analysis of a dielectric resonator antenna in a cylindrical conducting cavity: hem modes

Analysis of a dielectric resonator antenna in a cylindrical conducting cavity: HEM modes

X. Di A.W. Glisson K.A. Michalski

Indexing term: Dielectric resonator antenna, H E M modes

Abstract: Numerical modelling has been per- formed for the hybrid HEM modes of a flush- mounted dielectric resonator antenna in a cylindrical conducting cavity. The antenna is formed by a conducting cavity with a dielectric post and radiates through an annular slot in the upper ground plane. Coupled aperture magnetic current integral equations are used to model the slot and the effect of the post resonator and the cavity are included via Green’s function techniques. The moment method is applied to the coupled integral equations and the complex resonant frequencies for the antenna structure are found by searching for the zeros of the determi- nant of the impedance matrix in the complex frequency plane. The radiation Q factor, slot field distribution and radiation field are subsequently computed at the resonant frequency. The HEM modes with the lowest resonant frequencies have been investigated for different geometrical configurations.

1 Introduction

In this work a dielectric resonator antenna configuration is rigorously modelled to determine the hybrid HEM,, resonant modes of the structure. The radiating structure comprises a dielectric-post resonator located in a cylindrical cavity with an annular slot in the surface that forms the upper plate, which is of infinite extent, as illustrated in Fig. 1. In the authors’ previous work [l], the formulation and analysis was limited to the +-independent TM,, and TE,, modes. It was found that certain TE and TM modes are good radiating modes and that a low Q factor can be achieved for the antenna for those modes. However, since neither the TE nor the TM modes radiate in the broadside direction, the use of the antenna operated in those modes would be restricted to situations in which low elevation radiation is desired. It is expected that the most suitable mode of this radiating structure for

0 IEE, 1994 Paper 9773H (El I), first received 19th January and in revised form 10th June 1993 X. Di is with the National Center for Physical Acoustics, University of Mississippi, University, MS 38677, USA A.W. Glisson is with the Department of Electrical Engmeering, Uni- versity of Mississippi, University, MS 38677, USA K.A. Michalski is with the Department of Electrical Engineering, Texas A&M University, College Station, TX 77843, USA

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many applications would be a lower order hybrid HEM,, mode with m = 1 [l-lo], and an extensive investigation has been undertaken for these modes [l I]. Aside from the general advantages of the dielectric resonator cavity antenna configuration in applications requiring antennas with small dimension, low profile, conforming shape, and simple structure, perhaps most important aspect of the dielectric resonator cavity antenna oper- ating in the HEM,, mode is its broadside radiation pattern.

Fig. 1 upper plate

Dielectric post resonator cavity antenna with annular slot in the

Although the discussion in this paper is limited to the hybrid mode with m = 1, the formulation presented is general and complete. It covers all the possible modes, including HEM, TE, and TM cases. This research is an extension of the work presented in Reference 1, which considers the TE and TM modes. The historical back- ground and additional motivation for this research can be found in References 1 to 14.

2 Formulation

As in Reference 1, an equivalent problem to that of Fig. 1 is constructed as illustrated in Fig. 2. The surface magnetic current M,(p, 4,O) = z x E@, +, 0) is placed over the short-circuited slot for p l < p < p2 in the interior region, while - M , is placed on the exterior side of the short-circuited slot for the top plate [lS]. The formulation for hybrid HEM,, modes is then developed in the same manner as in Reference 1. The resulting approach is general and includes the TE and TM modes for the special case in which m is chosen to be zero. Thus the procedures employed here are essentially the same as that

This work was supported in part by the National Science Foundation under Grant No. ECS- 9015328.

I E E Proc.-Microw. Antennas Propag., Vol. 141, N o . I , February 1994

for the TE and TM modes [l]. Unlike the situation in Reference 1, however, the integral equations for the HEM,, modes are coupled equations that involve both p and 4 components of the equivalent current M , .

‘ t

Fig. 2 placed over the aperture region

Equivalent problem with the slot shorted and magnetic current

The exterior tangential aperture magnetic field is readily obtained using image theory as in Reference 1

and where a is the post radius, ko is the exterior region wavenumber. s = j k , a , ‘lo = ( p , / ~ , ) ” ~ , u(p’) = p’M,(p’), U’($) = au/ap’, and m is the azimuthal mode number. Note that (1) and (2) reduce to the corresponding equations in Reference l if m = 0.

Inside the cavity, to represent the more general hybrid mode fields, both vector potentials A ( p , z) = iY‘(p, z) and F(p, z) = iY’(p, z) are assumed for the HEM modes [16, p. 1291. As in Reference 1, the Green’s functions of the potentials Y: ‘(p, z), where the subscript m specifies the order of eigenvalues with respect to 4, can be represented as

m

G: ’(p, z) = C:,!’(p)Zz. ‘(z) “ = O

where

IEE Proc.-Microw. Antennas Propag., Vol. 141, No. I , February 1994

The explicit expressions for C;,!’(p) are given in the Appendix. Integration of eqn. (5) over the aperture (with respect to p’) then produces the potentials ‘Y: ’ (p , 2). Finally, the interior tangential magnetic fields HFm(p) and H&(p) can be obtained from the potential expressions using eqn. 6 in Reference 1.

3 Numerical procedure and results

The method of moments is applied to the coupled integral equations by utilising a pulse function expansion of the current M+,(p’) and a triangle function expansion of the p-component U,,,@’). Subsequently, the coupled integral equations involving the p- and +-components of the magnetic field are tested with pulse function and 6- function testing sets, respectively [17]. As a result, a set of simultaneous equations, which may be represented in matrix form as

ZIM) = 10) (7) are obtained. Due to the radiating nature of the structure, the resonant frequencies of the structure are complex quantities. Therefore, k , and k should be replaced by k , - jk” and ( ~ , ) ” ~ ( k , - jk”), respectively (including in the Appendix). Consequently, the normal- ised complex resonant frequency s should have the form s = j (k , -jk”)a. As in Reference 1, Z is the moment matrix, which is a function of s, and IM> is a column vector containing the surface magnetic current coeff- cients to be determined. Once a resonant frequency is determined such that eqn. (7) is satisfied, the quality factor Q is given by Q = k0/2k”. In addition to the singu- larity extraction techniques and the series acceleration techniques employed in References in 1 and 13, a new type of acceleration technique has been used here for a series with an asymptotic form of e-“x/nz to enhance the convergence of the series, where x = (n/h) I p - p’ I and n is the series index number. As an example, for a series with general term a, and asymptotic form e -nx /n2 , the series is accelerated as

2 + e - = - e - x ) + - 4

After implementing the series acceleration techniques described here and in Reference 1 and 13, we have found that all the series used in the cavity converge within 25 terms for all cases considered. For most cases, the series converge for around 10 to 15 terms. For large arguments, it is necessary to use the asymptotic forms for the modified bessel functions I, and K , . Also it is necessary to evaluate the product of I, and K , for large arguments, instead of evaluating them separately. The equations in the Appendix are arranged in a concise form, rather than in the form most suitable for numerical evaluation. Finally, we note that for the numerical results presented here, 10 to 20 pulses per wavelength have been used for the expansion of the slot magnetic current in the compu- tations.

Numerical results are presented for several HEM mode cases with an azimuthal index m = 1. Fig. 3 shows the migration in the complex frequency plane of the

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lowest HEM,, mode for several cases in which E, = 8.9, a = 3 mm, and b = 20 mm. The migration is shown as the antenna slot is opened from a fixed inner slot edge position at the dielectric post edge (3 mm from the post axis) to an increasing outer slot edge position for four different cavity heights: h = 15, 10, 6 and 4mm. Thus for each curve the slot increases in width from a small width to a maximum width of 17 mm, in which case the slot extends from the post edge to the cavity vertical wall. When the slot is very small, each curve approaches the resonant frequency of the lowest no-slot TM,,, mode ( k o a = 0.55). For the long post radiators ( h = 10 and 15 mm), the resonant frequency is significantly lowered when the slot width is large. At the maximum slot width both of these structures resonate at about k o a = 0.285 with a very low Q factor (about 1). For the shorter post structures (h = 6 and 4 mm), the resonant frequency increases as the slot becomes larger and the Q factors change from high to low to high and finally to low again for the widest slot, at which point the Q factor is about 6. From the mode chart of the closed cavity (not shown), the resonant frequencies of the second and third lowest resonant modes (HEM,,, and TM,,, or vice versa) increase as the post shape becomes flatter. Since the field may be represented as a joint contribution of all modes, the resonant frequency is expected to increase for a flatter post shape as the resonant frequency of the higher order modes also now becomes higher. With respect to the Q factor fluc- tuation, it is noted that the resonant frequency is domi- nated by the TM,,, mode of the no-slot cavity while the slot is small, therefore the Q factor initially decreases as the slot width becomes larger. When the slot reaches a certain width, and the resonant frequency rises to a certain level, the aperture current is no longer entirely positive or negative. The cancellation effect of the aperture current then makes the Q factor higher, and finally, as the slot width becomes still larger, postive addition of another lobe of the current distribution occurs and pulls the Q factor back down. A similar effect is well known for the dipole antenna.

r

/

r” 0 2 8 1 :3-20 , 1 3 - 2 0 , -018 -015 -012 -009 -006 -003 0

real s

Fig. 3 different heights h E, = 8 9 , n = 3 mm, b = 20mm ~ h = 1 5 m m -- h = l O m m

H E M , , mode migration (IS a function of slot width for four

h = 6 m m h = 4 m m

_ _ _ _

0 no-slot TM,,, mode

In Fig. 4, two HEM,, mode migration curves for the long shape post (h = 15 and 10 mm) are given. The dimensions and the variations of the slot opening are the same as in Fig. 3. When the slot width decreases the res-

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onant frequencies approach those of the no-slot HEM, modes for the corresponding cavity heights, which are the second lowest no-slot resonances with k , a = 0.621 and 0.7242, respectively. The lowest Q factors achieved for these cases are 19 and 5 for h = 15 and 10 mm.

0 5 5 ’ -006 -005 -004 -003 -002 -001

real, s

Fig. 4 different heights h e, = 8 9 , a = 3 mm, b = 20mm ~ h = 1 5 m m -- h = 1 0 m m

H E M , , mode migration as a function of slot width for two

0 no-slot H E M , , , mode

When the transition from a long post geometry to a flat post is observed, it can be seen in Figs. 3 and 4 that the resonant frequencies and Q factors of the flat-post HEM,, mode and the long-post HEM,, mode fall in the same region and migrate in the same general way when the slot is widely open, extending from the post edge to the neighbourhood of the cavity wall. The migration of a mode for a fixed width slot as the post height h changes can also be seen in these two figures. For example, if one refers to the end of the curve where the slot is open from 3 to 20 mm for h = 15 mm in Fig. 4 and to the same end of the curve for h = 4 mm in Fig. 3, the resonant frequency k , a migrates from 0.583 to 0.646 and the Q factor from 19.4 to 7.4. The resonant frequency changes very slowly for such a dramatically changed post height. For the radiator with larger cavity radius, the resonant frequency changes even more slowly when the post height is changing. A simple magnetic wall theory is helpful here to understand this migration of the resonant frequency. In Long’s experiment report [4], he investigates the radiation of a dielectric post on an infinite conducting ground plane. The lowest mode found there is HEM,,, and the resonant frequency was almost identical to the simple magnetic wall prediction. For Long’s experiment, a post radius of a = 3 mm and a relative dielectric constant of E, = 8.9 was used. When the post height h changed from 15 to 10 mm, the resonant frequency k , a moved from 0.626 to 0.637 using the magnetic wall theory prediction, which agreed very well with their experiment. In the present research, a similar approach has been taken, considering now that the slot is wide open from the post edge to the cavity wall which is relatively far away from the post, especially for a flat post. If a magnetic wall model for a dielectric post with a conducting cap seated on a conducting ground plane is used, the resonant frequency can be expressed as

1.84118 k o a = ~

(4”2 (9)

IEE Proc.-Microw. Antennas Propag., Vol. 141, No. I , February 1994

for the lowest HEM,, mode. For E, = 8.9, the magnetic wall model result (9) predicts koa = 0.6172 independent of the height h, but is valid only for relatively flat posts. As an approximation, this prediction is not bad, except that no Q factor information can be determined from it. In the complex frequency plane, however, one notes that there is a transition region in which the HEM,, and HEM,, modes mix for a specific post height h. This occurs around h = 9.6 to 9.7 mm (not shown) for the present geometry [ll]. In this transition region, the present mode designation for this kind of mode migration is really not meaningful, and only for a particular geometry, can HEM,, or HEM,, be specified with clarity. Similar behaviours have been found for radiators with other cavity radii. For radiators with b = 15, a mode migration chart is presented in Fig. 5 for cases where the inner edge of the slot is fixed at the post edge and slot width varies. The HEM,, mode migrations for four different post heights h have been plotted as a function of outer slot edge position. A transition from ‘long post modes’ to ‘flat post modes’ occurs at h = 6.3 mm.

O r

0 3l I -028 -024 -02 -016 -012 -008 -004

real, s

Fig. 5 diflerent heights h e, = 89, B = 3 mm, b = 15 mm - h = 3 5 m m _ _ h = 6 m m

- _ _ ~ h = 7 m m

H E M , , mode migration as a function of slot width for four

h = 1 4 m m

For h = 14 mm (long post), the resonant frequency migrates from the no-slot HEM,, , mode (koa = 0.64 and Q = CO), when the slot is small, down to koa = 0.4 and Q = 1.25 when the slot is open from the post edge to the cavity wall. For h = 7 and 6 mm (square post cross- section), a similar migration has been observed except that when the slot is very small, no resonant frequency in this range has been found. When the slot is very wide, the resonant frequency of the h = 7 mm case decreases to approximately that of the h = 14 mm case, but the resonant frequency of the h = 6 mm case increases for a very wide slot. For the flat post mode (h = 3.5 mm), the resonant frequency behaves in much the same manner as in the b = 20mm case. The lowest Q factor that can be obtained occurs for a square post cross-section (post height h equals its diameter) and its value is about 1 for a very wide slot.

Next, we investigate the effect of the positioning of a slot with a fixed width of 3 mm, as the slot is moved from the post edge to the cavity wall. With other parameters being the same as in Fig. 5, two migration curves for h = 6 and 7 mm are plotted in Fig. 6. Contrary to the results shown in Fig. 5, no transition from ‘long-post’ to ‘short-post’ modes is evident in Fig. 6. The resonant fre-

quencies decrease continuously as the slot moves away from the post. The Q factor has a value of 6.1 when the slot is placed against the post, it decreases to 4.6 when the slot is 3 mm away from the post, and finally rises to 7.6 when the slot reaches the cavity wall. If one takes into account that the slot aperture area is proportional to the slot radius, it is concluded that the slot of same aperture area positioned near the post gives a better radiation.

, ’>, , -008 -007 -006 -005 -004 -003 -002 -001 0

real, s

Fig. 6 post edge to the cavity wall E, = 89,a = 3 mm, b = 15 mm ~ h = 6 m m _ _ h = 7 m m

H E M , , mode migration as a 3-mm wide slot movesfrom the

The effect of the dielectric constant E, has also been investigated. Fig. 7 shows the resonant frequency migration for a 3 mm width slot as a function of E , . As the value of E, changes from 8.9 to 20, the resonant frequency decreases from k , a = 0.550 to 0.479 and correspondingly the Q factor increases from 4.553 to 18.02. It is worth noting here, however, that the Q factor does not always increase with an increase of the value of cr when E, is small. For example, we have found that the Q factor of an HEM,, mode for one case (not shown) does not change significantly until E, exceeds 15, instead of continuously increasing as E, increases. This implies for such cases that, when E, is small, fields are really stored in the whole cavity rather than being confined primarily within the dielectric post. As a rough estimation, our results indicate that better radiation effects will be obtained when the value of E, is smaller than 20, which is also suggested by the study of other dielectric post radiators, for instance Reference 4.

0541 0 52

‘O 0 501

0 46‘ I -007 -006 -005 -004 -003 -002 -001 0

real, 5

Fig. 7 H E M , , mode migration as a Junction o f t , a = 3 mm, h = 7 mm, b = 15 mm, slot location = 5-8 mm

IEE Proc.-Microw. Antennas Propag., Vol. 141, No. 1, February 1994 11

In the hybrid mode case, the aperture current consists of both M, and M, components. A typical current distribution is shown in Fig. 8 for the lowest HEM,, mode. Also for the HEM,, mode, a set of four radiation patterns is plotted in Fig. 9 for four different slot widths,

\ , . -_-- - - '- I

0 - \

- - - - - - - - - \

-0 2 I

5 6 7 8 slot, mm

a

.- c - E 2

slot. rnm

b

Fig. 8 s=(-0.06268,0.55197),&,=8.9,a= 3mm,h= 10mm.b = 15mm ~ M (real) _ _ M (imaginary)

H E M , , mode current distribution

8.0' 8.0"

0" 0" 0 -10 -20 -30 06-30 -20 -10 -0 0 10 20 30 OB 30 20 10 0

0 b

AZO' 8.0. . .

0' 0 -10 -20 -30 OB -30 -20 -10 0 0 -10 -20 -x) OB -30 -20 -10 0

C d

Fig. 9 H E M , , modejar Efield E (- -) and E , (- - -) for E, = 8.9, a = 3 mm, h = 6 mm, b = 20 mm o h o t from 3 to 8 mm. s = (-0.0165.0.5325) b Slot from 3 to 12 mm, s = (-0.0170,0.5727) c Slot from 3 to 14.5 mm, s = (-0.0094,0.5957) d Slot from 3 io 20 mm, s = ( -0.0397.0.6272)

IL

where the antenna configuration is same as in Fig. 3 (with h = 6 mm). Clearly, r a d i a t i o n is in the broadside direction and the radiation beam becomes narrower when the slot becomes large. For the largest slot width (17 mm), the 3 dB beam width is about 40" for E , and 20" for E , , and a sidelobe at about 60" with a strength of - 15 dB emerges.

4 Conclusion

Together with Reference 1, the dielectric resonant antenna in a cylindrical conducting cavity has been modelled rigorously to determine all of its modes: TE, TM, and Hybrid HEM modes. In this paper, the investigation has concentrated on the lowest HEM,, mode, along with the HEM,, mode. As in the cases of TE and TM modes, the resonant frequencies for the HEM modes of the slotted structure are found to be closely related to those of the no-slot structure and they approach those of the no-slot structure as the width of the slot is reduced. The computed resonant frequencies have also shown good agreement with the magnetic wall theory prediction for the case of a relatively flat post. For the HEM,, mode, a very low Q factor in the neighbourhood of 1 to 6 has been obtained for some configurations with E, = 8.9. Radiation is normal to the ground plane for this mode, which is more desirable for a radiator in many situations. The resonant frequency of the structure can be adjusted easily over a fairly large range by adjusting the radiator's dimensions, the slot position, or the slot width.

5 References

1 DI, X., GLISSON, A.W., and MICHALSKI, K.A.: 'Analysis of a dielectric resonator antenna in a cylindrical conducting cavity - TE and TM modes', IEE Proc. H, 1992.139, (5), pp. 453-458

2 KAJFEZ, D., and GUILLON, P. (Eds.): 'Dielectric resonators' (Artech House, Dedham, MA, 1986)

3 KAJFEZ, D., GLISSON, A.W., and JAMES, J.: 'Evaluation of modes in dielectric resonators using a surface integral equation formulation'. IEEE MTT-S Int. Microwave Symp. Dig., 1983, pp. 409- 41 1

4 LONG, S.A., McALLISTER, M.W.. and SHEN, L.C.: 'The resonant cylindrical dielectric cavity antenna', IEEE Trans. Antennas Propag., 1983, AP-31, pp. 406-412

5 GLISSON, A.W., KAJFEZ, D., and JAMES, J.: 'Evaluation of modes in dielectric resonators using a surface integral equation formulation', lEEE Trans. Microw. Theory Tech., 1983, MIT-31, pp. 1023-1029

6 KAJFEZ, D., GLISSON, A.W., and JAMES, J.: 'Computed modal field distributions for isolated dielectric resonators', IEEE Trans. Microw. Theory Tech., 1984, M'lT-32, pp. 1609-1616

7 McALLISTER, M.W., LONG, S.A., and CONWAY, G.L.: 'Rec- tangular dielectric resonator antenna', Electron. Lett., 1983, 19, pp. 218-219

8 McALLISTER, M.W., and LONG, S.A.: 'Resonant hemispherical dielectric antenna', Electron. Lett., 1984, 20, pp. 657-659

9 KISHK, A.A., AUDA, H.A., and AHN, B.A.: 'An accurate prediction of the radiation patterns of dielectric resonator antennas', Elec- tron. Lett., 1988, 23, pp. 1374-1375

IO KISHK, A.A., AUDA, H.A., and AHN, B.A.: 'Radiation character- istics of cylindrical dielectric resonator antennas with new applications', IEEE Antennas and Propagat. Soc. Newsletter, 1989, 31, (l) , pp. 6-16

I 1 DI, X.: 'Analysis of a dielectric resonator cavity antenna'. PhD dis- sertation, University of Mississippi, University, MS, 1990

12 BUTLER, C.M., and KESHAVAMURTHY, T.L.: 'Investigation of a radial, parallel-plate waveguide with an annular slot', Radio Sci., 1981,16, pp. 159-168

13 MICHALSKI, K.A., DI, X., and GLISSON, A.W.: 'Rigorous analysis of the TM,, modes in a dielectric-post resonator with an annular slot in the upper plate', IEE Proc. H, 1991, 138, (9, pp. 429-434

14 KOBAYASHI, Y., and TANAKA, S.: 'Resonant modes of a dielectric rod resonator short-circuited at both ends by parallel conduct-

IEE Proc.-Microw. Antennas Propog., Vol. 141, No. I , February 1994

ing plates’, IEEE Trans. Microw. Theory Tech., 1980, MlT-28, pp. 1077-I084

15 VAN BLADEL, J., and BUTLER, C.M.: ‘Aperture problems’, in SKWIRZYNSKI, J.K. (Ed.): ‘Theoretical methods for determining the interaction of electromagnetic waves with structures’ (Stijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1981), pp. 117-172

16 HARRINGTON, R.F.: ‘Time harmonic electromagnetic fields’ (McGraw-Hill, New York, 1961)

17 GLISSON, A.W., and WILTON, D.R.: ‘Simple and efficient numerical techniques for treating bodies of revolution’, Technical report no. 105, RADC Contract No. F30M)2-78-C-0120, Engineer- ing Experiment Station, University of Mississippi, March 1979, pp. 3-13

6 Appendix

The coefficient functions of the Green’s function expansion for a magnetic current loop located inside the post at P’ (and z’ = 0) are obtained as

mp’k2 oa,

+ - M,,(P’

where the three ‘stacked‘ terms in eqns. (10) and (11) are valid in the regions p < p’, p’ < p < a, and p > a, respectively. For the source loop outside the post, one obtains

(12)

13 I E E Proc.-Microw. Antennas Propag., Vol. 141, N o . I , February 1994

where the three ‘stacked’ terms in (12) and (13) are valid for the regions p < a, a < p < p’, and p > p’, respectively. In eqns. (10) to (13) I , and K , are modified Bessel functions and E, is the Neuman number defined in eqn. (6). Also,

E , - 1 aza,2

F’(x) = ~ e(b, a)h(b, a ) ~ ’

I’

14 I E E Proc.-Microw. Antennas Propag., Vol. 141, No. I , February 1994

analysis of a dielectric resonator antenna in a cylindrical conducting cavity: hem modes

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