IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-33, NO. 10, OCTOBER 1985

Frequency-Dependent Characteristics ofMicrostrip Discontinuities

in Millimeter-WaveIntegrated Circuits

PISTI B. KATEHI AND NICOLAOS G. ALEXOPOULOS, SENIOR MEMBER, IEEE

~bstract —A theoretical approachfor the representation of microstrip

disecmtbruities by equivalent circuits with frequency-dependent parametem

is presented. The model aeeounts accurately for the substrate presence and

associated surface-wave effects, strip finite thickness, and radiation losses.

The method cah also be applied for the solution of microstrip components

in the millimeter frequency range.

I. INTRODUCTION

T HE LITERATURE on the theory of microstrip lines

and microstrip discontinuities is extensive but, almost

without exception, the published methods do not account

for radiation and discontinuity dispersion effects. Micro-

strip discontinuity modeling was initially carried out either

by quasi-static methods [1]-[12] or by an equivalent wave-

guide model [13]–[22]. The former approach gives a rough

estimate of the discontinuity y parameters valid at low fre-

quencies, while the latter gives some information about

dispersion effects at higher frequencies. However, the ap-

plicability of the latter model is also of limited value since

it does not account for losses due to radiation and surface-

wave excitation at the microstrip discontinuity y under inves-

tigation. Therefore, it is reasonable to assume that the data

obtained with this model are accurate only at the lower

frequency range, i.e., before radiation losses become sig-

nificant.

In this paper, three types of microstrip discontinuities

are presented by equivalent circuits with frequency-depen-

dent parameters (see Fig. 1). The implemented method

accounts accurately for all the physical effects involved

including surface-wave excitation [23], [24]. The model

developed -in this paper also accounts for conductor thick-

ness and it assumes that the transmission-line and reso-

nator widths are much smaller than the wavelength. The

latter assumption insures that the error incurred by neglect-

Manuscript received March 13, 1985; revised May 31, 1985. Thisresearch was supported in part by the U.S. Army under Research Con-tract DAAG 29-84-K-O067, and in part by the Northrop Corp. underResearch Grant 84-110-1006.

P. B. Katehi is with the Department of Electrical Engineering andComputer Science, University of Michigan, Ann Arbor, Michigan 48109.

N. G. Alexopoulos is with the Electrical Engineering Department,University of California, Los Angeles, CA 90024.

Y

I

1029

G

t,l Tl;:?Hq!!?gCg

(b) ‘Ps Cp Cp ‘P

w1

I I

11 12

1I

EiZZl ‘7”

“ + h+ s-

(c)

%>

%

Fig. 1. Microstnp discontinuities and equivalent circuits. (a) Open-cir-cuit microstrip line. (b) Microstrip gap. (c) Coupled rnicrostrip reso-nator.

ing the transverse vector component of the current distri-

bution on each conducting strip is a second-order effect.

For each type of discontinuity, the method of moments is

applied to determine the current distribution in the longi-

tudinal direction, while the longitudinal current depen-

dence in the transverse direction is chosen to satisfy the

edge condition at the effective width location [28]. Upon

determining the current distribution, transmission-line the-

ory is invoked to evaluate the elements of the admittance

matrix for the open-end and the gap discontinuities (Fig.

l(a) and (b)). The same method is also applied for evalua-

tion of the resonant frequency of the coupled microstrip

resonator (Fig, l(c)). The equivalent circuits for the firsttwo discontinuities are evaluated and compared with the

results obtained by a quasi-static method based on the

concept of excess length and equivalent capacitance. Al-

though the quasi-static model does not include the discon-

0018-9480/85 /1000-1029$01 .00 W985 IEEE

1030 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-33, NO. 10, OCTOBER 1985

tinuity’s radiation conductance in the equivalent circuits, it

yields results which at low frequencies are in good agree-

ment with previously published data [5]. However, a com-

parison of the quasi-statically obtained results with those

of the dynamic model developed in this paper shows the

inadequacy of the quasi-static approach.

II. ANALYTICAL FORMULATION

A. Current Distribution Evaluation

The current distribution on the transmission-line sec-

tions for the discontinuities considered here radiates an

electric field given by Pocklingion’s integral equation

i(;) =JJF(7/7’).@) ds’ (1)s

~here ~(~ is the total electric field at the point 7 = ( r, 0, o),

G(7/F’) is the dyadic Green’s function and ~(7’) is the

unknown current distribution at the point 7’= (r’, O‘ =

v/2, #). The dyadic Green’s function is given by the ex-

pression

——F(7/7)= ~m[k:;+vv ].(Jo(A17– P]) F(A))dA

(2)

with ~ the unit dyadic, k. = 2 V/ A ~, and F(A) a known

dyadic function of the form

(3)

In (3), c, is the relative dielectric constant of the substrate,

h is the substrate thickness, and ~1, ~z are analytic func-

tions of their variables [29], [30].

The integrand in (2) has poles whenever either one of the

functions ~1( A, ~,, h), ~z( A, c., h) becomes zero. The contri-

bution from these poles gives the field propagating in the

substrate in the form of TE or TM surface waves [30].

Particularly, the zeros of ~1(~, c,, h ) correspond to TE

surf ace waves, while the zeros of ~z( A, c,, h) to TM surface

waves.

Since the widths of the microstrip sections are fractions

of the wavelength in the dielectric, it can be assumed that

the currents are unidirectional and parallel to the x-axis.

Therefore, the current vector in (1) may be written in the

form

7(7’) =.lf(x’]g(y’) (4)

where ~(x’) is an unknown function of x‘ and g( y‘) is

assumed to be of the form

dY’) = +(%3’-1’2“)In (5), We is the effective strip width given by We= w.+ 28,

where 8 is the excess half-width and it accounts for fring-

ing effects due to conductor thickness. Formulas for effec-

tive width exist in the literature [25], [26] and they have

been adopted in this formulation [28].

In order to solve (1) for the current density ~, the

method of moments is employed. Each section of the

microstrip is divided into a number of segments, and the

current is written as a finite sum

7(7’) =ig(y’) fi Infn(x’) (6)~=1

where N is the total number of segments considered and

the expansion functions fn (x’) have been chosen to be

piecewise sinusoidal functions given by

1sin[kO(x’–x._l)]

~_l<.x’<.xnsin(kOIX) ‘ x

fn(x’) = sin[k,(x~+l - x’)]xnsxf<xn+~

sin(kOIX) ‘

(0, otherwise

(7)

with 1X being the length of each subsection.

If the electric field is projected along the axis y = O, z = O

using as weighting functions the basis functions (Galerkin’s

method), (1) will reduce to a matrix equation of the form

[Zmn] [In] = [Vm] (8)

NxN(Nx1) (Nxl)

where [In ] is the vector of unknown coefficients and [J&] is

the excitation vector which depends on the impressed feed

model. [ Z~~ ] is the impedance matrix with elements given

by

zmn=8(y)8(z)/w’2dy’

[012y’2 1/2

‘“’2 1– —we

“~cdx~cdx’{k:~.. +s(~x.-~))f~(x)f)(x()’)

(9)

where

~..=2(%)im(fl::i))Jo(~p)e-u0(lo)

and

.JO(Ap)e-”O’dA. (11)

In (9) and (10)

u,= [A2-k;] ’/2 u= [A2-c,k;]l/2

(12)

f,(~, ~,, h)=uosinh(uh)+ ucosh(uh) (13)

f’(~, ~,, ~)=~,uocosh(uk) +usinh(~h) (14)

and

P= [(x–x’)2+ (y–y’)2]1’2. (15)

XATEFD AND ALEXOPOULOS: MICROSTRIP DISCONTINUITII?S

w.

--4.0 L------ -

~

% Zd=z0,

1031

ZO(ohms)A ,109

(a) 80 -

60 - _//*’z” ~ 3.0

—o

(b) 40 -+s-’-0)0”9 “ 2“8

//=’

20 - 2,6

(c)

10 20 30 40 50 f(GHZ)

Fig. 3. Characteristic impedance 20 and guide wavelength A8/ Aoof amicrostrip line as a function of frequency (6, = 9.6, w/h =1, h = 0.6mm).

(d)

respect to the characteristic impedance, is given by [28]

Fig. 2. Modeling of the excitation mechanism for a microstrip transmis-sion tine.

Using this form for the evaluation of the elements of the

impedance matrix, one can solve the matrix equation shown

in (8) to find the unknown coefficients for the current.

B. Excitation Mechanism — Equivalent Impedance

One difficulty always encountered in this type of prob-

lem is the implementation of a practical excitation mecha-

‘ nism which can be included in the mathematical modeling.

In most applications, the rnicrostrip line is kept as close to

the ground as possible and is excited by a coaxial line of’

the same characteristic impedance. As a result, a unimodal

field is excited under the transmission line, reflection at the

excitatiori end is minimized, and the current distribution

on the line beyond an appropriate reference plane forms

standing waves of a transverse electromagnetic (TEM)-like

mode. Thus, this microstrip line can be approximated by

an ideal transmission line of the same characteristic imped-

ance 20 terminated to an unknown equivalent impedance

Zd (see Fig. 2). It can be shown (see Appendix A) that the

reflection coefficient does not change in amplitude and

phase if the coaxial line is substituted with a voltage gap

generator and the line is left open at the excitation end

(see Fig. 2(c)). This excitation mechanism is adopted for

the application of moments method in the solution of

Pocklington’s integral equation and results in an excitation

vector [ V~] = [8,M] with

{

1 at the position of the gap generator (xi= x~ )aim =

O everywhere else (xi # x~)

The equivalent ideal transmission line for this type of

excitation is shown in Fig. 2(d) where Zd = ZOCfor the caseof an open-end rnicrostnp discontinuity. The quasi-TEM

mode considered has a wavelength A.g equal to the domi-

nant spatial frequency of the amphtude of the current

which is derived by the method of moments.

If the origin of the x coordinate is taken at the position

of Zd, then the equivalent impedance, normalized with

I+r(o)

‘~= 1 – r(o)

where “

SWR– I‘(0) = – SWR+l

eJmkJ

(16)

(17)

and x~= is the position of a maximum.

From (16) and (17), one can see that the accurate

determination of equivalent circuits for different discon-

tinuities depends on the accuracy of evahtating the char-

acteristic impedance 20 and guided wavelength A J ~ =

2T/Ag). The characteristic impedance of the transmissionline of Fig. 2(d) is given by (see Appendix B)

1 20= l–z‘O= 11~=1 1 – Zoc “

e~fllx~ml + _.__.JEe-JBlxm.l1 + Zoc

(18)

where x ~m is the position of a maximum and 11~=1 is the

maximum amplitude of the current. ZOC is the normalized

equivalent impedance of an open-circuited microstrip line

of length 1 = (n /2)Ag(n > 8) and the gap generator is

placed at a position Ag/4 from one end. The characteristic

impedance and guided wavelength are shown as functions

of frequency in Fig. 3. These results are in excellent agree-

ment with already existing data [25].

III. NUIWRICAL RESULTS

A. Microstrip Open-Circuit Discontinuity

The equivalent circuit from an open-end discontinuity,

as shown in Fig. 1, consists of a capacitance CO, in parallel

with a conductance GO=,which is proportional to radiation

losses. Using the method presented previously, values forthe normalized capacitance COc/ w in pF/m and for the

conductance GOC (in mmhos) are plotted as functions of

frequency for a microstrip line with w/h= 1 on a 0.6-mm

Alumina substrate (see Fig. 4). From these data, one can

conclude that the radiation conductance increases with

frequency while the capacitance decreases.

1032 IEEE TRANSACTIONSON

‘“’”S)~ ““”’8 - 80

0 Goc

e=.6

.=0

\* 0/“ - 60

4 x

/0/0 “i. ,.,,: 40

-02 - 20

10 20 30 40 50 f(GHz )

Fig. 4. Radiation conductance GOCand normalized capacitance CO,/ w

of an open-circuited nucrostnp line as functions of frequency (c, =9.6, w/h =1, h= 0.6 mm).

I 11 12

elect,, <

Iwall

‘+’”%2e

“o

v ““i’’v”nwy. Y.

Fig. 5. Gap discontinuity and eqmvalent circuits.

B. Microstrip Gap Discontinuity

(b)

(c)

The equivalent circuit for a microstrip gap is shown in

Fig. 1. For the evaluation of GP, CP, Gg, and Cg, both

sections of the microstrip are excited by gap generators

which are either in phase (see Fig. 5(a)) or out of phase

(Fig. 5(c)). The former case is equivalent to the presence of

a magnetic wall in the middle of the gap, while the latter to

an electric wall at the same position. The equivalent cir-

cuits for these two excitations are shown in Fig. 5(b) and

(c) and give

Y,= G,+ joCa = Y,, + Y12 (19)

and

Y.= G.+ juC~ = Yll – Y12 (20)

where Y,, Y~ are the equivalent normalized admittances for

the case of the electric and magnetic wall, respectively.

From (19) and (20), the conductance and capacitances of

a p-type equivalent circuit are given by

GP= G. (21)

Cp = cm (22)

bg=*(Ge– Gm) (23)

MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-33, NO. 10, OCTOBER1985

(mmhm ) I ! r 1

10 .

5 -

1

0.5

Cp

Cplw ..●.<—-9‘9

%.& .GP ‘---./

(PF/m)

100

50

I10

5

.,.l~l

10 20 30 40 50 60 f(GHz)

Fig. 6. Gap-discontinuity radiation conductance Gg, GP, and normal-ized capacitances C8/ w, CP/ w as functions of frequency ({r =9.6, w/h =1, h = 0.6 mm, s/h= 0.3762).

I 11.0

0.8

0,6

0.4

02

Phase

.( degrees)

90

80

70

60

10 20 30 40 50 60 70 f(GHz)

Fig. 7. Gap-discontinuity admittance matrix elements as functions offrequency (c, = 9.6, w/h= 1,h = 0.6 mm, s/h = 0.3762).

and

Cg=+(ce–cm) (24)

Values of GP, Cp, Gg, and C8 are plotted as functions of

frequency for a rnicrostnp line with w/h= 1 on a 0.6-mm

Alumina substrate (Fig. 6). From the values for the gap

and open-end conductance (Figs. 4 and 6), one can see

that as the frequency increases, the radiation losses become

higher and, therefore, the inter-circuit coupling through

space and surface waves becomes a dominant factor in thedesign of printed circuits. From (19)–(24), the elements of

the admittance matrix of the discontinuity considered as a

two port can be found in amplitude and phase as functions

of frequency (see Fig. 7).

C. Excess Length and Equivalent Capacitances

Another method of deriving equivalent circuits is based

on the evaluation of excess length. The method developed

here results in values for the excess length in such a way so

as to take into account dispersion and radiation losses. The

dominant reason for loss of accuracy at high frequencies

by this method is the way the equivalent capacitance is

KATEH1 AND ALEXOPOULOS: MICROSTRIP DISCONTINUITIES 1033

,,x

h (PFI”!)

I 1

0.8

tCoclw

Ie.-

0.6 .<:. 60~’ ?-%

“y.

0.4A&

\ C;clwI_Lo-oJL_o-aJO\L

4C

.—. — ._. _“++,

0.2._-_e exact method ‘\ 20.—. ,,,., s length method

\

----- q.=~$t.ti. aPPF..Ch

J 1 [ Ldward’s measurements IWIh = o 9)

10 20 30 40 50 6G

Fig. 8. Open-circuit normalized excess length and normalized capaci-tances as functions of frequency (c, = 9.6, w/k =1, h = 0.6 mm).

evaluated. The excess length Ald is measured from the

standing waves of the amplitude of the computed current

(25)

where A? is the guided wavelength and dm= is the position

of the fu-st maximum of the current amplitude from the

open end. The discontinuity capacitance C; is evaluated as

a function of Ald from the following relation [25]:

(26)

where c,ff is the effective dielectric constant and 20 the

characteristic impedance. Equation (26) gives quasi-static

values for the capacitance and, therefore, is accurate for

low frequencies only.

In Fig. 8, the open-end normalized equivalent capaci-

tance C~c/ w in pF/m is plotted as a function of frequency

and is compared to the values derived with the exact

method. As shown, the values of the two capacitances

agree at the lower part of the frequency range and they

shift away as the frequency becomes higher. In addition,

the capacitances CP, Cg of the p-type equivalent circuit

derived with the two methods are compared and they show

a big discrepancy at high frequencies where (25) becomes

much less accurate (Fig. 9).

D. Coupled Microstrip Resonator

For this discontinuity, the transmission-line model is

used for the evaluation of the normalized equivalent im-

pedance Z, as a function of frequency (see Fig. l(c)). The

normalized impedance Z, is measured at A ~/4 from the

open end. In Fig. 10, the real and imaginary parts of Z, are

plotted as functions of frequency for a microstrip line with

w/h =1, t = 0.0001A. on a 0.05 A0-thick Alumina substrate

(c, = 9.9), and for a gap s = 0.01 XO. The lengths have+beenmeasured in terms of the free space wavelength X ~ at a

specified frequency ~O. The resonator length L, varies

between 0.031 A. and 0.033 XO. Fig. 10 implies that there

exists a particular length L, for which the VSWR on the

transmission line at the resonant frequency f, becomes

unit y. Fig. 1l(a) indicates that the resonant frequency

,

(pF/nl) -

I 00

50

<c””a, -’

\

‘w< m C;lw

‘----

‘-%‘ -9 C;lw

10

Cptw

r

5 ● —.———.exact method.---4 excess length method ‘

1 10 20 30 40 50 60 f(GHz)

Fig. 9. Gap-discontinuity normalized capacitances CP/ w, Cg/ w asfunctions of frequency (c, = 9.6, w/h= 1, h = 0.6 mm, s/h= 0.3762).

Fig. 10. Normalized equivalent impedance 2, as a function of frequency~ and gaps (c, = 9.9, w/h =1, h = 0.05A0, L. ==0.032AO).

decreases as the resonator length becomes larger. It is also

very interesting to see how the resonant frequency changes

as a function of the length of the gap. Fig. n(b) shows that

the resonant frequency increases as the gap @OrneS larger,’

while the resonator length was kept constant and equal to

0.032 Ao;

IV. CONCLUSION

The representation of rnicrostrip discontinuities byequivalent circuits or admittance matrices has been treated

by an effective method. The current distribution is com-

puted by the method of moments in the longitudinal

direction of the microstrip discontinuities, while in the

transverse direction it is chosen as the Maxwell distribu-

tion, thus satisfying the edge condition. Upon determining

1034 IEEE TRANSACTIONS ON M3CROWAVE THEORY AND TECHNIQUES, VOL. MTT-33, NO. 10, OCTOBER 1985

(a)

0.96 ~

0.03 0031 0.032 0.033 0.034 L,(>O)

“(’0)l—————l1.01

u

./”--(b)

1,005

1 /.

0.01 0,02 0,03 0,04 0,05 gap (l. )

Fig. 11. Resonant frequency ~, as a function of the resonator length andtie gap.

the current, a transmission-line model is used for the

representation of the unimodal field excited under the

microstrip line.

Therefore, the normalized admittance matrix is evaluated

in terms of equivalent admittances. This method takes into

current is given by

d’l(x)~ +&I(x)= jflYovoa(x).

From (1A) to (3A), one can conclude that the voltage and

the current in this transmission line are of the form

1(x < ) = Ble-J~x + B2@’ (X< O) (4A)

1(x > ) = Ale-~~X + A2eJ~x (x> O) (5A)

T’(x < )+= Zo(A1e-j~x– A2e~6x) (Xs O) (6A)

and:

V(x > )’= Zo(B1e-J6x– B2eJflx) (x> O).

(7A)

The boundary conditions for this problem are

v(l) z

~= d

v(–~o) =2

l(– XO) 0’

(8A)

(9A)

l(x<)=l(x>)atx=O

and (1OA)

dI(x>) _ dl(x < ).,

dx dx= jpYovo . (11A)~=o ~=f)

From (4A) through (11A), one can find that

l+zdI(X>)=– T

cos (flxo) – jZOcsin(/3xo)

o (ZOC-Zd)cos [P(l-xo)] +j(ZOCZd -l)sin[~(l-xo)]

“( 1 – Zoce–M-O + —e+J13(x–

1 + 20=1))

account dispersion and radiation losses, provides us with

equivalent circuits which are an accurate representation of

the discontinuities under consideration, and does not have

any frequency limitations. Furthermore, this method was

compared with results obtained through a quasi-static as

well as a waveguide model and was found superior. The

accuracy of the method depends on the accuracy of

evaluating A ~, x~a, and VSWR. This implies that if more

subsections are considered in the method of moments, the

error will become smaller. In the present derivations, the

estimated error is up to two percent.

APPENDIX A

The transmission line of Fig. 2(d) is considered and it isassumed that the voltage generator is at the position x = O

and that Zd + ZOC. The coupled differential equations for

the voltage and current in this transmission line are of the

form

all(x)— — = jpYo(x)dx

and

dV(x )— —=jpzoI(x)+voa(x).dx

(1A)

(2A)

(3A)

(12A)

From (5A) and (12A), the current reflection coefficient is

given by

A2 1 – Zoc~i=x=me-’jflf. (13A)

Oc

If the case is considered where the left end of the

microstrip line is matched (see Fig. 2(b)), then B1 = O and

{I“(x > ) = – & e-j~’+

l–z~ eJ$Xe–2J$Io 1 + 20= )

(14A)

with

1 – Zoc _’,b,P7’=l_z e . (15A)

Oc

From (13A) and (15A), one can conclude that the ampli-tude and phase of the reflection coefficient do not change

when the line is matched at the generator end.

APPENDIX B

If Xo, 1 are equal to Ag/4 and q(Ag/2), then

l-xo=yAg (Bl)

and

1 + Zd Zoc

(

l–zI(x>)==zoczd–l e ‘J& +

)IT&?’px “

From (1A) and (2A), the differential equation for the (B2)

XATEHI AND ALEXOPOULOS: MICROSTRIP DISCONTINUITIES

At x = 1 – Xm= for Zd = ZOC, the absolute value of the

current is given by

(B3)

where x~= is the position of a maximum measured from

the equivalent impedance Zd. Equation (17) is derived

from (B3).

IU3FERf3Nt.2ES

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[s]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

A. Farrar and A. T. Adams, “Computation of lumped microstnpcapacitances by matrix methods: Rectangular sections and endeffects,” IEEE Trans. Microwave Theoty Tech., vol. MT”l-19, pp.495-497, 197LM. Maeda, “Analysis of gap in microstrip transmission lines,”IEEE Trans. Microwave Theory Tech., vol. MTT-20, 1972.T. Itoh, R. Mittra, and R. D. Ward, “A method of solving discon-tinuity problems in rnicrostrip lines,” in 1972 IEEE-GMTT Znt.Microwave Symp. Dig.

P. Silvester and P. Benedek, “ Equiwdent capacitittce of microstripopen circnits~’ IEEE Trans. Microwave Theory Tech. vol. MTT-20,1972.P. Benedek and P. Sylvester,, “Equivalent capacit~ce for microstripgaps and steps,” IEEE Trans. Microwave Theory Tech., vol. ~TT-20,1972.

P. Silvester and P. Benedek, ‘<Microitnp discontinuity capacitancesfor right-angle bends, T-junctions and crossings,” IEEE Trans.

Microwave Theory Tech., vol. MTT’-21, 1973.R. Horton, “The electrical characterization of a right-angled bendin microstrip linefl IEEE Trans. Microwave Th;ory T;ch., vol.MTT-21, 1973.R. Horton, “Equivalent representation of an abrupt impedance stepin rnicrostrip line,” IEEE Trans. Microwave Theoiy Tech., vol.MTT-21, 1973.A. F. Thompson and A. Gopinath, “Calculation of microstripdiscontinuity inductances; IEEE Trans. Microwave Theory Tech.,vol. MTT-23, 1975.A. Gopinath er al., “ Equivalerit circuit parameters of microstripstep change in width and cross junctions,” IEEE Trans. MicrowaueTheory Tech., vol. MTT-24, 1976.R. Garg, and I. J. Bahl, “ Microstrip discontimritiesj’ Int. J. Elec-tron. vol. 45, July 1978.C. Gupta and A. Gopinath, “ Eqti&mlent circuit capacitance ofmicrostrip step change in width,” IEEE Tram. Microwave TheoryTech., vol. MTT-25, 1977.T. Itoh, “Analysis of microstrip resonators,” IEEE Trans. Micro-

waoe Theory Tech., vol. MTT-22, 1974.I. Wolf, G. Kompa, and R. Mehran, “ C~culatioh method formicrostrip discontinuities and T-junctions,” Elecrron. Lett., vol. 8,

1972. -G. Kompa and R. Mehran, “Planar waveguide model for calculat-ing microstrip components,” Electron. Lett., vol. 11, 1975.G: Kompa, “’S-m~tnx computation of micm$tnp’ discontinuitieswith a planar waveguide model,” Arch. Elek. Ubertragung., vol. 30,1976.R. Mehran, “The frequency-dependent scattering matrix of micro-s@p right-angle bends, T-junctions and crossings,” Arch. Elek.

Ubertragung., vol. 29, 1975.

W. Merrzel and I. Wolf, “A method for calculating the frequencydependent properties of microstrip discontimrities~ IEEE Trans.

Microwave Theoiy Tech., vol. MTT-25, 1977.I. W. Stephenson and B. Easter, ” Resonant techniques for establish-ing the equivalent circuits of small discontinnities in microstep,”Electron. Lelt., vol. 7, pp. 582-584, 1971.B. Easter, “The equivalent circuits of some microstrip discontinni-ties,” IEEE Trans. Mierowaue Theory Tech., vol. MTT-23, pp.655–660, 1975.J. R. James and A. Henderson, “High frequency behavior ofrnicrostrip open-circuit tetinaticms,” Proc. Inst. Elec. Eng., Mi-crowaves, Optics and Acoustics, vol. 3, no. 4, pp. 205–218, Sept.1979.

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

1035

R. Mehran, “Computer-aided design of microstnp filters consider-ing dispersion loss and discontinuity effects,” IEEE Trans. Micro-waue Theory Tech., vol. MIT-27, pp. 239–245, Mar. 1979.P. B. Katehi and N. G. Alexopoulos, ‘; on the effect of substratethickness and permittivity on printed circuit dipole properties;’IEEE Trans. Antennd.s Propagat,, vol. AP-31, Jan. 1983.N. G. Afexopmrlos, P. B. Katehi, and D. B. Rutledge, “Substrateoptimization for integrated circuit antennas,” IEEE Trans. Micro-waue Theory Tech., vol. MTT-31, July 1983.K. G. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines.

Dedham, MA: Artech House, 1979.T. C. Edwards, Foundations for Microstrip Circuit Design. NewYork: Wiley, 1981.P. B. Katehi and N. G. Alexopoulos, “ C)n the theory of printedciicttit antennas for millimeter waves,’: in Conf. Dig., Sixth Int.

Conf. Infrared and Millimeter Waves, Dec. 1981, pp. F.2.9-F.2.1O.P. B. Katehi and N. G. Alexopoulos, “On the modeling of electro-magnetically coupled micro&p antennas-The printed strip di-pole,” ZEEE Trans. Antennas Propagat., vol. AP-32, Nov. 1984.P. B. Katehi, “A generalized solution to a class of printed circuitantennas,” Ph.D. dissertation, Univ. of California, Los Angeles,1984.N. G. Alexopoulos, D. R. Jackson, and P. B. Katehi, “ Criteria fornearly omnidirectional radiation patterns for printed antennas,”IEEE Trans. Antennas Propagat., vol. AP-33, Feb. 1985.P. B. Katehi and N. G. Alexopoulos, “Real axis integration ofSornmerfeld integrals with atmlications to minted circuit antennas.”J. Ma/h. Phys., ~ol. 24(3), Mar. 1983. ‘

Pisti B. Katetri was born in Athens, Greece. Sliereceived the B.S.E.E. degree from the NationalTechnical University of Athens, Greece, in 1977and the M. S.E.E. and Ph.D. degrees from theUniversity of California, Los Angeles in 1981and 1984, respectively.

She is currently an Assistant Professor in theDepartment of Electrical Engineering and Com-puter Science, University of Michigan, AnnArbor. Her research interests are in the theoreti-cal modeling and measurements of circuit anten-

nas and rnicrostrip integrated-circuit components.

*

Nic61aos G. Alex@oulos (S’68-M69– SM82) wasborn in Athens, Greece, in 1942. He graduatedfrom the 8th Gymnasium of Athens, Greece, andsubsequently obtained the B.S.E.E., M. S.E.E., andPh.D. degrees from the University of Michigan,Ann Arbor, MI, in 1964, 1967; and 1968, respec-tively.

He is currently a Professor in the Departmentof Electrical Engineering, university of Cali-fornia, Los Angeles, the President of NGA Scien-tific Consultants, and a Consultant with

Northrop Corporation’s Advanced Systems Division. His current researchinterests are in electromagnetic theory as it applies in the modeling ofintegrate,d-circuit components and printed circuit antennas for microwaveand millimeter-wave applications, substrate materials and their effect onintegrated-circuit structures and printed antennas, integrated-circuit an-tenna arrays, and antenna concealment studies. He is the Associate Editorof the IEEE TRArisAcnorw ON ANTENNAS AND PROPAGATION,Electrwmagnetics Journal, and A lta Frequenza.