction heating system using the same line current density by removing one of the inductors along with its backing iron. Table 4 shows the computed parameters for both systems using FEM and the layer theory approach. Again the results correlate well as dis-cussed in Subsection 3.1.

Figure 4 and Figure 5 show respectively the variation of normal and tangential (axial) flux density components along magnetic gap length. The maximum deviation be-tween the results of both methods is found to be 5.2%.

Figure 6 and Figure 7 show the variation of normal and tangential (axial) flux density components along dis-tance normal to the strip. Again both methods correlate well within 4%. In both systems the axial flux density component in the air gap is greater than the normal component, this may be attributed to the fact that the pole pitch is much greater than the air gap length in both sys-tems and in this case these systems are considered as axial flux machines. It is clear from these figures that the axial component of magnetic flux density is decreased within the strip due to skin effect which is not effectively pronounced in the normal component to the strip.

Table 4. Computed parameters for traveling wave induc-tion heating systems

Parameter FEM Value

Layer TheoryValue

Per phase DSIH strip resistance (Ω) 0.0497 0.0513 Per phase DSIH reactance (Ω) 0.014 0.017 DSIH strip power (kW) 1210.3 1250.2 DSIH axial force (N) 23914.5 26045.3 Per phase SSIH strip resistance (Ω) 0.0125 0.012 Per phase SSIH reactance (Ω) 0.0087 0.0086 SSIH strip power (kW) 305.06 292.8 SSIH axial force (N) 6300 6099.9

70 80 90 100 110 120 130115

120

125

130

135

140

145

Normal Distance (mm)

No

rma

l Flu

x D

en

sity

(m

T)

layer theoryfinite element

magnetic gap

Figure 4. Variation of normal flux density component along normal distance to strip for double sided induction heating system



409

70 80 90 100 110 120 1300

50

100

150

200

250

300


Axi

al F

lux

De

nsi

ty (

mT

)

layer theoryfinite element

stripthickness

magnetic gap

Figure 5. Variation of axial flux density component along normal distance to strip for double sided induction heating system

finite elementlayer theory

70 80 90 100 110 1200

10

20

30

40

50

60

70

80

90

100


No

rma

l Flu

x D

en

sity

(m

T)

finite elementlayer theory finite elementlayer theory finite elementlayer theory

strip thickness

inner yokesurface

Figure 6. Variation of normal flux density component along normal distance to strip for single sided induction heating system

70 80 90 100 110 1200

50

100

150

200

250

300


Axi

al F

lux

Den

sity

(m

T)

finite elementlayer theory finite elementlayer theory finite elementlayer theory

inner yoke surface

stripthickness

Figure 7. Variation of axial flux density component along normal distance to strip for single sided induction heating system

4. Conclusions

The layer theory approach has been used for the analysis of single sided, double sided traveling wave and single phase induction heating systems. This method has been applied to compute electrical parameters of various in-duction heating systems with ferromagnetic and non-magnetic thin strips.

The results show clearly that the theoretical results correlate well with finite element method results in addition to experimental one. This may be considered as fair jus-tification to the analysis method proposed in this paper.

5. Acknowledgements

The author is deeply indebted to Professor Daniel Erni, his host and co-worker at the University of Duisburg-Es- sen for advice and encouragement.

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411


Pierre Hillion

Institut Henri Poincaré, Bis Route de Croissy, Le Vésinet, France. Email: [email protected] Received February 9th, 2010; revised April 21st, 2010; accepted May 3rd, 2010.

ABSTRACT

Electromagnetic wave propagation is first analyzed in a composite material mde of chiral nano-inclusions embedded in a dielectric, with the help of Maxwell-Garnett formula for permittivity and permeability and its reciprocal for chirality. Then, this composite material appears as an homo-geneous isotropic chiral medium which may be described by the Post constitutive relations. We analyze the propagation of an harmonic plane wave in such a medium and we show that two different modes can propagate. We also discuss harmonic plane wave scattering on a semi-infinite chiral composite medium. Then, still in the frame of Maxwell-Garnett theory, the propagation of TE and TM fields is investigated in a periodic material made of nano dots immersed in a dielectric. The periodic fields are solutions of a Mathieu equation and such a material behaves as a diffraction grating. Keywords: Composite Materials, Maxwell-Garnett, Constitutive Relations, TE TM Fields

1. Introduction

Nanotechnology is blossoming with in particular the in- clusion of nano-particles (nano-dots) in some specific support [1,2]. Then, to analyze electromagnetic wave propagation such as light or X rays in these composite materials, we need a theory able to calculate average, ma- croscopic values from their granular microscopic properties. This job is performed by the Maxwell-Garett theory [3-6].

In this work, the support is a dielectric with permittivity 1, permeability and we consider two situations according that nano-dots are chiral or periodically distributed along a direction of the structure.

In the first case (chiral nano-dots) the permittivity of the composite material is according to the Maxwell- Garett formula

[ – 1][ + 21]1 = f [2 – 1][2 + 21]

1 (1)

f is the filling factor of inclusions (their volume fract- ion) in the host material, the subscripts 1, 2 corresponding to host and inclusions respectively and we get from (1)

= 1 (1 + 2f )(1 f ) ,

= (1 2)(1 2), (2)

Permeability µ is assumed the same for nano-dots and dielectric.

The relation (2) has been generalized [7,8] to chirality when both inclusions and host materials are chiral. But

here, the situation is different since only inclusions have this property and the relation (1) with 12instead of 12 has no meaning when 1 = 0. To cope with this difficulty, we introduce a reciprocal Maxwell-Garnett relation obtained by applying to (1) the transformation (12f) (f2, 1 1/f) which gives

[ – f2][ + 2f2]1 = f [1 – f2][1 + 2f2]

1 (3)

reducing for 1 = 0 to

= f2(1 f)(1 + 2f) (4)

From now on, we assume f << 1, 1 > 0, 2 < 0, > 0 and 2 < 0 so that the 0(f 2) approximation of (2) and (4) gives

= 1(1 + 3f ) > 0 (5)

= f2 > 0 (6)

So, this composite material made of nano-chiral particles included in a dielectic may be hand-led as an homogeneous chiral medium with permittivity and chirality (5) and (6) and permeabiity µ > 0 assumed to be the same for inclusions and dielectric.

In the second situation (periodically distributed nanorods), the relation (1) is still valid with f changed into a periodic function f (x). Assuming f (x) = f cos(2ax), we write the permittivity (x) in the following form reducing to (5) to the 0(f 2) order

(x) = 1exp[3fcos(2ax)] (7)

Using (5)-(7), we shall analyze harmonic plane wave



412

propagation in both composite materials, chiral and periodic.

2. Harmonic Plane Wave Propagation in a Chiral Composite Medium

We suppose this chiral medium endowed with the Post constitutive relations in which , have the expressions (5) and (6) [9,10]

D = E + i B, H = B/µ + i E, i = 1 (8)

This choice is not arbitrary because the Post constitutive relations, in their general form, are covariant under the proper Lorentz group as Maxwell’s equations which guarantees a consistent theory with a simple mathematical formalism, in agreement with the statement that only covariant mathematical expressions have a physical meaning.

Plane wave scattering from a semi-infinite chiral medium was discussed some time ago by Bassiri et al [11], also using the Post constitutive relations, but we proceed differently from these authors working with the Fresnel reflection and transmission amplitudes.

2.1 Refractive Indices

We consider harmonic plane waves with amplitudes E, B, D, H

(E,B,D,H)(x,t) = (E,B,D,H)(x,t) (9)

in which

(x,t) = exp[i(t + nsin x/c + ncos z/c)] (10)

in which n is a refractive index to be determined. Substituting (9) into the Maxwell equations

E + 1/c t B = 0, . = 0

H 1/c t D = 0, .D = 0 (11)

and taking into account (10) give the equations for the amplitudes E, B, D, H

ncos Ey + Bx = 0,

n(cos Ex sin Ez) + By = 0,

nsin Ey + Bz = 0, (12)

n cos Hy + Dx = 0

n(cos Hx sin Hz) Dy = 0

nsin Hy Dz = 0 (13)

with the divergence equations

sin Bx + cos Bz = 0, sin Dx + cos Dz = 0 (14)

We get at once from (8) and (14), the divergence equation satisfied by the electric field

sin Ex + cos Ez = 0 (15)

Substituting (8) into (13) gives

ncos (By / + iEy) + Ex + i Bx = 0

ncos (Bx / + iEx) nsin (Bz / + iEz) Ey iBy = 0

nsin (By / + iEy) Ez iBz = 0 (16)

Taking into account (12), these equations become ncos By / + Ex + 2iBx = 0

ncos Bx / nsin Bz/ Ey 2iBy = 0

nsin By / Ez 2iBz = 0 (17)

Then, eliminating B between (12) and (17) gives the homogeneous system of equations in which = 2n

(n2/ )Ex icos Ey = 0

(n2/ )Ey i(sin Ez cos Ex) = 0

(n2/ )Ez + isin Ey = 0 (18)

This homogeneous system has nontrivial solutions if its determinant is null and a simple cal-culation gives

(n2/ )[(n2/ )2 2 ] = 0 (19)

Deleting (n2/ ) = 0 which would correspond to an a-chiral medium, we get from (11) two modes (n±

2/ ) = ± in which which = 2n so that the refractive index depends not only on permittivity and permeability but also on chirality with the positive expressions

n+ = + (22 )1/2, (20)

n = + (22 )1/2 (21)

Changing the square root into its opposite gives negative refractive indices.

Consequently, two modes with respectively the refractive indices n+, n can propagate in the metachiral slab, they are independent as long as the medium is infinite, otherwise they become coupled at boundaries. The amplitudes of the field components in these two modes have now to be determined.

2.2 Electromagnetic Fields

1) We first suppose n+2/ = and n+ = + (22

)1/2 with and µ > 0: fields and parameters are characterized by superscripts or subscripts + respectively.

Then, we get at once from (18) and (12) in terms of E+y

E +x = icos+ E+

y, E+

z = isin+ E+y,

B+x = n+ cos+ E+

y,

B+y = i n+ E

+y, B+

z = n+ sin+ E+y (22)

and substituting (22) into (8)

D+x = icos+ +E+

y,

D+y = +E+

y,

D+z = isin+ +E+

y (23)

H+x = cos+ +E+

y,

H+y = i +E+

y,

H+z = sin + +E+

y (24)

in which



413

+ = + n+, + = n+/ = ( 2 /)1/2 (25)

2) For n2/ = and n = + ( 2 2 )1/2,

we get at once with now super-scripts and subscripts :

Ex = i cos Ey, E

z = i sin E

y,

Bx = n cos E

y, B

y = i nE

y,

Bz = n sin E

y (26)

and substituting (26) into (8)

Dx = icos E

y, Dy = E

y,

Dz = isin E

y (27)

Hx = cos E

y, Hy = i E

y,

Hz = sin E

y (28) with

= n, = n/ + = ( 2 /)1/2 = + (29)

Then, according to (9) and (10), the electromagnetic field of the plus and minus modes, each depending on an arbitrary amplitude E+

y , E

y, is

(E±, B±, D

±, H±) (x,t) = (E±, B

±, D±, H

±) ±(x, t) (30)

with the amplitudes given by (22)-(24) and (26)-(28) and the phase functions

±(x,t) = exp[i (t + n± sin± x/c + n± cos± z/c)] (31)

2.3 Plane Wave Scattering from a Semi-Infinite Chiral Composite Medium

We suppose that the chiral composite material fulfills the half space z < 0 on which impinges from z > 0 on the interface z = 0 an harmonic plane wave characterized by the phase factor i

i = exp[in0(xsini + zcosi)] (32)

n0 is the refractive index in z > 0 and the components of the incident electromagnetic field are [12] with two amplitudes Mi, Ni:

Eix = cosi Mi

i, Eiy = Ni

i), Eiz = sini Mi

i

Hix = n0cosi Ni

i, Hiy = n0Mi

i),

Hiz = n0sini Ni

i (33) The reflected field in the half-space z > 0 has a similar

expression with (Mi, Ni, i) changed into (Mr, Nr, r) while the refracted field in z < 0 is supplied by (30).

According to (31) and (32), also valid for the reflected wave, the continuity of the phase at z = 0 implies the Descartes-Snell relations

n0 sini = n0 sinr = n+ sin+ = n sin (34)

The continuity of the components Ex,y, Hx,y, at z = 0 supplies four boundary conditions to de-termine in terms of Mi, Ni the amplitudes Mr, Nr of the reflected field and those E +

y, E

y of the refracted field. According to (22), (26) and (33) and taking into ac-

count (34), we get for the Ex,y components

cos i(Mr Mi ) = icos + E +y icos E

y Nr + Ni = E +

y + E y (35)

while for Hx,y, according to (24), (28) and (33), we have since = + (= )

n0 cosi(Nr Ni ) = (cos+ E +y + icos E

y)

n0( Mr + Mi) = (E +y E

y ) (36)

To make calculations easier, we introduce the notations

Mr + Mi = M, Nr + Ni = N,

Mr Mi = M’, Nr Ni = N’ (37)

and

a = n0/ (cos+ + cos) (38)

Then, we get at once from (36)

E +y = a(cosi N’ + cos M)

E y = a(cosi N’ cos+ M) (39)

and, substituting (39) into (35) gives

cosi M’ = a11 N’ + a12 M

N = a21 N’ + a22 M (40)

in which

a11 = iacosi (cos + + cos), a12 = 2iacos + cos a21 = acos i, a22 = a(cos + cos + ) (41)

Taking into account (37) the system (40) becomes

(cos i a12)Mr + a11 Nr = (cos i+ a12) Mi a11 Ni

a22Mr (1 a21) Nr = a22Mi + (1 + a21)Ni (42)

from which we easily get the amplitudes Mr, Nr of the reflected field and consequently M’, N’ according to (37) to obtain finally the amplitudes Ey

± of the refracted field from (39).

One has a simple result for a normal incidence i = r = ± = 0 since the Equations (35) and (36) reduce to

Mr Mi = i(E +y E

y), Nr + Ni = E +y + E

y

n0 i = (E +

y + E y), n0(Mr + Mi) = (E

+y E

y) (43)

with the solution

Mr = ( + in0) ( in0) Mi, Nr = (1 + 2n0 /)Ni (44)

E +y = n0( in0)Mi n0/ Ni,

E y = n0( in0)Mi n0 / Ni (45)

Remark 1. If the angles + obtained from (34) are real, the plus and minus modes propa-gate in the chiral medium. If they are both purely imaginary, we get from (34)

cos(±) = i[(n0/±)2 sin2i 1]1/2 (46)

the negative sign in front of the square root in (46) corresponds to the physical situation: refracted waves are evanescent and, incident waves undergo a total reflection,



414

with as consequence for beams of plane waves a Goös- Hanken lateral shift and a Imbert-Fedorov transverse shift [13]. Of course with a single angle pure imaginary, only one mode propagates, the other mode giving rise to an evanescent wave.

Remark 2. At the expense of more intricacy, the present formalism may be generalized to wave propagation in a chiral slab located between z = 0 and z = d. Then, two more fields exist respectively reflected at z = d inside the slab and refracted outside in the z < d region, supplying four supplementary amplitudes matched by the boundary conditions at z = d. But, instead of a 4 4 system of equations to get the amplitudes of the electromagnetic field, we have to deal with a 8 8 system more difficult to solve.

3. Harmonic Plane Wave Propagation in a Two Dimensional Nano-Periodic Medium

With B = µH, D = (x)E, and exp(it) implicit, the Maxwell equations are for E(x,z), H(x,z)

zEy ic Hx = 0, zHy + i (x)c Ex = 0

zEx – xEz + ic Hy = 0,

zHx – xHz – i (x)c Ey = 0

xEy + ic Hz = 0, xHy – i(x)c Ez = 0 (47)

with the divergence equations

[’ + x]Ex + zEz(x,z) = 0, xHx + zHz = 0 (48)

giving rise to TE (Ey, Hx, Hz) and TM (Hy, Ex, Ez) waves.

3.1 TE Wave Propagation

Assuming f << 1, we work with the Maxwell-Garnett 0(f2) approximation of (7)

(x) = 1 f cos(2ax), = 31 (49)

The component Ey satisfies the Helmholtz equation in which ∆ = x

2 z2

[∆ + 2 (x)/c2]Ey(x,z) = 0 (50)

We look for the solutions of this equation in the form, A being an arbitrary amplitude

Ey(x,z) = A exp(ikzz) (x) (51)

Substituting (51) into (50) and taking into account (49), gives the differential equation satisfied by (x)

[x2 + k0

2 + f ke2 cos(2ax)] (x) = 0 (52)

in which

k02 = 21c2 kz

2, ke2 = 2 c2 (53)

Using the variable = k1 x , Equation (52) becomes a Mathieu equation [14,15]

[2 + 2 + f cos(2a/ke)] () = 0, 2 = k02/ke

2 (54)

with solutions in the form [14,15,16] where has to be determined

() = ∑m=- cm exp([i( + 2m) a/ke)] (55)

Substituting (55) into (54) gives the following recurrence relation [15] for the coefficients cm

cm + m() (cm + cm) = 0 (56)

with

m() = f /2 [(2m + )2 2] (57)

Now, the main difficulty [14,15] is to get in terms of f and , but f being small, the infinite determinant of the system (56) supplies to the 0(f 3) order [15]

cos(π) = cos(π) + πf 2 [4 2(1 2)1/2] sin(π) (58)

Once known, the cm coefficients may be obtained by numerical methods based on the recurrence relations (36) or on some variant of it. It is shown [15] how for mod- erate values of and f, these relations can be transformed into convergent continued fractions Rm() = cm/cm, Lm() = cm/cm.

So, according to (51) and (55), Ey(x,z) = Ey(x + π/a,z) and

Ey(x,z) = A exp(ikzz) ∑m=- cm exp[i( + 2m)ax)],

0 ≤ x < π/a (59)

and taking into account the Maxwell Equation (47), the other two components Hx, Hz of the TE field are obtained from zEy and xEy respectively. Writing (59)

Ey(x,z) = A∑m=- cm exp(ikzz + ikmx),

km = ( + 2m)a, 0 ≤ x < π/a (60)

Ey(x,z) appears as a periodic beam of plane waves propagating in the directions defined by the wave vectors with components (kz, km), their amplitude being weighted by the coefficients cm.

3.2 TM Wave Propagation

For TM waves (Hy, Ex, Ez), we start with the expression (7) of (x). Then, according to the Maxwell Equation (47) the component Hy satisfied the equation

[∆ + 2 (x)/c2 ’(x)(x)x] Hy(x,z) = 0 (61)

We look for the solutions of (61) in the form

Hy(x,z) = A exp(ikzz) (x) (62)

(x) = u(x) (x), (x) = exp[ f1/2 cos(2ax)],

f1 = f (63)

A simple calculation gives the first and second derivative of (x)

’(x) = [u’/u af1 sin(2ax)] (x)

’’(x) = [u’’/u 2a u’/u f1 sin(2ax) 2a2 f1 cos(2ax) + a2f1

2 sin2 (2ax)] (x) (64)



415

and since ’ = 2a f1 sin(2ax), we get to the 0( f12) order

’’ ’ ’ = [u’/u 2a2 f1 cos(2ax)] (x) + 0( f12) (65)

so that

[’’ ’ ’]Hy (x,z) = [u’/u 2a2 f1 cos(2ax)]Hy(x,z)

(66)

Then, according to (62) and (66), we get from (61), the differential equation satisfied by u(x)

[x2 + 2(x)/c2 kz

2 2a2 f1 cos(2ax)]u(x) = 0 (67)

which becomes with the Maxwell-Garnett approximation (49) of (x)

[x2 + k0

2 + f kh2 cos(2ax)]u(x) = 0 (68)

with k02 given by (53) while

kh2 = 21c2 2a2 (69)

The comparison of (52) and (68) shows that, to the 0(f 2) order, one has just to change ke into kh to go from TE to TM waves so that all the calculations of Subsection 3.1 can be repeated mutatis mutandis.

3.3 TE Wave Scattering in a Semi Infinite Nano-Periodic Material

The granular material, made of nano dots immersed in a dielectric, lies in the z < 0 half-space and we suppose that a TE harmonic plane wave (Ey

i, Hxi, Hz

i) impinges from the upper half-space z > 0 with refractive index and permeability on the z = 0 interface.

The components Eyi, Ey

r of the incident and reflected waves are

Eyi(x,z) = Ai exp[ic (x sini + z cosi)]

Eyr(x,z) = Ar exp[ic (x sini z cosi)] (70)

and according to the Maxwell Equation (47), the components Hx

i, Hxr involved in the boundary conditions are

Hxi(x,z) = cosi Ey

i(x,z),

Hxr(x,z) = cosi Ey

r(x,z) (71)

Now, the refracted periodic field in z < 0 has the form (59)

Eyt(x,z) = At exp(ikzz) ∑m=-

cm exp[i( + 2m) ax],

0 ≤ x < π/a (72)

and, still using (47)

Hxt(x,z) = Ey

t(x,z), = ckz / (73)

the boundary conditions impose the continuity on z = 0 of Ey and Hx, that is, according to (70)-(73)

(Ai + Ar) exp(i c x sini)

= At ∑m = – cm exp[i( + 2m)ax],

0 ≤ x < π/a (74)

cosi (Ai Ar) exp(i c x sini)

= At ∑m=- cm exp[i( + 2m)ax],

0 ≤ x < π/a (75)

Let us write (74)

Ai + Ar = At ,

= ∑m=- cm exp[i(km ki) ßπ] (76)

in which according to (60) and (70)

km = ( + 2m)a, ki = c sini (77)

with 0 ≤ ß < 1 since 0 ≤ x ≤ π/a. So, taking into account (60), the granular periodic

semi-infinite material behaves as a diffraction grating: the beam of plane waves propagating in the directions defined by the wave vectors with components (kz, km) have their amplitudes modulated by the coefficients cmexp(iki ßπ). And, acccording to (76), the relations (74) and (75) become

Ai + Ar = At , cosi (Ai Ar) = At , 0 ≤ ß < 1 (78)

from which we get in terms of the incident amplitude Ai

Ar = ( cosi) ( + cosi)Ai,

At = 2 cosi ( + cosi) Ai (79)

So, the amplitude At is not constant on the interval (0, π/a).

4. Discussion

The relation (6), leads to a consistent formalism but further work is needed to prove or to amend it. In any case, two different modes of harmonic plane waves propagate in these chiral materials. The Post constitutive relations used to characterize such media, allow to get exact ana- lytic expressions for the amplitude of the electromagnetic field in each mode, a note-worthy property due, as noticed in the introduction, to the covariance of Post’s relations under the proper Lorentz group. An excellent review of chiral nano-technology may be found in [17] with a discussion of two topics: nanoscale approaches to chiral technology and, corresponding to the situation considered here, nanotechnology that benefits from chirality. In particular, a section is devoted to chiral carbon nanotechnology and the authors conclude “possible applications of such materials in the field of biomedecine and biotechnology range from prepara-tion of novel antibact- erial, cyclotonic and drug delivery agents to catalysis and materials science applications”.

Remark: The analysis of Section 2 may be performed in left-handed chiral materials with negative : just change into ||||.



416

Granular periodic materials are currently used in mechanical engineering and, with the ob-jective to appraise their properties, theoretical studies have been devoted to acoustic wave propagation in these structures [18]. In electrical engineering, photonic crystals [19,20] are the main illustration of periodic nanomaterials and they take an increasing importance in today technology. But, they are not composite with inclusions immersed in a dielectric structure. For instance, a one-dimensional photonic crystal with a permittivity periodic in the direction of propagation may be described by an expansion in which U is the unit step function

(z) = 1 ∑n [U(z 2na) U(z 2n + 1a)]

+ 2 ∑n [U(z 2n + 1a) U(z 2n + 2a)] (80)

and, the solutions of Maxwell’s equations are the Bloch functions ∑m ck,m exp(ikz+2iπmz/a) to be compared with (59) (and (80) with (49)). Incidently, (80) has a simple expression in terms of the square-sine function

(z) = + sin(az) / |sin(az)| U(z),

1 = + , 2 = (81)

which suggests to work with the Laplace transform of Maxwell’s equations since tanh(πp/2a) is the Laplace transform of the square-sine function [21]. People fluent with the Laplace transform, could think in terms of p instad of z as they use to do with instead of t.

In opposite to photonic crystals, composite granular materials with a continuous filling factor have no lattice structure and, as shown in Section 3.3, they rather behave as a smooth dielectric grating [22]. Some of the restri- ctive assumptions on the filling factor f could be somewhat released at the expense of more intricacy:

1) It would be interesting to check what happens when a higher order approximation than 0( f 2) is used;

2) When f(x) = f cos(2ax) is changed into f (x) = ∑0

fm(cos2max), the Mathieu equation becomes a Hill equation [14,15] with solutions similar to (55) but the recurrence relations bet-ween the coefficients cm is more in- tricate;

3) Finally a generalization to a two-dimensional filling factor f (x,y), periodic in x and y would approach more closely a real physical situation.

To sum up, the application of the Maxwell-Garnett theory to nano composites deserves further research, taking into account the innocuity or not of such materials in biomedecine [23]. This theory is also used to analyze, in the frame of surface plasmon polaritons, the scattering of TE, TM light waves from a composite material made of metallic nano spherical particles immersed inside a metallic structure such as Ag particles in a Sio2 matrix [24].

The 0( f 2 ) Maxwell-Garnett approximation of the periodic permittivity in the nanodoped medium of Section 3

implies that TE, and TM fields are solutions of the Ma- thieu equation as if they were diffracted from a dielectric grating [25].

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“Nanorotors Using Asymmetric in Organic Nanorods in an Optical Trap,” Nanotechnology, Vol. 17, No. 11, 2006, pp. S287-S290.

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Guihua Mei1,2, Yuanzhang Sun1, Yancun Liu2

1Department of Electrical Engineering, Tsinghua University, Beijing, China; 2Department of Electrical Power System Technique, Guang Dong Electric Power Research Institute, GuangZhou, China. Email: meiguihua, [email protected], [email protected] Received February 20th, 2010; revised April 16th, 2010; accepted April 25th, 2010.

ABSTRACT

This paper focus on the Modeling and Calculation of DC current distribution in AC power grid induced under HVDC Ground-Return-Mode. Applying complex image method and boundary element method, a new field-circuit coupling model was set up. Based on the calculation result with complex image method, this paper derived the modification fac-tor for induced earth potential from practical measurement, which increased the accuracy of calculation. The modifica-tion method is helpful for evaluation on the effect of means used for blocking the dc-bias current in transformer neutral and also useful for the forecast of the DC current distribution when the power grid is in different line connection mode. The DC distribution character in Guangdong power grid is shown and suggestion is proposed that the mitigation of dc-bias should start from those substations whose earth-potential is highest. Keywords: HVDC, DC-Bias, DC Distribution, Ground-Return Current, Simulation

1. Introduction

As an emergency mode, it’s inevitable that HVDC transmission system operated in Ground-Return Mode (GRM) occasionally. When HVDC operate in GRM, as a result of current injection into soil through grounding electrodes, part of the return current will flow through the coils of the transformer whose neutral connected to the earth, and it will lead to wave distortion, noise and overheating etc. Sometimes it will lead to the damages of transformer or/and capacitor bank, disturbing the normal operation of power grid.

For evaluation on the range and degree of the influ-ence by the HVDC in GRM, it is necessary to calculate the dc current distribution in ac network, summarizing characteristics of dc current distribution and providing data for the application of dc current blocking devices (DCBD). It’s also need to evaluate the effects after using DCBD for better location.

In the related topic, most researches are focus on the calculation and analysis of earth potential distribution induced by ground return current of HVDC before 2003. Article [1] described a methodology that allows suitable calculations of the electric field as well as the potentials and current densities due to current in the ground electrode of a HVDC system in any point of nonhomogeneous soils and air media. Paper [2] presen-

ted a method for the theoretical evaluation of the soil surface potentials induced in ground return mode and giving rise to soil models amenable to mathematical analysis. Paper [3] presented a method to evaluate the electric voltage and the current density in the immedi-ate vicinity of a toroidal grounding installation of DC substations.

After 2005, related researches [4-14] report a lot. 2006, Dr Zhang Bo [4] used numerical methods to

calculate the DC current distribution in AC power sys-tem caused by a HVDC system. Moment method is applied to calculate the electric fields in complex earth structure caused by all the grounding systems including DC grounding electrodes, AC substation grounding systems, and the long metal pipe lines. The circuit equations are coupled to the moment method to combine the AC transmission lines with the grounding systems.

Based on specified model, [5] and [10] calculated DC currents through earthed neutral transformers by using field circuit coupling method and the resistance network algorithm. It concluded that the resistance network algorithm can be applied to engineering instead of the using of field-circuit coupling method.

Paper [9] derived the Green’s function of horizontal- vertical-composite structure soil. The DC potential and current distribution caused by HVDC transmission’s ground return mode was computed and DC current



419

through transformers’ earthed neutral under different soil structure was analyzed. Based on the physical meaning of Green’s function, [6] solved the potential expression for horizontal-vertical-composite structure soil by applying the image method. In the same year, [8] studied DC bias problem of transformers of nuclear power plant and the character of DC current distribu-tion, potential distribution of vertical grounding electrode and the DC capabilities of transformers were also carried out.

In the research report [7] by China Southern grid technology center and Tsinghua University, the field- circuit model was brought forward. This model considered inhomogeneity of soil. It was divided into two parts: the grounding network and the transmission line network. Applying this model, DC current distribution in AC power grid under HVDC transmission’s ground return mode was analyzed. DC bias simulation model for three-phase compact transformer, also three-phase and three-limb transformer, three-phase transformer with five-limb core, is presented. The analyzed method for 3 dimensions for eddy field is put forward. The influence of HVDC transmission’s ground return cur-rent to transformers is analyzed. Some rules about the difference of different structure transformer and miti-gation transformer’s DC bias were proposed.

Although lots of researches have been made, there are still many questions to be investigated. The diversity and complexity of geological condi-

tion leads to low accuracy of simulation and the simulation result will be unauthentic.

The simulation result should be checked out by lots of measuring data.

Theoretical and practical research of the 220kV and 500kV substations in Guangdong power grid was car-ried out on the base of the online measuring system of DC current through transformers’ neutral. Further re-search of modification for the theoretical model by the data of measuring system was accomplished.

2. Calculation Model

The model for power network above ground was de-scribed by electrical circuit method while the model for underground part was described by the field model which was shown in the electrical circuit way. By these models, we combine the circuit network above ground with the field effect underground to solve the problem.

The underground model of substation, which is ex-pressed by a DC ground resistance in series with an earth potential source, is shown in Figure 1.

R is the equivalent resistance between the transformer’s neutral and the zero-potential point (the farthest point).

P is the earth potential induced by HVDC ground return current and the grounded current of other substation.

In this paper, the solution method for P was complex image method or boundary element method on the as-sumption that P are related with the HVDC ground return current, the position of the substation and DC grounding electrode, geological condition, but independent of op-eration mode of power grid.

The whole model of dc distribution was shown in Fig-ure 2.

By nodal analysis of circuit theory, we have

GV SP (1)

In (1), G is the conductance matrix, V is the nodal volt-age matrix, P is the earth potential of substation, S is the transfer matrix of earth potential and nodal current.

The relationship between nodal current and earth po-tential was shown as follow:

DCP YI MI (2)

In (2), Y is the mutual resistance matrix between sub-stations, I is the injected current matrix of substations, M

Figure 1. The network model of substation

Figure 2. The whole model of dc distribution



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is the mutual resistance matrix between substations and dc electrode, IDC is the injected current of dc electrode.

The relationship between nodal voltage and injected current was shown as follow:

I XV (3)

In (3), X is the transfer matrix of nodal voltage and in-jected current.

From (1)-(3), we have

DCG SYX V SMI (4)

Equation (4) shows injected current drives DC distri-bution in ac power grid and dc distribution is decided by the DC network parameters.

3. Comparison between Complex Image Method and Boundary Element Method

Complex image method (CIM) and boundary element method (BEM) are of different characters when applied in the large scale grounding problem.

CIM is relatively simple. mountain, river and ocean are not took into account, only the equivalent horizontal stratified structure soil is needed, regardless of the subdi-vision of boundary element. The program is relatively simple, the consumption of CPU time and PC memory are much less than BEM.

In theory, BEM is the most proper method for the complexity and difference of geological condition. But sub-division of boundary element, also the coupling of dif-ferent elements is needed. When the number of boundary element becomes large, CPU time and memory are great- ly consumed. There are some questions when large scale BEM program operated in personal computer. By exam-ple of CDEGS of SES Corporation, the upper limit for the number of boundary element is about 20000 when the software used in PC. For the problem in this paper, 20000 boundary elements can not fully expressed the complexity and difference of geological condition in Gu- angdong (the area of Guangdong is about 180000 km2).

XinAn HVDC system was taken as example for comparison of the two methods. By validation of measuring data, we will find the better method to solve this problem.

4. Complex Image Model

The horizontal stratified structure soil is shown in Figure 3. h is the thickness of the soil while ρ is the resistivity, and I is the current point.

Applying Prony method [15], the Green’s function of horizontal stratified structure soil is:

2 2

2 2 2 21

1,

4

4 ( ) ( )

Mi i

ii i

ρIr z

r za cρI

r z b r z d

(5)

where: ai and ci is the amplitude of complex image, bi and di is the position of complex image.

For approximation, all the DC and AC grounding electrodes were viewed as point source. So the earth potential of substation induced by current point source could be got by (1).

The earth potential induced by XinAn HVDC ground return current with 600 A could be obtained and the earth potential curve is shown in Figure 4. The soil parameter is listed in Table 1.

Figure 4 shows that earth potential decrease rapidly while d < 20 km. Potential difference between line-connected substations may lead to dc bias of transformers.

Figure 3. Model of multi-layered soil

Table 1. 4-layer horizontal stratified soil parameter

layer number Resistivity (Ω·m) thickness (m)

1 235 30

2 5900 1000

3 14100 50000

4 120 inf

0 20 40 60 80 100 120d (km)

250

200

150

100

50

0

p(V

)

Figure 4. Earth potential



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5. Method of Model Modification

Because of the complexity and difference of geological condition, the approximation will bring down the verac-ity of the model and even the simulation result will be-come invalid.

This paper presented a feasible method to overcome this shortage. Applying the measuring data, provided by the online monitoring system of DC current through the neutral of transformer, we modify the parameter of the model.

The core of this method is to modify the earth poten-tial P by the measuring data. The question of modifying the earth potential could be solved by the least-square method.

Supposed the injecting current of DC electrode is x. there are m substations’ earth potentials to be modified. Let

, 1,i i ir x I x J x i m (6)

where Ii(x) are the measuring data, Ji(x) are the computa-tional result by CIM.

2

1

1( ) ( ), ,

2

mn

ii

f x r x x R m n

(7)

Let

1 2( ) ( ( ), ( ), , ( ))Tmr x r x r x r x (8)

The objective function is:

1( ) ( ) ( )

2Tf x r x r x (9)

This equation could be solved by numerical analysis of Gauss-Newton.

In many practical problems, if *x is the locally solu-

tion and corresponding objective function value *( )f x

close to 0, when the iterative point close to *x , or the

curve ( )ir x is almost a straight line ( 2 ( ) 0ir x ), the

Gauss-Newton method can produce a good effect. But

when ( )( )kr x is large, or ( )ir x ’s curvature is large,

2 ( )( )kf x ignores 2

1

( ) ( )m

i ii

r x r x

which is not ne-

glectable, so the Gauss-Newton method can’t get the right answer.

Obviously, ( ) ( )( ) ( )k T kJ x J x is a Positive semidefinite

matrix. If Jacobi matrix ( )( )kJ x is column full rank,

matrix ( ) ( )( ) ( )k T kJ x J x is positive definite matrix. So ( )kd , getting from the equations, is agree with the de-

scending direction of ( )f x , but ( 1) ( )( ) ( )k kf x f x

isn’t confirmed, then additional one-dimension search is needed as the Newton method.

6. Examples

XinAn HVDC system consists of a 1200km long bipolar ±500 kVdc transmission line with transmitting capacity of 3000MW in Southern China. When HVDC system operates in ground return mode, 3000 A direct current will inject into the earth.

Under the transmission lines connection of Guangdong power grid in Apr 2007 and suppose that 600 ADC current injected into the earth by HVDC, The result of trans-former neutral dc current at some substation by CIM and BEM was listed in Table 2. The geological condition was approximate to horizontal 4 layers soil for complex image method. The BEM boundary condition was deter-mined by the geologic exploration result.

From Table 2, we can see, 1) The result of YingDe-PaJiang-ChongHua by CIM is

close to measuring data, but BeiJiao’s computation result was near 3 times of measuring data. As a whole, horizontal 4 layers soil is suitable for the local geological condition.

2) The difference between computation and measuring for QingYuang, KangLe, SangShui and LuoDong was large. The reason is another HVDC system (TianGuang) was then under bipolar operation mode with unbalanced power of 400MW, which influence the earth potential of the substations on the path of QingYuan-KangLe-Sang- Shui-LuoDong, when the measuring data were recorded. The event leads to the measuring data stand for not only the influence of XinAn HVDC system and the error. But however the correction of the method presented in this paper do not be influenced.

Although actual geological condition of Guangdong province was considered and LAPACK for matrix algo-rithm is used, problems were still found that BEM de-mands huge CPU time and occupied big size memory.

Table 2. Computational result and measuring data

Name Complex Image

Method (A) Measuring Data (A)

Boundary Element Method (A)

PaJiang 25.7 25.9 8.25

YingDe # 1T 11.5 12.0 3.90

ChongHua 12.2 12.9 3.51

SanShui 17.8 3.0 25.52

KangLe 33.8 7.5 37.23

BeiJiao 17.5 6.0 23.08

LuoDong 1.1 9.0 1.62

QingYuan 21.4 3.0 33.77



422

Boundary element’s number N is limited as computing time grows proportionally to N3, which leads to the boundary division being rough and the geological condi-tion could not be simulated properly, especially for rivers and ocean. Another side, the lacking of geological condi-tion parameter in large area for DC current conducting also leads to simulation error.

6.1 Example for the Method of Model Modification

Based on the operation mode of Guangdong power grid in Apr 2007, the modification model for complex image method was used to solve DC distribution when XinAn HVDC system was under commissioning in 2007. Com-parison between computation result and measuring data is shown in Table 3.

From the result above we could see that in large area, the simulation result of the transformer neutral dc current are more close to the measuring data without appearance of big error. The modification factor getting by the field measuring data will be helpful for these simulations, such as changing of transmission line connection, being built of new transmission line or the effect evaluation of DCBD using, etc.

7. Summary on the DC Distribution in Guangdong Power Grid

According to the operation mode of Guangdong power grid in the first half of 2008, the correction model was used to analyze the DC bias caused by the 4 HVDC transmission projects in Guangdong power grid. Some conclusions are listed as follows.

When HVDC system is under ground return mode, the transformer, whose location is close to the location of electrode of HVDC system, is easier suffering from DC bias. The transformers in the transmission line path, which connect the substation with higher earth potential and the substation with lower earth potential, might also suffer from DC bias. Generally the substations far away from the electrode location, especially those that close to river or ocean are of the lower earth potential. The electrode

Table 3. Comparison between computation result and measuring data when XiAn HVDC system operated in GRM (600ADC current injected into the earth)

Name Modification factor Current (A) Measuring data (A)

QingYuan 0.63 0.58 3.0

KangLe 1.93 7.50 7.5

SanShui 2.16 2.41 3.0

XianXi 1.10 2.26 1.0 LuoDong 2.27 14.56 9.0 PaJiang 0.92 27.13 25.9 YingDe 0.95 7.92 12.0

ChongHua 0.90 13.17 12.9

BeiJiao 0.67 5.35 6.0

locations of the four HVDC projects in Guangdong province are illustrated in Figure 5.

For TianGuang HVDC system, LuoDong, SanShui, XianXi, GuoTang are close to the DC grounding elec-trode of TianGuang HVDC system. Because of the high amplitude earth potential of the above substations, the DC current through the transformers’ neutral are relative big by the monitoring data. Meanwhile the high earth potential at those substations leads to DC current goes through the transmission line to other substations with lower DC earth potential. Most of the transformers, which connected with the transmission lines of SanShui→ KangLe → QingYuan or LuoDong → HongXing →

ZiDong→XiJiang→FenJiang, are in DC bias. Another typical line is LuoDong→XiJiang→JiangMen→KaiPing→TaiShan→TangMei, which begins from the substation near the HVDC grounding electrode and end at the sub-station near the coast of South China Sea. Especially TangMei near the coast suffered severe DC bias for the DC earth potential close to coast is near to zero.

There are 4 ± 500 kV HVDC systems which transmit power from western China to Guangdong power grid in 2008. When different HVDC system is in ground return mode, there is similar rule for the transformers which might be in dc-bias by the result of simulation and moni-toring measures.

From the rule of DC current distribution, we could find that the substations with higher induced dc earth potential should be installed the DCBD first when miti-gation of dc-bias is under schedule.

8. Conclusions

1) Comparison between complex image method and B- EM shows BEM isn’t suitable for the solution of grounding problem in large scale complex geological condition for the limits of PC hardware. Although complex image method couldn’t take geological condition into account completely, proper choice for the soil structure and pa-rameter still could simulate the local geological condition precisely.

Figure 5. DC grounding electrodes in Guangdong



423

2) Utilizing the measuring data, the correction model was presented. This method is significant for the accu-racy improvement of simulations, such as changing of transmission line connection, or the effect evaluation of DCBD installation, etc.

3) When HVDC system is under ground return mode, the transformer, whose location is close to the ground electrode of HVDC system, is easier suffering from DC bias. Another case, the transformers which located far away from the ground electrode of HVDC system, with lower dc earth potential, especially those that close to river or ocean, and connected to the substation with higher dc earth potential might also easier suffer from DC bias.

4) The substations with higher induced dc earth poten-tial should be installed the DCBD first when mitigation of dc-bias is under schedule.

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Field and Potential Distributions into Soil and Air Media for a Ground Electrode of a HVDC System,” IEEE Trans-actions on Power Delivery, Vol. 18, No. 3, 2003, pp. 867- 873.

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[10] H. S. He, “Computational Method Study on DC Current Distribution in AC System when HVDC Operating in Ground-Return Mode. Power Technology Development and Energy Conservation,” The 9th Symposium on Youth Essays in Chinese Society for Electrical Engineering, November 2006.

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Polynomial-Based Evaluation of the Impact of Aperture Phase Taper on the Gain of Rectangular Horns Konstantinos B. Baltzis

Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece. Email: [email protected] Received March 28th, 2010; revised May 12th, 2010; accepted May 18th, 2010. ABSTRACT The aperture phase taper due to quadratic phase errors in the principal planes of a rectangular horn imposes signifi-cant constraints on the on-axis far-field gain of the horn. The precise calculation of gain reduction involves Fresnel integrals; therefore, exact results are obtained only from numerical methods. However, in horns’ analysis and design, simple closed-form expressions are often required for the description of horn-gain. This paper provides a set of simple polynomial approximations that adequately describe the gain reduction factors of pyramidal and sectoral horns. The proposed formulas are derived using least-squares polynomial regression analysis and they are valid for a broad range of quadratic phase error values. Numerical results verify the accuracy of the derived expressions. Application examples and comparisons with methods in the literature demonstrate the efficacy of the approach. Keywords: Microwave Antennas, Rectangular Horn, Gain, Quadratic Phase Error, Linear Regression

1. Introduction Horns are among the simplest and most widely used microwave antennas. They occur in a variety of shapes and sizes and find application in areas such as wireless communications, electromagnetic sensing, radio frequency heating, and biomedicine. They are commonly used as feed elements for reflector and lens antennas in microwave systems and as high gain elements in phased arrays. Moreover, they serve as a universal standard for calibration and gain measurements of other antennas [1].

Among the microwave horns, the rectangular horn is the simplest and most reliable one. This is a hollow pipe of a rectangular cross section that is flared to a larger opening in the E- or H-plane direction (sectoral horn) or in both directions (pyramidal horn). Rectan-gular horns are useful tools in science and engineering due to their simplicity in construction, ease of excitation, versatility, and high gain.

A classical expression for the gain of a pyramidal horn is the Schelkunoff’s horn-gain formula. This for-mula calculates the on-axis far-field gain of the horn as the product of the directivity of a uniform dominant mode rectangular waveguide without flares and the gain reduction due to the amplitude and phase taper across the horn aperture [2]. Its main assumptions are

that the horn operates at the dominant TE10 waveguide mode and it is well-matched to the feeding waveguide; moreover, it neglects the contribution of the fringe currents caused by the discontinuity of the aperture and the mutual interaction between the aperture edges [2,3]. The formula includes the geometrical optics of the rad- iated field and the singly diffracted fields of the aperture edges and represents the monotonic gain component. However, it omits multiple diffraction and diffract- ted fields reflected from horn interior; therefore, it is adequate for pyramidal horns but calculates errone- ously the gain of sectoral ones [4]. In [5], Schelkun-off’s formula was extended by involving an additional term that accounts for the influence of the edge effect on the on-axis gain and included sectoral horns and open-ended rectangular waveguides. In general, the ex-pressions presented in [2,5] give adequate results and are commonly used in the literature [6-11]. Compari-sons between calculated results and measured data showed an uncertainty ±0.5 dB for frequencies below 2.6 GHz and ±0.3 dB for higher ones [12]. Several so-lutions with increased accuracy can be found in the published literature, e.g. [13-16]. However, their com-plexity and computational cost are worthwhile only if we require very accurate results.



425

In both horn-gain formulas [2,5], the calculation of the gain reduction factors involves Fresnel integrals and it is made numerically. However, approximate but simple closed-form expressions are often required [11, 17,18]. In this paper, we extend the analysis in [17] to include a broader range of aperture phase error values. The approximate formulas in [17] are valid for aperture phase errors up to the optimum gain condition ones. Here, we provide improved approximate polynomial expressions for the gain reduction factors of a rectangular horn. These formulas were obtained from the application of least squares polynomial fitting over the range of aperture phase error parameters from zero to one (typical values for practical applications [19,20]). We further investigate the impact of the polynomial order on the approximation error and give representative examples that show the merits of our proposal. Co- mparisons with methods in the literature and results derived from professional antenna design software [21] validate the formulation.

The rest of the paper is organized as follows: Section 2 discusses some theoretical background. Section 3 presents and evaluates the proposed formulation. In Sect- ion 4, representative examples show the merits of our proposal. Finally, Section 5 concludes the paper.

2. The Schelkunoff ’s Classical and Improved Horn-Gain Formulas

Figure 1 shows the geometry of a pyramidal horn with throat-to-aperture length P and aperture sizes A and B. The inner dimensions of the feeding rectangular waveguide are a and b. When A = a or B = b, we get the E- or the H-plane sectoral horn, respectively. Next, in Figure 2, we give the cross-sectional views of the horn in the two principal planes.

We assume a lossless pyramidal horn that it is well- matched to the rectangular waveguide and operates in the dominant TE10 mode. In this case, the on-axis far-field gain of the horn is1 [2,18]

2

32E H

ABG L Lλ

=π

(1)

where λ is the free-space wavelength and LE and LH are the gain reduction factors that represent the impact of the aperture phase taper due to the quadratic phase errors in the principal planes calculated [22] from:

222

110

2 exp 1 1 dB

EyL jkR y

B R

= − + − ∫ (2)

A

a

P

Figure 1. Pyramidal horn geometry

E-plane view H-plane view

Figure 2. Cross-sectional views of a pyramidal horn antenna

2

22

220

cos exp 1 1 dA

Hx xL jkR x

A A R

π π = − + − ∫

(3)

with 2k λ= π and 1j = − . Notice that (2) and (3) do not include the path long error approximation increasing the accuracy of the results. The gain reduction factors can also be written as functions of the aperture phase error parameters in the E- and H-plane that are given by

( )8

B B bs

λP−

= (4)

and

( )8

A A at

λP−

= (5)

respectively, as [6,17]

( ) ( )2 22 2

4E

C s S sL

s

+= (6)

22

2

1 8 1 864 4 4

1 8 1 84 4

Ht tL C C

t t t

t tS St t

π + − = −

+ − + −

(7) 1Usually, it is assumed that the overall efficiency (i.e. the product of the reflection, conduction, and dielectric efficiencies) of a rectangular horn is one. In this case, the on-axis far-field gain and the directivity of the horn are identical [1].



426

where ( )C ⋅ and ( )S ⋅ are the cosine and sine Fresnel integrals [23], respectively.

Equation (1) can be extended by incorporating the edge effect and the impact of the fringe currents at the aperture edges. In this case, the gain-formula becomes [5,7]

2

232 11 1

2 E HAB kG L L

βλ

= + − π (8)

where ( )21 2β k a= − λ is the TE10 mode propaga-tion constant [24]. The improved formula is valid for both pyramidal and sectoral horns. It reduces to (1) for large apertures ( 1β k ≈ ) and calculates the gain values of the E- and H-plane sectoral horns by setting 1HL ≈ and 1EL ≈ , respectively.

3. Polynomial Description of the Gain Reduction Factors

In [17], Aurand provided the following first- and second- order approximations for the gain reduction factors:

( )

( )

1

1

1.032462 0.813696

1.033320 0.567302E

H

L s

L t

= −

= − (9)

( )

( )

2 2

2 2

1.001633 0.07082 2.97150

1.002535 0.07341 1.31704E

H

L s s

L t t

= − −

= − − (10)

that are valid for 0.25s ≤ and 0.375t ≤ . Despite their accuracy in the given range of values, these approximations can not describe the gain reduction factors for phase errors far from the optimum gain condition ones (see next Section).

In this paper, we extend Aurand’s proposal and produce closed-form expressions for the gain reduction factors LE and LH by polynomial regression curve fitting [25-28] of (6) and (7). The fitting curves are linear polynomials calculated with the least squares method [25-27]. In this method, curve-fitting involves the minimization of the sum of the squared residuals, i.e. the squared differences between the exact LE (LH) value and the LE (LH) value that is computed from the curve-fit equation for the same aperture phase error. In order to get the best fit, we use the R2 goodness-of-fit statistics metric (this is the square of the sample correlation coefficient between the data values and the calculated ones from the fitting polynomial). The fit improves as R2 values approach unity.

In practice, we approximate LE and LH with nth-order polynomials, i.e. it is

( )

( )

,0

,0

, 1

, 1

nn i

E E n ii

nn i

H H n ii

L l e s s

L l h t t=

=

≈ = ≤

≈ = ≤

∑

∑ (11)

Let en and hn be vectors with elements the polynomial coefficients en,i and hn,i, i = 0,1...n, respectively. In this case, we formulate the least squares problem as:

( )( )2

, ,0

find : Minimize N

nn E j E j

jl L

=

−∑e (12)

and

( )( )2

, ,0

find : Minimize N

nn H j H j

jl L

=

−∑h (13)

The subscript j in (12) and (13) denotes that the specific values are calculated at s or t equal to j N (the maximum value of the two aperture phase error parameters is one, see (11)). Each curve is evaluated at 10001 points in steps of 10-4 in the range [0,1], i.e. N = 104. Derivation of en and hn gives the nth-order polynomial approximation of LE and LH, respectively2.

Recall that the choice of the best fit approximation uses the R2 goodness-of-fit statistics metric. Table 1 gives the calculated R2 values for the best fit nth-order polynomial approximation of (6) and (7) for n = 1,2…10. The corresponding polynomial coefficients, en,i and hn,i, are given in Tables 2 and 3. In Table 1, we also give the F-statistic values [25-27] of each approximation (F-statistic value goes toward infinity as the fit becomes more ideal). Notice that LH is adequately approximated from a polynomial with order lower than the one that is required for LE. All the results were checked and validated using Matlab R2008a curve fitting routines [29].

Table 1. Goodness-of-fit values

E-plane H-plane n

R2 F-statistic R2 F-statistic

1 0.95754249 225507.05 0.99012258 1002310.4

2 0.97988824 243562.02 0.99051231 521894.17

3 0.99871152 2582915.7 0.99988122 28051368

4 0.99993194 36715144 0.99998119 1.328863·108

5 0.99999478 3.8304487·108 0.99999961 5.1801471·109

6 0.99999993 2.2592440·1010 0.99999999 1.2452233·1011

7 0.99999999 2.3209471·1011 1 4.1892924·1012

8 1 3.7006211·1013 1 2.8864097·1014

9 1 3.7613197·1014 1 8.9753437·1015

10 1 1.2875745·1017 1 1.1346748·1018

2A further discussion on this issue is beyond the scope of the paper; the interested reader can find additional information about the development and implementation of least squares algorithms in the proposed literature [25-27].



427

Table 2. Polynomial coefficients en,i

i n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10

0 1.0336239 1.1457623 1.0240209 0.9888856 0.9976955 1.0004342 1.0000973 0.9999895 0.9999976 1.0000002 1 –1.1374395 –1.8103371 –0.3490752 0.3539478 0.0894666 –0.0256761 –0.0067831 0.0009942 0.0002653 –2.414 x10-5

2 — 0.6728975 –2.9804397 –6.1444123 –4.2926737 –3.1409013 –3.3960639 –3.5322412 –3.5161931 –3.5083720 3 — — 2.4355582 7.3574573 2.4191175 –2.1885480 –0.7707180 0.2281648 0.0783339 –0.0120796 4 — — — –2.4609496 3.0948216 11.734669 7.8352862 4.0889891 4.8195421 5.3734495 5 — — — — –2.2223085 –9.8255266 –4.2101716 3.5826714 1.5369186 –0.4574140 6 — — — — — 2.5344060 –1.5211957 –10.613170 –7.2033875 –2.7711778 7 — — — — — — 1.1587434 6.7253379 3.3850412 –2.7660781 8 — — — — — — — –1.3916486 0.3829062 5.5730509 9 — — — — — — — — –0.3943455 –2.8292557 10 — — — — — — — — — 0.48698211

Table 3. Polynomial coefficients hn,i

i n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10

0 1.0744003 1.0639483 1.0033308 0.9962335 0.9995997 1.0001198 1.0000154 0.9999976 0.9999997 1.0000000 1 –0.8163131 –0.7535950 –0.0260029 0.1160082 0.0149499 –0.0069168 –0.0010603 0.0002270 3.6932 × 10-5 –4.7892 × 10-6

2 — –0.0627181 –1.8817891 –2.5209139 –1.8133644 –1.5946318 –1.6737272 –1.6962670 –1.6920827 –1.6909553 3 — — 1.2127140 2.2069413 0.3200012 –0.5550385 –0.1155393 0.0497937 0.0107279 –0.0023056 4 — — — –0.4971136 1.6257471 3.2665367 2.0578052 1.4377260 1.6282047 1.7080531 5 — — — — –0.8491443 –2.2930680 –0.5524193 0.7374358 0.2040421 –0.0834507 6 — — — — — 0.4813079 –0.7758482 –2.2807329 –1.3916927 –0.7527678 7 — — — — — — 0.3591875 1.2805586 0.4096356 –0.4770785 8 — — — — — — — –0.2303428 0.2323408 0.9805257 9 — — — — — — — — –0.1028186 –0.4538229 10 — — — — — — — — — 0.0702009

In order to describe simple but accurately the gain redu-

ction factor, we have to estimate the minimum required polynomial order. In practical terms, the goodness-of-fit statistics may not provide an efficient way to estimate the degree of error. In this case, a graphical inspection of the

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

Gai

n re

duct

ion

fact

or

LE

lE(1)

lE(2)

lE(3)

lE(4)

s Figure 3. E-plane gain reduction factor: Exact and approximate curves

fitting curves ensures the suitability of the proposed approximations. Figures 3 and 4 show the exact and the (approximate) fitting curves of the gain reduction factors. Notice that the curves that describe ( )4

El and ( )3Hl are

almost similar to the exact solution.

0.0 0.2 0.4 0.6 0.8 1.00.2

0.4

0.6

0.8

1.0

1.2

Gai

n re

duct

ion

fact

or

LH

lH(1)

lH

(2)

lH(3)

t Figure 4. H-plane gain reduction factor: Exact and approximate curves



428

In order to further investigate the relation between the approximation error and the polynomial order, we used two well-known error metrics, the average absolute error and the rms error. Figure 5 shows the variation of the two metrics as a function of the polynomial order of the approximate formulas. We see that LH is adequately approximated with fewer terms than LE; for example, ( )3

Hl and ( )5

El yield errors less than 1%. We also notice that the average absolute error is always slightly smaller than the rms error.

4. Application Examples In order to show the efficacy of our approach, we give three representative examples.

Example 1: Let us consider some typical X-band pyramidal horns, see Table 4. Horns operate at 10 GHz and they are fed from WR-90 waveguide. Their aperture phase errors are calculated from (4) and (5). Figure 6 illustr- ates the gain values of the horns. With G0,1 and G0,2 we present the values that are calculated from (1) using Aur- and’s first- and second-order approximations, respective- ely; G1, G2, G4, and G6 are the results obtained from (1) and the proposed first-, second-, fourth-, and sixth-order approximation. G0 denotes the gain values that are calculated from (1) with adaptive quadrature integration of (2) and (3). Finally, Gacc are the exact gain values calculated with the professional antenna design software ORAMA [21].

As it was expected, (9) and (10) are adequate only for small values of s and t (moreover, G0,2 takes complex values for great values of s; these are not shown in Figure 6). In any case, the fourth-order approximations give results almost identical to the numerically calculated ones. The results are also in good consistency with the exact values calculated with ORAMA. The small gain values in cases 3 and 5 are due to the fact that the far-field gain is not maximized at the horn’s axis.

2 4 6 8 1010-6

10-5

10-4

10-3

10-2

10-1

100

101

102

H-plane

Perc

enta

ge o

f err

or (%

)

average asbolute error rms error

E-plane

n Figure 5. Average absolute and rms error versus the order of the fitting polynomials

Table 4. Horns specifications

ID A(cm) B(cm) P(cm) s t

1 14.6 11.48 20 0.250 0.375 2 10 10 20 0.187 0.161 3 10 20 20 0.792 0.161 4 20 10 20 0.187 0.739 5 20 20 20 0.792 0.739

1 2 3 4 510.0

12.5

15.0

17.5

20.0

22.5

Gai

n (d

Bi)

iDs

G0,1

G0,2

G1

G2

G4

G6

G0

Gacc

IDs Figure 6. Calculated gain values (in dBi)

Example 2: We consider an E- and an H-plane sectoral

horn that operate at 10 GHz and are fed from WR-90 waveguide. In the first case, B = 20 cm; in the second one, it is B = 20 cm. In both cases, the throat-to-aperture length is 20 cm.

First, we calculate the gain values from (8) with adap-tive quadrature integration of (2) and (3). The exact horns’ gains are 9.1 dB (E-plane sectoral horn) and 10.15 dB (H-plane sectoral horn). The gain values that are ob-tained from (1) and (9) are 14.83 and 11.52 dB, respec-tively; Aurand’s second-order approximation gives worst results. On the other hand, our formulation gives (the subscript denotes the order of the fitting polynomial) that G1 = 10.19 dB, G2 = 10.22 dB, G4 = 9.03 dB and G6 = 9.09 dB (E-plane sectoral horn). In the case of the H-plane sectoral horn, the approximation error is smaller. The calculated gains are G1 = 10.37 dB, G2 = 10.39 dB, G4 = 10.15 dB, and G6 = 10.15 dB. Again, the fourth- order approximations give adequate results.

Example 3: In [11], Selvan proposed a design method for pyramidal horns of any desired gain and aperture phase error. However, the accuracy of his method is strictly related to the accuracy of the approximations of (6) and (7).

Let us consider the design examples in Table 5. Table 6 gives the calculated horns dimensions with

the method proposed in [11] using (10) (Aurand’s app-



429

roximation) and the proposed second-, fourth-, and sixth- order approximations. Next, we used this data and calculated the exact gain values with ORAMA, see Table 7. Finally, Figure 7 shows the absolute relative error between the computed and the desired gain for each case. We notice that Aurand’s approximation is adequate at small values of s and t (IDs 1 and 3). However, as the aperture phase error parameters increase (IDs 2 and 4) these approximations lead to erroneous results. In any case, the proposal in [11] is an accurate design method when a polynomial approximation with polynomial order at least equal to four is used for the description of the gain reduction factors.

Table 5. Horns design examples

ID f (GHz) Waveguide type Gdes (dBi) s t

1 1 WR-975 15.45 0.2 0.3

2 1 WR-975 15.45 0.4 0.6

3 34 WR-28 24.58 0.25 0.375

4 34 WR-28 24.58 0.5 0.75

Table 6. Calculated hors dimensions

Aurand’s appr. 2nd order appr. ID

A (cm) B (cm) P (cm) A (cm) B (cm) P (cm)

1 74.046 55.901 50.681 77.829 59.027 57.359

2 128.095 100.331 91.916 112.992 87.949 69.228

3 6.601 5.273 14.690 6.951 5.558 16.384

4 24.200 19.646 107.371 11.980 9.667 25.500

4th order appr. 6th order appr. ID

A (cm) B (cm) P (cm) A (cm) B (cm) P (cm)

1 74.130 55.971 50.826 74.122 55.964 50.812

2 112.13 87.242 68.029 111.989 87.126 67.834

3 6.591 5.264 14.642 6.580 5.255 14.587

4 12.510 10.099 27.880 12.525 10.112 27.951

Table 7. Desired and calculated gain values (in dBi)

ID Desired Aurand’s appr. 2nd order appr.

4th order appr.

6th order appr.

1 15.45 15.62 16.09 15.63 15.63

2 15.45 16.75 15.64 15.57 15.56

3 24.58 24.67 25.12 24.66 24.64

4 24.58 30.34 24.22 24.60 24.61

1 2 3 40

10

20

30

40

270

280

abso

lute

rela

tive

erro

r (%

)

iDs

Aurand's appr. 2nd order appr. 4th order appr. 6th order appr.

IDs Figure 7. Relative gain errors

5. Conclusions In this paper, we presented a set of nth-order polynomial approximate expressions for the gain reduction factors of pyramidal and sectoral microwave horns. The formulas were derived with polynomial regression curve fitting techniques. Comparisons with methods in the published literature and results calculated with commercial antenna design software verified the accuracy of the proposed formulation and demonstrated the benefits of the approach. We also explored the relation between the polynomial order of the derived formulas and the approximation error. It was found that that a third-order polynomial approximation of the gain reduction factor in the H-plane is adequate; in order to obtain accurate results in the E-plane, a fourth-order approximation is required. This paper ex-tends previous work in the literature and applies to horns with large values of aperture phase errors. The proposed formulation is a useful tool in the analysis and design of rectangular horns, especially when simple closed-from expressions are required.

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3rd Edition, John Wiley & Sons, Inc., Hoboken, 2005. [2] S. A. Schelkunoff, “Electromagnetic Waves,” David Van

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[4] E. Jull, “Errors in the Predicted Gain of Pyramidal Horns,” IEEE Transactions on Antennas and Propagation, Vol. 21, No. 1, January 1973, pp. 25-31.

[5] K. T. Selvan, “An Approximate Generalization of Schel- kunoff’s Horn-Gain Formulas,” IEEE Transactions on



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[18] J. F. Aurand, “Pyramidal Horns, Part II: A Novel Design Method for Horns of Any Desired Gain and Aperture Phase Error,” Proceedings of the 1989 IEEE Antennas and Propagation Society International Symposium, San Jose, Vol. 3, 26-30 June 1989, pp. 1439-1442.

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431

Inf luence of Magnetic Field Intensity on the Temperature Dependence of Magnetization of Ni2.08Mn0.96Ga0.96 Alloy

Kharis Y. Mulyukov, Irek I. Musabirov

Bulk nanostructures and nanotechnology, Institute for Metals Superplasticity Problems Russian Academy of Sciences, Ufa, Russia. Email: [email protected] Received April 20th, 2010; revised June 7th, 2010; accepted June 12th, 2010.

ABSTRACT

Results of investigation of the temperature dependence of magnetization of Ni2.08Mn0.96Ga0.96 alloy in the magnetic fields of various intensities are reported. An abrupt change in magnetization at transformation of low temperature phase to the high temperature one is observed. Magnetization increases during the phase transition in the magnetic field having intensity below 500 kA/m and decreases at higher intensities. The explanation is based on zigzag configuration of do-mains in twinned structure. In the Curie temperature region the ferromagnetic ↔ paramagnetic phase transition occurs sharply at low field strength, while at higher field strength the transition is smooth. It is concluded that the increase in flatness of the curve σ = f (T) and the increase of ferromagnetic state destruction temperature with increase of the in-tensity of the magnetic field is indicative of the main role of Mn in magnetization of the alloy. Keywords: Shape Memory Alloy, Magnetization, Ni-Mn-Ga, Martensitic Transformation, Heusler Alloys

1. Introduction

Owing to the fact that martensitic phase transition occurs in Ni-Mn-Ga alloys in room temperature region, they exhibit shape memory effect (SME) [1]. It is also known that phases forming in Ni-Mn-Ga system alloys before and after martensitic transformation are ferromagnetic [2]. Such feature of these materials allows controlling the effect of shape memory by means of the external mag-netic field [3]. In particular, under the magnetic field there occurs a change in crystal linear dimensions up to 9 % in single crystal Ni48.8Mn29.7Ga21.5 sample [4]. The possibility to control sample dimensions expands practi-cal application of these alloys as functional materials. Observation of similar effect for polycrystalline samples is very promising for applications but this requires a de-tailed study of physical properties of the material.

In spite of a large number of works considering physical properties of Ni-Mn-Ga system alloys, the mech- anisms of changes of some of their properties during the phase transformation are still not fully understood. For example, measurement made in the magnetic field of low intensity has revealed an abrupt increase in magnetization on the temperature dependence of magnetization for Ni2MnGa alloy. Traditionally this increase

is attributed to the change in magnetocrystalline anisotropy constant at phase transformation [5-7]. Some authors consider that magnetic moments of Ni and Mn in low temperature phase are antiparallel and after transition to high temperature phase they become parallel [8,9].

In the existing literature, the authors usually do not pay attention to the behavior of the temperature dependence of magnetization in the vicinity of the Curie temperature in spite of the facts, that also in this region the effect of the magnetic field is rather essential. Magne- tic field reveals the role of the magnetic moment of Mn atoms in the magnetization of the alloy. Previously its role was confirmed by the neutron diffraction studies [10,11].

In order to obtain additional data on the change of alloys physical properties during the phase transitions we carry out the investigation of temperature dependence of magnetization of polycrystalline Ni2.08Mn0.96

Ga0.96 alloy in magnetic fields of various intensities. An explanation of the obtained results is proposed.

2. Materials and Experimental Procedure

The Ni2.08Mn0.96Ga0.96 alloy was prepared by vacuum arc melting [12]. The sample with 1 mm × 1 mm × 6 mm

Influence of Magnetic Field Intensity on the Temperature Dependence of Magnetization of Ni2.08Mn0.96Ga0.96 Alloy


432

dimensions was cut from alloy cast ingot by electrospark method. The selection of this alloy was motivated by the absence of intermartensitic transformation and the oc-currence of direct transition of the low temperature phase to the high temperature one. Therefore during measuring the temperature dependence of magnetization in low in-tensity magnetic field only one abrupt changing of mag-netization in phase transition region was observed.

The data of temperature dependence of magnetization have been recorded by means of a vibrating-coil magne-tometer using a sample cut from the ingot processed for former studies of thermal expansion [13]. In measurements, magnetic field of intensity not greater than 1 MA/m was applied along the long side of the sample. To simp- lify the task, heating curves for all values of intensity are shown in Figure 2, and heating and cooling curves for two values of field intensity in Figure 3. Measurements were made within the temperature range 225-425 K at a rate of 5 K/min.

The sample surface was polished at 340 K for the purpose to expose of microrelief of the low-temperature phase. At this temperature the alloy is in the height temperature phase. Martensitic transformation that takes place on cooling of the sample results in the appearance of microrelief on its surface visible by optical microscope. Investigation of microstructure was carried out by optical metallographic microscope AXIOVERT-100A with es-pecial attachment for cooling the sample down to 230 K.

3. Results and Discussion

Figure 1 shows a fragment of the low temperature phase microstructure of the sample processed at 260 K. Evi-dently the polycrystalline sample has sufficiently coarse grains (grain size ~ 360 μm) and each is divided into many twin plates with a width varying from 1 to 30 μm. Since the sample is polycrystalline the distribution of twins is chaotic. After repeated heating and cooling the distribution of twin structure does not change. The essen-tial influence of magnetic field with intensity below 916 kA/m on twin plate dimensions and theirs distribution in low temperature phase formed during martensitic trans-formation without applying of magnetic field has not been revealed. Figure 2 shows the temperature depend-ence of magnetization (σ/σS (T), where σS is saturation magnetization measured at 223 K) for the selected alloy, the measurement being made in the magnetic field of different intensity. From Figure 2 one can conclude that the influence of magnetic field on the σ/σS (T) depend-ence is most apparent in phase transition regions. In par-ticular, in the region of martensite transformation it causes an abrupt changing of magnetization while in the region of magnetic phase transition it increases the tran-sition temperature of ferromagnetic to paramagnetic state. From curve 1 in Figure 2 (H = 24 kA/m) it is seen that magnetization of low temperature phase remains equal to

0.10 × σS until 271 K. Then with increasing temperature a value of magnetization increases abruptly and reaches 0.46 × σS at 315 K. This abrupt increase is obviously due to the transition to high temperature phase. On further heating magnetization of high temperature phase starts to decrease slowly at first and then rapidly drops to zero at 375 K.

The character of curve 2 (Н = 80 kA/m) is almost similar to curve 1.The difference is in the value of magnetization which is higher by 16%. The transition of the alloy to the paramagnetic state begins at 380 K and also has an abrupt character.

The character of curve 3 (Н = 409 kA/m) slightly dif-fers from that of curve 2. The increase in magnetization is not so abrupt and is about 4%. Moreover, the declinea- tion of σ/σS (T) dependence curve to zero is smoother. In this case ferromagnetic state is destroyed at 391 K.

Figure 1. Optical micrograph of low temperature phase Ni2.08Mn0.96Ga0.96 alloy

220 250 280 310 340 370 400 430

1.0

0.8

0.6

0.4

0.2

0.0

σ/σ s

, arb

, uni

t

T, K

1

2

3

4

Figure 2. Curves of temperature dependence of magnetiza-tion Ni2.08Mn0.96Ga0.96 alloy, recorded during the heating in magnetic field follow intensity: 1) Н = 24 kA/m, 2) Н = 80 kA/m, 3) Н = 409 kA/m, 4) Н = 916 kA/m



433

220 250 280 310 340 370 400 430

1.0

0.8

0.6

0.4

0.2

0.0

σ/σ s

, arb

, uni

t

T, K

1

2

Figure 3. Curves of temperature dependence of magnetization Ni2.08Mn0.96Ga0.96 alloy, recorded during the heating and cooling in magnetic field follow intensity: 1) Н = 80 kA/m, 2) Н = 916 kA/m

The character of curve 4 (Н = 916 kA/m) is absolutely

different. Unlike the previous cases, characterized by the abrupt increase in magnetization, a decrease in magneti-zation by 3.5% is observed here. In the region of para-magnetic transition the σ/σS (T) curve is flatter and it drops to zero at 396 K. The measurements σ = f (H) have shown that in this magnetic field magnetization almost reaches saturation.

We think that the most interesting result is an abrupt increase in magnetization in the field of low intensity. The known explanation of this phenomenon [5-9] requires a deeper analysis. The present work proposes a dif-ferent explanation of the observed effect presented in what follows.

It is known that in ferromagnetic crystals magnetic moments of domains are aligned with defined direction relative to crystallographic axes, named easy magnetic axes. Easy magnetic axis is tilted by 45° relative to the plane of twinning [14,15]. In the absence of the magnetic field 180°- domains of similar width are formed in twins. It is typical for uniaxial ferromagnets. Such distribution of domains in twinning structure of low temperature phase is shown in Figure 4(a). Consequently at transition from one twin to another MS vectors will form a zigzag type line. In the case of the magnetic field of low inten-sity the domains orientated favorably to the field direction start to expand while adjacent ones shrink. Then magnetization of the sample in this field is the sum of pro- jections of differences ΔMS of wide and narrow domains on the direction of external magnetic field. Obviously, due to the zigzag type orientation of domains, the vector sum ΔMS is less than the algebraic sum. This results in a smaller value of magnetization in low temperature phase as compared to its value in high temperature phase. This explanation is similar to the one proposed in 1996 by Vasil’ev A.N. et al. [16]. However, it has not become widespread.

Below it has been shown that with the increase of magnetic field intensity up to 80 kA/m the curve σ/σS (T) increases only slightly (by 16%) and it practically does not change. So one can conclude that in the field with such intensity the orientation of vectors are controlled by magnetic crystalline anisotropy while the increase in σ is due to the change in the width of the domains orientated favorably to the applied field. The distribution of do-mains in this case is shown in Figure 4(b). The signifi-cant increase in σ and a decrease in magnetization jump in the case of H = 409 kA/m testify that the effect of the external field is commensurable with that of crystalline anisotropy. Under such conditions the magnetization of the sample occurs not only due to the shift of domain boundaries but also due to the rotation of MS toward the direction of the external field (Figure 4(c)). A small in-crease in σ indicates that there remains the influence of anisotropy.

The essential growth of magnetization in the field of saturation (curve 4, Figure 2) testifies that under the ef-fect of such a field MS of domains in all twins are aligned along the magnetic field irrespective of their previous orientation, as in Figure 4(d). Some decrease in a value of saturation magnetization at phase transformation is evidently attributed to the occurring distortion of crystal lattice symmetry.

Before explaining the character of the curves σ/σS (T) in the region of magnetic phase transformation (ferromagnetic ↔ paramagnetic), one should note a significant influence of magnetic field intensity on both the transition temperature in paramagnetic state and the gradient of its drop to zero. One can expect the effect of magnetic field on these characteristics because Mn atom has a great magnetic moment. In high temperature phase of Ni2MnGa

(a)

(b)

(c)

(d)

Figure 4. Scheme of domain structure in twins with mag-netic fields various intensities



434

magnetic moment of Mn (2.3 μB) is larger than that of Ni (0.22 μB) by a factor of 10 [11]. Therefore atoms of Mn play the major role in forming the alloy magnetization.

The value of magnetization of ferromagnetic in the magnetic field at each temperature is a result of competition between the exchange energy (Wex.en.) and interaction energy of magnetic moments of atoms with magnetic field (Wmagn. int.) on one hand and thermal energy on the other hand.

The exchange interaction and the interaction of atom magnetic moments with external field tend to keep the ferromagnetic order in the sample whereas the thermal fluctuations tend to disorder the arrangement of atom magnetic moments. In ferromagnetic, where the magnetization is created by relatively small magnetic moments, the relation Wmagn. int. << Wex.en. should be fulfilled. For this reason, the intensity of the applied field in this case does not noticeably affect the character of the curve σ/σS (T) in the region of magnetic transformation. When mag-nitude of magnetic moments is large, the values Wmagn. int. and Wex.en. are comparable and their joint action should maintain the ferromagnetic order at higher temperatures. As a result, the σ/σS (T) curve in the region of the Curie temperature should be flatter and the ferromagnetic state should be destroyed at higher temperature.

The σ/σS (T) dependencies obtained on heating and on cooling of the sample in the magnetic field with the intensities 80 kA/m and 916 kA/m are shown in Figure 3. The heating curves have been described above. As for the cooling curves, one can say the following.

Apparently the curves σ/σS (T) obtained on heating and on cooling in the magnetic field of 80 kA/m intensity are similar. The difference is that cooling curve is shifted to the low temperature region. The shift is 19 K in the structure transition region, and 7 K in the magnetic transformation region. Such difference in temperature hystere-sis evidently testifies to the fact that magnetic transformation (second order phase transition) occurs easier, while structural transition (first-order phase transformation) occurs with more difficulties, since it requires more essential overcooling. Magnetization of the low temperature phase after cooling down to 223 K remains slightly higher its initial value. Most likely, this is related to the fact that ferromagnetic undergoes stronger magnetization when cooling takes place in magnetic field.

From the cooling curve in the magnetic field of 916 kA/m intensity it is seen that temperature hysteresis in the magnetic transition region is 9 K. In the region of structure transformation the temperature hysteresis is not observed and after cooling magnetization of low tem-perature phase remains lower than its initial value.

4. Conclusions

In can be concluded that in the region of structural phase transformation the curves σ/σS (T) is affected by the

magnetic field of low intensity, while in the region of magnetic phase transformation it is affected by the mag-netic field of high intensity. The abrupt decrease in mag-netization at the transition from the high temperature phase to the low temperature one in magnetic fields of low intensity is attributed to a zigzag change of vector MS in the domains of twin structure. The increase in the intensity of the magnetizing field leads to a noticeable increase of the ferromagnetic ↔ paramagnetic transition temperature.

5. Acknowledgements

The authors would like to thank Professor V. Shavrov and V. Koledov for providing us with the alloy for the present study.

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and V. V. Kokorin, “Large Magnetic-Field-Induced Strains in Ni2MnGa Single Crystals,” Applied Physics Letters, Vol. 69, No. 13, 1996, pp. 1966-1968.

[2] A. N. Vasil’ev, V. D. Buchel’nikov, T. Takagi, V. V. Khovailo and E. I. Estrin, “Shape Memory Ferromagnets,” Phys-ics-Uspekhi, Vol. 46, No. 6, 2003, pp. 559-588.

[3] A. A. Cherechukin, I. E. Dikshtein, D. I. Ermakova, A. V. Glebov, V. V. Koledov, D. A. Kosolapov, V. G. Shavrov, A. A. Tulaikova, E. P. Krasnoperov and T. Takagi, “Shape Memory Effect due to Magnetic Field-Induced Thermoe-lastic Martensitic Transformation in Polycrystalline Ni– Mn–Fe–Ga Alloy,” Physics Letters A, Vol. 291, No. 2-3, 2001, pp. 175-183.

[4] A. Sozinov, A. A. Likhachev, N. Lanska and K. Ullakko, “Giant Magnetic-Field-Induced Strain in NiMnGa Seven- Layered Martensitic Phase,” Applied Physics Letters, Vol. 80, No. 10, 2002, pp. 1746-1748.

[5] P. Lazpita, J. M. Barandiaran, J. Gutierrez, M. Richard, S. M. Allen and R. C. O’Handley, “Magnetic and Structural Properties of Non-Stoichiometric Ni-Mn-Ga Ferromag-netic Shape Memory Alloys,” European Physical Journal: Special Topics, Vol. 158, No. 1, 2008, pp. 149-154.

[6] D. Kikuchi, T. Kanomata, Y. Yamaguchi, H. Nishihara, K. Koyama and K. Watanabe, “Magnetic Properties of Fer-romagnetic Shape Memory Alloys Ni2Mn1−xFexGa,” Jour-nal of Alloys and Compounds, Vol. 383, No. 1-2, 2004, pp. 184-188.

[7] J.-H. Kim, F. Inaba, T. Fukuda and T. Kakeshita, “Effect of Magnetic Field on Martensitic Transformation Tem-perature in Ni–Mn–Ga Ferromagnetic Shape Memory Al-loys,” Acta Materialia, Vol. 54, No. 2, 2006, pp. 493-499.

[8] V. D. Buchelnikov, M. A. Zagrebin, S. V. Taskaev, V. G. Shavrov, V. V. Koledov and V. V. Khovaylo, “New Heusler Alloys with a Metamagnetostructural Phase Transition,” Bulletin of the Russian Academy of Sciences: Physics, Vol. 72, No. 4, 2008, pp. 564-568.

[9] V. D. Buchel’nikov, S. V. Taskaev, M. A. Zagrebin and P. Entel, “Phase Diagrams of Heusler Alloys with Inversion



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of the Exchange Interaction,” Letters to Journal of Ex-perimental and Theoretical Physics, Vol. 85, No. 11, 2007, pp. 560-564.

[10] A. Ayuela, J. Enkovaara, K. Ullakko and R. M. Nieminen, “Structural Properties of Magnetic Heusler Alloys,” Jour-nal of Physics: Condensed Matter, Vol. 11, No. 8, 1999, pp. 2017-2026.

[11] P. J. Brown, A. Y. Bargawi, J. Crangle, K.-U. Neumann and K. R. A. Ziebeck, “Direct Observation of a Band Jahn–Teller Effect in the Martensitic Phase Transition of Ni2MnGa,” Journal of Physics: Condensed Matter, Vol. 11, No. 24, 1999, pp. 4715-4722.

[12] A. A. Cherechukin, I. E. Dikshtein, D. T. Ermakov, A. V. Glebov, V. V. Koledov, D. A. Kosolapov, V. G. Shavrov, A. A. Tulaikova, E. P. Krasnoperov and T. Takagi, “Shape Memory Effect due to Magnetic Field-Induced Thermoelastic Martensitic Transformation in Polycrystal-line Ni–Mn–Fe–Ga Alloy,” Physics Letters A, Vol. 291,

No. 2-3, 2001, pp. 175-183.

[13] K. Y. Mulyukov and I. I. Musabirov, “Effect of a Mag-netic Field on the Thermal Expansion of Ni2.08Mn0.96Ga0.96 Alloys,” Technical Physics, Vol. 53, No. 6, 2008, 802- 803.

[14] Y. Ge, O. Heczko, O. Soderberg and V. K. Lindroos, “Various Magnetic Domain Structures in a Ni–Mn–Ga Martensite Exhibiting Magnetic Shape Memory Effect,” Journal of Applied Physics, Vol. 96, No. 4, 2004, pp. 2159-2163.

[15] O. Heczko, “Magnetic Shape Memory Effect and Mag-netization Reversal,” Journal of Magnetism and Magnetic Materials, Vol. 290-291, 2005, pp. 787-794.

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Sergey Katenev Katenev, He Shi

Theoretical Radiophysics Department, V. N. Karazin Kharkov National University, Kharkov, Ukraine. Email: Katenev, [email protected] Received May 26th, 2010; revised June 14th, 2010; accepted June 18th, 2010.

ABSTRACT

H0i-eigenwave characteristics of a periodic iris-loaded circular waveguide (PICW) are examined, as concerns the ei-genmode behavior vs arbitrary variations of the geometric parameters and the Bragg bandwidths vs the parameter of filling ld / extremums. Keywords: Periodic Structure, Pass/Stop Band, Periodicity Dispersion, Partial Waves

1. Introduction

The periodic iris-loaded circular waveguide, Figure 1, has long since found its several important applications, e.g. in the particle acceleration field [1], and thus stimu-lated its electromagnetics studies. Despite this even its eigenwave characteristics available are not to be regarded as generally satisfactory [1,2]; foremost theoretically and a good deal so [2], whereas exactly knowing the ropes wouldn’t do any harm in all respects.

Certain conceptual points as to the eigenwave propagation in PICW are given in [2] to get those waves theory building started. As the next step and immediate continuation, this paper is concerned with characteriza-tion of one of the PICW particular wave types - its H0i-eigenwaves.

It is not that only the PICW asymmetric and sym-metric E0i-waves, in view of their acknowledged com-plexity [1,3], cannot be properly perceived except by rigorous computations. Any simplified modeling, e.g. as that of l 0, d 0 in [3], and others like it, are

a b

0

r

z

-l -d 0 d l

(I)

(II)

Figure 1. Periodic iris-loaded circular waveguide

rather unsatisfactory, concerning even the simplest guided wave type of H0i-waves. And in fact, there is no other way at all for dealing adequately with the PICW eigenwave problem except via rigorous computations; which is certainly one of the major difficulties in their investigation.

This way, the H0i-waves are generally looked at on the dispersion side of their electromagnetics; and all of the necessary terms, notions and ways employed are intro-duced and discussed in detail in [2].

2. Arbitrary Geometric Parameters

As some work model of PICW to be employed through-out this investigation [2], and in this section in particular, radius b is held constant b 3, the long period l = 3 and the short one l = 0.75 are examined in detail, as one of the wide and one of the narrow cells are considered, and radius a is optimally varied.

The multi-mode Brillouin diagrams is the most suitable instrument for the purpose.

The PICW dispersion curves are drawn below with so- lid lines, those of the regular waveguide b = 3 with dot-ted lines, and those of the regular waveguide r = a with dashed ones.

2.1 Period l = 3

At the narrow iris for d = 2.8, the effect of radius a varia-tions is represented in Figure 2 for the junior 12 modes and a 2.8, 2.4, 2, 1.2, 0.4.

The initial periodicity dispersion (i.p.d.) is quite in ef-fect at a = 2.8, and H01, H04 are the regular PICW modes originated in accordance with the regular waveguide r =



437

3 modes rr HH 0201, , respectively. All the other eigen-

modes are the periodicity ones generated by the former: H02, H03 and H011, H012 by H01 ( rH01 ), H05, H06 and H09,

H010 by H04 ( rH02 ). The modes H011, H012 are the most

complex ones due to the effect of rH03 mode involved.

Down to a = 2, all the senior modes of those presented are clearly piecewise composed. Ultimately, at a = 0.4, the closed- off H05, H06 and H011, H012 get in very close vicinities in between.

There are three regular waveguide r = b modes riH0 , i

= 1,2,3, in the bandwidth. And as radius a decreases, a monotonous growth of all of the eigenfrequences for

,0iH i = 1,…,12, occurs, except in the regular frequen-cies: rrr HHHHHH 0301102050101 ,, =0.5 for 3 > a > 1.2, rrr HHHHHH 0301002040101 ,, =0.5 for

02.1 a . In the waveguide with a fairly thick iris, e.g. d = 0.3,

the effect of radius a variations is represented in Figure 3, the junior 12 modes, a2.8, 2.4, 2, 1.2, 0.8, 0.4.

Here, the regular waveguide r = a i.p.d. effect is valid up to a = 2 for all of the modes, except in a few of the Bragg bands. At a = 2.8, = 0, the modes H01, H05 are

the regular ones (by rr HH 0201, , respectively), the mode H05 being only a slightly composed one (the fragment f-1, Figure 4); H02, H03; H04, H06; H011, H012 and H07, H08; H09, H010 are the periodicity modes by rH01 and rH02 re-spectively. The fragments f-1,2,3, Figure 4, demonstrate, in particular, a significant localization of the periodicity partial-wave effect closely around the Bragg wave-points; as well as some other exact details of the mode forming. For example, in f-2, 5.0 , the modes H07, H010 are formed after rH01 , the modes H08, H09 after rH02 , and the corresponding Bragg bands are one inside the other. In f-3, ,0 H07, H08 are formed after rH02 and H09, H010 after rH01 , and the two Bragg bands go one by one.

A certain regular-waveguide r = a modeling may be in some validity in this case, whereupon the eigenfrequency equals the regular model’s one for the upper boundaries

ui of the appropriate Bragg bandwidths i so that

ri

ui H0 = 0.5.

2.2 Period l = 0.75

At the wide cell d = 0.65, radius a variations are demon-strated in Figure 5, 12 modes, a 2.8, 2.4, 2, 1.2, 0.4.

κ

κα

κ

κα

κ

κα

(a) (b) (c)

κ

κα

κ

κα

(d) (e)

Figure 2. l = 3, d = 2.8; the effect of radius a variations (a) a = 2.8; (b) a = 2.4; (c) a = 2; (d) a = 1.2; (e) a = 0.4



438

κ

κα

κ

κα

κ

κα

(a) (b) (c)

κ

κα

κ

κα

κ

κα (d) (e) (f)

Figure 3. l = 3, d = 0.3; radius a variations (a) a = 2.8; (b) a = 2.4; (c) a = 2; (d) a = 1.2; (e) a = 0.8; (f) a = 0.4

l = 3, d = 0.3, a = 2.8 l = 3, d = 0.3, a = 2.8 l = 3, d = 0.3, a = 2 l = 0.75, d = 0.65, a = 2.8

f-1 f-2 f-3 f-4

l = 0.75, d = 0.65, a = 1.6 l = 0.75, d = 0.2, a = 2 l = 0.75, d = 0.2, a = 1.6

f-5 f-6 f-7

Figure 4. Some particularities of the eigenmode formation as radius a varies

(a) (b) (c) (d) (e)

Figure 5. l = 0.75, d = 0.65; radius a variations (a) a = 2.8; (b) a = 2.4; (c) a = 1.6; (d) a = 1.2; (e) a = 0.4



439

At a = 2.8, the modes H0i, i = 1,2,3,6,11, are the regu-

lar ones in one-to-one correspondence with riH0 , i =

1,2,3,4,5, consequently. Of the rest modes, H04, H05 (by rH01 ), H07, H08 (by rH02 ), H09, H010 (by rH03 ) and H012

(by rH04 ) are the periodicity ones. Eventually, at a=0.4,

the closed-off H04, H05 and H07, H08 and H011, H012 are very close in between.

The piece-wise mode composition due to a lot of the inner Bragg wave-points and the wave propagation up to rather small radius a values, characterize the waves. Two particular cases as to the mode forming are shown in detail in the fragments f-4 and f-5, Figure 4.

The regular-waveguide r = b modeling scheme is not relevant in this case, even to the extent it has been in l = 3, d = 2.8 event; much less is the r = a scheme.

At the thick iris d = 0.2, the effect of radius a varia-tions is demonstrated in Figure 6 for the junior 12 modes, a 2.8, 2.4, 2, 1.2, 0.4; with two detailed fragments on the particularities of the mode forming, f-6 and f-7, Figure 4.

As radius a goes down, the i.p.d. is still mainly in ef-fect up to a = 1.2; which is evidenced by a fairly straight geometry of the dispersion curves.

The regular-waveguide r = a modeling scheme as that in the previous thick-iris event, Figure 3, principally holds true in this case also , and even more accurately.

The fragments f-1 to f-7, Figure 4, exhibit some par-ticular features of the eigenmode formation and trans-formation in the waveguide. As the values of d and a parameters vary, the standard i.p.d. scheme of the perio-dicity mode origin in pairs at 5.0,0 , and their

further forming at 0 < < 0.5, somewhat changes to include at least three interacting eigenmodes. As it is in f-1, H05 being the regular mode ( = 0); in f-4, H06 the regular mode, in f-6, H05 the regular mode (0 < < 0.5); in f-7, H07 the regular mode ( = 0.5). In f-2, f-3, ( = 0.5), mentioned above, all of the modes involved are the periodicity ones which, at least after the disper-sion way of analysis, quite conform to the standard i.p.d. scheme [2].

3. The Bragg Bandwidths Extremums

Another view on the H0i-eigenwave behavior is via their Bragg bandwidths )(i extremum characteristics vs

the parameter of filling 0 < = d/l < 1 [4]. In essence, this is the d-parameter variation in the waveguide in ef-fect, looked at under a quite promising aspect as to the PICW characterization. For one thing, such graphic rep-resentation of those bandwidths behavior as that, e.g., in Figure 7, enables to look simultaneously at both stop and pass bandwidths characteristics. And second, the other PICW eigenwave types do display a good deal of analo- gical behavior, with certain peculiarities of their own [4].

In this section, the period values considered are l = 5, 3, 1.8, 1, 0.75. According to the classifications in [2], l = 5, 3, 1.8 are the long periods, l = 1, 0.75 are the short ones; and thus, some borderline set of the period values is ex-amined below.

The general rule for the periodicity modes originated by a given regular one in PICW (after the i.p.d.) is that

)(i , = 0, 0.5, have i maxima and i-1 minima over the interval 0 < < 1; while )(i 0 as 0

(a) (b) (c)

(d) (e) (f)

Figure 6. l = 0.75, d = 0.2; radius a variations (a) a = 2.8; (b) a = 2.4; (c) a = 2; (d) a = 1.2; (e) a = 0.8; (f) a = 0.4



440

(a) (b)

(c) (d)

Figure 7. l = 5, a = 2.8; the Bragg bandwidths 1iΔ ( ) = 1, 2, 3, 4, 5, 9,11ω θ , i

(infinitesimally thin slot) and )(i w > 0 as

1 (infinitesimally thin iris). In Figure 7, l = 5, d = 4.8, a = 2.8, there are the Bril-

louin diagram for 12 junior modes, Figure 7 (a), and seven of its Bragg bandwidths )(1 i , i = 1,2,3,4,5,9,

11, represented via their upper and lower boundaries ub

and lb vs , l

biubii , Figures 7 (b), (c) and (d).

The bandwidths )(),( 19

12 and )(),( 1

1114

are of a similar origin by their regular “parent” modes: )(),( 1

412 are originated by rH01 , )(),( 1

1119

by rH02 . And while the partial-wave interactions for the

bandwidths )(1 i , i = 1,2,3,4,5, are originally entirely

symmetrical, they are not so for )(1 j , j = 9,11. Be-

cause the nonsymmetrical partial-wave interactions in the appropriate inner B. w.-p. (0 < < 0.5) do have their effects regarding )(),( 1

1119 bandwidths, though

quite slightly there. The graphic representation of the PICW pass ),(j

5,4,3,2j , and stop ,5,...,2,1),( ii bandwidths of Figure 7 (b) (and every vertical line const there, yields us those in PICW) is equivalent to the continuum of the Brillouin diagrams of Figure 7 (a) for ,1,0 iH i

5,...,2 , d [0,1]. In view of the relationship of equiva-

lence between the wave and the dispersion equations [see, e.g. 2], Figure 7 (b) has, in its way, everything on the

iH0 -waves, i = 1,2,…,5, as a function of d. The effects of radius a variation for l = 3, )(1 i , i =

1,2,3, are shown in Figures 8 (a)-(c), accordingly. The bandwidths )(),( 1

512 , l = 1.8, a = 2.8, are

presented in Figures 9 (b) and (c). )(),(),(),( 1

1026

111

16 , l = 1, a = 2.9, are

presented in Figures 10 (b)-(e), accordingly. The band-widths )(),( 1

1026 for the inner B. w.-p.s are

much harder to examine, because their eigenvalue B shifts as varies. Nevertheless certain extremums of the bandwidths are obviously available in this case also.

And finally, )(),( 19

14 , l = 0.75, a = 2.8, are

given in Figures 11 (b) and (c). The presence of the inner Bragg wave-points for all of the eigenmodes on the short [2] periods (wherein l = 1, l = 0.75 are such ones), with their asymmetrical partial-wave interactions, does distort the regularity of the max/min pattern of above; which can be seen in )(1

4 case, Figure 11 (b). Yet for )(19 ,

Figure 11 (c), these extremums are still clearly available.

4. Conclusions

Under the fundamental primary-causal influence of the



441

( ), . , . , . , . , . , , . 1

1 2 9 2 8 2 6 2 4 2 2 2 1 8 a ( ), . , . , . , . , . , , . 1

2 2 9 2 8 2 6 2 4 2 2 2 1 8 a ( ), . , . , . , . , 1

3 2 9 2 8 2 6 2 4 2 a

θ θ θ

(a) (b) (c)

Figure 8. l = 3; the Bragg bandwidths 1iΔ ( ) = 1, 2, 3ω θ , i , as radius a varies

(a) (b) (c)

Figure 9. l = 1.8, a = 2.8; the Bragg bandwidths 1iΔ ( ) = 2, 5ω θ , i

(a) (b) (c)

(d) (e)

Figure 10. l = 1, a = 2.9; the Bragg bandwidths 1iΔ ( ) = 6,11ω θ , i and 2 1

6 10Δ ( ) Δ ( )ω θ , ω θ



442

(a) (b) (c)

Figure 11. l = 0.75, a = 2.8; the Bragg bandwidths 1iΔ ( ) = 4, 9ω θ , i

period value, in particular, in setting the number of eigenmodes, with all the consequences of the i.p.d. network thus produced [2], and further variations of d and a pa-rameters, the PICW H0i-eigenwave characteristics can be seen are quite complex; even without any of their power-flow treatment, illustrated in [2].

These waves are not to be satisfactorily interpreted by some regular-waveguide modeling schemes, though the latter may be in some validity to this case.

A monotonous response of the H0i-eigenfrequencies to both d and a variations, 0/,0/ ad , is a major characteristic feature of those waves. Wherein,

0)(lim i , as 0 (the i.p.d. of the regular r = a waveguide via the narrow cell), 0)()(lim awi , as 1 , )(aw monotonously grows as a decreases from b downwards (the regular r = b waveguide model-ing, with the narrow-iris l-d effect in the waveguide). As a result, each H0i, is stable (approximately constant) vs a at its upper iegenfrequency u

i0 , i.e. either at 5.0 or 0 . In fact u

i0 monotonously and rather slightly grows as a decreases.

Since the PICW eigenwaves originate principally due to interactions in the Bragg wave-points (e.g., after the partial-wave model [2]), the Bragg bandwidths )(i

extremum law of i/i-1 maxima/minima at 0.5, 0, presented here in brief, can be treated as the general pe-

riodicity law of the Bragg bandwidths variation vs . The limits and specificity of its holding true as radius a varies, are different for different wave types [4].

It needs a special power flows investigation in order to further physically interpret this law in proper detail and understanding.

And finally, the upper-and-lower-boundary represen-tations of the pass and stop bandwidths, ),(),( ii

like those in Figure 7(b), are instrumental and informa-tive enough, as regards 10 variations, to be in their way some 3rd full-right member of the relationship of equivalence in the matter, see, e.g., [2].

REFERENCES [1] O. A. Valdner, N. P. Sobenin, B. V. Zverev and I. S.

Schedrin, “A Guide to the Iris-loaded Waveguides,” in Russian, Atomizdat, Moscow, 1977.

[2] S. K. Katenev, “Eigenwave Characteristics of a Periodic Iris-Loaded Circular Waveguide. The Concepts,” Progress in Electromagnetic Research, Vol. 69, 2007, pp. 177-200.

[3] Y. Garault, “Etude D’Une Classe D’Ondes Electromag-netique Guidées: Les Onde EH. Application aux Dèflec-tuers Haute Frèquence de Particules Rapides,” Annales de Physiques, Vol. 10, 1965, pp. 641-672.

[4] S. K. Katenev and H. Shi, “Stop Bandwidth Extremums of a Periodic Iris-Loaded Circular Waveguide,” 6th In-ternational Conference on Antenna Theory and Tech-niques, Sevastopol, 17-21 September 2007, pp. 471-473.



443

List of Notations Pertaining to the Problem

1. ),( BB ≡ B),( — Bragg wave-number and

its ordinate on the Brillouin plane ),( , i.e., the

Bragg wave-point (B. w.-p.);

2. B — Bragg band, i.e., a (locally) forbidden band;

kjjii ,...,2,1,, — the i-mode propagation band

and all of its possible Bragg bands (the mode being be-neath those);

3. periodicity dispersion — the first one of the two

factors — periodicity and diffraction — responsible for the waveguide dispersion forming;

4. initial periodicity dispersion (i.p.d.) — the waveguide dispersion at infinitesimal irises;

5. regular mode — the PICW eigenmode in one-to-one correspondence to that of the smooth waveguide;

6. periodicity mode — the PICW eigenmode originat-ing due to the periodicity effect;

7. partial waves — the independent ingredients of a PICW eigenwave.




Abhilasha Mishra1, Ganesh B. Janvale2, Bhausaheb Vyankatrao Pawar3, Pradeep Mitharam Patil4

1Department of Electronics Engineering, North Maharashtra University, Jalgaon, India; 2 Department of Computer Science and Information Technology, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, India; 3Department of Computer Science, Mumbai University, Mumbai, India; 4Department of Electronics & Telecommunications, Vishwakarma Institute of Technology, Pune, India. Email: [email protected], [email protected], [email protected], [email protected] Received April 1st, 2010; revised May 23rd, 2010; accepted May 27th, 2010.

ABSTRACT

The paper presents the Quasi Newton model of Artificial Neural Network for design of circular microstrip antenna (MSA). In this model, a closed form expression is used for accurate determination of the resonant frequency of circular microstrip patch antenna. The calculated resonant frequency results are in good agreement with the experimental re-sults reported elsewhere. The results show better agreement with the trained and tested data of ANN models. The re-sults are verified by the experimental results to produce accurate ANN models. This presents ANN model practically as an alternative method to the detailed electromagnetic design of circular microstrip antenna. Keywords: Circular Microstrip Antenna (CMSA), Artificial Neural Network (ANN), Quasi Newton (QN)

1. Introduction

The MSA is an excellent radiator for many applications such as mobile antenna, aircraft and ship antennas, remote sensing, missiles and satellite communications [1]. It consists of radiating elements (patches) photo etched on the dielectric substrate. Microstrip antennas are low profile conformal configurations. They are lightweight, simple and inexpensive, most suited for aerospace and mobile communication. Their low power handling capa-bility posits these antennas better in low power transmis-sion and receiving applications [2]. The flexibility of the Microstrip antenna to shape it in multiple ways, like square, rectangular, circular, elliptical, triangular shapes etc., is an added property.

The rectangular and circular patches (Figure 1) are the basic and most commonly used designs in microstrip antennas. Their designing methods are numerous, yet getting the actual data for developing real prototypes for experiment is found to be difficult. ANN offers a viable solution to obtain the design parameters. Hence, in this paper we have tried to develop the Quasi Newton algo-rithm for the design of circular patch antennas.

ANN is the most powerful optimizing tool in the field of computational electromagnetic. An ANN consists of interconnected processing units that store experimen-tal knowledge. Such; this knowledge is acquired by a learning process and is stored in the form of parameters

of the ANN [3]. The basic characteristics of ANN is its ability to learn and generalize, fault tolerance, non- line-arity, and adaptivity.

The learning in ANN can be unsupervised or supervised. When an ANN undergoes learning in an unsuper-vised manner, it extracts the features from the input data based on a predetermined performance measure. When an ANN undergoes learning in a supervised manner, it is presented with the input patterns and the desired output patterns. The parameters of the ANN are adapted such that the application of an input pattern results in the de-sired pattern at the output of the ANN [4]. The Quasi Newton is one of the proven universal approximator in ANN design.

The design of CMSA as a closed form expression is given in Section 2. The Section 3 of the paper contains the description regarding the QN algorithm and design of the microstrip antenna as the analysis and synthesis model.

h

y

x

a

Ground Plane

Figure 1. Circular microstrip antenna (CMSA)



445

The results and conclusions are described in Section 4 and 5 respectively. ANN models are developed by using NeuroModeler 1.5 tool [5].

2. A Closed Form Expression for CMSA Design

The performance of circular microstrip antennas has been studied extensively, both analytically and experimentally. Consider the circular microstrip antenna with radius a, height h and permittivity constant εr whose resonant fre-quency in the dominant TM11 mode as explained by Guney [6], is given by (1).

And 1

21 11 12

2 2r r

eff

h

a

(2)

where αnm- the mth zero of the derivatives of the Bessel function of order n

c - the velocity of light. a - Radius of the circular patch fr - resonant frequency of circular patch. h - Height of dielectric substrate εr - Permittivity of dielectric substrate

1.84118nm c (for n = m = 1)

The six patches of CMSA are designed by considering height h as 0.235 and εr as 4.55 with variation of radius a.

The resonance frequency fr calculated by the Guney’s equation was tested in four different and independent experiments. Their results when compared showed very little variation [7-10]. The deviation for resonant frequency of CMSA developed and measured by Abboud [7], Ho-well [8], Wolff [9] and Derneryd [10] is 0.19122, 0.0723,

0.0788, 0.238559 respectively, which is very low as pre-sented in Table 1. That explains the reason why the re-searchers chose Guney’s equation for use in the present study of ANN modeling.

3. ANN Modeling by Using QN Algorithm

3.1 QN Algorithm

This is an advanced training algorithm in which second order derivative information is used. It approximates the inverse of the Hessian matrix of the training error func-tion. The convergence of this method is almost similar to Newton’s method when approaching the solution. The algorithm involves matrix operations. As such, computa-tion effort becomes intensive as the size of the neural network increases. Hence, this training algorithm proves more useful for small and medium scale neural networks. The Quasi-Newton training may follow Conjugate Gra-dient to further reduce the training error. The formula under reference approximates the inverse Hessian matrix during optimization.

Here

( ) ( 1)g E epochs E epoch

Quasi-Newton condition is:

H w g or w B g

where H is Hessian Matrix,

( )rHw

T

T

E

w

B is the approximation of inverse H. Quasi-Newton method uses history of Δw and ΔE to

approximate B: ( 0)B epochs I

Table 1. Analysis of deviation of measured result from Guney’s equation

1

2

1.84118

22 1 1.44 1.77 0.268 1.65

2

r

eff r rr

cf

h a ha ln

a h a

(1)

Fr (GHz) [6] Abboud [7] Deviation Howell [8] Deviation Wolff [9] Deviation Derneryd [10] Deviation

5.434885 4.945 0.489885 5.353 0.081885 5.308 0.126885 4.848 0.586885

4.090526 3.750 0.340526 3.963 0.127526 3.950 0.140526 3.661 0.429526

2.15468 2.003 0.15168 2.061 0.09368 2.067 0.08768 1.965 0.18968

1.438828 1.360 0.078828 1.379 0.059828 1.384 0.054828 1.332 0.106828

1.078434 1.030 0.048434 1.037 0.041434 1.042 0.036434 1.009 0.069434

0.863001 0.825 0.038001 0.833 0.030001 0.836 0.027001 0.814 0.049001

Average Deviation 0.1912256 0.0723923 0.0788923 0.238559



446

( 1) ( ) 1T T

T T

T T

T

w w g B gB epochs B epoch

w g w g

B g w w g B

g w

(BFGS Formula) (3)

The speed of QN is high and the memory space re-quired for it is N2w.

3.2 Analysis Design of Circular Microstrip Antenna by Using QN Algorithm

In this model, the accurate value of resonant frequency has been calculated by using Equation (1) and (2). The input parameters are permittivity εr, the height of sub-strate h and patch dimension in terms of radius a.

The NN structure for the analysis of CMSA is shown in Figure 2. The Quasi-Newton algorithm has been con-sidered for ANN modeling. The QN-ANN model con-sists of three layers i.e. input layer, hidden layer and output layer. The neurons for input layer are 3 and for hidden layer are 12; while the output layer has only one as shown in Figure 3. The epochs given for training the model were 200 from which it took only 80 epochs. For the purpose of training, 4397 data are generated for ANN modeling by application of Equation (1), from which 2199 data are selected while the remaining data are se-lected for testing the ANN model. The performance graph is displayed of training and testing in Figure 4 and Fig-ure 5 respectively. As the experimental values of only 6 patches designed by [7-10] are given hence the result of only six patches is shown in Table 2. The output of ANN

a

h

εr

fr

Figure 2. Analysis model of ANN

a

h

εr

fr

Input Layer Hidden Layer Output Layer

Figure 3. QN structure of analysis design of CMSA

Training Error by using QN

Tra

inin

g E

rror

Epochs

2.5

2

1.5

1

0.5

00 10 20 30 40 50 60 70 80 90 100

Figure 4. Performance result of QN algorithm developed for training analysis model of CMSA

Neural Model Output vs Test Data (QN)

Out

put N

euro

n #1

Sample Number

10

9

8

7

6

5

4

3

2

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Model:OutputNeuron#1

Data:OutputNeuron#1

Figure 5. Performance result of testing QN-ANN model of CMSA

Table 2. Forward modeling for the Prediction of Resonant frequency

Input Parameters Target QN

a (radius) h εr fr (GHz) ANN O/P

(GHz) Error (GHz)

0.77 0.235 4.55 5.434885 5.441164 0.006278

1.04 0.235 4.55 4.090526 4.086281 –0.00425

2 0.235 4.55 2.15468 2.154862 0.000182

2.99 0.235 4.55 1.438828 1.439156 0.000328

3.975 0.235 4.55 1.078434 1.078185 –0.00025

4.95 0.235 4.55 0.863001 0.866268 0.003267

Average Value 2.510059 2.510986 0.000926



447

model is compared with the target data where QN-ANN model shows as little error as 0.000926, thereby laying a claim to be the best ANN analysis model for the design of CMSA.

3.3 Synthesis Design of Circular Microstrip Antenna by Using QN Algorithm

There are various methods available for the calculation of resonant frequencies of different patch antennas. But reverse calculation of radius from the inputs fr, heigh h and permitivitty constant εr is not available in the litera-ture. The solution for this is reverse modeling of ANN. The reverse model is also called as synthesis model which predicts the value of radius of circular patch as shown in Figure 6.The synthesis model consists of three layers- input layer with three neurons and hidden layer with 12 neurons and output layer with one neuron. The hidden layer uses sigmoid function while output layer uses linear function as shown in Figure 7.

The model is trained with 2199 data and training error graph is presented in Figure 8. The trained synthesis model is tested with 2198 data from which only six results are verified as its practically measured values are available in the literature [6-10] and it is given in Table 3. It takes only 101 epochs out of 200 for training synthesis model. The performance graph of testing synthesis model is presented in Figure 9. The actual error in values is given in Table 3 with very less error 0.0006 for the predic-tion of radius.

a h

εr

fr

Figure 6. Synthesis model of ANN

ah

εr

fr

Input Layer Hidden Layer Output Layer

Figure 7. QN structure of synthesis design of CMSA

Training Error by using QN

Tra

inin

g E

rror

Epochs

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 20 40 60 80 100 120

Figure 8. Performance result of QN algorithm developed for Training CMSA

Neural Model Output vs Test Data (QN)

Out

put N

euro

n #1

Sample Number

5

4.5

4

3.5

3

2.5

2

1.5

1

0.50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Model:OutputNeuron#1

Data:OutputNeuron#1

Figure 9. Performance result of testing QN-ANN model of CMSA

Table 3. Reverse modeling for the prediction of radius

Input Parameters Target QN

fr (GHz) h εr a (radius) ANN O/P

(GHz) Error (GHz)

5.434885 0.235 4.55 0.77 0.76673 0.00327

4.090526 0.235 4.55 1.04 1.043208 –0.00321

2.15468 0.235 4.55 2 2.001889 –0.00189

1.438828 0.235 4.55 2.99 2.988124 0.001876

1.078434 0.235 4.55 3.975 3.976177 –0.00118

0.863001 0.235 4.55 4.95 4.945272 0.004729

Average Value 2.620833 2.620233 0.0006



448

Table 4. Analysis of computed, measured & ANN predicted resonant frequencies

a (radius) h εr fr (GHz) Abboud Howell Wolff Derneryd ANN Model

0.77 0.235 4.55 5.434885 4.945 5.353 5.308 4.848 5.441164

1.04 0.235 4.55 4.090526 3.750 3.963 3.950 3.661 4.086281

2 0.235 4.55 2.15468 2.003 2.061 2.067 1.965 2.154862

2.99 0.235 4.55 1.438828 1.360 1.379 1.384 1.332 1.439156

3.975 0.235 4.55 1.078434 1.030 1.037 1.042 1.009 1.078185

4.95 0.235 4.55 0.863001 0.825 0.833 0.836 0.814 0.866268

Average Deviation 0.1912256 0.0723923 0.0788923 0.238559 0.000926

0

1

2

3

4

5

6

1 2 3 4 5 6

Circular Patches

Res

on

ant

Fre

qu

ency

By Guney

By Abboud

By Howell

By Wolff

By Derneryd

By ANN Model

Figure 10. Comparison of measured fr by [6-10] and QN-ANN output

4. Result

The results demonstrate the excellent capacity of the neural model approximation which has been analyzed from Ta-bles 1, 2, 3 and 4. The good agreement between the neu-ral model answers and the fabricated patches as showed in Figure 10, demonstrates a good generalization capac-ity of the model through the Quasi-Newton modular structure. Besides, the QN structure, directly trained by means of measured/simulated data through the EM- ANN technique, is very flexible. It has the potential to be substituted, as models, mainly when new components/ technologies for microwaves circuits come up.

5. Conclusions

The neural models presented in this work have been found to possess high accuracy and requires no complicated mathematical functions. Using these models one can calculate resonant frequency of circular microstrip antenna accurately. The second ANN model of synthesis has uni- que characteristic of predicting radius of circular patches which is not available in the literature. If more data set is used for the training, the NN model gives more robust results. The analysis and synthesis models of ANN, gives

better result for CMSA design which is found to compare well with the fabricated and measured values. So, it can be concluded that both the models are efficient for the prediction of resonant frequency and radius of the circu-lar patch for all the practical purposes.

6. Acknowledgements

Authors acknowledge the support and valuable guidance; they have received from Prof. Q. J. Zhang, Professor and Chair, Department of Electronics, Carleton University, Ottawa, Canada in preparation of this paper.

REFERENCES [1] G. Kumar and K. P. Ray, “Broadband Microstrip Anten-

nas” Artech House, London, 2003.

[2] G. Garg, P. Bhartia, I. Bahl and A. Ittipiboon, “Microstrip Antenna Design Handbook,” Artech House, Canton, 2001.

[3] B. Yegnanarayana, “Artificial Neural Networks,” Pren-tice-Hall of India, Delhi, 1999.

[4] Q. J. Zhang, K. C. Gupta, and V. K. Devabhaktuni, “Arti-ficial Neural Networks for R.F. and Microwave Design: from Theory to Practice,” IEEE Transactions on Micro-wave Theory and Techniques, Vol. 51, No. 4, April 2003, pp. 1339-1350.



449

[5] Q. J. Zhang, “NeuroModeler Version 1.5 Software,” Carleton University, Ottawa, 2004.

[6] K. Guney, “Resonant Frequency of Electrically-Thick Circular Microstrip Antenna,” International Journal of Electronics, Vol. 77, No. 3, 1994, pp. 377-386.

[7] F. Abbound, J. P. Damiano and A. Papiernik, “New De-termination of Resonant Frequency of Circular Disc Mi-crostrip Antenna: Application to Thick Substrate,” Elec-tronics Letters, Vol. 24, No. 17, 1988, pp. 1104-1106.

[8] J. Q. Howell, “Microstrip Antenna,” IEEE Transactions

on Antennas and Propagation, Vol. 23, No. 1, 1975, pp. 90-93.

[9] I. Wolff and N. Knoppik, “Rectangular and Circular Mi-crostrip Disk Capacitors and Resonators,” IEEE Transac-tions on Microwave Theory and Techniques, Vol. 22, No. 10, October 1974, pp. 857-864.

[10] A. G. Derneryd, “Microstrip Disc Antenna Covers Multi-ple Frequencies,” Microwave Journal, Vol. 21, May 1978, pp. 77-79.



Return Signal Intensity Ratio Modulates the Impact of Background Signal on Ozone DIAL Night Time Measurement in the Troposphere

Nianwen Cao1, Tetsuo Fuckuchi2, Takashi Fujii2, Zhengrong Chen1, Jiansong Huang1

1School of Atmospheric Physics, Nanjing University of Information Science and Technology, Nanjing, China; 2 Electrical Physics Department, Komae Research Laboratory, Central Research Institute of the Electric Power Industry, Tokyo, Japan. Email: [email protected] Received December 14th, 2010; revised March 22nd, 2010; accepted April 15th, 2010.

ABSTRACT

This paper discusses the uncertainty of ozone differential absorption lidar (DIAL) measurements due to the impact of background signal. The impact of background signal on ozone concentration profiles is proportional to the background

intensity and the ratio of return signal intensities at “on” and “off” wavelength (),(

),(

off

on

zp

zp

) (hereinafter we call it the

return signal intensity ratio). Analysis suggests that an appropriate return signal intensity ratio can make the impact of background signal very small, negligible. The simulations based on the analysis coincide with the experimental results. The experimental results show that the impact of background signal is negligible at an appropriate return signal inten-sity ratio of 0.96 at wavelength pair (280,285 nm). In case of unknown background intensity, we can adjust the laser pulse energy levels at the two wavelengths to obtain an appropriate return signal intensity ratio on the oscilloscope to suppress the impact of background signal and ensure the accuracy of night time ozone measurements. Keywords: Differential Absorption Lidar (DIAL), Ozone

1. Introduction

Measurements of ozone in troposphere by differential absorption lidar (DIAL) should be corrected for the im-pact of aerosol and other species [1-6]. The uncertainty of ozone DIAL measurements is related to aerosol opti-cal properties and aerosol loading [7]. The uncertainty due to impact of other species can be removed by opti-mizing the wavelength pairs [8]. In order to get rid of both of the impacts of the aerosol and other species, multi-wavelength DIAL and Raman DIAL have been developed [9-10]. The uncertainty due to SO2 impact in UV region is considerable in case of high SO2 concentra-tion from volcanic emission; but it is very small, negligi-ble, in case of background SO2 value, 1 ~ 2 ppb [8].

Although DIAL measurements of ozone in troposphere has been developed and performed for tens years, how to treat the background intensity is still an open question. In general, background intensity is very small, and difficult to ascertain. Usually, the PMT (Photo-mul- tiplier tube) signal before laser shots is taken as back-ground intensity. Some researchers also take the signal

at the most distance as background intensity [11]. The uncertainty of ozone DIAL measurement is sensitive to background intensity; and unknown background intensity can cause huge error. In ozone measurements the meas-urement wavelengths is always set as close to 300 nm (UV region). Wavelength pair (280 nm, 285 nm) is just an example. In night time ozone measurements, the background value wavelength dependence is negligible and the background is caused by dark current of PMT, or thermal noise. Therefore, investigation on how to reduce the impact of background signal is valuable. This paper focuses on discussion of the impact of back-ground signal on ozone measurement. It includes theo-retical analysis, simulations, and experiments on this issue. The aerosol and other species (such as SO2) im-pact on ozone measurement are not discussed here.

2. Theoretical Analysis

The results of ozone DIAL measurements are suscepti-ble to the background intensity. During data processing, the background intensity should be subtracted from the



451

signal. The resulting ozone concentration is very sensi-tive to the magnitude of the subtracted background intensity. Unknown background intensity can cause larger measurement error. How to determine the background intensity precisely is important. The impact of background signal is analyzed in this section. The focus is placed on the factors that determine the impact of background signal and the way to suppress the impact. This analysis is based on very weak background signal at night time ozone measurement.

According to lidar equation, ozone concentration can be calculated by [11]

]),(

),(

),(

),(ln[

2

1

0 off

off

on

on

zp

zzp

zzp

zp

zn

(1)

where, n is the ozone concentration, z is the range resolution, 0 is differential absorption cross-section,

),( zp is the return signal intensity. Because we can

consider 0z as constant for a given DIAL instru-

ment, ozone concentration only depends on the later part of the equation, which we define as

]),(

),(

),(

),(ln[

off

off

on

on

zp

zzp

zzp

zp

(2)

Thus, we focus on instead of n in the following analysis.

None of the background value criterion is made. PMT signal before laser shots can be taken as background in-tensity; PMT signal at high altitude also can be taken as background signal, it due to the measurements condition. Assume on and off as background intensities for the re-turn signals at “on” and “off” wavelength, so the uncer-tainty for can be explained as

( )

( , )( , )ln[ ]

( , ) ( , )

( , )( , ) ln[ ]

( , ) ( , )

( , )( , )ln[

( , ) ( , )

1( , )

1

( , )

off offon on

on on off off

offon

on off

offon

on off

on

on

on

on

p z zp z

p z z p z

p z zp z

p z z p z

p z zp z

p z z p z

p z

p z z

1( , )

]

1( , )

( , )( , ) ln[ ]

( , ) ( , )

off

off

off

off

offon

on off

p z z

p z

p z zp z

p z z p z

]),(

1ln[]),(

1ln[

]),(

1ln[]),(

1ln[

off

off

off

off

on

on

on

on

zpzzp

zzpzp

(3)

where )( is the ozone concentration including the

impact of background signal, and is the real ozone concentration without the impact of background signals.

In night time ozone measurement; usually background intensity is much smaller than lidar signal at lower dis-tance. Therefore, we have

),(),(

),(),(

off

off

off

off

on

on

on

on

zpzzp

zzpzp

(4)

Wavelength separation between the “on” and “off” wavelength is sufficiently small (usually, differential wavelength between two wavelengths pair is governed as less than 5nm) so that the background intensities and receiver optical efficiency at the “on” and “off” wave-lengths can be considered to be approximately equal, respectively. Thus, single background intensity can be adopted in the calculation of the uncertainty. At night time measurement, the background intensities at two UV wavelengths with small wavelength separation can be

assumed as identical , with assumption 1p

. Us-

ing the value obtained from the PMT signal before laser shots, Equation (4) is reduced to

)),(

1

),(

1

),(

1

),(

1(

offoff

onon

zpzzp

zzpzp

(5)

Theoretically, we have zezzp

zp

2

),(

),(

where,

is attenuation at . In case of zz , Equation (5) becomes

)1(),(

1

)1(),(

1

2

2

z

off

z

on

off

on

ezp

ezp

(6)

We introduce A and B to represent the two terms zone 21 and zoffe 21 , respectively. Usually, dur-

ing the measurement time the variation of atmospheric temperature is neglected, even although the absorption cross section dependence on the temperature exists. Con-



452

ventional lidar measurements represents the background value of ozone concentration profile as 45 ~ 60 ppb with 10% measurement error, and as stable over the range from 1000-4000 km [11,12]. Therefore, the extinction coefficient is almost constant over the range between 1000 km ~ 4000 km, The ozone measurements are always carried out in clear fine day, the aerosol loading is very small, especially in Japan. Most of extinction coefficient is caused by molecular. In Japan, in lidar measurements, the range correct signal (I*r2) is always smooth, and decay at constant ratio, it due to the clear fine weather condition, aerosol loading is weak and almost homogeneous. If the return signal not smooth, inhomogeneous aerosol loading occur, the ozone measurement data is not reliable, it due to aerosol error. The aerosol error topic is out of the scope of this paper. The range resolution z is usually taken as 75 m or 150 m [11,13]. The extinction coefficients de-pendence on the range is very small than the return signal. So we treated A and B as constants.

Therefore we have

)),(),(

(offon zp

B

zp

A

(7)

It shows that the impact of background signal depends on background intensity (), and the quantity in the pa-rentheses. In order to reduce the impact of background signal, we can make the background intensity smaller or make the ratio of return signal intensities at “on” and

“off” wavelength ),(

),(

off

on

zp

zp

close to the value of B

A.

A and B can be calculated with the attenuation on and off which are given as [13]

onaonmonOon n ,,3 )( (8)

offaoffmoffOoff n ,,3 )( (9)

where, n is the ozone number density (concentration), O3(on), O3(off) is ozone absorption cross section at “on” and “off” wavelength, respectively; m,on and m,off are molecule extinction at “on” and “off” wavelength, respectively; a,on and a,off are aerosol extinction at “on” and “off” wavelength, respectively.

According to Equations (6)-(9) with assumption of homogeneous background value of ozone concentration over range 1000 ~ 4000 km and homogeneous aerosol loading, the value of A/B dependence on range is small (~1%), and negligible.

Combine Equations (1), (2) and (7) we obtain the impact of background signal:

)),(

),((

),(2

1

0 off

on

onbackground zp

zp

B

AzpB

z

(10)

From Equation (10), it can be seen that the impact of background signal depends on two factors, one is signal

to noise ratio ( ),( onzp

) (or background intensity), the

other is return signal intensity ratio (),(

),(

off

on

zp

zp

). When

the return signal intensity ratio equals A/B, the impact of background signal tend to be zero, in case of faint the background intensity. At high backgrounds and low sig-nal levels the background count rate does matter even if the above condition is met. The value of A/B vary with the wavelength pairs, in other words, it is variable at dif-ferent wavelength pairs (this can be seen in the following simulation with Table 1 and Figure 1), it is 1 at null pro-file measurement ( offon ) to investigate the system-

atic uncertainties associated with measurements. So we

can make return signal intensity ratio (),(

),(

off

on

zp

zp

) close

to A/B to minimize the impact of background signal, even though background intensity is not known precisely. If the signal is much larger than background signal then the background signal has little impact on the ozone re-trievals, especially, in the night time measurement.

3. Simulation

Based on the theoretical analysis, we simulate the impact of background signal for four different wavelength pairs. In this simulation, we assume the error due to homogeneous aerosol loading, the statistic error due to atmospheric condition and laser stability, and the instrument syste- matical error due to beam alignment are all small, negligible. The simulation parameters are listed in Table 1. The cross sections were taken from value in the literature [14], even although the absorption cross sections are tempera-ture dependent, usual ozone measurements by DIAL re-gard the variation of atmospheric temperature as negligi-ble during the measurement time over the range from

1000 km - 4000 km [11,13]. ),(zp

or Signal to Noise

Ratio (SNR) is assumed as 1000 at 1000 m, 500 at 1250 m,

Table 1. Simulation parameter

Wavelength pairs A/B Absorption cross section

280 ~ 285 nm 0.806 280 nm ~ 3.96×10-22m2 285 ~ 290 nm 0.835 285 nm ~ 2.35×10-22m2 290 ~ 295 nm 0.885 290 nm ~ 1.30×10-22m2 295 ~ 300 nm 0.915 295 nm ~ 7.01×10-23m2

300nm ~ 3.49 ×10-23m2 z 150 m nO3 60 ppb

P(z,)/ (SNR) 1000 at 1000 m; 500 at 1250 m; 285 at 1500 m; 100 at 2000 m



453

285 at 1500 m, 100 at 2000 m. The value of 1000, 500, 285, 100 decay as the same rate of 0.28, 0.14, 0.08, 0.02 which taken from actual lidar return signal as height se-quence of 1000 m, 1250 m, 1500 m, 2000 m.

The results are shown in Figure 1(a) and (b). Figure 1(a) present the relation between effects of background signal ratio at wavelength pair and SNR at different height. The lines represent the impact of background signal as a function of the return signal intensity ratio

(),(

),(

off

on

zp

zp

) and Signal to Noise Ratio (SNR). The differ-

ential wavelength between wavelength pair is set as less than 5 nm, therefore, the aerosol scatter property of the two wavelengths should be almost same, and the differ-ential intensity between two return signals is only due to the differential ozone absorption, the decay ratio of the return signal intensity at two wavelength dependence with height should be identical, in this case, the return

0 1 2-0.10

-0.05

0.00

0.05

0.10

zero error

0.806 0.915

A/B: 0.806 280-285 nm A/B: 0.835 285-290 nm A/B: 0.885 290-295 nm A/B: 0.915 295-300 nm

SNR 1000heigh:1000 m

Effe

ct o

f b

ack

gro

un

d s

igna

l (p

pb

)

0 1 2-1

0

1

zero error

0.806 0.915


SNR 100height: 2000 m

0 1 2-0.2

-0.1

0.0

0.1

0.2

zero error



0.806 0.915Effe

ct o

f b

ack

gro

un

d s

igna

l (p

pb

)

0 1 2-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

zero error



0.806 0.915

( , )onp z / ( , )offp z ( , )onp z / ( , )offp z

(a)

0 1 2 3 4 5 6-6

-5

-4

-3

-2

-1

0

1zero error

1 SNR 1000 height: 1000 m



4

3

2

1


Effe

ct o

f ba

ckgr

ound

sig

nal

(p

pb)


( , )onp z / ( , )offp z

(b)

Figure 1. (a) Simulation results: the relation between effects of background signal and return signal intensity ratio at wave-length pair and Signal to Noise Ratio (SNR) at differential height. Simulation parameter is as follows: SNR 1000, height 1000m; SNR 500, height 1250 m; SNR 285, height 1500 m; SNR 100, height 2000 m; (b) Simulation result corresponding to Figure 1(a), combination of four panels of Figure 1(a).



454

signal intensity ratio at wavelength pair ),(

),(

off

on

zp

zp

sho-

uld be constant, almost independent of height. Lidar re-turn signal intensity at two wavelengths is impossible to be zero. Before ozone measurement, we do beam align-ment associated with oscilloscope to ensure the laser beam located in the field of view (FOV) of telescope. If the lidar return intensity at any wavelength of the DIAL pair is zero, it means the beam misalignment happens, the laser beam is out of the FOV of telescope, or the laser energy drop, in this case the measurement should be meaningless, the measurement result should be nothing, it should be stopped and redo the beam alignment until the optimum lidar return occur on oscilloscope, unequal to zero. The lines all cross the x axis (y = 0) at A/B. Note that the definition of y is the impact of background signal, we call the point (x = A/B) as zero error point. This means that when the return signal ratio approaches to A/B, the impact of background signal becomes minimum (close to zero). The value of A/B is close to 1, equals 1 at null profile measurements. Therefore, we can adjust laser power levels at two wavelengths to make the return sig-nal intensity ratio close to (A/B) to suppress the impact of background signal. In Figure 1(a) and (b), it is shown that the background error will become larger when return signal intensity become very smaller, but in case of very weak return signal the measurement result should be nothing (no physical meaning), in this case we talk about error at none measurement result is meaningless at all.

The simulation result also shows that the impact of background signal is related to the choice of wavelength pairs, the value of A/B is changeable at different wave-length pairs. Figure 1(b) corresponds to Figure 1(a), combination of four panels of Figure 1(a). The four group lines 1, 2, 3 and 4 decay at different slope which vary

with SNR at different height.

4. Experimental Result

The DIAL system incorporates two tunable dye lasers pumped by two Nd: YAG lasers and a 50-cm –diam Newtonian telescope. The Nd: YAG lasers operate at a repetition rate of 10 Hz, and each dye laser can emit two wavelengths ( a , b ) on alternate pulses, tunable within

oscillation range of the dye. For ozone measurement, the wavelengths 280 nm and 285 nm were used for the on and off wavelengths, respectively. Each measurement consisted of a sequence of five profiles, each with a 2- min integration time for a measurement time of 10 min. The measurement was carried out at Komae Research Laboratory, Central Research Institute of Electric Power Industry, in Komae City, Japan (35o, 38’N, 139o, 35’E), selected from 24hours measurements data in November 2000.

The range-resolved ozone concentration profile was obtained by Equation (1). The absorption cross section of ozone was taken to be 3.96 × 10-22 m2 at 280 nm and 2.35 × 10-22 m2 at 285 nm, so 2221016.1 m .[14] The obtained return signal are processed as followings:(1) the background intensity, which was obtained from the PMT signal before laser shots, was subtracted, (2) all signals were averaged over 20 time bins corresponding to a range resolution of mz 150 , and (3) the concentra-tion was calculated by Equation (1).

Figure 2 is the return signal for ozone measurement at 280 nmon and 285 nmb . In Figure 2(a), the

intensities of return signals at on and off wavelengths almost equal. Corresponding to Figure 2(a) and Figure 2(b) is enlarged over the range from 1000m to 2000 m. In Figure 2(b) the peak return signal intensity ratio

(a) (b)

Figure 2. Return signals of on and off wavelengths at 280nm and 285nm. (a) Return signal intensity ratio is 0.96; (b) Corre-sponds to (a), enlarged during the range from 1000m to 2000m. (This measurement was carried out at Komae Research Laboratory, Central Research Institute of Electric Power Industry, in Komae City, Japan (35o, 38’N, 139o, 35’E), selected from 24hours measurements data in November 2000.)



455

(),(

),(

off

on

zp

zp

) is 0.96. The background intensity is uncer-

tain. We tried to take different value as background in-tensity for data processing. For example, the PMT signal before laser shots or the signal at the most distance have been used as the background intensity. In Figure 2(a) and (b), the intensities of the PMT signal before the laser shots were almost same as 0.009 mV, and the signal at the most distance (5 km) were about 0.0093 mV. Thus, based on Figure 2(a) and (b), we take 0.009 mV, 0.00915 mV, 0.0093 mV as background intensity. Figure 3 is the ozone concentration profiles calculated by the Equation (1) corresponding to Figure 2(a) and (b). The right panel of the Figure 3 corresponds to the left one, and is enlarged over the range from 1000 m to 2000 m. In Figure 3, the ozone concentration profiles represented by symbols (a), (b) and (c) correspond to background intensity at 0.009 mv, 0.00915 mv, and 0.0093 mv, respectively. And the profile shown by symbol (d) was ob-tained without subtraction of background intensity. Pro-files (a), (b), (c), (d) is colored differently for easy to distinguish. From the right panel of Figure 3, we can see the four profiles represented by symbol (a), (b), (c), and (d) are almost identical. They overlap over the range from 1000 m to 2000 m. It means that in case of an ap-

propriate return signal ratio (),(

),(

off

on

zp

zp

, 0.96) the impact

of background signal is small, negligible. An appropriate return signal intensity ratio makes the impact of back-ground signal on ozone concentration profiling small, and negligible, even if background intensity in unknown. In other words, an appropriate return signal intensity ra-tio ensures the accuracy of the ozone measurements. We adjust the laser energy to make the return signal ratio at wavelength pair (280, 285) close to 0.96 by oscilloscope;

the impact of background signal is suppressed. It is showed in Figure 3. The background error is negligible (the measurements result almost has nothing to do with background value) at appropriate return ratio, therefore, in Figure 3 the ozone profiles are almost same whether background is subtracted or not.

5. Discussion and Implementation Methodology

The impact of background signal on ozone concentration profiling depends on the background intensity and return

signal intensity ratio (),(

),(

off

on

zp

zp

). The impact of back-

ground signal is related to the choice of the wavelengths pair, we can suppress the impact of background signal according to the value of BA / associated with the wavelengths pair. The small wavelengths separation minimizes the impact of hard to characterize aerosols and a jud- icious wavelength pair minimizes the impact of SO2 [12]. Usually, the wavelength separation is governed as less than 5 nm, and the background of SO2 concentration is about 1~2 ppb [8], very weak, in this case the SO2 im-pact on ozone measurement can be neglected. If in day time ozone measurement, as wavelength drops below ~300 nm the solar background rapidly falls off which has a major impact of background. This issue is not dis-cussed in this paper and will be considered in future re-search. In night time ozone measurements, the back-ground intensity is weak and also the dependence on wavelengths is negligible. Therefore, the background intensities at two wavelengths are assumed as identical.

Using an appropriate return signal intensity ratio, the impact of background signal on ozone concentration pro-filing is small, and negligible.

Figure 3. Effect of background signal for ozone concentration profiles at appropriate return signal intensity ratio corre-sponding to Figure 2(a) and (b). (The ozone concentration profiles represented by symbols (a), (b), and (c) correspond to background intensity at 0.009 mv, 0.00915 mv, and 0.0093 mv, respectively. And the profile shown by symbol (d) was ob-tained without subtraction of background intensity. Profiles (a), (b), (c) and (d) is colored differently for easy to distinguish.)



456

In real lidar operation, firstly fine beam alignment ma-

kes the return signals at two identical wavelengths over-lap completely on the oscilloscope, it does the null profile measurement to calibrate the lidar system, then change one wavelength according to the DIAL pair by computer setup, and adjust the laser pulse energy levels at the two wavelengths to obtain appropriate return signal intensity ratio on the oscilloscope close to A/B to suppress the im-pact of background signal and ensure the accuracy of ozone measurements. The return signal intensity ratio on the oscilloscope includes the weak background signal ,

and it is

),(

),(

off

on

zp

zp. If ),(,),( offon zpzp ,

),(

),(

),(

),(

off

on

off

on

zp

zp

zp

zp

exist. Actually, the return sig-

nal intensity ratio on the oscilloscope represents the re-turn signal ratio in the above analysis.

The practical criterion of work is the night time ozone profile in the lower troposphere is homogeneous and stable at background value. Conventional ozone measurements present the background value of ozone 45 ~ 60 ppb with 10% measurement error, except the case of volcanic emission and high concentration air pollution. Calcula-tion by Equations (8) and (9) shows the differential value of A/B at the variation of ozone concentration from 45 ppb to 60 ppb is about 1%. Therefore, we can adjust the laser pulse energy levels at two wavelengths to obtain appropriate return signal ratio close to A/B, and suppress the background error. In case of inhomogeneous ozone profile, this methodology should be modified in future research.

Except background impact, the real measurement error also includes aerosol and other species impact, instru-ment error due to null profile measurement, and statisti-cal error due to atmospheric conditions variation, these error have been discussed in references [8,11-14].

REFERENCES [1] E. V. Browell, S. Ismail and S. T. Shipley, “Ultraviolet

DIAL Measurements of O3 Profiles in Regions of Spa-tially Inhomogeneous Aerosols,” Applied Optics, Vol. 24, No. 17, 1985, pp. 2827-2836.

[2] Z. Wang, H. Nakane, H. Hu and J. Zhou, “Three-Wave-length Dual Differential Absorption Lidar Method for Stratospheric Ozone Measurements in the Presence of Volcanic Aerosols,” Applied Optics, Vol. 36, No. 6, 1997, pp. 1245-1252.

[3] Y. Sasano, “Simultaneous Determination of Aerosol and Gas Distribution by DIAL Measurements,” Applied Op-tics, Vol. 27, No. 13, 1988, pp. 2640-2641.

[4] Y. Zhao, “Simplified Correction Techniques for Back-scatter Errors in Differential Absorption Lidar Measure-ments of Ozone,” Optical Remote Sensing of Atmosphere, Vol. 18, 1991, pp. 275-277.

[5] A. D. Attorio, F. Masci, V. Rizi, G. Visconti and E. Bo- schi, “Continuous Lidar Measurements of Stratospheric Aerosols and Ozone after the Pinatubo Eruption. Part I: DIAL Ozone Retrieval in Presence of Stratospheric Aerosol Layers,” Geophysical Research Letters, Vol. 20, No. 24, 1993, pp. 2865-2868.

[6] W. Steinbrecht and A. I. Carswell, “Correcting for Inter-ference of Mt. Pinatubo Aerosols on DIAL Measurements of Stratospheric Ozone,” In: M. P. McCormick, Ed., Pro-ceedings of the 16th International Laser Radar Confer-ence, Cambridge, 20-24 July 1992, pp. 27-30.

[7] V. A. Kovalev and J. L. McElroy, “Differential Absorp-tion Lidar Measurement of Vertical Ozone Profiles in the Troposphere that Contains Aerosol Layers with Strong Backscattering Gradients: A Simplified Version,” Applied Optics, Vol. 33, No. 36, 1994, pp. 8393-8395.

[8] T. Fujii, T. Fukuchi, N. W. Cao, K. Nemoto and N. Ta-keuchi, “Trace Atmospheric SO2 Measurement by Mul-tiwavelength Curve-Fitting and Wavelength-Optimized Dual Differential Absorption Lidar,” Applied Optics, Vol. 41, No. 3, 2002, pp. 524-531.

[9] A. Parayannis, G. Ancellet, J. Pelon and G. Megie, “Mul-tiwavelength Lidar for Ozone Measurements in the Tro-posphere and the Lower Stratosphere,” Applied Optics, Vol. 29, No. 4, 1990, pp. 467-476.

[10] Z. Wang, J. Zhou, H. Hu and Z. Gong, “Evaluation of Dual Differential Absorption Lidar Based on Raman- Shifted Nd: YAG or KrF Laser for Tropospheric Ozone Measurements,” Applied Physics B: Lasers and Optics, Vol. 62, No. 2, 1996, pp. 143-147.

[11] T. Fukuchi, T. Nayuki, N. W. Cao, T. Fujii and K. Nemoto, “Differential Absorption Lidar System for Simultaneous Measurement of O3 and NO2: System Development and Measurement Error Estimation,” Optical Engineering, Vol. 42, No. 1, 2003, pp. 98-104.

[12] N. W. Cao, S. Li, T. Fukuchi, T. Fujii, R. L. Collins, Z. Wang and Z. Chen, “Measurement of Tropospheric O3, SO2 and Aerosol from a Volcanic Emission Event Using New Multi-Wavelength Differential-Absorption Lidar Tech-niques,” Applied Physics B, Vol. 85, No. 1, 2006, pp. 163-167.

[13] N. W. Cao, T. Fujii, T. Fukuchi, N. Goto, K. Nemoto and N. Takeuchi, “Estimation of Differential Absorption Li-dar Measurement Error for NO2 Profiling in the Lower Troposphere,” Optical Engineering, Vol. 41, No. 1, 2002, pp. 218-224.

[14] L. T. Molina and M. J. Molina, “Absolute Absorption cross Sections of Ozone in the 185- to 350-nm Wave-length Range,” Journal of Geophysical Research, Vol. 91, 1986, pp. 14501-14508.

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