Response of coplanar transmission lines to an incident EM field

Response of Coplanar Transmission Linesto an Incident EM Field

Hamid Khodabakhshi, Ahmad Cheldavi

College of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran

Received 22 June 2006; accepted 3 December 2006

ABSTRACT: In the present article, a new concept is being proposed to find the voltages

and currents induced by external EM fields on an arbitrary transmission line like coplanar

lines. In this new method, a 3D problem of the ‘‘transmission line excited by external EM

field’’ is subdivided into two problems. The first problem called ‘‘forced terms problem’’ is a

2D problem of the ‘‘cross section of the transmission line excited by external EM field.’’ The

second problem is a one-dimensional problem of the ‘‘transmission line excited by external

EM field’’ called ‘‘TL problem.’’ The first problem can be solved using a two-dimensional

analysis. The second problem can be described by a pair of simple nonhomogeneous differ-

ential equations of the transmission lines with forced terms. The forced terms of the differ-

ential equations are obtained solving the first problem. The method is fast and validated

with conventional 3D full wave analysis software like HFSS. VVC 2008 Wiley Periodicals, Inc. Int J

RF and Microwave CAE 18: 157–167, 2008.

Keywords: common mode; differential mode; field decomposition; forced terms; Green’s func-

tion; moments method; Sommerfeld integral

I. INTRODUCTION

In recent years, the utilization of planar and coplanar

integrated circuits has become increasingly important

in microwave and millimeter wave applications. So,

the effect of external incident wave on different

structures is a very important topic in EMC research

groups around the world. There are two categories of

transmission lines. The first category is the TL in ho-

mogeneous structures like multiconductor TLs above

a ground plane. For this category the transmission

line approximation leads to good results.

The second category is the TLs in nonhomogene-

ous media. These structures with some degree of

approximation can be solved using the quasi-TEM

approach. The transmission line methods can only

solve the problem of field coupling to the first cate-

gory of TLs. For example, the response of a multi-

conductor wire transmission line (homogeneous

structure) illuminated by external EM fields has been

considered in several papers [1–3]. Also, the nonuni-

form wire transmission line (homogeneous structure)

above ground plane has been considered in [4]. The

exact solution for the field coupling to some kind of

transmission lines in nonhomogeneous structures can-

not be obtained easily. The quasi-TEM approach and

some concepts are used in [5] to find the effect to uni-

form and single microstrip structures. The problem of

multiple coupled nonuniform microstrip transmission

lines is solved in [6].

In the present article, the induced voltage and cur-

rent on the transmission line is divided into common-

mode and differential-mode parts. The common

mode part is obtained using a two-dimensional analy-

sis. This part is used as the forced terms (the right-

Correspondence to: A. Cheldavi; e-mail: [email protected] 10.1002/mmce.20252Published online 4 February 2008 in Wiley InterScience (www.

interscience.wiley.com).

VVC 2008 Wiley Periodicals, Inc.

157

hand side) for the partial differential equations of the

transmission lines (differential mode part). The solu-

tion of these differential equations gives the differen-

tial-mode part of the induced voltages and currents.

The method can be used for any transmission line

structure.

Here the method is applied to a coplanar transmis-

sion line structure. Finally, the method is evaluated

with conventional full wave analysis software like

HFSS.

II. MATHEMATICAL CONCEPTS

The method can be used for any nonhomogeneous

TL structure with an arbitrary cross-section excited

by an external EM field. Here without loss of general-

ity we consider a simple coplanar microstrip structure

whose cross-section is shown in Figure 1, excited by

an external EM wave.

The induced voltages and currents along the line

consist of two parts, the common mode and the dif-

ferential mode parts. The common mode part which

is the antenna mode part is excited when the line is

infinite in length. For a line with electrically small

cross section, we can assume each termination as a

node in circuit theory. Therefore, the total current at

the terminations should be zero (differential mode).

So that the common mode current not predicted by

TL equations is zero at the terminations and is there-

fore of no importance in predicting the terminal

response of TL [7]. The differential mode part is the

total solution only in the terminals.

So, the nonhomogenous equations of transmission

line (differential mode part) will provide the com-

plete prediction of the terminal response of a trans-

mission line. Also, when there is no termination, and

the distance of the signal trace and its ground is small

compared to wavelength of the incident EM wave,

just the common mode or antenna mode is excited.

Therefore, the common mode part can be obtained

solving the problem of the cross section of the trans-

mission line.

In this paper, a new concept in EMC computation

is being proposed. In this new method, a 3D problem

of the ‘‘transmission line excited by external EM

field’’ is subdivided into two problems. The first prob-

lem called ‘‘forced terms problem’’ is a 2D problem

of the ‘‘cross section of the transmission line excited

by external EM field.’’ The second problem is a one-

dimensional problem of the ‘‘transmission line

excited by external EM field’’ called ‘‘TL problem.’’

The first problem can be solved using a two-dimen-

sional analysis. The second problem can be described

by a pair of simple nonhomogeneous differential

equations of the transmission lines with forced terms.

For simplicity it is assumed that the conductors

and substrate are lossless, the dimensions of the

ground plane are much larger than the wavelength,

and the transverse dimensions of the line, including

the substrate height (h), the width of the signal trace

(w), and the gap between the signal trace and the

ground plane, are much smaller than the wavelength.

With such assumptions, the external EM wave

will cause a quasi-TEM mode to propagate along the

line. The induced voltages and currents along the line

are obtained solving the cross section of the problem

(eliminating the z dependence of the problem).

The incident field and the phase constant of the

incident field in the z direction are (Fig. 2)

H0z ¼ �E0

gcosw sin h ð1aÞ

bincz ¼ �binc cos h ð1bÞ

w determines the type of polarization, where w ¼ 0

produces a TEz polarization, w ¼ p=2 produces a

TMz polarization. It will be shown that the voltage

Figure 1. A simple coplanar microstrip structure.

Figure 2. Definition of incident wave angles.

158 Khodabakhshi and Cheldavi

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

and current of the line satisfy the usual transmission

line equations including forced terms on the right.

Now, the transmission line equations of the structure

are obtained.

III. THE TRANSMISSION LINE EQUATIONS

Consider coplanar transmission line as shown in Fig-

ure 1. Integrating the Faraday equation over the path

shown in Figure 3 and using Stokes’s theorem results

in (demonstrated in Appendix A):

d

dz

Z w=2þd

w=2

Ex x; 0; zð Þdx ¼ �jxl0

Z w=2þd

w=2

Hy x; 0; zð Þdx

ð2Þ

A similar equation can be obtained for the left-

side gap:

� d

dz

Z �w=2�d

�w=2

Ex x; 0; zð Þ dx

¼ �jxl0

Z �w=2

�w=2�d

Hy x; 0; zð Þ dx ð3Þ

Now, we consider the continuity equation and

integrate it over the volume shown in Figure 4. Using

the divergence theorem we have (demonstrated in

Appendix B)

d

dz

Z w=2

�w=2

Jszðx; zÞ dx ¼ �jxe0

�Z w=2

�w=2

Eyðx; 0þ; zÞ

3 dx� er

Z w=2

�w=2

Ey x; 0�; zÞ dx�

ð4Þ�

For this structure, an additional condition for the

frequency and the angles of the incident wave should

be considered. The phase difference of the incident

wave over the two gaps, Dux ¼ kx d þ wð Þ, should be

negligible.

We now define the following relations for the

induced voltage and current:

VðzÞ ¼ 1

2

�Z w=2þd

w=2

Exðx; 0; zÞ dx

�Z �w=2

�w=2�d

Ex x; 0; zð Þ dx�

ð5Þ

IðzÞ ¼Z w=2

�w=2

Jsz x; zð Þ dx ð6Þ

Setting (5) and (6) in (4), (3), and (2), we get

dVðzÞdz

¼ 1

2

��jxl0

Z w=2þd

w=2

Hy x; 0; zð Þ dx

þ jxl0

Z �w=2

�w=2�d

Hy x; 0; zð Þ dx�

ð7Þ

dIðzÞdz

¼ �jxe0

�Z w=2

�w=2

Eyðx; 0þ; zÞ dx�er

3

Z w=2

�w=2

Ey x; 0�; zð Þ dx�

ð8Þ

These are the differential equations describing the

induced voltage and current along the line. Note that

the fields in the right-hand sides of (7) and (8) cannot

be obtained simply.

IV. FIELD DECOMPOSITION

To obtain the fields on the right-hand sides of (7) and

(8), we decompose them into two parts, the ‘‘differen-

tial mode fields’’ and the ‘‘common mode fields.’’

Also, it is obvious that currents and voltages along

the transmission line can be decomposed as the com-

Figure 3. The integration path for eq. (2).

Figure 4. Elementary volume for integrating (4).

Response of Coplanar Lines to an EM Field 159


mon mode and differential mode parts. The common

mode voltages and currents are excited by a part of

the fields which we call ‘‘common mode fields.’’ In

a similar way the differential mode voltages and

currents are related to a part of the fields which we

call ‘‘differential mode fields.’’ In an externally

excited infinite long line (with no terminations) for

quasi-TEM mode, the induced voltages and currents

are just in common mode form. In other words, the

differential mode currents and voltages are the

results of the terminations. Once the differential

mode currents and voltages appear in the line, the

differential mode fields are produced. Therefore, the

fields can be decomposed into common and differ-

ential mode parts as

~E ¼ ~Ec þ~E

d

~H ¼ ~Hc þ ~H

dð9Þ

This helps us decompose the three-dimensional

problem into

1. A two-dimensional problem (infinite long line

problem) for common mode fields, voltages,

and currents.

2. A one-dimensional problem consisting of an

excited transmission line described by a pair

of inhomogeneous differential equations.

We now define the per unit length inductance and

capacitance parameters of the line as

L ¼ l0

R w=2þdw=2

Hdy x; 0; zð Þ dx�

R�w=2

�w=2�d Hdy x; 0; zð Þ dx

2IðzÞð10Þ

C ¼ 2e0

R w=2

�w=2Edy x; 0þ; zð Þ dx� er

R w=2

�w=2Edy x; 0�; zð Þ dxR w=2þd

w=2Edxðx; 0; zÞ dx�

R�w=2

�w=2�d Edxðx; 0; zÞ dx

ð11Þ

Setting (9), (10), and (11) in (7) and (8) yields

dV zð Þdz

þ jxLI zð Þ ¼ 1

2jxl0

Z �w=2

�w=2�d

Hcyðx; 0; zÞ dx

�Z w=2þd

w=2

Hcyðx; 0; zÞ dx

!ð12Þ

dI zð Þdz

þ jxCV zð Þ ¼ þjxC

2

Z w=2þd

w=2

Ecx x; 0; zð Þ dx

�Z �w=2

�w=2�d

Ecx x; 0; zð Þ dxÞ � jxe0

Z w=2

�w=2

Ecy x; 0þ; zð Þ

3 dx� er

Z w=2

�w=2

Ecy x; 0�; zð Þ dxÞ ð13Þ

Equations (12) and (13) describe the problem of a

one-dimensional externally excited transmission line

represented by an inhomogeneous differential equa-

tion. It should be mentioned that the right-hand sides

of the above equations are all functions of common-

mode fields, which all have a exp �jbincz z

� �z-depend-

ence. Combining (12) and (13), we obtain a second-

order differential equation for the line voltage

d2

dz2V þ b2V ¼ K exp �jbinc

z z� �

ð14Þ

where K ¼ �jabincz � jxLb, b ¼ x

ffiffiffiffiffiffiLC

pand a, b pa-

rameters are obtained as

a ¼ þ 1

2jxl0

�Z �w=2

�w=2�d

Hcyðx; 0; 0Þ dx

�Z w=2þd

w=2

Hcyðx; 0; 0Þ dx

!ð15Þ

b ¼ þjxC

2

Z w=2þd

w=2

Ecxðx; 0; 0Þ dx

�Z �w=2

�w=2�d

Ecxðx; 0; 0Þ dxÞ � jxe0

Z w=2

�w=2

Ecyðx; 0þ; 0Þ

3 dx� er

Z w=2

�w=2

Ecy x; 0�; 0ð Þ dx

!ð16Þ

which are only functions of the geometry and inci-

dent wave properties.

Equation (14) has a simple solution of the form

V zð Þ ¼ Ae�jbz þ Beþjbz þ K

b2 � binc2z

e�jbincz z ð17Þ

Applying the terminal conditions, A and B will be

obtained:

V 0ð Þ ¼ �RSI 0ð ÞV lð Þ ¼ RLI lð Þ



where l is the length of the line exposed to the EM

wave. The terminal conditions can be used directly

with the form of the TL equations in (12) and (13)

which are in terms of the total voltages. So, the appro-

priate terminal constrains must be in terms of the total

voltages [7, p. 404]. The terminal conditions are

shown in Figure 5.

The capacitance (C), inductance (L), characteristic

impedance (Z0), and the effective relative permittivity

(ere) of the line are given in [8].

V. FORCED TERMS

Our main problem is to find the right-hand side

(forced terms) of the aforementioned differential

equations. These forced terms can be obtained by

solving the two-dimensional problem for the common

mode fields.

This problem can be modeled using the induced

magnetic current density on the gaps as shown in Fig-

ure 6. Since most of the incident Ez in the gaps is

choked for the presence of the Perfect Electric Con-

ducting, the total E-field in the gaps is mostly x-

directed. Since Mx over the gaps is related to Ez,

which is already small compared to Ex, so with a

good approximation, Mx can be ignored, compared to

Mz. Also, Ex is continuous at the boundary of air and

the dielectric substrate. So, the magnetic current den-

sity ~Ma;b in the free space and �~Ma;b in the dielectric

is assumed to exist over the gaps. Using equivalence

theorems [9], the boundary in the problem of Figure

6a can be replaced by a perfect electric conductor

(Fig. 6b). Finally, the fields in y � 0 and y � 0

regions can be obtained using two equivalent prob-

lems shown in Figure 6c.

The magnetic fields in Figure 6c can be obtained

using [10]

~H1 ¼ 2

Zgaps

G¼ free�spaceH � ~M dr0 þ ~Hinc

þ ~Hinc;imaged; y > 0 ð18Þ

~H2 ¼ �2

Zgaps

G¼dielectricH � ~M dr0 y < 0 ð19Þ

Figure 5. The longitudinal cross section of the line.

Figure 6. Modeling the gaps using magnetic current densities.



where G¼H is the dyadic Green’s function for the H

field, in free space and dielectric. Since ~M x; zð Þ is z-directed, only Ghzz x; y; x

0; y0ð Þzz is needed to help us

derive an integral equation.

VI. CALCULATION OF THEGREEN’S FUNCTIONS

Ghzz x; y; x0; y0ð Þ is the Hz field of an infinitely long

magnetic line current, which only has a exp �jbincz z

� �variation and is obtained by first calculating the vec-

tor potential as [10]

Ghzz x; y; x0; y0ð Þ ¼ 1

xleb2z � x

� �e4H

2ð Þ0 bq q� q0j j� �

ð20Þ

To find Gdhzz x; y; x

0; y0ð Þ, we perform a simple trans-

formation of coordinate axes (Fig. 7). So, we have

Gdhzz x; y; x

0; y0ð Þold¼ Hy x; z; x0; z0ð Þnew ð21Þ

The three-layer medium is modeled using a linear

filter. This filter can simulate the TE and TM plane

wave propagation through three-layer structure. Sum-

merfield integral is then used to expand the field of a

simple magnetic dipole as a sum of infinite number of

plane waves. The response of the filter is then computed

for each plane wave. Finally, the total field is obtained

as an integral of the responses to each plane wave.

A. Transfer Function ofThree-Layer Media

Consider the case that the observation point and

source point are inside structure. One can design the

flow diagram shown in Figure 8 for incident and

reflected waves. Now, using this flow diagram and

the Mason’s Law [11], one can obtain the transfer

function relating the output electric field to the input

electric field as

TTE;TM sð Þ ¼ Et2

E0

¼ exp �c2 z� z0j jð Þ þ C21 exp �c2 zþ z0 þ 2hð Þð Þ1 � C21 exp �2c2 zþ hð Þð Þ � C23 exp �2c2 h� zð Þð Þ

ð22Þ

where c2 ¼ �jk2

ffiffiffiffiffiffiffiffiffiffiffiffi1 � s2

p; s ¼ sin h. Also, note that

to find each component of the electric and magnetic

field the proper reflection and transmission coeffi-

cients should be used.

B. Expansion of Dipole Sourceas Plane Waves

Consider an arbitrary oriented small magnetic dipole

as

~M ¼ M au; au ¼ uqaq þ uuau þ uzaz

Note that uq and uu are functions of q and u. It is

simple to show that the field of this simple magnetic

dipole can be expanded as the sum of the TEz and

TMz waves as [12–14]

Eq

Eu

Ez

Hq

Hu

Hz

26666664

37777775¼

Eq

Eu

Ez

Hq

Hu

Hz

26666664

37777775

TE

þ

Eq

Eu

Ez

Hq

Hu

Hz

26666664

37777775

TM

ð23Þ

where

Figure 7. Calculation of the Green’s function of a three-

layer medium. [Color figure can be viewed in the online

issue, which is available at www.interscience.wiley.com.]

Figure 8. The flow diagram obtained for analysis of

three-layer medium.



Eq

Eu

Ez

Hq

Hu

Hz

2666666664

3777777775

TE

¼ E0

Z 1

0

3

�juusJþðk0sqÞjuqsJ�ðk0sqÞ � uz

s2ffiffiffiffiffiffiffiffi1�s2

p J1ðk0sqÞ0

�j 1g0uqs


pJ�ðk0sqÞ þ 1

g0uzs

2J1ðk0sqÞ

�j 1g0uus


pJþðk0sqÞ

1g0uqs

2J1ðk0sqÞ � j 1g0uz

s3ffiffiffiffiffiffiffiffi1�s2

p J0ðk0sqÞ

266666666664

377777777775

3 e�jk0

ffiffiffiffiffiffiffiffi1�s2

pz ds ð24Þ

Eq

Eu

Ez

Hq

Hu

Hz

2666666664

3777777775

TM

¼ E0

Z 1

0

3

�juusJ�ðk0sqÞjuqsJþðk0sqÞ

uus2ffiffiffiffiffiffiffiffi1�s2

p J1ðk0sqÞ�j 1

g0uq

sffiffiffiffiffiffiffiffi1�s2

p Jþðk0sqÞ�j 1

g0uu

sffiffiffiffiffiffiffiffi1�s2

p J�ðk0sqÞ0

26666666664

37777777775e�jk0

ffiffiffiffiffiffiffiffi1�s2

pz ds ð25Þ

in which E0 ¼ g0k30

M4pl0

, J�ðk0sqÞ ¼ J0 k0sqð Þ�J2 k0sqð Þ2

.

The transfer function of the three-layer structure is

a function of several variables like position of the

source and observation points, polarization of the

incident field, and the angle of the incident wave. To

find the Green’s function, we need to integrate the

transfer function over all angles of incidence(s).

Also, the Green’s function is written as a function of

position of the source and observation points (~r;~r 0).So for simplicity of notation we call the transfer func-

tion for TE and TM polarizations as TTEð~r;~r 0; sÞ and

TTMð~r;~r 0; sÞ, respectively.

Also, note that for each polarization and field com-

ponent the proper reflection coefficient G should be

used [10].

The Green’s function of the multilayer structure

for magnetic dipole can be obtained as

GmEqð~r;~r 0Þ

GmEuð~r;~r 0Þ

GmEzð~r;~r 0Þ

GmHqð~r;~r 0Þ

GmHuð~r;~r 0Þ

GmHzð~r;~r 0Þ

2666666664

3777777775¼Z 1

0

TTMðsÞ

3

�juusJ�ðs~qÞE0

juqsJþðs~qÞE0

juus2

c J1ðs~qÞE0

uqsc Jþðs~qÞH0

uusc J�ðs~qÞH0

0

2666666664

3777777775e�ck2z0 dsþ

Z 1

0

TTEðsÞ

3

�juusJþðs~qÞE0

jupsJ�ðs~qÞ � juzs2

c J1ðs~qÞ� �

E0

0

�uqscJ�ðs~qÞ þ uzs2J1ðs~qÞ

� �H0

�uuscJþðs~qÞH0

ups2J1ðs~qÞ þ uz

s3

c J0ðs~qÞ� �

E0

266666666664

377777777775e�ck2z0 ds ð26Þ

where the magnetic dipole is located in z ¼ zo and

~q ¼ k2q, H0 ¼ E0=g0.

In our problem, only Hq component contributes in

Gdhzz x; y; x

0; y0ð Þoldand can be obtained as

Hq ¼ H0

Z 1

0

TTM sð Þ scJþ sk2qð Þ expð�ck2hÞ ds

�

�Z 1

0

TTE sð ÞscJ� sk2qð Þ expð�ck2hÞ ds

ð27Þ

where c ¼ffiffiffiffiffiffiffiffiffiffiffiffis2 � 1

p. To find desired Green’s function,

eq. (28) should be integrated over y-axis.

VII. SOLUTION OF THE INTEGRALEQUATION

To satisfy the continuity of tangential magnetic fields

at the boundary, one has (Fig. 6)

y 3 ~H1 ¼ y 3 ~H2 at y ¼ 0

orZgaps

Ghzz x; 0; x0; 0ð Þ þ Gdhzz x; 0; x0; 0ð Þ

�M x0ð Þ dx0

¼ �Hincz x; 0; 0ð Þ ð28Þ



This is the desired integral equation for calcula-

tion of the equivalent magnetic current density over

the gaps, for the 2D problem. Once the magnetic

current densities are obtained for the two gaps, all

fields and the forced terms can easily be obtained.

To find the magnetic current density, the method of

moments has been used [10]. The width of each gap

is subdivided into N equal intervals. The basis func-

tions are chosen as

fiðxÞ ¼1 xi � D=2 < x < xi þ D=2

0 otherwise

�ð29Þ

where D ¼ d/N. The unknown magnetic current den-

sities are now expanded as

MaðxaÞ ¼XNi¼1

aifiðxaÞ

MbðxbÞ ¼XNi¼1

bifiðxbÞð30Þ

The magnetic field in each subinterval of the gap

is as

Hincz ðxi; 0; 0Þ ¼ Hinc

z exp �jbincz xi

� �ð31Þ

The weighing functions are chosen to be Delta

functions. Finally a system of linear equations will be

built, solution of which yields the 2N unknown ai’sand bi’s.

VIII. EXAMPLES AND RESULTS

To validate the proposed method, consider a coplanar

structure with the parameters er ¼ 2.54, d ¼ 2 mm,

w ¼ 6 mm, h ¼ 15 mm. Here a simulation has been

done using HFSS software [15] with capability of

modeling incident plane wave. A wave front with the

properties of Einc ¼ 1V=m; h¼ 70�; u¼ 70�; w¼ 30�

illuminates a 20-cm-long coplanar transmission-line,

which is terminated in 50 X load resistors.

The voltages along the coplanar line are calculated

using the HFSS and the proposed method. Figure 9

shows the voltages along the coplanar line at the fre-

quency of 3.0 GHz. The execution time of the pro-

posed method (N ¼ 16), using MATLAB software,

on a desktop computer with Pentium4 processor and

512MB RAM, is about 24 min while the running

time of the HFSS is �17 h and 34 min. It is about

Figure 10. Induced voltage on the terminal resistor at

z ¼ 0. [Color figure can be viewed in the online issue,

which is available at www.interscience.wiley.com.]

Figure 11. Magnitude of the induced magnetic current

densities over the gaps. [Color figure can be viewed in the

online issue, which is available at www.interscience.

wiley.com.]

Figure 9. Induced voltage along the coplanar line at the

frequency of 3 GHz. [Color figure can be viewed in the

online issue, which is available at www.interscience.

wiley.com.]



36 times faster than the 3D full wave analysis soft-

ware such as HFSS. It is obvious that this method is

very efficient in terms of the running time and the

computation load.

Also, Figure 10 shows the induced voltage on the

source resistor at z ¼ 0, versus frequency. It is

observed that there is an acceptable agreement

between the results obtained using a full wave

method and the proposed method.

The magnitude of the induced magnetic current

densities over the gaps has been shown in Figure 11,

for three frequencies.

As it was mentioned, phase difference of the inci-

dent wave over the two gaps, Dux ¼ kx(d þ w), has a

considerable effect on the validity of the model. For

the model to be valid, this phase difference has to be

negligible. A suitable measure for the error can be

defined as the relative difference between two electric

fields or equivalently between two induced magnetic

currents at the center of the two gaps. This defined

error has been plotted in Figure 12 for the line consid-

ered in this example with er ¼ 2.54. As it is seen in

this figure, the smaller the phase difference the better

the validity of the model. In this paper, the properties

of the incident wave have been chosen so that the rela-

tive difference in the phase of the impinging wave will

not affect the validity of the proposed model.

To investigate the effect of relative permittivity on

the induced voltage, consider the second structure with

a relative permittivity er ¼ 10 (other parameters are

the same). In Figure 13a, the voltage induced along

the line of the first structure (er ¼ 2.54) are shown for

y ¼ 458 and y ¼ 708. For y ¼ 458, the voltage magni-

tude is greater than in the previous case.

For the second structure, the magnitude of the

induced voltage for the same incident angles is givenin Figure 13b. With respect to the results of Fig-

ure 13a, a noticeable reduction in the induced voltage

can be observed in both cases. The existence of a

high dielectric substrate reduces the electromagnetic

coupling of the incident wave to the line. The spatial

distribution varies more rapidly than in the first struc-

ture since the substrate of higher permittivity reduces

the wavelength of the coplanar line [5].

IX. CONCLUSION

A new approach has been proposed to calculate the

effect of an external wave on transmission lines in

nonhomogeneous medium. The induced voltage and

current on the transmission line is divided into com-

mon-mode and differential-mode parts. To obtain the

forced terms using magnetic current densities over

the gaps, a simple formulation for Green’s function

Figure 12. Relative difference between magnitude cur-

rent values in the gaps in terms of phase difference of the

incident wave over the two gaps.

Figure 13. (a) Voltage magnitude induced along the first

coplanar line (er ¼ 2.54) and (b) voltage magnitude

induced along the second structure (er ¼ 10). [Color figure

can be viewed in the online issue, which is available at

www.interscience.wiley.com.]



for three-layer medium has been derived. Using the

proposed method, the total voltage and currents can

be obtained along the line, and also on the terminals.

The 3D full wave software can be employed to simu-

late such structures, but usually the dimensions of the

problem might render such methods inefficient in

terms of computational cost. The proposed method

can give quick quantitative knowledge of a coplanar

structure’s immunity from external EM waves. Fur-

ther work can be done to apply the method to the

other nonhomogeneous transmission line structures.

APPENDIX A: DETAILSPERTAINING TO (2)

We integrate the Faraday equation over the path

shown in Figure 3 and use the Stokes’s theorem

results in

Il

~E � d~l ¼ �jxl0

ZS

~H � n ds

¼ �jxl0DzZ w=2þd

w=2

Hy x; 0; zð Þ dx ðA1Þ

The tangential electric fields along paths 2 and 4

are zero. So, after some simple calculations, (A2) is

obtained

Z w=2þd

w=2

Ex x; 0; zþ Dzð Þ dx�Z w=2þd

w=2

Ex x; 0; zð Þ dx

¼ �jxl0DzZ w=2þd

w=2

Hy x; 0; zð Þ dx ðA2Þ

Dividing (A2) by Dz and taking the limit as Dz ?0, we obtain

d

dz

Z w=2þd

w=2

Ex x;0; zð Þ dx¼�jxl0

Z w=2þd

w=2

Hy x;0; zð Þ dx

ðA3Þ

APPENDIX B: DETAILSPERTAINING TO (4)

We consider the continuity equation and integrate it

over the volume shown in Figure 4. Using the diver-

gence theorem we have

IS

~J � n ds ¼ �jxIS

e~E � n ds

or

Z w=2

�w=2

Jsz x;zþDzð Þ�Jsz x;zð Þ½ �dx¼�jxDze0

3

Z w=2

�w=2

Ey x;0þ;zð Þdx�er

Z w=2

�w=2

Ey x;0�;zð Þdx !

ðB1Þ

Dividing (B1) by Dz and taking its limit as Dz ?0, we have

d

dz

Z w=2

�w=2

Jsz x; zð Þ dx ¼ �jxe0

Z w=2

�w=2

Ey x; 0þ; zð Þ

3 dx� er

Z w=2

�w=2

Ey x; 0�; zð Þ dx!

ðB1Þ

REFERENCES

1. A.K. Agrawal, H.J. Price, and S.H. Gurbaxani, Tran-

sient response of multiconductor transmission lines

excited by a nonuniform electromagnetic field, IEEE

Trans Electromagn Compat EMC-22 (1980), 119–

129.

2. C.R. Paul, Frequency response of multiconductor

transmission lines illuminated by an electromagnetic

field, IEEE Trans Electromagn Compat EMC-18

(1976), 183–190.

3. C.D. Taylor, R.S. Satterwhite, and C.W. Harrison Jr.,

The response of a terminated two-wire transmission

line excited by nonuniform electromagnetic field,

IEEE Trans Antennas Propag AP-13 (1965), 987–

989.

4. M. Omid, Y. Kami, and M. Hayakawa, Field cou-

pling to non-uniform and uniform transmission lines,

IEEE Trans Electromagn Compat EMC-39 (1997),

201–211.

5. P. Bernardi, R. Cicchetti, and C. Pirone, Transient

response of a microstrip line excited by an external

electromagnetic source, IEEE Trans Electromagn

Compat EMC-24 (1992), 100–108.

6. A. Cheldavi, S. Safavi-Naeini, and G. Rezai-rad,

Field coupling to nonuniform coupled microstrip

transmission lines, RF Microwave CAE 13 (2003),

215–228.

7. C.R. Paul, Analysis of multiconductor transmission

lines, Wiley, New York, 1994, Chapter 7.

8. R.E. Collin, Foundations for microwave engineer-

ing, 2nd ed., McGraw Hill, Singapore, 1992, Chap-

ter 3.

9. A.A. Omar and Y.L. Chow, Complex image Green’s

functions for coplanar waveguides, Symp Antenna Propag

AP-S 3 (1992), 1496–1499.



10. C.A. Balanis, Advanced engineering electromagnetic,

Wiley, New York, 1989, Sections 5.5 and 12.2.

11. D.M. Pozar, Microwave engineering, 1st ed., Addi-

son-Wesley, Boston, MA, 1990, pp. 245–250.

12. T. Yamaguchi, M. Mizusawa, and Y. Amemiya,

Shielding effectiveness of the metal sheet for the

field of a magnetic dipole parallel to the sheet,

Trans IEICE J73–B-II (1990), 927–930 (in Japa-

nese).

13. A. Nishikata and A. Sugiura, Analysis for electromag-

netic leakage through a plane shield with an arbitra-

rily-oriented dipole source, IEEE Trans Electromagn

Compat EMC-34 (1992), 284–291.

14. A. Sommerfeld, On the propagation of waves in wire-

less telegraph, Ann Der Phys 81 (1926), 1135–1153

(in German).

15. Ansoft Corp., HFSS: http://www.ansoft.com/products/

hf/hfss/.

BIOGRAPHIES

Hamid Khodabakhshi was born in Ker-

manshah, Iran, in 1979. He received the

B.S. and M.S. degrees in electrical engi-

neering from University of Tehran, UT, in

2000 and 2002, respectively. He is currently

working toward the PhD degree in electrical

engineering at the Iran University of Sci-

ence and Technology, IUST. His research

interests include microstrip antennas, electromagnetic theory,

EMC/EMI, and smart antennas.

Ahmad Cheldavi was born in Ahwaz, Iran,

in 1966. He received the B.Sc. degree (with

honors) from Iran University of Science and

Technology (IUST) in 1992 and the M.Sc.

and PhD degrees (with honors) from the

College of Electrical Engineering, Univer-

sity of Tehran, Iran, in 1994 and 2000,

respectively. He is currently an Associate

Professor at the College of Electrical Engineering, IUST. He is

the author of more than 100 journal and conference papers in the

field of EMC/EMI, and microwave transmission lines, genetic

algorithms, and coupled lines characterization.



Response of coplanar transmission lines to an incident EM field

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