The Application of Macromodels to the Analysis of a Dielectric Resonator Antenna Excited by a Cavity

Backed Slot Andrzej A. Kucharski#1, Piotr M. Słobodzian#2

#Radiocommunications and Teleinformatics Department, Wrocław University of Technology Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

1Andrzej.Kucharski@pwr.wroc.pl 2Piotr.Slobodzian@pwr.wroc.pl

Abstract— The analysis of cavity backed, slot excited dielectric resonator antennas (DRA) is accelerated with the use of a macromodel within the framework of the IE-MoM formulation. The macromodel can be regarded as a projection of complete information about the DRA onto the slot in a wide frequency range, and thus it provides huge reduction in the total number of unknowns. Consequently, this enables a considerable decrease in computation time and makes it possible to curry out fast optimization of a feeding network structure enclosed in a cavity without the need for handling large systems of linear equations associated with the DRA.

I. INTRODUCTION Dielectric Resonator Antennas (DRAs) become popular in

the microwave antenna community because of their interesting features including low cost, temperature stability, compact size and relatively wide frequency bandwidth [1]. One of common feeding mechanisms, is an excitation through a slot in the ground plane, on which the DRA is positioned [2]–[4]. Usually, the feeding system is made in microstrip technology [4], and is placed below the ground, possibly in a metal cavity, in order to suppress electromagnetic coupling with other microwave devices. Such a structure can be rigorously analysed with the use of the IE-MoM approach, as it was already demonstrated by the authors, for example, in [5] and [6]. In such a case, the IE-MoM approach involves modelling both the DRA and the feeding system embedded inside the cavity. The cavity and microstrip circuitry were treated by means of the IE-MoM formulation that is normally used to analyse shielded microwave structures [7]. In turn, the DRA was modelled with the use of the equivalence principle leading to the popular electric/magnetic surface current formulation within the IE-MoM approach [8]. The slot itself, which provides a coupling between the cavity and DRA, was modelled by means of an unknown surface magnetic current [9]. The bottleneck of this approach is the necessity of solving relatively large systems of linear equations at each frequency of interest, where the majority of unknowns usually describe the DRA rather then the feeding system embedded in the cavity. In the situations where the feeding system must be optimised such approach leads to an unacceptable computational burden, since every minor modification on the “cavity” side provokes long-lasting computations.

DRA

Slot

Cavity

Microstripcircitry

Fig. 1 Slot excited DRA over a cavity containing microwave circuitry.

In this paper we propose a remedy to the aforementioned

drawback, and the main idea is based on the Integral Equation Macromodels (IEMs) [10], which combine the domain-decomposition method (DDM) with the fast frequency sweep technique, i.e. the asymptotic waveform evaluation method (AWE). The method consists of the following steps. First, we construct the classical IE/MoM formulation based on equivalence principles in free space for DRA and the rectangular cavity Green’s function formulation for the feeding system. Then, the domain decomposition method is applied to enable independent computations of the slot excited DRA. Next, the wideband approximation to the resulting sub-matrices is obtained using AWE-like procedure. Finally, the system of linear equations with much smaller number of unknowns, related mainly to the description of the feeding system rather than the original problem, is successively solved.

In this paper we provide the example proving the correctness and efficiency of the approach.

II. CLASSICAL FORMULATION The problem under consideration is depicted in Fig. 1. This

structure can be analysed with subsequent application of the equivalence principle, which consists in sub-dividing the original problem into several equivalent ones. To this end, fictitious surface equivalent currents are introduced, which

978-2-87487-006-4 © 2008 EuMA October 2008, Amsterdam, The Netherlands

Proceedings of the 38th European Microwave Conference

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produce correct fields in corresponding sub-regions. Integral formulas for fields due to these currents, together with the boundary conditions allow us to formulate integral equations of the following form [5]:

( ) ( )[ ] =−+−+− tddsd MHJHMH )2( ( ) ( )[ ]tee MHJH += , on Sd (1)

( ) ( )[ ] =−+−+− tddsd MEJEME )2(

( ) ( )[ ]tee MEJE += , on Sd (2)

( ) ( )[ ] =−+−+− tddsd MHJHMH )2(

( ) ( )[ ]tincccsc HJHMH ++= , on Sa (3)

( )[ ] 0)( =++ tinc

ccsc EJEME , on Sc (4)

In (1)-(4), J and M denote the equivalent currents flowing on the air/dielectric interface and its image, reproducing correct fields in the upper half-space region (their negatives, together with the negative magnetic current over the slot, reproduce correct fields inside DRA). The positive magnetic current Ms flowing over the slot, together with the electric currents Jc inside the cavity produce correct fields inside the cavity. The slot magnetic current in (1)-(3) is doubled, again as a result of the application of the image theory. The subscript “t” stands for “tangential component”, whereas subscripts d, e, c denote the environment in which the currents radiate (respectively: inside the DRA, external to the DRA and inside the cavity). The surface of conductors (metallization) inside the cavity is denoted by Sc, Sa denotes the surface of the slot/aperture, and Sd is the surface of the DRA and its image. Finally, Einc denotes the electric incident field inside the cavity. Obviously, determination of the fields in (1)-(4) due to respective surface currents requires the knowledge of suitable Green’s functions.

The presented system of integral equation may be easily solved using Method-of-Moments (MoM), with suitable basis and testing functions. Here, we apply the well-known roof-top basis functions spanned over pairs of triangles (for the DRA surface) and rectangles (for the surface of the slot and metallization inside the cavity). The application of the MoM to (1)-(4) results in the following set of linear equations:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+++++

incc

sccEJc

ccEMs

csHJc

csHMs

dsHMs

dsHM

dsHJ

ddHMs

edHM

ddHM

edHJ

ddHJ

ddEMs

edEM

ddEM

edEJ

ddEJ

E000

00

00

JMMJ

ZZZZZZZ

ZZZZZZZZZZ

(5) where J, M, Ms, Jc now have the meaning of the expansion coefficients describing respective currents, ij

PQZ denotes elements corresponding to the field P due to the current Q radiating in the environment i, tested over the surface j.

Now, let us rewrite (5) in a more compact notation, grouping the sources and fields on the surface of DRA:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+inc

c

sccJc

ccMs

csJc

csMs

usMs

usK

udMs

udK

EJMK

ZZZZZZ

ZZ00

0

0, (6)

where K is a column of coefficients describing together J and M, while “u” simply denotes “upper” part of the problem (both “dielectric” and “external” operators contribute to corresponding matrices). Obviously, equation (5) may be solved directly using typical methods of linear algebra, as it was done, in fact, in [5] and [6]. Such a method has however the following limitations:

1. Relatively large system of equations has to be solved for each frequency of interest leading to long-lasting computations.

2. Each change (optimization) of the feed inside the cavity enforces new computations, although the part of the problem introducing the majority of unknowns, i.e. the DRA, remains unchanged.

In fact, the number of unknowns describing the DRA may be two orders of magnitude larger than that corresponding to the rest of the problem (i.e. to the slot and metallization inside the cavity). The listed limitations can be eliminated by means of the aforementioned macromodel.

III. MACROMODELS The time required for solving (6) in the wide frequency

band could be considerably reduced if one applied fast frequency sweep techniques [11], instead of step-by-step calculations. This technique can be easily applied to the “upper” part of the problem (i.e. DRA) [12]. However, application of techniques like the Asymptotic Waveform Evaluation (AWE) requires calculating derivatives of the Green’s functions, which may be not a trivial task for the multilayered medium inside the cavity. Moreover, a large system of equations is still involved in the solution of the whole problem.

In this section, we will present how a combination of AWE with the Domain Decomposition Method (DDM), introduced by one of the authors in [10], can be used to solve the described problem very effectively. The outline of the procedure is as follows:

1. First, we eliminate K from (6) using the fact that certain entries in the moment matrix are zeros (some parts of the problem do not interact), so we get:

( )⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ +⎭⎬⎫

⎩⎨⎧ −

−

incc

s

ccJc

ccMs

csJc

csMs

usK

udK

udMs

usMs

EJM

ZZ

ZZZZZZ 01

(7)

2. Then, we take the term in curly braces, namely:

520

( )⎭⎬⎫

⎩⎨⎧ −=

− usK

udK

udMs

usMs

M ZZZZZ1

(8)

3. And finally, we find the Padé approximation to the

entries of ZM using the procedure given in [10].

In general, the procedure of point 3 consists in multiple application of the AWE method for different excitations of the slot, and then projecting the solutions again onto the aperture, to find the Taylor expansions of ZM around chosen frequency points. From Taylor expansion we obtain the Padé approximation using the standard procedure, described for example in [11]. The wideband approximation to ZM is called the macromodel. Once computed for a given frequency interval, it may be re-used any number of times to solve equations (7), provided that only cavity or microstrip circuitry configuration can be modified. Note that solving (7) is fast, as the number of unknowns is usually small when compared to (5).

IV. APPLICATION EXAMPLE The procedure outlined in the previous section has been

applied to the analysis of the rectangular DRA, which was earlier analysed using the classical approach (the results of calculation and measurements were published by the authors in [6]).

In order to use the presented approach first, the macromodel of the combination of the DRA and the excitation slot has been built. The macromodel is independent on the rest of the structure, since the cavity “sees” the DRA only through the slot. For the frequency interval of interest it was enough to take a single expansion point at f0=2.45GHz. The degrees of the numerators and denominators of the Padé approximations have been chosen as 2 and 3, respectively. Obviously, the computation time necessary to build the macromodel depends on the discretization of the DRA surface, but the final size of the macromodel (i.e. the number of the involved unknowns) depends only on the number of unknowns related to the slot. In our case the slot was discretized into 3x8 cells, each forming a rectangle, which led to 37 unknowns. With such number of unknowns it takes less then one second to incorporate information generated by the macromodel in the final solution of the whole problem. However, before application of the macromodel we need to tune and test its performance.

DRA

Slotand excitation

Infiniteground plane

Fig. 2 Slot excited DRA above an infinite ground plane.

The quality of the macromodel was first tested without joining it with the sophisticated cavity excitation. The first cycle of computations was done for simple delta gap excitation placed in the middle of the slot and air-filled half-space on the opposite side of the ground plane, as shown in Fig. 2. In this case the final system of equations to be solved reduces to [10]:

incs

csMss

M HMZMZ =⋅+⋅ (9)

For consistency with the previous formulas, we have left in (9) superscript c describing the environment below the slot (although there is no cavity in this case). Also note the change of excitation to the magnetic type, as now it is the slot that is excited, not the metallization. In order to validate the macromodel we have computed the input impedance of such excited DRA using classical solution and the macromodel. The results of computations are shown in Fig. 3.

-200

-100

0

100

200

300

400

500

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

Frequency [GHz]

Impe

danc

e [O

hm]

Re

Im

macromodel

classic

Fig. 3 Input impedance of the DRA shown in Fig. 2. The parameters of the DRA and the slot are the same as in the next example.

Next, we have incorporated the obtained macromodel into

the original problem, which is shown in Fig. 4. In this case, wideband computations simply consisted in solving the following matrix equation:

⎥⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +inc

c

sccJc

ccMs

csJc

csMs

M

EJM

ZZZZZ 0

(10)

where values of ZM describe the macromodel that was obtained from the previously computed example and stored out-of-core data describing the remaining elements of the problem. The results of calculations exploiting the macromodel are shown in Fig. 5. As we can see, the results obtained with and without the macromodel are almost indistinguishable, while the computations using macromodel were about 20 times faster. This constitutes considerable decrease in the computational burden, and hence it makes the proposed method suitable for optimisation procedures.

521

25.8 mm

Dielectric resonator (Eccostock HiK 500)hDR=9 mm, r=16, tg 0.002

L1

L3

L2SMA

Slot

Strip

40.0 mm

Layer description:

L1: Rohacell 31HFh1=5 mm

r1=1.045tg =0.00163

L2: Ultralam 2000h2=0.762 mm

r2=2.4tg =0.002

L3: Airh3 =44.9 mm

r3=1tg =0

40.0 mm

Ground plane

200x200 mm

Cavity outline

Slot

4.0 mm

2.1 mm

27.0 mm

Strip

Fig. 4 The structure and dimensions of the cavity backed, slot excited DRA used for comparisons [6].

-16

-14

-12

-10

-8

-6

-4

-2

0

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

Frequency [GHz]

S11

[dB]

classic solutionmeasurement

macromodel

Fig. 5 The comparison between the results of classic solution and the solution with the macromodel incorporated (measured results are also shown).

V. CONCLUSION In the paper, we have presented an efficient method for

speed-up the analysis of slot excited dielectric resonator antennas. It was shown that the application of the macromodel approach leads to the procedure that may be more than order of magnitude faster than the original one (classic one).

The important feature of the proposed approach is that once constructed and tested macromodel may be easily used for many different configurations and environments. This feature makes it possible to pre-compute a database of macromodels of typical DRAs (or other types of slot excited antennas).

One obvious limitation of the approach is that the obtained macromodel is strictly related to the particular slot size, position and discretization (presently, the authors work on removing this restriction). Also an interesting way of further developments seems to be a combination of the developed IE-MoM macromodels with other techniques of computational electromagnetics.

ACKNOWLEDGMENT This work was supported by Polish Ministry of Science and

Higher Education, from resources for the science for the period 2007-2009, under Grant N N517 1650 33.

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[2] K.Y. Chow, K.W. Leung, “Theory and experiment of the cavity-backed slot-excited dielectric resonator antenna”, IEEE Trans. EMC, vol.42, pp.290-297, Aug. 2000.

[3] S. Hashemi-Yeganeh, C. Birtcher, “Theoretical and experimental studies of cavity-backed slot antenna excited by a narrow strip”, IEEE Trans. Antennas Propagat., vol.41, pp.236-41, Feb. 1993.

[4] G.P. Junker, A.A. Kishk, and A.W. Glisson, “Input impedance of aperture-coupled dielectric resonator antennas,” IEEE Trans. Antennas Propagat., vol. 44, no. 5, pp. 600–607, May 1996.

[5] A. Kucharski, P. Słobodzian, Analysis of a Slot Excited DRA above a Cavity Filled with a Multilayered Medium, International Conference on Electromagnetics in Advanced Applications, ICEAA-2005, 12-16 Sept., Torino, Italy, pp.253-255.

[6] R. Borowiec, A.A. Kucharski, and P. Slobodzian, ”Slot excited dielectric resonator antenna above a cavity - analysis and experiment,” in Proc. 16th International Conference on Microwaves, Radar and Wireless Communications, MIKON-2006. Krakow, 2006, pp. 824-827.

[7] P. Slobodzian, Electromagnetic analysis of shielded microwave structures. The surface integral equation approach, Wroclaw, 2007.

[8] R.F. Harrington, Time-harmonic electromagnetic fields. McGraw-Hill Book Company, 1961.

[9] R.F. Harrington, J.R. Mautz, Electromagnetic Transmission Through an Aperture in a Conducting Plane, Archiv-fur-Elektronik-und-Uebertragungstechnik, vol.31, Feb. 1977, pp. 81-87.

[10] A.A. Kucharski, "Efficient Solution of Integral-Equation Based Electromagnetic Problems With the Use of Macromodels," IEEE Trans. on Antennas Prop., vol. 56, May 2008, pp. 1482-1487.

[11] W.C. Chew, E. Michielssen, J.M. Song, J.M. Jin. Fast and efficient algorithms in computational electromagnetics. Boston, London: Artech House, 2001.

[12] A.A. Kucharski, “AWE accelerated analysis of slot-excited dielectric resonator antennas,” in Proc. of the International Conference on Computer as a Tool. EUROCON 2007, Warsaw, September 9-12, 2007, pp. 41-44.

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