Mutual coupling between rectangular microstrip patch antennas

Gi,(hl) of the transverse scalar potential V for a grounded single-layered substrate. The method is based on the separation of a the Green's function into a regular part and a singular part in the spectral domain as well as in the space domain. The corresponding Green's functions are deduced in the spectral domain and calculated for the space domain by means of Bessel transformation. The quadrature of the Sommerfeld integral is discussed in detail and the results of numerical computation are given. The static approximation of the scalar Green's function GI, in the near field is shown to b e only valid in a limited range that strongly depends on the choosen parameters.

ABSTRACT A comprehensive study o f rhe mutual coupling between two recran-

giilar rnicroslrip parch antennas is presented. Using rhe cavity model. nirrnerical results f o r both murual impedance and mutual coupling paramerer are given f o r the E-plane, H-plane, diagonal, and perpendicular orientarions. T h e effects of substrare thickness. sirbstrute permirtivitv, and feed positions are discussed. ce 1991 Johti M ' I k \ R. Som.

IIlC.

1. INTRODUCTION

Mutual coupling between microstrip dipoles and microstrip patch antennas has received considerable theoretical and experimental investigation [l-41. For microstrip dipoles, theoretical results on mutual impedance for a number of configurations (broadside. collinear, and echelon) were given. In the case of rectangular microstrip patches, extensive studies were carried out for the E- and H-plane configurations shown in Figures l ( a ) and l(b), with little attention paid to other orientations. Moreover, results for the E- and H-plane couplings were given in terms of the mutual coupling parameter, C,, = 20 loglS21/, with no information on mutual impedance ( Z 2 *values. )

ACKNOWLEDGMENT

The authors wish to thank the Deutsche Forschungsgemeinschaft ( D F G ) for its financial support of this work. REFERENCES R. F Harrington. Time-Harnionic Electromagnetic Fields, McGraw-Hill. New York. 1961. A . Sommerfield. Prirtielle Differeritial~leiciiungen der Physik. Akadeniischc Verlagsge~ellschaftGeest & Portig K. G . . Leipzig. 1947. K . A . Michalski. "On the Scalar Potential of a Point Charge Associated with ii Time-Harmonic Dipole in a Layered Medium." l E E E TrnnJ. Antennas Propagat. Vol. AP-35. N o . 11. 1987. pp. 1299-1301. Juan R. Mosig. "Integral Equation Technique." in Tatsuo Itoh. Ed.. ,Yurnerical Trchniqirrs f o r Microwai,e and Millirnerer- W a w P a n i w Srt-uctitres. Wile!. New York. 19x0. A . Ertcza and B. K. Park. "Nonuniqueness of the Resolution of Hertz Vector in Presence of a Boundary. and the Horizontal Dipole Problem." IEEE Trans. A n r m n a s Propagat.. Vol. AP-17. 1969. pp, 376-37x. J . A . Kong. Electrutnognerrc W a w Tllror!.. Wiley. New York. 1986.

J . R . Mosig and F. E. Gardiol. "Analytical and Numerical Technique3 in the Green's Function Treatment of Microstrip Antennas and Scatterers." I E E Proc .. P I H : Xlicrowve.~.Opr. Antennrts. Vol. 130. 1983. pp. 175-182

Receii.ed 4-73-92 Microwave and Optical Technology Letters. 5 : 11. 566-572 C' 1992 John Wile). & Sons. Inc. CCC 0595-7477 9z/$4.00

MUTUAL COUPLING BETWEEN RECTANGULAR MICROSTRIP PATCH ANTENNAS

(C)

Tan Huynh Decibel Products Dallas Texas 75247 Kai-Fong Lee and Siva Rao Chebolu Department of Electrical Engineering The University of Toledo Toledo Ohio 43606

on U

R. Q. Lee NASA Lewis Research Center Cleveland Ohio 441 35

KEY TERMS Parch anrcnna. muru(i1 coupling, micro7rrp

572

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS

(d)

Figure 1 ( a ) €-plane (radiating edge) coupling. (b) H-plane (nonradiating edge) coupling. (c) A triangular-grid array. (d) A CP subarray consisting of four sequentially rotated linearly polarized patches

Vol 5 No 1 1 , October 1992

While the E- and H-plane couplings are the dominant COUpling for rectangular grid arrays, the coupling between two diagonally oriented patches is also of interest, particularly in arrays with triangular grid arrangements. Similarly, in the design of circularly polarized subarrays utilizing four sequentially rotated linearly polarized patches [Figure l(d)], the mutual coupling between two perpendicular patches is of interest. In this communication, the main objective is to present the calculated mutual coupling results for the diagonal and perpendicular orientations and to compare them with the Eand H-plane orientations. In addition to the coupling parameter C,,, the mutual impedance Zzl = Rz, + jXzl is also given for a number of cases. In Section 11, the formulas for the calculation of mutual impedance and mutual coupling parameter based on the cavity model are presented. Numerical results obtained under the condition that the antenna is matched to the transmission line are given in Section 111. The effect of mismatch is discussed in Section IV. II. FORMULATION

The geometry of two arbitrarily oriented rectangular patches is shown in Figure 2. The two patches are identical: of dimensions a and b and fabricated on a substrate of thickness I . relative permittivity er, and loss tangent tan 6. The coaxial feeds are located at (xi, y i ) and ( x i , y i ) , respectively. We assume that the thickness is much less than wavelength so that coupling occurs mainly through space wave. Using the cavity model the formulas for the mutual impedance calculation are given below:

From Reference [5]. we have

X

sin(nz 7rA/2a) (m 77A/2a) i x i?,

(2)

where i

ab Eop =

i

mrx nry cos -, a b

1, 2,

(3)

p = 0, pf0, (4)

and A is the effective width of the current ribbon modeling the feed. The magnetic field H(')is calculated using the formula

where

The transmission coefficient SZlis given by where I , and I z are the currents at the feeds of the patches. H") is the magnetic field on antenna No. 2 due to antenna No. 1. M'?) is the linear magnetic current density on antenna No. 2 when it is self-excited. Integration is over the perimeter of antenna No. 2 .

where Zo is the characteristic impedance of the transmission lines and Z,, is the input impedance of each antenna with the other open circuited. In this article we assume that Zll is not appreciably different from the input impedance of the isolated patch. The mutual coupling parameter C, is defined as

c,

=

20 log

/&,I.

(9)

111. NUMERICAL RESULTS

I

(x;,Y;)

Ftgure 2 Geometry of two arbitrarily oriented rectangular patch

antennas

We have used the formulas of the previous section to compute the mutual impedance Z?, and the mutual coupling parameter C,, for two rectangular patches with the following parameters: a = 4.5 cm, b = 3.0 cm, er = 2.32, tan 6 = 0.001, I = 20 mil. The feed points are located at x i = x i = 1.7 cm, y ; = y i = 0.87 cm. For this feed position, the isolated input impedance for both the TMnl and the TMlo modes is approximately 50 R, which is the characteristic impedance of the transmission lines connecting the antennas. When the antennas are excited at the resonant frequency of the TMol mode, the mutual impedances as a function of spacing (in units of free-space wavelength A") are shown in Figures 3(a)-3(d) for the four orientations indicated. Figure 4 shows the coupling parameter C, for the four cases. As expected, E-plane coupling is the strongest and coupling between perpendicularly oriented patches is the weakest. The curves for the diagonal configuration and the H-plane configuration cross at dlh,, = 0.5.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 5, No. 11, October 1992

573

10

I

I

I

I

5

I

I

I

I

-

’ h J a I

I

I

0.5

1.0

1.5

-20 0

1

2.0

2.5

-15‘ 0

I

I

I

0.5

1.0

1.5

d/>

TM0 1 I

2.0

2.5

d/Ao

(a) E plane oriented

( b ) H plane oriented

1.o 0.5

-

i-:

0

1

I

1

0.5

1.0

1.5

I

2.0

J

0

-0.5

2.5

0

0.5

1.0

1.5

2.0

2.5

d /?,

0

Diagonally oriented Figure 3 Mutual impedance Z - , = R., + j X 2 , kersus spacing for t (C)

h = 3 0 cm.

E. =

2 32.

I =

~ rectangular o patches excited in the TM,,, mode a = 4 5 cm,

10 mil

Figure 5 compares the coupling parameter for two values of substrate thicknesses and for two values of substrate relative permittivity for the TMOIradiating edge coupling. It is seen that coupling via space wave increases with substrate thickness. Except for a narrow range of diA,,, the coupling decreases as increases. (For these examples. tih,, = 0.005 for = 2.32 and tih,, 0.003 for E , = 9.8 so that surface wave effects are negligible. Figure 6 shows the coupling parameter when the antennas are excited in the TMI,,mode. In this case, the H-plane coupling curve interesects both the E-plane curve and the curve for two diagonally oriented patches, at dlho = 0.7 and dih,, = 1.0. respectively. For brevity, the mutual impedance curves are omitted. 2-

IV. EFFECT OF MISMATCH

Although the antennas should be matched to the transmissionline characteristic impedance, some mismatch may occur in practice. Moreover. some authors report the coupling param-

574

( d ) Perpendicularly oriented

eter under mismatched conditions (e.g., Reference [3]). It is therefore of interest to see how a mismatch affects the mutual coupling. For this purpose, we have chosen three feed positions to illustrate the effect. These are: (a) x‘ = 1.7 cm, y ’ = 0.87 cm; (b) x ’ = 2.0 cm, y ’ = 1.1 cm, and (c) x ’ = 0. y ’ = 0. The isolated input impedances for these feed positions are: (a) 50 R (TMol and TMlllmodes); (b) 22 R (TM,,,) and 12 R (TMLI,);(c) 135 R (TM,,) and 376 R (TMlo). The nonradiating edge coupling parameters for the TMol mode are shown in Figure 7. It is seen that the coupling for the TMol mode is not very sensitive to feed position. The differences among the three cases are within 3 dB. O n the other hand, we find that the nonradiating edge coupling for the TMlo mode varies by about 8 dB as the feed moves from - 1.7 cm, y’ = 0.87 cm to x ’ = 0, y’ = 0. This is mainly because the input impedance of the TMlo mode when x ‘ = 0. y ’ = 0 is much higher than that of the TMol mode. Similar results are obtained for the case of radiating edge coupling and for other configurations. For brevity, we d o not display the results here.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol 5,No 1 1 , October 1992

-10

I

TMol

I

I

1

0

1

I

I

I

1

t

MODE

I4 -20

m

T 3

- -30-

- -30

d

d N

N

cn rn

I

D

0

-40-

0

-40

rl 0 N

0 N

II

-50-

II

V

a-50

.'

Va

-60

-70

-80' d/ho

I

0.5

0

I

I

1.5

1.0

I

2.0

I

2.5

d/i

Figure 4 Mutual coupling parameter versus spacing for two rectangular patches excited in the TMoI mode. a = 4.5 cm, b = 3.0 cm, c, = 2.32, t = 20 mil. (a) E-plane orientation (solid line), (b) diagonal orientation (dashed line). (c) H-plane orientation (crosses), (d) perpendicular orientation (dot-dashed line)

Figure 6 Coupling parameter of two rectangular patches excited in the TMlo mode. a = 4.5 cm, b = 3.0 cm, e, = 2.32, t = 20 mil. (a) E-plane orientation (solid line), (b) diagonal orientation (dashed line), (c) H-plane orientation (crosses). (d) perpendicular orientation (dot-dashed line)

-15 TMOl NODE -15 f' -20

-

-20

-25

cn m

-25

4

rn - 3 0

N

0

4

5

w- - 3 5 m rl

-30

0 N

0

rl

-40

I1

0

a

N

-35

I1

a-45

U

-50

.\

-40

-55

- Y Y

0

0

0.5

1.0

1.5

2.0

2.5

0.5

1.0

1.5

2.0

2.5

aho

d/ho

Figure 5 Comparison of the coupling parameter of two E-plane oriented rectangular patches excited in the TMol mode. a = 4.5 cm, b = 3.0 cm. (a) t = 20 mil, cr = 2.32. (b) t = 10 mil, c, = 2.32, (c) I = 20 mil, c, = 9.8

Figure 7 Illustrating the effects of mismatch on the coupling parameter of two H-plane oriented rectangular patches excited in the TMol mode. a = 4.5 cm, b = 3.0 cm, c, = 2.32. f = 20 mil. (a) x' = 1.7 cm, y ' = 0.87 cm. Zll = 50 R (matched case). (b) x ' = 2.0 cm, y ' = 1.1 cm, Z , , = 22 R . (c) x' = 0, y' = 0, Z , , = 135 R

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS I Vol. 5, No. 11, October 1992

575

V. CONCLUDING REMARKS

1. INTRODUCTION

In this article a comprehensive study of the mutual coupling between two rectangular microstrip patch antennas has been presented. Using the cavity model, numerical values for both mutual impedance and mutual coupling parameters are given for the E- and H-plane. diagonal, and perpendicular orientations. The dependences on substrate thickness and substrate permittivity, as well as feed position, are also discussed.

Shown in Figure 1 is the structure of a general asymmetric dielectric-slab waveguide, and the structures of step discontinuities in dielectric-slab and dielectric-loaded waveguides are shown in Figures 2(a) and 2(b). respectively, assuming that the incident wave contains the dominant TE mode only. The power components transmitted and lost at the discontinuity in dielectric-slab waveguide in Figure 2(a) can be eval-

ACKNOWLEDGMENT

This research is partially supported bv an Ohio Supercomputer Center Start-up Grant. REFERENCES 1. N . G. Alcxopoulos and I . E. Rana. "Mutual Impedance Computation between Printed Dipoles." I E E E Trans. Atitentias PropU R ( I / . . Vol. AP-2Y. NO 1. pp. 1981. 106-1 11. 2. R. P. Jsdlicka. M. T Poe. and K. R . Carver. "Measured Mutual Coupling abetween Microstrip Antennas." l E E E Truns. Anreririas Propugur.. Vol. AP-29. No. 1. 1981. pp. 147-149. 3. E. Penard and J . P. Daniel. "Mutual Coupling between Microstrip Antennas." Electron. Lrtt.. Vol. 18. 1982. pp. 605-607. 4. D . M. Pozar. "Input Impedance and Mutual Coupling of Rectangular Microstrip Antennas." I E E E Trafls. Afireririas Propagur., Vol. AP-30. 1982. pp. llYl-llY6, 5 . W. F. Richards. Y. T. Lo, and D . Harrison. "An Improved Theory for Microstrip Antennas and Application." I E E E T r a m . Ati1etirui.s Propugut.. Vol. AP-29. 1981. pp. 38-46.

i n52

Figure 1 The structure of a general asymmetric slab dielectric waveguide ( a i i t y = 0)

Microaave and Optical Technology Letters. 5 ' 1 1. 572-576 L 1992 John Wile) K Sons. Inc ccc 0895-2377 911$1 00

Y

kz

ANALYSIS OF STEP DISCONTINUITIES IN DIELECTRIC-SLAB AND DIELECTRIC-LOADED WAVEGUIDES USING GEOMETRICOPTICS APPROACH Cheng-Cheh Yu

Electronic Engineering Department National Taipei Institute of Technology Taipei Taiwan Tah-Hsiung Chu

Electrical Engineering Department National Taiwan University Taipei Talwan KEY TERMS Step disconrittuiry, dielectric-slab n.uwguide, dielecrric-loaded n'ai'eguide. geomerric opricr ABSTRACT 111 / h i 5 article colculatiori~Nf'pa!.ver components rransmir/ed und reflected rir rhe siep disconririuiries in dielecrric-slab and dieleciricloaded ~ m ' e g i t i d e sare presetired using the geometric-oprics approach. Srep discontinuities regarding rhe dielectric slab rhickness arid refracrive irides are corisidered. Nunierical results are shown 10 be bi good agreement wirh [hose of orher rigorous rnerhod5. T h e computation involved in rhis approach is eficienr since the associared rnnrheniurical forniularion can be arialyrically expressed f r o m the geomelric oprics poinr of r'iew. 6 1992 John Wile! & Sons. Iric.

576

(b) Figure 2 Step discontinuities in (a) dielectric-slab waveguide and (b) dielectric-loaded waveguide

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol 5 No 1 1 , October 1992