acked circular

F. Zavos h J .T. A be r I e

Indexing terms: Cavity-backed microstrip patch an tenm, Shorting posts, Green k functions, Vector Hankel transform

Abstract: The paper presents a theoretical investigation of post-loading techniques as applied to cavity-backed circular microstrip patch antennas. The effects of the number and location of shorting posts on the polarisation and operating frequency of the antenna are analysed. A number of practical designs using shorting-post loading are also presented.

1 Introduction

The microstrip patch antenna has been shown to be a practical radiator for many applications that require only a narrow bandwidth. However, in certain applica- tions, the linear polarisation of a microstrip antenna is a limitation. There are a variety of techniques for achieving circular polarisation with microstrip anten- nas, most of which involve permanent physical changes to the antenna. As a result, they are not amenable to electronic control of the antenna’s performance. Other techniques, such as the use of varactor diodes or varia- ble length transmission lines, add additional design complications, and require space outside the basic microstrip antenna’s boundaries. One simple approach at achieving polarisation agility is the use of shorting posts (such as switching diodes) at appropriate points within the microstrip boundary [l]. By choosing the appropriate number and locations of the shorting posts, greater polarisation control can be achieved. In this paper, the concept is applied to cavity-backed cir- cular patch antennas.

Another limitation of microstrip patch antennas is operation at UHF and lower frequencies. To achieve lower operating frequency requires prohibitively large patch sizes, making the antenna impractical for many aerospace and mobile communication applications. One common solution to this problem is to use a high dielectric permittivity substrate. However, a higher die- lectric constant confines most of the fields within the substrate, resulting in an inefficient radiator. In addi- tion, since the higher dielectric constant material has higher density, this results in greater weight which is usually undesirable. Shorting posts can also be used to

~ ~ _ _ _

0 IEE, 1996 IEE Proceedings online no. 19960036 Paper first received 22nd March 1995 and in revised form 15th August 1995 The authors are with the Telecommunication Research Center, Anzona State University, Tempe, AZ 85287-7206, USA

achieve greater frequency agility, without greatly affect- ing the bandwidth and other characteristic of the antenna [l, 21. This is primarily achieved by exciting modes which have a substantially lower resonant fre- quency than the conventional patch (TM,,) mode. The- oretical results of this design are also presented here.

Y

X

circular patch

shorting pins Cavity-backed probeyefed circlar patch with two shorting posts Fig. 1

MS > Fig. 2 Phr’cal equivalence model of a post-loaded, probe-fed cavity- backed pate antenna

2 Theory

For the antenna geometry shown in Fig. I , an integral equation (IE) solution, based on Green’s function/ Galerkin’s technique, yields the most computationally efficient solution. To obtain the IE model of the cavity- backed microstrip patch antennas, it is necessary to obtain the dyadic Green functions for the structure. In the analysis presented here, the Vector Hankel Trans- form (VHT) [3-51 is utilised to greatly simplify the der- ivation of Green’s functions. Furthermore, through the use of the physical-equivalence principle, the aperture in the problem is closed off, and replaced with equiva- lent magnetic surface currents just above and below the

IEE Proc -Mccrow Antennas Pvopag , Vol 143, No 1, February 1996 18

aperture, as shown in Fig. 2. This effectively divides the problem into two separate but coupled regions, namely, the exterior (semi-infinite free space) region and the interior (cavity) region. The physical-equiva- lence principle is also used to replace the probe feed and the shorting posts within the cavity with equivalent electric surface currents (see Fig. 2). Integral equations for the equivalent currents arise from enforcing the boundary conditions that the tangential magnetic field must be continuous across the aperture, and the tan- gential electric field must vanish on the probe feed and the shorting posts. In order to obtain these integral equations, the dyadic Green functions for the structure, as well as an appropriate set of basis functions for the unknown magnetic and electric surface currents are required.

2.1 External-region Green functions In the exterior region of the structure (Q,), the only equivalent source is the tangential magnetic current in the aperture. Thus only four components of the dyadic Green function for the magnetic field need to be evalu- ated. These components are, GAM, GAM, GGM and

Through the use of the VHT, the spectral domain form of Green’s functions for the tangential magnetic field, due to an arbitrarily directed tangential magnetic current can be obtained as:

G h M .

. 27T { ~ ( P P o ) [ ;] .Ip6.) P dP (2)

Eqns. 1 and 2 are Green’s functions for the tangential magnetic field at z due to infinitesimal p- and=@- directed magnetic currents at zo, respectively, and GHM (p, zlzo) is given in Appendix 7.1. Since only the mag- netic fields within the aperture are required, the above Green functions only need to be evaluated at z = 0.

2.2 Internal-region Green functions The procedure discussed in the previous Section must be repeated for the internal regions (Q,) of the antenna structure. However, due to the presence of the probe feeds, additional equivalent vertical electric-current sources contribute to the fields in the cavity. As a result, four components of the dyadic Green function for the magnetic field and three components of the dyadic Green function for the electric field must be evaluated.

Since the cavity is finite in the p-direction, the stand- ard VHT cannot be applied in solving this problem. A transform is used here which takes into account the discretisation of the solution in the p-direction. Such a transform has an infinite-sum dependence for both the @ and p variation of the solution, and can be viewed as a discrete vector Hankel transform (DVHT) [3].

Using the DVHT along with the appropriate bound-

IEE Proc.-Microw. Antennas Propag., Vol. 143, No. I , February 1996

ary conditions, we can solve for the desired internal- region Green functions, yielding:

M M

p = - m q=o

. j E J ( p , ~1.0) . Jp(Ppo)e-j”0

Eqns. 3 and 4 are Green functions for the tangential magnetic field at z due to infinitesimal p- and @- directed magnetic currents at zo, respectively. Eqn. 5 is the Green function for the z-directed electric field at z due to infinitesimal p- and @-directed magnetic currents at zo. Eqns. 6 and 7 are the Green function for the tan- gential magnetic field at z and the z-directed electric field due to an incnitesimal z-directed electric current at zo. g(Pp, p’p), H T ( P p , p’p), BHM, gEM, g””, and gEJ are given in Appendix 7.1.

2.3 Basis functions Since the matrix elements in the MoM/Galerkin’s for- mulation are to be evaluated in the spectral domain, and owing to the regular geometry of the structure of interest, entire-domain basis functions are best suited for this analysis.

A suitable set of vector basis functions for expanding the unknown magnetic current in the annular aperture of the antenna may be derived from the TEM, TM and TE eigenmodes of a coaxial waveguide. This basis set, defined at z = 0 for RI 5 p 2 R2, can be represented by

for TE modes with even symmetry, and

M , =YA(PmnRi) 6 P m n J A ( P m n ~ ) sinm4

+4;Jm(DmnP) cosm4 I [

- Jh(PmnR1) [i;PmnYA(PmnP) sinm4

+4;Ym(PmnP) cosm41 (12)

-

for TE modes with odd symmetry, where a,, and Pmn are the nth solutions to

and Jm(afi~)Ym(afi~) - Jm(C&)ym(aR~) = 0 (13)

Jk(aR1)YA(aRz) - J~(cY.Rz)YA(o!RI) 10 (14) Both the VHT and the DVHT of the above basis set

can be evaluated in closed form, adding to the overall efficiency of the formulation.

In the analysis presented here, the cavity height (c) is assumed to always be c/3Lo 2 0.08. Therefore a single- pulse expansion mode will be sufficient to model the probe feed and the shorting posts. In addition, assum- ing that the radius of the probes are very much less than the free-space wavelength, a filamentary approxi- mation can be made, yielding pulse basis functions of the form - J ( z , y, Z ) = 2 6 ( ~ - ~f)S(y - y f ) -C 5 z 5 0 (15)

where xf and yf are the probe coordinates.

2.4 Integral-equation solution To satisfy the boundary conditions that H,,, must be continuous in the aperture, and that Etan must vanish on the probe feed and the shorting posts, a set of inte- gral equations is developed.

For the scattering case, the integral equations can be written as --n -a2 - Ht&(-%s) -pF:n(zs) +Zy:n(Tsl) + . ' . +H,,n( J s N )

- = Hian + zun in aperture (16)

and -a -a2 - E&(-%,) + Etun(J,i) + . . + Ef:n(Ts~) = 0

on shorting posts (17)

20

and - E 2 (-Zs 1 +z",:, (Tfd ) + ' . . +E?:", ( J S N ) +evL 6(Z +e) = 0

on probe feed (18)

where rfi' is the magnetic field of the incident-plane wave, i? is the magnetic field of the reflected-plane wave that would exist in the absence of the aperture, VL is the voltage drop across the load, and the other quantities are the electric or magnetic fields arising from the specified unknown (&fX or rfl ... rsN) in the specified region (a1, Cl,). The right-hand side of eqn. 16 depends on the polarisation of the incident wave. In general, an arbitrary polarised incident plane wave can be resolved into the vector sum of a TM" (or 'hard' polarisation) component and a TEz (or 'soft' polarisation) component.

For the radiation case, the integral equations can be written as -0 H,,:,(-li?,)-H,,:,(Ms)+~~~(Jsl)+".+~2,",(Tsw)=o -Q -

E,,:(-%,) + Et,,(JSl) + . ' . + z,:n(;rsN) = 0

- Z2(-Zs) - E f a n ( J s I ) - . I ' - E t a n ( J s N ) = zan

in aperture (19) and

-Q 0 2 - (20)

on shorting posts and

-02 - Q2 -

on probe feed (21) where is the incident electric field due to a one-volt delta-gap generator source placed at the base of the probe. Greater accuracy can be achieved using a mag- netic frill feed model. However, the delta-gap model used here is sufficiently accurate and substantially sim- plifies the analysis.

Expanding the unknown magnetic current in the aperture and the electric current on the probe feed and shorting posts into a set of vector basis functions with unknown coefficients, and applying Galerkin's proce- dure, yields a matrix equation which is solved numeri- cally for the unknown coefficients.

For the scattering case, the matrix equation can be written as

["".; +

For the radiation case, the matrix equation is given by

The corresponds to the self and mutual admit- tance terms of the-magnetic-current basis functions in the aperture. The Tg: and T@& are the cross-wupling between electric and magnetic currents, and Z2j are the electric-current self and mutual impedance terms. The Vis are the coefficients of the magnetic current

IEE Proc-Microw. Antennas Propag., Vol. 143, No. I , February 1996

basis functions; Zl ... IN-~s are the unknown coefficients of the electric current on the shorting posts, and ZN is the coefficient of the current on the probe feed. The ejs are the excitation terms. Due to the circular symmetry, the external matrix elements can be simplified into a single integral form, and the internal matrix elements can be simplified into a single summation form. These are given by:

my* =HM

yth,, = 2rcm3 6m,m3 1 F , ( P ) (p,z=oIzo=o) -

* p , (P)P dP (24)

m 9 -20-

' -30- E .

-40-

-50-

U ) .

M

n=O

n = O

sin(-rni+pp,) T M O ~ ~ / T E " " " "

C O S ( T D , ~ ~ ~ ~ ) TMewen /TEodd ( 2 7 )

m=O n=O

. Jm (Ppp, COS m (4% - 43 1 ( 2 8 ) Here, the Fs repesent the VHT (DVHT) of the basis functions, the GHM and CHM are as given in Sections 2.1 - 2.2, and the gs are integrals of the spectral-domain dyadic Green functions given in Appendix 7.2. The above expressions are a numerically efficient represen- tation of the MoM/Galerkin's matrix elements, and yield a computationally efficient numerical model of the antenna.

In order to achieve accurate results for both the scat- tering and radiation analysis of the structures consid- ered here, we used between 40 and 80 entire-domain basis functions to expand the magnetic current in the antenna aperture. The number of basis functions required for convergence is strongly dependent on the dielectric constant of the substrate, the electric size of the aperture, and the proximity of the feed or shorting posts to the edge of the patch. In the examples consid- ered, the number of basis functions are increased, until the coefficient matrix elements are converged to within four significant figures. Since the field variation in 4 is rapidly converging, it was found that three modes in m (4 variation) and nine or ten modes in n (p variation) are adequate for convergence.

Once M, has been determined via eqn. 22, the RCS can be evaluated by removing the ground plane and replacing it with 2M, radiating in free space. Further- more, the input impedance of the antenna can be eval- uated by using eqn. 23 and

zm = 1/IN ( 2 9 )

3 Results

In this Section, theoretical results for scattering and radiation characteristics of post loaded, probe-fed cav-

IEE Proc -Microw Antennas Propag., Vol 143, No 1, February 1996

ity-backed patch antennas are presented. A number of design examples are given to demonstrate the polarisa- tion diversity and frequency agility achieved with this technique.

Table 1: Parameters for the probe fed cavity-backed cir- cular patch antenna with two shorting posts (case 1)

R, = 3.1 cm R2 = a = 4.0cm

xp = -1.6cm yp = 0.80cm fop = 0.065cm

xspl = 2.39cm Ysp1 = 0 rasp, = 0.065crn XSD2 = 0 ySD2 = -2.39cm roso2 = 0.065cm

c = l.Ocm

E, = 2.5 per= 1.0

A conventional circular patch antenna fed with a sin- gle coaxial feed radiates linearly polarised waves. The polarisation can, however, be selectively altered by properly locating shorting posts along the patch. To demonstrate the polarisation agility achieved by using shorting posts, the antenna of Table 1 is considered. The r0s represent the feed probe and post radii, and the other parameters are as shown in Fig. 1.

A'

1.0 2.2 2.4 frequency, GHz

Monostatic RCS of a post-loaded cavity-backed patch against fie- Fig. 3 quency with open circuit termination @ = 00, e = 600 - %e

Omm

"""' o,$e

-1 0 O r

1.9 2.0 2.1 -601 5 J ' v , , , , ,

1.6 1.7 1.8 frequency, GHz

Return loss of a post-loaded cavity-barked patch against f ie- Fig.4 quenry

The four components of the monostatic RCS are shown in Fig. 3 . The antenna RCS exhibits large cross- pol components over the range of 1.6 - 1.9GH.z. The dual resonance nature of the oee component is due to the presence of the shorting posts, and is not normally observed in conventional circular patches. Dual reso- nance can also be observed in the 04@ component, but to a lesser degree. This difference is primarily a func- tion of the shorting-post configuration. Fig. 4 illus-

21

trates the return loss of the antenna near its first resonance. The antenna is conjugate matched to the impedance at the centre frequency (Zmatch = 54 + j41 Q), and exhibits an intrinsic bandwidth of 9.2% for a 2 : 1 VSWR, The input impedance over the same range is shown in Fig. 5. Notice that the dual resonance charac- teristic exhibited in the RCS pattern is also evident in the input impedance. To see if the antenna is radiating circularly polarised radiation near resonance the axial ratio is considered, as shown in Fig. 6. The antenna exhibits circular polarisation over a 2.3% band centred at the antenna resonance, as indicated by an axial ratio of less than 3dB. Fig. 7 illustrates the axial ratio versus 0 in three different planes at the resonant frequency of 1.85GHz. Notice that the antenna maintains an axial ratio of less than 3dB up to 0 = 50" in all planes. This indicates that the antenna radiates circular polarisation over a fairly wide range of angles. Note that through the proper choice of shorting-post locations, a diverse polarisation performance, including right and left circu- lar, and horizontal and vertical linear polarisation can be achieved.

,I j

20 1.6 1.7 1 .8 1.9 2.0 2.1

Input impedence of a post-loaded cavity-backed patch against fre- frequency , GHz

Fig. 5 quency - Rm

x,,

1.6 1.7 1 .8 1.9 2.0 2 .1 frequency.GHz

Fig. 6 Axial ratio of a post-loaded cavity-backedpatch against frequency

An additional flexibility offered by shorting posts is frequency tuning. This is particularly useful in substan- tially lowering the operating frequency of the patch antenna. A relatively simple design for lowering the operating frequency of a circular patch, is to load it with two shorting posts positioned symmetrically about the centre of the patch. The feed position is varied until the best return-loss performance is achieved. As an example, the antenna with parameters given in Table 2 is considered.

Table 2: Parameters for the probe fed cavity-backed cir- cular patch antenna with two shorting posts (case 2)

RI = 3.0crn R2 = a = 4.0cm c = l.Ocm

E,= 1.0 p,= 1.0

xp = 0.18cm yp = 0.18cm rap = 0.065cm

x,,, = 0.18cm Ysp1 = 0 rasp, = 0.065cm

r,,,,, = 0.065cm xsD2 = -0.18cm ysD2 = 0

The monostatic RCS (oQ8) of the antenna versus fre- quency, with and without the shorting posts, is illus- trated in Fig. 8. The antenna without the posts exhibits a resonance at 2.15GHz, which corresponds to the TMI1 mode of the patch. However, the antenna with the shorting posts exhibits an additional narrow-band resonance at 940MHz, which is nearly 44% below the resonant frequency of the TMll mode. Fig. 9 shows the monostatic RCS versus angle at a frequency of 940MHz. There is a null in the RCS pattern at broad- side, indicating that the patch behaves like a top-loaded monopole at this frequency.

, 0 10 20 30 40 50 60 70 80 90

Fig. 7 Q = 0, 45. 90 plane - 8 = 0 "

Axial ratio of a post-loaded cavity-backed patch against angle

Q = 45" ... ...

Q = 90"

-40 ti \ E

: ,R2, , , , , 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

frequency, GHz Monostatic RCS of a post-tuned cavity-backed patch against fre- Fig. 8

quency with open circuit termination Q = O", 8 = 60" - with post

without post

The antenna return loss versus frequency, near the post-tuned resonance mode, is shown in Fig. 10. The antenna is matched to 50Q and exhibits a VSWR of less than 2:1 over a 1.2% bandwidth. The E8 and E@ directivity patterns of the antenna in the 4 = 0 plane are shown in Fig. 11. Notice that the antenna E, pat-

IEE ProcMicrow. Antennas Propag., Vol. 143, No. 1, February 1996 22

tern is similar to that of a top-loaded monopole with a deep null at broadside.

A clearer understanding of the effect of tuning-posts on the resonance of the antenna can be achieved, by studying the effect of the number of posts on the mon- ostatic RCS of the antenna. Fig. 12 illustrates oB8 for the antenna of Table 2 for a varying number of posts positioned along the patch. The substrate permittivity is changed to 2.5, to illustrate the effect more dramati- cally. The posts are positioned symmetrically along $ = 0 in each case, and are spaced equally apart. As indi- cated by the results, the resonant frequency varies con- siderably as the number of posts are varied. There is nearly a shift of 450MHz in the RCS peak from the 2- post to the 6-post case. As a practical design, a centre feed patch, consisting of three or more pairs of switch- ing diodes, can be built to achieve multiple frequency resonance over a wide band. The pairs of switching diodes can be placed symmetrically about the feed, and electronically switched on or off based on the operating frequency desired. The impedance bandwidth at each operating frequency is narrow, but thc range of fre- quencies over which the antenna operates is substan- tial. The disadvantage of this design is that some form of matching circuit would be required, in order to achieve acceptable impedance values at all operating frequencies.

Ill

u -

C 3 - L

c

m

- -10-

-15- 3!

0,deg. Fig.9 angle (open circuit termination)

Monostatic RCS of a post-tuned cavity-backed patch aguinst

I

-20b.90 ' 0.92 ' 0.94 ' 0.96 ' ' 0.98 ' ' 1.00 frequency, GHz

Fig. 10 quency

Return loss of a post-tuned cavity-backed patch against fre-

3.1 Computational time A set of Fortran 77 codes were implemented on an HP 735/125 platform for this analysis. Two different linear system solvers are considered. The first is a full LU solver from the numerical recipes package. The second

IEE Proc-Microw. Antennas Propag., Vol. 143, No. I , February 1996

solver considered is QMRPACK, which is a sparse iter- ative solver, utilising the minimal residual method with matrix preconditioning [6-81, available from Netlib. Due to the orthogonality of the basis set, the overall fill in the problem ranges from 15 to 25%, which results in a relatively sparse linear system. In addition, since the system is diagonally dominant, the iterative solver gen- erally yields a solution in only one iteration. A compar- ison of the computational times for the two solvers for various numbers of unknowns is given in Table 3. As can be seen, when a small number of unknowns are used, the total computational time for the two solvers are nearly equal. This is because the matrix-fill time is much greater than the matrix-solve time. However, as the number of unknowns is increased, the matrix-solve time becomes substantial, and substantial improvement in performance can be achieved by using the iterative solver.

directivity patterns 0

180 Fig. 11 patch - Eo and E$ directivity patterns of a post-tuned cavity-bucked

-40

-50 0.6 0.8 1.0 1.2 1.4

c I

-401 I , I

i i i

; i I

' i : _.#. -. =.=:7.:::--.- .-._ i , ....,, ..,j -50 i ,

.... . . : i

0.6 0.8 1.0 1.2 1.4 frequency,GHz

Fig. 12 quency for varying number of shorting posts $I = O", 8 = 60" - = 2 posts

= 4 posts = 6 posts

Monostatic RCS of a post-tuned cavity-backedpatch against fre-

. . . . . . .

4 Conclusion

A full wave analysis of post-tuned cavity backed circu- lar patches has been presented. The analysis demon-

23

strates the polarisation agility achieved through proper positioning of shorting-posts on the patch. In addition, the results indicate that shorting posts can be used to substantially lower the operating frequency of the microstrip patch. Also, a multiple frequency operation can be attained through the use of multiple pairs of shorting posts positioned symmetricaliy about the cen- tre of the patch.

Table 3: Computational time as a function of unknowns for LU and ~ ~ ~ ~ A ~ K

Number of CPU time (s) unknowns Full LU QMRPACK

10 3 3 50 4 4

100 6 5

250 21 12

500 98 20

This work was supported by the Advanced Helicopter Electromagnetics Industrial Associates Program (AHE), NASA Grant NAG-1-1082, and NASA Grant NAG-1-1630.

eferences

SCHAUBERT, D.H., FARRAR, F.G., SINDORIS, A.R., and HAYES, S.T.: ‘Post-tuned microstrip antennas for frequency- agile polarization-diverse applications’, Technical report, US Army Electronics Research and Development Command, Harry Diamond Laboratories, Adelphi, MD, March 1981 DELAVEAUD, C., LEVEQUE, P., and JECKO, B.: ‘New kind of microstrip antenna: the monopolar wire-patch antenna’, Elec- tron. Lett . , 1994, 30, (l), pp. 1-2 ABERLE, J.T., and ZAVOSH, F.: ‘Analysis of probe-fed circular microstrip patches backed by circular cavities’, Electromagnetics,

ZAVOSH, F., and ABERLE, J.T.: ‘Infinite phased arrays of cav- ity-backed patches’, IEEE Trans., 1994, AP-42, (3), pp. 390-398 CHEW, W.C., and HABASHY, T.: ‘The use of vector transforms in solving some aelectromagnetic scattering problems’, IEEE Trans., 1986, AP-34, (7), pp. 871-878 BARRETT, R. et al.: ‘Templates for the solution of linear sys- tems: building blocks for iterative methods’, Supported by DARPA and ARO, 1991 FREUND, R.W., and NACHTIGAL, N.M.: ‘A quasi-minimal residual method for non-hemitian linear systems’, Numer. Math., 1991, 60, pp. 315-339 FREUND, R.W.: ‘A transpose-free quasi-minimal residual algo- rithm for non-hermitian linear systems’, SIAM J. Sci. Comput., 1993, 4, pp. 470482

1994, 14, (2), pp. 239-258

7 Appendixes

7.7 Appendix A The dyadic Green function parameters are as follows:

(35) 2 cos IC,(zo + c) 1 p y p , z = 0120) = - a2 IC, sin k,c [JL(z,q)]z

7.2 Appendix 6 The MoM/Galerkin matrix element parameters are:

24 IEE Proc-Microw. Antennas Propag., Vol. 143, No. I , February 1996