Resonant frequencies of post-wall waveguide cavities

J.R. Bray and L. Roy

Abstract: A new method is presented for predicting the resonant frequencies of post-wall rectangular waveguide cavities, in which the narrow walls of the guide are composed of rows of vias. By interpreting the fields inside the guide as a superposition of plane waves, the surface impedance of a row of wires, when illuminated by a plane wave, is used to model the narrow walls of the waveguide. The derivation leads to equivalent rectangular Waveguide dimensions that are then used to calculate the resonant frequencies of the cavity. Theoretical results are compared with the observed T E , h resonant frequencies of several post-wall waveguide cavities fabricated in low temperature cofired ceramic (LTCC). The results confirm that the method improves the resonant frequency prediction when the pitch of the vias is at least ten times greater than their radius.

1 Introduction

The post-wall Waveguide (PWWG), shown in Fig. I , closely resembles a dielectric-filled rectangular waveguide (RWG) except that the narrow walls of the guide consist of rows of vias instead of contiguous metal [I]. This type of transmission medium is also known as laminated waveguide [2] and substrate integrated rectangular waveguide [3]. The construction allows for three-dimensional waveguide com- ponents, such as resonators and antennas, to be fully embedded within common substrate material [MI. Grow- ing interest in the medium has provided the impetus to characterise the dispersion of post-wall waveguide, which is similar to that of a rectangular waveguide when the via pitch p is kept sufficiently small [ I , 2, 7. The lower limit of the via pitch is ultimately set by the fabrication laboratory and may or may not satisfy the condition. It is therefore important to be able to predict the behaviour of PWWG components when the pitch is relatively large.

Although methods of determining the propagation constant of an arbitrary PWWG transmission line have

recently been presented, they are fairly complex and require numerical electromagnetic computation [ I , q. Further, they do not address PWWG cavities, in which the end walls of the guide, again composed of vias, also affect the resonant frequency. In this paper, a simple analytical model using an equivalent impedance for a row of vias is used to predict the resonant frequencies of PWWG cavities.

2 Methodology

It is well-known that the TE mode fields within a RWG are identical to the interference pattem produced by two perpendicularly polarised plane waves, each travelling at an angle 0 with respect to the z-axis, as illustrated in Fig. 2. When one plane wave lags the other by IXO", the complete expression for the electric field is

E = y 2 ~ ~ e - ' ~ ~ ~ ~ s i n ( ~ sin 0) (1) where Eo is an arbitrary field magnitude and k= 2x/i is the wavenumber in the medium. Spatial nulls exist colinear to the z-axis, where kxsin0 = 0, kx, &2?r, etc. The TE10

Fig. 1 a, via radius r, and via pitch p

Post-wall wuueguide illustrating viu centre-to-centre width

0 IEE, 2003 IEE Proeeedbtgs online no. 20030772 doi: lO.I049/ipmap:2M)30772 Paper f iot received 11th O n o k r 2002 and in revisd form 29th May 2003. Online publishing date: 29 August 2003 The authors arc with the Department of Electroics, Carleton University, I125 Colonel By DCve, Ottawa. ON, KIS 586, Canada

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Fig. 2 Interfering plane walies propagating ut un angle 0 with respect to the z-a.rir

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mode corresponds to the space between the neighbouring nulls at kxsin0 = 0 and n. Conducting walls may be placed along electric field nulls without perturbing the fields, hence the width of the guide must be chosen as

U = n/ksinO (2) whereas the broad walls of the RWG may be plaoed arbitrarily since they are perpendicular to the electric field.

2.1 Grid impedance method The problem of a RWG having a row of wires for its narrow wall has been dealt with before, but in the wntext of narrow wall (Transvar) couplers, in which the row of wires is used to couple two colinear waveguides [SI. Given the equivalence, the circuit model of a row of wires illuminated by a perpendicularly polarised plane wave may be used to represent the narrow walls of the guide. When the radius of the wires r is much smaller than their pitchp, the wires may be modelled as a single inductor, as first proposed by Macfarlane in 1946 [9]. When this restriction is not met, the more general model proposed by Marcuvitz may be used, shown in Fig. 3 [IO]. Note that the equivalent circuit is referenced with respect to propagation in the x-direction, where the angle of incidence is denoted by 4 = n/2-S and the characteristic impedance is given by 2, = q/cos 4, where q=,/&/t;) is the wave impedance of the medium. The equations for the circuit elements, originally obtained using the Rayleigh-Ritz variational method, are repeated here as (3) and (4) [IO]:

z* i 2nr 2"-

2

(4)

The equations are valid over the rangep(l +sinq$)/A<l, or more restrictively, p/i<0.5. Further, the equations are strictly valid for small obstacles only, r<< 2. and p > lor, over which range the error of the model is estimated to be less than 10% [IO].

T 7

Fig. 3 wires w d ifs equivalent circuit

Perpendicularly polarisedplune u'uve incident on a row of

Once the reactance values of the T-network are known, network analysis is applied to determine the Z,-normalised input impedance to the vias, ZL, which is defined at the plane passing through the centres of the vias, denoted by T in Fig. 3. The surface impedance of a conductive RWG narrow wall has the form Z,= Rs(l +]I, where R, is the surface resistance, hence the complex zL value may be interpreted as the input impedance of a conducting narrow wall that is offset by some distance Au from the T-plane, as

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a - = a + Z A a 14 I4 a 'I 'I

Fig. 4 Equiualenf rectwgulur wuveguide luzving an effective width U = U + 2 A a

shown in Fig. 4. The normalised input impedance is given by

where z, = rLI +I), r, = RJZ,, and A = tanpxAu. The real and imaginary parts of ( 5 ) yield the simultaneous equations r = rs[ 1 + A(r + x)] and r,(rA-xA- 1) = A-x, which, when combined, lead to the quadratic equation:

A2(x+ r) + A ( I - r2 -x2) + ( r -x) = 0 (6) the solution of which leads directly to the evaluation of Au:

( 7 ) arctan(A) - arctan(A) Aa=---

Pr ksin 9

The sign of Au is usually positive forp> lor, indicating that the effective width of an equivalent RWG, U* = a + 2Aa, is larger than the physical width of the PWWG. When the new width value is used in (2), it alters the launch angle 0 so the above procedure must be iterated until the effective width converges. Fortunately, only a few iterations are required before the change in Au becomes imperceptible.

2.2 Other characterisation methods Although the above grid impedance method is applicable within the limit p r lor, another rule may be used when the via pitch violates this inequality. Using a boundary-integral numerical method, Cassivi has obtained a simple curve- fitted expression for calculating the effective width of a PWWG transmission line when p<8r, repeated here as (8) ~71:

which is quoted as yielding errors of better than f 5% over the range p/%<O.5 and p<8r [A. Note that the effective width of the guide is smaller than its physical width when p<8r. Equation (8) obviously fails when p> I Or because, as we have shown, the effective width of the guide is larger than its physical width over this range. It is of interest to note that (8) is in good agreement with yet another numerical analysis of a post-wall waveguide that uses Galerkin's method of moments [I]. In [I], the results indicate that, although the equivalent waveguide width is smaller when p < lor, it nonetheless becomes larger as the pitch is increased and is very nearly equal to the actual waveguide width U' = a when p = lor.

2.3 Extension to PWWG cavities The grid impedance method may be extended to the analysis of post-wall waveguide cavities. Using four interfering plane waves instead of two, it may be shown

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that the resulting net electric field in space obeys

E =y4E~sin(kzcos0)sin(lc:sinB) (9) Again, nulls OCCUI along lines that are colinear to the z-axis when kxsinB=O, f n , f2n , etc., hence the TElo mode is again enclosed by narrow walls according to (2). Nulls also occur colinear to the x-axis when kzcos0 = 0, f x, i 2n, etc. If the physical cavity exists along the positive z-axis, the T E j h mode requires that the length. of the guide in z be given by

m 0 m 4 -

s 2 -

Resonance occurs when (2) and (IO) are satisfied simulta- neously using the same launch angle 0, and the combination of these equations yields the resonance condition for 8:

A A A A

0 = arctan - ( r fa)

Once again, the effective RWG width extension Aa is found using the method described previously. In addition, an etkctive RWG length extension Ad is found simultaneously using the same (3) and (4) hut by replacing 4 with 0, since the incident angle on the narrow wall is 4 while the incident angle on the end wall is 0, as shown in Fig. 5. The new effective width a and length 6 are then substituted into (I 1) and the process is iterated until Au and Ad converge. Again, very few iterations are required.

Fig. 5 and length extensions

3

Various post-wall waveguide TElh cavities have been fabricated using Heraeus a 2 0 0 0 LTCC tape, the para- meters of which are listed in Table 1. A total of nine resonators have been fabricated, corresponding to all permutations of the d and p variables shown in Table I . Equation (8) is expected to be valid when p<8r , whereas the p = 10r value represents the lower limit of the small- obstacle range over which the proposed grid impedance

Post-~~all waveguide cavity with equitialent RWC width

Fabricated LTCC post-wall waveguide cavities

Table 1: Heraeus CT2000 post-wall waveguide cavity circuit parameters

Dielectric Guide Guide Via radius Via pitch Cavity constant height width r, mm p, mm length Er b, mm a, mm d, mm

0.233 p, = 0.50 d, =5.0

9.1 f3layersl 2.50 0.10 h=0 .75 d2=6.5

h=1.00 d3=10.5

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model is expected to yield small errors [IO]. The cavities are excited by a small waveguide probe that consists of a via descending into the guide through a small circular aperture in the upper broad waveguide wall, as illustrated in Fig. 6. The waveguide prohe emerges at the surface of the LTCC module where it is contacted by coplanar probes. Each cavity has two such probes, allowing both reflection and transmission measurements to be taken. Figure 7 shows the transmission response of the (d, p ) = (6.5 mm, 1 .OO mm) resonator over the 20-45 GHz hand. The nine cavities have yielded a total of 54 measured resonances over the 20- 45 GHz range. Figure 8 plots the resonant frequency error of the three analytical methods for the p = 5r case: using the grid impedance method, using (8) (for both the end walls and the narrow walls), and using no correction at all. The

--II

I , ‘ Id , I I

I I

Fig. 6 ‘Cross-sectional view of the waceguide excitation showing the top layer coplanar probe pads

frequency, GHz

Fig. 7 resonator

Trmsmir.yion coeficient of the (4 p ) = (6.5”. 1.00mm)

20 30 40 50 frequency, GHz

Fig. 8 Percentage error between the memured and calculated resonant frequencies for the three Hrraew CT2wD PWWG resonatar.7 having p = 5r

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pitch of p = 5r is well outside the range of the grid impedance method, which yields very poor estimates of the resonant frequency. However, the pitch is within the p<Xr limit of (X), which yields excellent estimates, well within

1 %. The calculated values without correction consistently underestimate the resonant frequency, indicating that the fields are contained in a region that is smaller than the centre-toantre spacing between the via posts. Figure 9 plots the errors for the intermediate pitch value of p = 7.5. Given that the grid impedance pitch condition is still violated, this method again yields poor values. Although its condition of p<8r is still satisfied, (8) no longer yields the best resonant frequency estimates, although its errors are still within the *5% range quoted in [fl. This time it is the uncorrected resonant frequency calculation that yields the best estimates, all within i 1 %, indicating that the fields inside the cavity are bound very nearly to the exact centres of the via posts. Finally, Fig. IO plots the errors for the p = 10r case. As expected, the grid impedance now offers the best resonant frequency estimate across the band, since p = 10r is now within this method's range of applicability

6

using grid impedance method

no Correction # 4

20 30 40 50

frequency. GHL

Fig. 9 Percentuqe error between the meanired and calculated resonant frequencies for the three H e r a m CROOO PWWG Ieson~lurs huliny p = 7 5

using grid impedance method

no Correction

I n

and is outside that of (8) @<8r) , which now yields the worst estimates. It should be noted, however, that the errors at low mode orders are comparatively high, well beyond the

1% previously achieved, which is likely due to thep> 10r condition being only exactly satisfied for this particular case.

4 Conclusion

Three simple methods have been used to predict the resonant frequencies of post-wall waveguide cavities: a closed-form equation based on integral method analyses, no correction, and the proposed grid impedance method. The calculated values have been verified by comparing them to the observed frequencies of nine different LTCC cavities having three different via pitch values. For values of p<7.5r, the integral method equation yields the best estimates, whereas the grid impedance method improves the resonant frequency prediction when the via pitch is at least ten times greater than the radius. In the intermediate range of 7.5r<p< lor , the physical centre-to-centre via width of the guide (using no correction) yields the best resonant frequency estimates.

5 Acknowledgment

The authors would like to thank the staff of the Manufacturing Technology Group of VIT Electronics, Oulu, Finland, for fabricating the LTCC circuits.

6 References

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4

".. 1-26?

3 Maas, S . , Delacueva, J., Li, 1.. and white, S.: 'A low wst cavity stabilized ~~ 5.8GHz oscillator nalired in LTCC', Microw,. J , 2001. 44. (4), pp. 13W134 Litzeenkrger. J., Clenet, M., Maen, G.A., and Antar, Y.M.M.: 'A new antenna implemented in laminated waveguide'. Presented at 9th hi. Symp. an Antenna icchnolom and apptid electramagnetics. ANTEMOZ, St. Hubert, Que&, Canada, July-August 2002,

4

".. 1-26? pp ,- <"<

5 Hill, M.J.. Ziolkowski, R.W.. and Papapolymeroq J.: 'Simulated and measured results from a Duroid-based planar MBG cavity resonator filter', IEEE Micrr,~ Cnrided Wme b i t . , 2wO. 10, (12). pp. 52t-530 Hill, M.J.. Ziolkowski, R.W., and Papapolymerou. J.: 'A highQ reconfigurable planar EBG cavity resonator', IEEE Microw. Gulded Woce k i t . , 2001, 11, (6). pp. 25S257 Cassivi, Y., Pemgeni, L.. Arcioni, P., Bressan, M., Wu, K.. and Gmdauro, G.: ' D q x s ~ m charactenstics of substrate integrated mtangular waveguide', IEEE Microw W i d Compos Lex, 2W2. 12,

Tomiyasu, K., and Cohn. S.B.: 'The transvar directional coupler', h o c . IRE, 1953, 41, (July), pp. 922-926 Macfarlane. G.G.: 'Surface impedance of an infinite parallel-\rire grid at oblique angles of incidence', J. IEE, 1946, 93, part IIIA, (In). pp. 152H527 Marcuvitz, N.: 'Waveguide handbook' (McGraw-Hill, New York,

6

I

(9), pp. 333-335 8

9

10 MI), VOI. in, pp. 2 8 ~ 2 8 9

~

20 30 40 50

frequency, GHr

Fig. 10 Percentage error between the measured and culculated resonant frequencies f o r the rhree Herueus CROOO PWWG reomtors Iiuainy p = IOr

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