IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 6, NOVEMBER 1982 1191

Input Impedance and Mutual Coupling of Rectangular Microstrip Antennas

DAVID M. POZAR, MEMBER, IEEE

Abstract-A moment method solution to the problem of input im- pedance and mutual coupling of rectangular microstrip antenna ele- ments is presented. The formulation uses the grounded dielectric slab Green’s function to account rigorously for the presence of the sub- strate and surface waves. Both entire basis (EB) and piecewise sinu- soidal (PWS) expansion modes are used, and their relative advantages are noted. Calculations of input impedance and mutual coupling are compared with measured data and other calculations.

I. INTRODUCTION

T[lL 121 HE INCREASING USE of microstrip antenna technology

requires analysis methods capable of accurately predicting the input impedance, mutual coupling, and radia- tion of these antennas. Two methods which have been some- what successful for calculating input impedance and radiation are the transmission line model [3], and the cavity model [ 4 ] , [5] . Although proven to be useful for predicting input imped- ance, the above two methods do not rigorously account for the presence of surface waves on the antenna substrate, and do not account for mutual coupling between closely spaced antenna elements. Also, they lack a convenient way of im- proving solution accuracy.

Recently, a moment method solution to the microstrip antenna problem was proposed [ 6 ] . In [ 61, image theory was used with an integral equation to solve for the antenna patch current; the dielectric slab was treated with equivalent cur- rents. This method, too, gave good results for input imped- ance, but suffered from the necessity of extremely accurate numerical evaluations of the impedance matrix elements.

Presented here is a moment method solution for rectangu- lar microstrip antenna elements, including mutual coupling. This approach uses the exact Green’s function for the di- electric slab, and thus rigorously accounts for surface waves and coupling to adjacent antenna elements. Since the ground plane is accounted for analytically by the Green’s function, no severe numerical problems are encountered as in [6], which relied upon image theory to treat the ground plane. The pres- ent method can handle edge (microstrip) or probe (coax) type feeds, and can be generalized to treat antenna elements of arbi- trary shape. The solution is similar in principle to the pre- viously treated case of printed dipoles [ 7 I , [ 8 I , and [ 91.

Two types of expansion modes are considered-entire basis (EB), and piecewise sinusoidal (PWS). The relative advantages and disadvantages of these modes are noted and discussed, and it is shown how accuracy can be improved by the appropriate choice and number of expansion modes.

Section I1 describes the theory of the method. The Green’s function and its efficient evaluation are discussed, and defini-

Manuscript received October 30, 1981; revised May 28, 1982. This work was supported in part by Grant NAG-1-163 between the National Aeronautics and Space Administration, Langley Research Center, Hamg ton, VA, and the University of Massachusetts, Amherst, MA.

The author is with the Department of Electrical and Computer En- gineering, University of Massachusetts, Amherst, MA 01003.

tions of the expansion modes and impedance matrix elements are given. Section 111 presents numerical results for input im- pedance and mutual coupling for various antenna geometries. These results are compared with measurements and with other solutions. Also presented is a design data curve for resonant re- sistance of a rectangular element with variable width and feed position.

11. THEORY

The Dielectric Slab Green’s Function

For the calculation of impedance matrix elements and volt- age vector elements, the electric field from a horizontal elec- tric current element on a grounded dielectric slab is needed. No generality is lost by having the current element directed in the ;-direction, since all impedance matrix elements can be found by orienting the test mode for current flow in the i- direction. So for the geometry shown in Fig. 1, the vector potential is [ 71 - [ l o ] (assuming d W t time dependence).

A , = 0

where

sin k l z zG1 =-

Tf?

G , = ( € r - l ) sin k l d COS k l z

Te Tm

Te = k l cos k 1 d + j k z sin k l d (5)

Tm = q k 2 cos k l d + j k l sin k l d

k 1 2 = f r k o 2 - D 2 , (Im k l < 0)

k Z 2 = ko2 - D 2 , (Im k2 < 0) (8)

D2 = k X 2 4- kY2 ( 9 )

ko2 = ~ ~ / l o € o , (10)

the field point is at ( x , y , z);thesourcelocationisat(x,,,y~,d). Note that dielectric loss is easily included by replacing e, by er(l - j tan 6 ) in (7), where tan 6 is the loss tangent of the substrate material. The electric fields are then found by

The integrals in (1 ) and (2) will be evaluated numerically, however this can be facilitated by changing to polar coordi-

0018-926X/82/1100-1191$00.75 0 1982 IEEE

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 6, NOVEMBER 1982

slab Green's function. The unknown surface current density on the microstrip antenna elements is expanded in a set

of N basis functions, N

n=l where Tn is t he n th basis function and In is its unknown am- plitude. The basis functions Jn may represent currents in the x or y directions. Use of the same set of functions as testing functions leads to a set of linear algebraic equations to be solved for the unknown I n :

X I \GROUND

PLANE

Fig. 1. Geometry for an infinitesimal dipole on a grounded dielectric slab.

nates P, a where kx = 6 cos a where

k, = p sin a. (13) The integration ranges are = 0 +. 00 and a = 0 + 27r. The terms T , and T , are functions of P (not of a), and represent transverse electric (TE) and Transverse magnetic (TM) surface wave poles, respectively. The first TM surface wave has a zero cutoff frequency, so there is always at least one surface wave pole (unless E, = 1). These poles occur for real values of /3 = Po such that ko < Po <&ko, if tan 6 = 0 (no loss). If dielectric loss is present, the poles move off the real /3 axis t o /3 = Po - f i , y > 0. The exact pole location Po - jy can be determined by using the Newton-Rhapson procedure to find the zeros of (5) and (6). To avoid numerical difficulties when numerically integrating on 0, the integration from Po - 6 to Po + 6 , where S - 0.001k0, is done analytically [ 7 ] . For example, if only one surface wave mode is present,

Vm = 1; Em j i dv .

In (19) and (20), E, is the electric field-due to current 7, in the presence of the dielectric slab, and Ji is the impressed (source) current. Since the same set of functions is used for testing and expansion (Galerkin), Z, , = Z n m .

Fig. 2(a) shows a possible layout of expansion modes for a microstrip antenna problem. The two expansion modes are shown on different antenna patches, but they could be on the same patch, and could have currents in the same direction or in orthogonal directions, Therefore, all impedance elements are of one of two possible types: Z x x (test current in x , expan- sion current in x ) , or Zxy (test current in x , expansion current in y). Using (1)-(4), (1 I), and (1 9) allows these impedance matrix elements to be written as

* Fx*(Jm)Fy*(J,) d k , d k ,

where

(16)

For the lossless case y = 0, and (1 6) reduces to

and F x , Fy are the Fourier transforms of the x and y depend- ences of J :

which is the same as the result obtained using residue theory. Higher order'surface wave modes can be treated in the same manner. The infinite integration in (14) can generally be termi- nated at 150 ko.

The Moment Method Formulation

The moment method solution used here is a Galerkin solu- tion of the electric field integral equation with the dielectric

J n X ( x ) e j k x X d x , Xn

POZAR: RECTANGULAR MICROSTRIP ANTENNAS 1193

PORT a 2

Fl f PORTi

Ix DIELECTRIC GROUNDED 2

(a) +-YE - -

(b)

Fig. 2. (a) Example of e o coupjed microstrip antenna elements and two expansion modes, J , and J,. (b) Definitions of twc-port volt- ages and currents.

where Jn(x , y ) has been factored into the product of a func- tion of x and a function of y :

J n ( x > U) =J,"(X)Jn'(Y). (26)

If the impressed source current is expressed as

J i = j s ( x o - x p y ( ~ o - ~ p ) , (27)

where the feed position is at ( x p , y p ) , then the voltage vector elements can be found by using (2)-(4), (1 l), and (20):

vm = [I Qu(kx, ky)Fx*(Jm)Fy*(Jm) -m

. ejkXxp+jkyYp dk, dk,, (28)

where

-jZo ,6*k,(€,.- 1) sinkld +jk,kl T, sin k l d 9" =-

4n2 ko -.

Tc Tm k l

(29)

Note that (21) and (22) consist of six integrations; however, the integrals for F , and F y can be done in closed form leaving only two numerical integrations. Equation (28) similarly re- quires two numerical integrations. As mentioned previously, the infinite k, , k , integrations can be facilitated by convert- ing to polar coordinates. Also, the even and odd properties of the integrand can be used to reduce the ru = 0 -+ 2n integration range t o Q! = 0 -+ n/2. Equations (21), (22), and (28) can then be written as

zm n xx =4 d*'2 la Q, Re [F,(J,)F,*(J,)I

- Re[F,(J,)F,*(Jm)IPd,6dQ! (30) m

z m n = -4 ['2 1 12, Im [ F, (J,)F, * (Jm

' Im [Fy(Jn)Fy*(Jm)IPdP dol (3 1)

Re [ Fy*(Jm)eikxYp]/3 do dru. (32)

Equations (30)-(32) are the final forms used for the computa- tion of the impedance matrix and voltage vector elements.

Feed Modeling A coaxial or probe-type feed can be modeled by using (27)

to represent a unit current source at the probe position ( x p , y p ) , and accounting for the probe self-inductance by adding j xp to the input impedance, where [ 1 ]

Z O xp =- (33)

A microstrip-type feed can also be modeled by using (27) to represent an equivalent i current source at the point where the feed line joins the antenna patch. Strictly speaking, the width of the feed line should be incorporated by using a rib- bon of current of width equal to that of the feed line, but it has been found that the line element of (27) gives an identical result for the narrow feed lines that are in common use. It has also been found [9 ] that the voltage term (32) should be mod- ified by the factor - to account for edge effects of the microstrip line. If W is the width of the feed line, the effective width W e is [ 1 ]

W e = W + 0.412 ( )( ) d , E , + 0.30 W + 0 . 2 6 2

e, - 0.258 W + 0.813d

where ee is the effective dielectric constant [ 1 1 , E , + 1 q.- 1 ..=-+-(l+?) - 112 .

2 2 (34)

Port Impedances

Consider the two-element microstrip antenna geometry shown in Fig. 2(a). The relation between the port voltages and currents (defined in Fig, 2(b)) is

[ PI = [ m [ p l , (3 5) where the superscript "p" is used to differentiate between these "port" quantities and the moment method quantities of (18). This distinction is an important one and necessary to avoid confusion and erroneous results. That these two sets of quantities are distinct can easily be seen by noting that [Z] of (18) can be of order N if N expansion modes are used on the two antenna elements, while [ZP] of (35) is always a 2 X 2 matrix (for a two-port geometry). The "port" impedance matrix [ Z p ] is the one of interest for determining input im- pedance and mutual coupling, so its relation to (18) is now described.

Z1 l p is the input impedance of element one with element two open-circuited, and can be written as

where 3') is the total electric fieltfrom the N expansion modes caused by the source current Ji ( ' ) at port one, and Ii is the terminal current of the source (1 A in this case). The use of (20) then gives

N z11p = - x 1, & ( I ) , (3 7)

n= 1

1194 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 6 , NOVEMBER 1982

where In are the expansion mode currents found from (18), and Y,(l) are the voltages due to source 1. Note that all I , for expansion modes on element 2 should be set to zero, since element 2 is open-circuited. Equation (37) also gives the input impedance of a single antenna element, and can be used for Z by substituting V,(’), the voltage due to source 2, for V,(h’and setting all I , on element 1 to zero.

The open-circuit transfer impedance between ports 1 and 2 can be written as

where E ( 2 ) is the total electric field caused by source current J”(2) at port 2. Using (20) gives

N

n= 1

where V,(’) is the voltage due to source 2, and all I , on ele- ment 2 are set to zero. Of course, ZIZP = Z2 l P .

Expansion Modes

Both EB and PWS expansion modes were used in this solution. The current density for the EB modes with current in the i-direction can be written as J(x, y ) = Jx(xvyb), where

and

for y’ -w/2 < y < yf -I- wj2, and x‘, y f are the coordinates of the center of the mode and a and w are the half-length and width of the mode, respectively. This mode is constant in the y-direction and has a sinusoidal variation in the x-direction, with zero current at the ends of the mode and continuous derivatives over the range of the mode. The constant m in (40a) determines the order (number of variations in x ) of the expansion mode, and assumes integer values starting at m = 1. Fig. 3(a) shows EB modes of order m = 1 and 2 arranged on a microstrip antenna element. Often symmetry arguments can be used to eliminate certain expansion modes, such as all odd modes or all even modes. As the number of expansion modes increases, the solution should converge and accuracy increase.

The EB modes have the useful property that many combi- nations of modes on the same antenna element are uncoupled (Zmn = 0), thus making evalution of the impedance matrix faster. For example, (30), (31), (25), and (40) can be used t o show the coupling and uncoupling of impedance elements as shown in Table 1.

In Table I mt and me are the values of m for the test mode and expansion mode, respectively, and Z,, is the impedance between two modes with current in the same direction, while Z,, is the impedance between two modes with current in orthogonal directions. This data are only valid for modes on the same antenna element (identical x’ , y’ for each mode); modes on different elements are not generally uncoupled.

(3) Fig. 3. (a) Example of layout of two EB expansion modes on an an-

tenna patch. @) Example of layout of three PWS modes on an an- tenna patch.

TABLE 1 COUPLING OF EB IMPEDANCE MATRIX ELEMENTS

mt me I zxx Z X V

1 1

2 0 = O

2 = O = O = O = O

2 # O # O

The current density for a PWS mode centered at X I , y ‘ is

J x (x ) = ( 4 0 ~ )

for X’ - a < x <xf -I- a, and a y-dependence as given by (40b). This mode is constant in y , piecewise sinusoidal in x , with zero current at the ends of the mode and a discontinuous derivative at x = x‘. Fig. 3(b) shows three PWS modes arranged on an antenna element. Accuracy is improved by increasing the number of overlapping PWS modes. In contrast to the EB modes, no uncoupled PWS modes exist, however symmetry be- tween PWS modes often reduce the number of matrix ele- ments which need to be calculated. For example, in Fig. 3(b) symmetry gives Z l l = ZZ2 = 223, and Z 1 2 = 2 ’ 3 . Another advantage comes from the fact that a voltage vector element V , is negligibly small unless the feed position lies within the bounds of mode M. Table I1 summarizes the computational cost of using EB or PWS modes for some typical mode layouts for a calculation of input impedance of a single antenna ele- ment.

In this table, a unit “cost” is defined as the effort required for evaluation of a double integration like that in (30H32) . Also, Table I1 assumes that the EB modes are used sequentially (no modes removed by symmetry considerations), and N,, N , refer to the number of expansion modes in the x and y-direct- tions, respectively. The wavenumber k for the PWS mode in (40c) can be chosen arbitrarily, however, a judicious choice will improve convergence. It has been found that setting k

s i n k ( a - l x - x ’ I )

sin ka

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 6 , NOVEMBER 1982

TABLE I1 COMPARISON OF COMPUTATIONAL COST FOR EB AND

PI'S MODES

Quantity cost using cost using Layout Calculated EB Modes PWS Modes

N , = 1 N y = 0

total 2 2 l l -

N x = 2 1 Z 2 1 N,, = 0 V 2 2

7 total

7 4 total 3 1 N y = O V 4 3 Z 4 3

Z 2 3 N y = 1 V 2 2

total 4 5

N x = 3 1 ",=lI

equal to the "effective" wavenumber for the substrate is a good choice:

k = k,=& (41) where e, is defined by (34). This choice for k was also made in [61.

Neither the EB or PWS modes included an edge condition; this simplification was justified on the basis of previous surface patch modeling work [ 1 11, in which it was discovered that, if sophisticated expansion modes are used (e.g., PWS or EB), the edge condition does not need to be enforced for accurate results.

111. RESULTS

In this section numerical results using the above-described method are compared with measurements and other calcula- tions of input impedance and mutual coupling for microstrip antennas.

Fig. 4 compares calculated input impedance of an edge-fed microstrip antenna with measurements from [ 12 J . Four sets of calculations are shown here, using both one and two expan- sion modes for both EB and PWS modes. As can be seen, agreement with measurements is very good for all cases, and improves as the number of modes used increases from one to two. Also, it can be seen that the EB results are generally closer to the measured values than the PWS results, for the same number of modes. The tradeoff here is that, as Table I1 shows, the PWS calculation is faster for N > 1.

Fig. 5 shows calculated input impedance of a coax-fed antenna using one EB mode compared with calculations and measurements from [ 6 ] . The present calculations compare well with the measured data and, for frequencies above reso- nance, appear to be more accurate than the calculations from [61. This is probably due to the fact that the method of [ 6 ] suffered from numerical sensitivities and an approximate treat- ment of the dielectric slab. (Note: The measured data point at f = 639 MHz appears to be incorrectly labeled as 640 MHz in [ 6 1 J

Fig. 6 shows calculated input impedance of a coax-fed antenna using one EB mode compared with measurements from Lo [ 131. Again the agreement is good.

Fig. 7 gives design data for the resonant resistance of a coax-fed microstrip antenna versus feed position and width of the patch. These data show the radiation resistance decreasing

1195

4.02 ~ 4 7 - c m -

0 Measured 0 C a l c u l a t e d ( I EB M o d e ) x C a l c u l a t e d ( I P W S Mode) 0 C a l c u l a t e d ( P P W S Modes) A C a l c u l a t e d ( 2 E8 M o d e s )

Fig. 4. Measured and calculated input impedance of an edge-fed micro- strip antenna. Calculations made using one and two EB and PWS modes.

0 M e a s u r e d [ 6 1

X C a l c u l a t e d C61 6cm 0 C a l c u l a t e d

Fig. 5 . Measured and calculated input impedance of a coax-fed micro- strip antenna compared with calculations from [6 ] .

1196 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 6 , NOVEMBER 1982

0 Measured ( L o )

0 C a l c u l o t e d

Fig. 6 . Measured and calculated input impedance of a coax-fed micro- strip antenna.

4 w = 6 . 9 8 c r n

._ W

a 400 c 0 ._ + 0 0 200 0 [L

0 0 0.1 0.2 0.3 0.4 0.5

x O I L

Fig. 7. Radiation resistance for a coax-fed microstrip antenna versus feed position and width. L = 13.97 cm, E? = 2.60, d = 0.1588 cm, tan 5 = .002.

as the feed approahces the center of the patch, and increasing as the width is made larger. The resonant frequency is in- dependent of feed position, but decreases from 666 MHz for W = 6.98 cm to 659 MHz for W = 20.45 cm.

Fig. 8 shows the calculation of mutual c o u p h g (SI 2 p a a m - eter) between two coax-fed microstrip antenna. . iompaed with measurements by Carver [ 141. Both g-plane and i?-plane couplings are shown. The E-plane coupling is larger due to a stronger surface wave being excited for this case. The good agreement demonstrates the accuracy and versatility of the method.

It should be noted that, due to the very narrow bandwidth of microstrip antennas, the good agreement between measured and calculated input impedance is somewhat fortuitous-the percentage tolerance on the substrate permittivity and patch size can be of the same order of magnitude as the antenna bandwidth (a few percent is typical). This error causes a shift ih resonant frequency, while the impedance level is unaffected. Thus, in Figs. 4-6, the substrate permittivity is nominally 2.55, but values of 2.55 t o 2.59 were used in the calculations to obtain best results.

0

- 10

m N - 2 0 0

- N - v) -

- 30

- 4 0

E - P l a n e H - P l a n e

D m Measured ( C a r v e r ) - C a l c u l a t e d

E - P l o n e

I I I I 0 0.25 0.50 0.75 1.00 1.25

SIXo Fig. 8. Measured and calculated mKtual coupling between two coax-

fed microstrip antennas, for both E-plane and H-plane coupling. W = 10.57 cm,L = 6.55 cm,d = 0.1588 cm, E ? = 2.55,fz 1410. MHz.

IV. CONCLUSION

This paper has presented a moment method solution for microstrip antennas using the rigorous grounded dielectric slab Green’s function. Both entire base and piecewise sinusoidal modes have been used, and their relative advantages and disad- vantages noted. Good agreement between calculated and meas- ured values of input impedance and mutual coupling has been shown.

REFERENCES K. R. Carver and J. W. Mink, “Microstrip antenna technology,” IEEE Trans. Antennas Propagat. ~ vol. AP-29, pp. 2-24, Jan. 1981. R. J . Mailloux, J. F. McIlvenna, and N. P. Kernweis, “Microstrip array technology,” IEEE Trans. Antennas Propagat., vol. AP-29, pp. 25-37, Jan. 1981. A. G. Derneryd, “Linearly polarized microstrip antennas,” IEEE Trans. Antennas Propagat., vol. AP-24, pp. 846-851, Nov. 1976. Y. T. Lo, D. Solomon, and W . F. Richards, “Theory and experi- ment on microstrip antennas,” IEEE Trans. Antennas Propagar., vol. AP-27, pp. 137-145, Mar. 1979. K. R. Carver and E. L. Coffey, ”Theoretical investigations of the microstrip antenna,” Univ. New Mexico Phys. Sci. Lab. Rep. PT-00929, Jan. 1979. E. H. Newman and P. Tulyathan, “Analysis of microstrip antennas using moment methods,“ IEEE Trans. Antennas Propagat.. VOl.

AP-29. pp. 47-53,Jan. 1981. N. K. Uzunoglu, N. G. Alexopoulos, and J. G. Fikioris, “Radia- tion properties of microstrip dipoles,” IEEE Trans. Antennas Propagat. , vol. AP-27, pp. 853-858, Nov. 1979. I . E. Rana and N. G. Alexopoulos, “Current distribution and input impedance of printed dipoles.” IEEE Trans. Antennas Propagat.. vol. AP-29. no. 1 . pp. 99-105. Jan. 1981. N. G . Alexopoulos and I . E. Rana, “Mutual impedance computa- tion between printed dipoles,” IEEE Trans. Antennas Propagat., vol. AP-29, no. 1, pp. 106-1 11, Jan. 1981. M. C. Bailey, “Analysis of the properties of microstrip antennas

AP-S Int. Symp. Digest. Seattle, WA, pp. 373-379. using strips embedded in a grounded dielectric slab,” in IEEE 1979

D. M. Pozar and E. H. Newman, “Analysis of a monopole mounted near or at the edge of a half-plane,” IEEE Trans. Antennas Propa- gar . . vol. AP-29. pp. 488495. May 1981. M. C. Bailey and M. D. Deshpande, “Resonant frequency of rectangular microstrip antennas,” in lEEE I981 AP-S Int. Symp. Digest , Los Angela, CA, pp. 3-6. Y . T. Lo, D. D. Harrison, D. Solomon, G. A. Deschamps, and F. R . Ore, “Study of microstrip antennas, microstrip phased arrays, and microstrip feed networks,” RADC Tech. Rep. TR-77-406, Oct. 21. 1977. R. P. Jedlicka, M. T. Poe, and K. R. Carver, “Measured mutual coupling between microstrip antennas.“ IEEE Trans. Antennas Propagat. , vol. AP-29, pp. 147-149, Jan. 1981.

David hl. Pozar (S‘74-M’80), for a photograph and biography please see page 350 of the May 1982 issue of this TRANSACTIONS.