6 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31, NO. I , JANUARY 1989

A Broad-Band 'Iransmission Line Model Rectangular Microstrip Antenna

Abstmet-A method has been developed using the transmission line model to predict the input characteristic of rectangular microstrip an- tennas over a wide band of frequencies. The method involves a series Combination of transmission lines to represent each transverse magnetic (TM) mode. An equivalent length, width, and offset dimension are intro- duced to model each mode. The results predicted by the model compared well with experimental results that had varying feed positions.

I. INTRODUCTION

HE USE of a transmission line to model rectangular mi- T crostrip antennas has been a natural progression from its

for a

experimental data showed the validity of the model but also the present inadequacy of the equivalent offset term.

11. THEORY

The model developed by Pues and Van de Capelle [5] has been chosen as the basis for this work as it is fairly complete and gives good results. It is based upon an aperture model and incorporates mutual coupling of the main radiating edges and the influence of the side edges. A three-port circuit representa- tion is used to determine the input characteristic (admittance) of the rectangular patch Yp&,, and with the help of Fig. 1 it is given by

use in modeling microstrip lines. The most noteworthy con- tributions to this technique were by Munson [l], Derneryd [2], [3], Lier [4], [ll], and Pues and Van de Capelle [5]. The transmission line model is used to predict the input char- acteristic of rectangular microstrip antennas, especially for computer-aided design (CAD) applications, due to its accu- racy and numerical efficiency. It also plays an important role in the modeling of arrays. Its great drawback is its inability to predict the input characteristic much beyond a fundamen- tal resonance. For some applications it is necessary to know the performance of the antenna at other frequencies. Other techniques can get around this problem, but this involves full wave techniques or the summation of a series which results in long calculation times. The work presented here shows how the transmission line model can still be used, with some minor adjustments, to determine the input characteristic over a large band of frequencies. Following a brief summary of the ba- sic transmission line model, the idea of an equivalent length, width, and feed offset will be introduced. These equivalent terms are used in the model to account for a particular mode. A comparison of the results produced by the final model with

Manuscript received February 3, 1988; revised April 22, 1988. The authors are with the Dienst Algemene Electriciteit (Department of

Fundamental Electricity), Vrije Universiteit Brussel (Free University of Brus- sels), Faculteit der Toegepaste Wetenschappen, Pleinlaan 2, 1050 Brussels, Belgium.

IEEE Log Number 8824698.

where

Y," = Y,' + Y i and A,,y = (2)

and YO is the characteristic admittance and y = a + j/3 is the complex propagation constant of the microstrip structure; see [6]. Y, is the self-admittance of the main radiating edges, and Ym is their mutual (radiative) admittance given in [5]. Coth (z) and csch (z) are the complex hyperbolic contangent and cosecant functions of argument z , respectively. &E will be termed the offset, as it is a measure of where the feed is relative to the center of the rectangular microstrip patch or transmission line. Note that the copper and dielectric losses of the patch antenna are accounted for by the attenuation constant of the transmission line a. However, the effects due to surface waves, diffraction at the substrate and ground plane edges, and contributions due to higher order modes are not included. The model has a very broad range of validity in terms of patch aspect ratio ( W/L) , substrate relative permittivity ( e r ) , and electrical thickness (h /yd) .

The task of improving the frequency applicability of the transmission line model is based upon the observation by Carver and Mink [7] concerning the modal-expansion cavity model. The input impedance could be represented by a series combination of parallel R-L-C networks for each transverse magnetic (TM) mode. Of course, an R-L-C network is a sim- plification and can be better replaced by a transmission line

OO18-926X/89/01OO-O6$01 .OO @ 1989 IEEE

7 DEARNLEY AND BAREL: BROAD-BAND TRANSMISSION LINE MODEL

APC-7 Plane Connector

Ground’ El Fig. 1. Geometry of experimental microstrip antennas.

as such a model is not only a function of frequency but also of the patch dimensions, as the patch dimensions determine at what frequency a mode is excited and the feed position determines what modes are excited. The TMm mode, due to the static capacitance and loss in the substrate, has little influ- ence on the input characteristic of the microstrip antenna and can be ignored. However, the TMlo mode is fundamental to the response of the rectangular microstrip antenna. Having no field variation in the y direction (congruous with dimension W ) and a cos (?rx/L) variation in the x direction (congruous with dimension L ) its input characteristic can be modeled by using (1). Similarly for the TMol mode, there is no field vari- ation in the x direction and a cos (TU/ W ) variation in the y direction, so (1) can again be used if L is replaced by W and xf replaced by y f . Note that it is assumed that L L W so that the TMlo mode frequency is equal to or lower than the TMol mode frequency. The transmission line model is also a har- monic model, so it will not only account for the fundamental modes, TMlo or TMol, but also at multiples of these fun- damental modes, i.e., TM20, TM30, . . . or TMo2, TMo3, . . . , respectively. Now the problem is how to model those modes where m and n are both nonzero, i.e., the TMII, TM2], TM12, . , modes. This means determining some equivalent physical length L , and equivalent feed position zeq that can be used in the transmission line model to cause an equivalent resonance at these nonzero modes.

In [8] the resonant frequencies of a rectangular microstrip antenna fbn were aptly predicted using a derivation of the rectangular cavity method. This relationship can also be used

to define a resonant length Lmn given by

where m , n are integers and A L , A W are line extension terms to account for fringing effects. This resonant length cannot be directly used as the transmission line model also extends the physical length of the microstrip antenna by a line extension term. This means that the resonant length Lmn must also con- tain a line extension term and so must be defined in terms of an equivalent physical length L , as

Lmn = Leq + 2ALeq. (4) By manipulating the right side of (3) ,

LW + 2[LAW + WAL] + 4ALAW l/m2[W + 2AWI2 + n2[L + 2ALI2 Lmn = * (5)

The problem now is how to split (5) to obtain an expression for L , and also to determine an equivalent width We, as the line extension term AL, is a function of this term. To do this, there are three basic boundary conditions to be met:

1) when m = 1 and n = 0, then Lm, = L + 2AL; so

2 ) when m = 0 and n = 1, then L,, = W + 2AW; so

3 ) expressions (4) and (5) should be equal.

Two simple solutions can easily be derived so that the first

L , = L and Weq = W ;

L , = W and W , = L ;

two constraints are satisfied and are

or LW + 2[mLAW + nWAL]

dm2[W + 2AWI2 + n2[L + 2ALI2 L , =

LW + 2[nLAW + mWAL] l/n2[W + 2AWI2 + m2[L + 2ALI2 Weq = * (7)

To see which solution best satisfied the third constraint, the ratio of Lmn over (Leg + 2AL,) was plotted (see Fig. 2) against the ratio W / L for m = 1, n = 1 (curves a and b); m = 2,n = 1 (curves c and d), and m = 1 ,n = 2 (curves e and f). An er = 2.34 and h = 1.568 mm were used. It is obvious from this figure that the curves determined using (6), i.e., curves a, c, and e, best satisfy condition 3.

The next step is to determine an equivalent feed position, i.e., offset term, to be used instead of the term AoR in (1). For the fundamental modes TMlo and TMol, this offset is determined by the actual physical position of the feed and will be defined as

W and A y y ~ = I - y f l , for TMol. (8)

8 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31, NO. 1, JANUARY 1989

0.90 1

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 W L

Fig. 2. Performance of effective length versus W / L . Using (6), for graph a, m = 1 , n = 1 ; for graph c, m = 2, n = 1; for graph e, m = I , n = 2. Using (7), for graph b, m = I , n = 1; for graph d, m = 2, n = 1; for graph f, m = 1,n = 2.

In determining an equivalent offset Aq off two basic conditions must be met:

4) when m = 1 and n = 0, then Aqoff = Axoff, and 5) when m = 0 and n = 1, then Aeqoff = Ayoff.

A similar sort of relationship that determines L, and W, could be used, i.e.,

Ax off Ay off

Aqoff = l/m2A&ff + n2A:off. (9)

Another possibility would be to use an equivalent feed posi- tion z , and determine the equivalent offset from it and the equivalent length. Therefore,

It was found experimentally that (9) gave better results than (lo), which in some instances gave totally wrong results.

Therefore, by using (1) as a basis to determine the contribu- tion of each mode excited in the microstrip antenna, the total input admittance of a rectangular patch can be determined. The most important modes, for frequencies below 3TM10, are the TMlo, TMol, TMll, TM21, and TM12 modes where the TM20, TMo2, TM30, and TM22 modes are harmonics of TMlo, TMol, TMlo, and TMll modes, respectively, and are already accounted for by the transmission line model. The mi- crostrip patch antenna input admittance will now be defined as I_

Ypatch = [ 1 + - + - + - + - 1 1 ' I - ' (11) yo, y11 y21 y12

where

(12) Yi + Y," + 2 Y, YO ~ 0 t h (yLq) - 2 Ym YO csch (yLq) Y,, = 2Yo

(Yi + Y:) ~ 0 t h (yLeq) + (Yi - Y,") cosh (2yA,) csch (yL) + 2YS Yo

(10) where L, is defined by (6). It can easily be seen that (9) and (10) are only equivalent when m = 0 or n = 0. There are, however, anomalies with (9) and (10) which are summarized as follows:

a) if n = 0 and Ayoff = 0, i.e., y f = W/2, then (9) will be undefined and (10) will give Aq = IL/(2m) -xf/mI;

b) if m = 0 and AXOR = 0, i.e., xf = L/2, then (9) will be undefined and (1 0) will give Aq = 1 W/(2n) - yf/nl.

with Y,",L,, and Aeq being defined by (2), (6), and (9), respectively, Yo and y , Y, , Y, have already been defined.

The experimental microstrip antennas were coaxially fed, so before (1 1) could be tested the influence of the coaxial feed must also be accounted for. A coaxial feed behaves like a simple inductive reactance Xfd in series with the patch ad- mittance. Therefore, the actual microstrip patch antenna input impedance Zbatch would be

(13) 1

Zbatch = - + jxfd. Ypatch

A number of methods have been proposed to describe this

9 DEARNLEY AND BAREL: BROAD-BAND TRANSMISSION LINE MODEL

TABLE I DIMENSIONS OF EXPERIMENTAL RECTANGULAR MICROSTRE' ANTENNAS

Mean Length Mean Width L (mm) k Standard W (mm) f Standard X feed Y feed

Patch Deviation Deviation Xf ("1 Yf (")

38.15 f 0.05 38.70 f 0.10 38.50 f 0.10 38.25 f 0.20 38.35 k 0.15 38.30 f 0.30 37.95 k 0.15 38.40 k 0.05 38.00 k 0.25

31.40 k 0.20 31.50 f 0.05 31.80 k 0.15 31.30 f 0.05 31.25 k 0.05 31.45 f 0.15 31.35 f 0.15 31.55 k 0.05 31.80 k 0.05

. 19.00 9.55 0.00 19.00 9.50 0.00 18.10 10.30 0.00

15.70 15.75 15.80 7.60 7.65 7.60 0.15 0.05 0.20

100 200 300 400 500 600 700 800 9 0 0 Frequency (GHz)

(a)

0, 3

8 2 120.0

.B 60.0

f

c d

0.0

-60.0

-120.0

- 180.0 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00

Frequency (Ob)

(b) Fig. 3. Measured and calculated reflection coefficient of patch 1. - Measured. . . . . . . 3 modes. - - - - - 5 modes.

~

10 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31, NO. 1, JANUARY 1989

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.M) 9.00 Frequency (GHr)

(a)

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 Frequency (GHz)

(b) Fig. 4. Measured and calculated reflection coefficient of patch 2. - Measured. . . . . . . 3 modes. - - - - - 5 modes.

feed reactance, the simplest being to model the probe of di- ameter d as a short-circuited lossless coaxial transmission line of length h . Harrington [9, p. 3781 derives the impedance of a coaxial to parallel-plate waveguide feed system, and a similar method was used by Fong et al. [lo], while Lier [4], [ l l ] has determined an expression for a rectangular patch resonant along the x axis. These methods all predict the same trend, that as frequency and substrate thickness increase so increases the reactance. The method given by Harrington [9] was used to incorporate the feed reactance as it is incorporated in the other methods and is

where { = 1.781072. . and is derived from Euler’s constant. 111. EXPERIMENTAL VERIFICATION

The method was compared with the measured input char- acteristics of nine patches whose dimensions and standard de-

viations are given in Table I. The antennas were fabricated on 3M CuClad 233 microwave substrate which is a polyte- trafluoroethylene (PTFE) woven glass laminate material. The relative dielectric permittivity and thickness of the substrate were specified by the manufacturer as 2.34 f 0.04 and 1.568 f 0.025 mm, respectively. The patches all had a 14 x 14-cm ground plane. The input characteristics were measured on a HP-84 10 automatic microwave measurement system. To study the speed of calculation and the influence that the higher order modes have on the result Ziatchr the method was tested using only the first three modes, TMlo, TMol, and TMll and then also with the TM2, and TM12 modes.

The results are shown in Figs. 3-11 as reflection coeffi- cients as all the measurements were carried out in this domain, where the effect of the connector has been removed. As can be seen, the calculated results obtained for frequencies be- low 6 GHz, i.e., the TMlo, TMol, TMI1, TMzo, and TMo2 modes, compare very well to the measured results. Only for

DEARNLEY AND BAREL: BROAD-BAND TRANSMISSION LINE MODEL 11

0,.

I -3-

Y c a

I

12 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31, NO. 1 , JANUARY 1989

0- O N

mz 2

0 2 W Y

x F:

o u

DEARNLEY AND BAREL BROAD-BAND TRANSMISSION LINE MODEL 13

c Y &

14 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31, NO. 1, JANUARY 1989

O.1° 1 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00

Frequency (GHz)

(a)

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

.\ ; '3.

i ,.. , i : I i'; , i. . .

i 7

9.00 Frequency (GHz)

(b) Fig. 11. Measured and calculated reflection coefficient of patch 9. - Measured. . . . . . . 3 modes. - - - - - 5 modes.

patches 4, 5, and 6 is the prediction incorporating the first three modes wrong, but this is because the TM21 mode also occurs around 6 GHz. The method using the first five modes does not predict the TMzl mode for patch 4 and overpredicts the TM21 mode for patch 5. Above 6 GHz the three-mode method cannot predict the TM12 mode, but it does approxi- mate the TM30 and TM22 modes. Its great weakness is that it often predicts a mode at around 8 GHz when it does not actually occur, i.e., for patches 2 and 4. The method using the five modes does not predict the TM12 mode for patches 2, 3, and 8, and the TMzl mode for patch 4 while it also incorrectly predicts a mode around 8 GHz for patches 2 and 4. The method using only the first three modes best predicted the deepness of each resonant dip. What is interesting is that for patch 7 a small dip occurs at around 4 GHz which was thought at first to be a measurement error; however, the the- ory confirms that the TMll mode is being slightly excited.

This can also be seen for patch 8 at around 5 GHz for the TMZ0 mode.

It was found that the equivalent offset value causes these faults, especially at frequencies higher than the TMll mode. This is especially evident for the modes where m or n are zero, both methods correctly predict the occurrence of the mode. However, when m and n are not zero, except for the TMll mode, both methods are often incorrect. For frequen- cies higher than the TMll mode the difference between actual dip deepness and that predicted using three modes can prob- ably be attributed to the fact that there is no contribution of higher order modes, while the actual occurrence of a dip is due to the uncertainty of the patch and feed position dimen- sions. This problem was expanded upon in [8] in determining the resonant frequencies of a rectangular microstrip antenna. Another point which must not be forgotten is that the self- and mutual admittance parameters were determined for a funda-

DEARNLEY AND BAREL: BROAD-BAND TRANSMISSION LINE MODEL 15

mental mode, i.e., TMol or TMlo, so their validity for higher order modes cannot be assumed.

IV. CONCLUSION The results so far show that the transmission line model

can be successfully used to predict the input characteristic of a rectangular patch antenna over a wide band of frequencies. Even though the model is conceptually simple, it still produces accurate results in a relatively short period of computing time. The concept of equivalent length, width, and feed offset has been validated by experimental results but has shown that the equivalent offset term needs some improvement. The number of modes incorporated in the model will depend on the fre- quency range over which the antenna must be modeled. Using three transmission lines gives good results up to the frequency at which the TMo2 mode is excited.

REFERENCES

R. E. Munson, “Conformal microstrip antennas and microstrip phased arrays,” IEEE Trans. Antennas Propagat., vol. AP-22, no. 1, pp. 74-77, 1974. A. G. Derneryd, “Linearly polarized microstrip antennas,” IEEE Dans. Antennas Propagat., vol. AP-24, no. 6, pp. &16-850, 1976. A. G. Demeryd and A. G. Lind, “Extended analysis of rectangular microstrip resonators antennas, ” IEEE Trans. Antennas Propagat. vol. AP-27, no. 6, pp. 846-849, 1979. E. Lier, “Improved formulas for input impedance of coax-fed mi- crostrip patch antennas,” Proc. Inst. Elec. Eng., vol. 129, pt. H, no. 4, pp. 161-164, 1982. H. Pues and A. Van de Capelle, “Accurate transmission-line model for the rectangular microstrip antenna,” Proc. Inst. Elec. Eng., vol. 131, pt. H, no. 6, pp. 334-340, 1984. E. 0. Hammerstad and 0. Jensen, “Accurate models for computer- aided design,” in Dig. 190 MTT-S Int. Microwave Symp., Wash- ington, DC, 1980, pp. 407-409. K. R. Carver and J. W. Mink, “Microstrip antenna technology,” IEEE Trans. Antennas Propagat., vol. AP-29, no. 1, pp. 2-24, 1981. R. W. Deamley and A. R. F. Barel, “A comparison of models to de-

termine the resonant frequencies of a rectangular microstrip antenna,” IEEE Trans. Antennas Propagat., this issue, pp, 114-118. R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. K. S. Fong, H. F. h e s , and M. J. Withers, “Wideband multilayer coaxial-fed microstrip antenna element,” Electron. Lett., vol. 21, no. 11, pp. 497-499, 1985. E. Lier, Proc. Inst. Elec. Eng., vol. 130, pt. H, no. 2, p. 190, 1983.

Russell W. Dearnley (S’76-S’83-M’83-M’85) was bom in Geelong, Australia, on September 15, 1955. He received the B.Eng. degree from the Gor- don Institute of Technology in 1977, the M.Sc. de- gree from Deakin University in 1980, and the Sij- zonder Licenciaat in de Technologie (Special Mas- ters degree in Technology) and the Doctoraat in de Toegepaste Wetenschappen (Doctor of Applied Sci- ence) in 1980 and 1987, respectively, from the Vrije Universiteit Burssels, Belgium, in 1987.

From 1978 until 1984 he studied at the Vrije Uni- versiteit Brussel on a scholarship from the Belgian Ministry of National Ed- ucation and Dutch Culture. In 1984 he became an Assistant and in 1987 a part-time Professor in the Department of Fundamental Electricity at the Vrije Universiteit Brussel. Since July 1988, he has been employed with Interscan International, Sydney, Australia. His interests lie in the areas of microstrip antennas, microwave measurement techniques, diffraction techniques, mi- crowave devices and circuits.

Alain R. F. Barel was born in Roeselare, Belgium, on July 27, 1946. He received the degree in electri- cal engineering from the Universite Libre de BNX- elles, Belgium, in 1969, the Postgraduate degree in telecommunications from the Rijks Universiteit Gent (State University of Ghent), Belgium, in 1974, and the Doctoraat in de Toegepaste Wetenschappen (Doctor of Applied Science) from the Vrije Univer- siteit Brussel in 1976.

He worked as an Assistant at the Vrije Univer- siteit Brussel (Free University of Brussels). Cur-

rently, he is a Professor in the Department of Fundamental Electricity at the Vrije Universiteit Brussel and teaches in the fields of electromagnetism and microwaves. His research interests lie in the areas of dielectric reflectome- try, the application of microstrip antennas in telecommunications, microwave hyperthermia, and NMR imaging.