722 II?l?E TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MIT-27, NO. 8, AUGUST 1979

Radiation from Microstrip Discontinuities

MOHAMAD DEEB ABOUZAHRA, MEMBER, IEEE, AND LEONARD LEWIN, SENIOR MF!MBER, IEEE

A6stract-Radicalfy different methods of calculation by Lewin and by

van der Pauw are compared. It is shown that the difference in the two sets

of results is due not to the dtiference in method$ but to ssaumptions onthe value of dielectric constant to be used when calculating the substratepolarisation. Curves ahow that dfffereneea of up to 30 pereent can arise

from this source, but the differences are mneh smaller for tbe larger vafumof dieketric constant.

I N AN earlier paper [I], radiation from rnicrostrip

discontinuities was calculated using the Poynting vec-

tor method, the fields being obtained from the strip cur-

rent and the dielectric polarization beneath the strip. In

order to account for leakage of the field into the air above

the strip, the effective dielectric constant c was used in

place of the actual constant c*. The substitution was made

both in the propagation constant, and also in the polariza-

tion part of the calculation. The reason for doing so in the

latter was that since some of the field lines leak into the

air, where they give no polarization contribution, the

polarization effect is reduced by an amount comparable

to the velocity reduction, so it seemed reasonable to use

the value e everywhere, and not just in the propagation

constant.

The radiated power was calculated for a unit incident

current wave, and took the form

P= 60(kt)21’(c) (1)

where k = 27r/~, t is the substrate thickness, and F(c) is

the radiation factor. F’(6) depends on the discontinuity,

and takes the form

.+1 (6-1)2 EV’+ 1F1(6)=—– — —10g ~1/2_ ~Ze3/2

(2)E

for an open circuit, and

from referring the radiation to a peak unit voltage maxi-

mum rather than to a unit incident current wave; but the

presence of both c and c“ elsewhere in the formula sug-

gested that the main difference stemmed not from the

radically different treatment, but from the use of c*,

rather than ~ in the calculation of the contribution of the

dielectric polarization to the radiated fields. To check out

this latter feature the calculation of [1] was repeated with c

replaced by C* in the polarization term. The Hertzian

vector consists of two components; IIz from the strip

current and IIX from the polarization. Both contain c in

the propagation factor, and this is left untouched. But the

polarization term contains an initial factor (~ – 1)/c and

this is replaced by (c* – 1)/c*. In support of this change,

it could be maintained that the needed quantity is indeed

EX(6* – 1), with Ex given by – aHY/az = jucOc*Ex. Since

HY is proportional to the strip current to which all quanti-

ties are referred, this gives Ex proportional to 1/6*, lead-

ing to the appearance of (E* — 1)/C* in the polarization

term. With this change the following modified forms for

the far-field are obtained from an open-circuited micro-

strip end:

(4)

E@= –j 120 ktc112[

sin2 /)+cos2 e(6*– 1)/c*

1

e –XrCos @—

● — COS2 e r

(5)

with r, 6, and @ spherical coordinates from the strip open

circuit. The Poynting vector can now be constructed and

integrated over an infinite hemisphere to give

e—lF’(c) = 1 – —

●1/2+1log —

2eV2 ~1/’_ 1{

(3) F,(c,c*)= ~ l+(c*2-4c*+2e+ 1)

for a matched termination.

Van der Pauw [2] has given a completely different type

of analysis, utilizing Fourier transforms and a rigoroustreatment of the microstrip configuration, from which

low-frequency approximations can be obtained based on

the assumption of an axial strip current uniform over the

strip width. The formula for the power radiated by the

open circuit differs from (2); in particular, it contains both

c and 6*, and reduces to (2) in the limit for large e, apart

from an initial factor c/2. The latter comes essentially

Manuscript received Deeember 11, 1978; revised Aprit 20, 1979.The authors are with The Electromagnetic Laboratory, Department

of Electrical Engineering, University of Colorado, Boutder, CO 80309.

“[ 1 1 ●1/’+l–—log—

2(C– 1) 4~1/2 ~1/2_l I(6)

On comparison with van der Pauw’s equation (21), using

the interpretation for his symbols of a = LC = q b = c*, it

is seen that the two results are the same, apart from the

above-mentioned initial factor of ~/2. Hence it may be

concluded that the two methods, so very different in

approach, can give essentially the same final result.

Whether or not it is more appropriate to use c everywhere,

0018 -9480/79 /0800-0722$00.75 01979 IEEE

ABOUZAHR4 AND L13WIN: RADIATION FROM MICROSTRIP DISCONTIkUITIES

16 —

14 —

t 12 —

*w

ylo —

ll_-

08 —

06 —

-

c—

Fig. 1. Graph of F’l(c, c“) versus c.

as in [1], is another matter, and cannot be resolved by a

simple comparison of the formulas.

A graph of (6) is given in Fig. 1. The value of e ranges

from C* for wide strips to (c*+ 1)/2 for very narrow

strips. It is also of interest to note that for large q (6) has

the asymptotic expansion

8F,(e, c”)-x

[/1+:–++.”” 1 (7)

so that the leading term 8/3e is identical to that obtain-

able from (2). The graph of (6) is given for FI(E, 6*) versus

q for a range of values of c*. It is seen that the modified

formula always gives a greater value than the original,

723

with differences of up to 30 percent when c = 2 and C*= 3.

In consonance with (7) the differences decrease greatly for

larger values of 6.

A similar calculation can be made for the st~ip

terminated in a matched stub. The electric-field compo-

nents are

and the radiation factor becomes

F2(6,6*)=CC*2+ C2— 2&

6*2(6 – 1)

– ~(c*’–2c*+ e) log -. (10)

This possesses the asymptotic expansion

[F2(e,c*)=& 1+; –++... 1 (11)

and, as in the case of the open circuit, reduces to the

leading term of the unmodified formula for large c.

ACKNOWLEDGMENT

Mr. Abouzahra would like to thank S. A1-Marzouk for

sponsoring him financially, and helping him to pursue ;his

graduate studies.

[1]

[2]

REFERENCES

L. Lewin, “Radiation from discontinuities in stripline: Proc. Im$t.

Elect. Eng., pp. 163-170, 1960.L. J. van der Pauw, “The radiation of electromagnetic power byrnicrostrip configurations: IEEE Trans. Microwave Theory Tech.,

vol. MTT-25, pp. 719–725, Sept. 1977.