A Mode-matching Approach for the Analysis and Design of ...

Frequenz, Vol. 65 (2011), pp. 287–292 Copyright © 2011 De Gruyter. DOI 10.1515/FREQ.2011.042

A Mode-matching Approach for the Analysis and Designof Substrate-integrated Waveguide Components

Jens Bornemann,1;� Farzaneh Taringou1 andZamzam Kordiboroujeni1

1 Department of Electrical and Computer Engineering,University of Victoria, Canada

Abstract. A mode-matching approach is presented that al-lows a fast and accurate analysis of substrate-integratedwave-guide components with rectangular/square via holes.Models for several discontinuities are discussed which in-clude microstrip as well as all-dielectric wave-guide feeds.The numerical technique is verified by comparison withcommercially available field solvers. An example of a four-pole dual-mode filter in substrate-integrated wave-guidetechnology illustrates the capabilities of the approach.

Keywords. Mode-matching techniques, substrate-integrated wave-guide technology, numerical modeling,computer-aided design.

PACS®(2010). 84.40.Az, 84.40.DC, 85.40.Bh.

1 Introduction

Numerical modelling approaches based on modal expan-sion techniques have long been known as reliable and com-putationally efficient analysis tools for waveguide-basedtransmission media [1, 2]. Especially with a reduced modeset, such as in H-plane waveguide circuits [3], E-plane[4] or iris-type configurations [5], the classical Mode-Matching Technique (MMT) presents a powerful methodfor the computer-aided analysis and design of air-filled ordielectric-slab-loaded waveguide components [6, 7].

More recently, substrate-integrated circuits emerged[8]. Especially the Substrate-Integrated Waveguide (SIW)presents a now widely accepted circuit compromise in thelower millimetre-wave regime where microstrip compo-nents are increasingly lossy and waveguides too bulky andtoo expensive. Thus a large variety of SIW circuits hasappeared in the literature covering frequency ranges from

Corresponding author: Jens Bornemann, Department of Electricaland Computer Engineering, University of Victoria, Victoria, B.C.,V8W 3P6, Canada; E-mail: [email protected].

Received: June 20, 2011.

5 GHz [9], over 30 and 60 GHz [10, 11], to above 100 GHz[12, 13].

SIW circuit designs are usually carried out within com-mercial field solvers such as CST Microwave Studio or An-soft HFSS. They provide accurate responses for the analy-sis of a given SIW structure within reasonable time frames.In optimization mode, however, the numerical effort ofsuch packages is tremendous and time-consuming, espe-cially for frequency-sensitive structures such as bandpassfilters. Thus other techniques, which are more geared to-wards waveguide-type modeling, are in demand. So far, theMMT is used for dispersion analysis [14], and the Bound-ary Integral-Resonant Mode Expansion (BI-RME) methodfor equivalent-circuit extraction [15]. Although the SIW isan H-plane circuit and would lend itself straightforwardlyto MMT analysis and design, the circular via holes appliedin common integrated-circuit fabrication techniques com-plicate the MMT formulation and thus increase computa-tional complexity. However, with the emergence of newfabrication techniques, especially laser drilling of via holes,more arbitrary via shapes become feasible [12], and rectan-gular/square via shapes have been implemented in [13].

Therefore, this paper presents a mode-matching ap-proach for the analysis and design of SIW components withsquare via holes. The numerical procedure follows fun-damental MMT principles in [16] and includes SIW feed-ing by microstrip transformers [17] or all-dielectric wave-guide [18].

2 Theory

Figure 1 shows a SIW consisting of ten pairs of equallyspaced square via holes. It is fed by a microstrip line anda short microstrip transformer [19] at the input and an all-dielectric waveguide port at the output.

In order to model such a circuit, we have to considerseveral different discontinuities. First, a microstrip-to-microstrip discontinuity to analyse the transformer whichwill be represented by staircasing in the MMT procedure;secondly, a microstrip-to-dielectric waveguide discontinu-ity to capture the step from the microstrip line to the di-electric waveguide containing the via holes; thirdly, the dis-continuity from the dielectric waveguide to an N -furcatedwaveguide, representing a via-holed section, and back to thedielectric waveguide; and fourthly, a discontinuity betweentwo dielectric waveguides to model the output port.

AUTHOR’S COPY | AUTORENEXEMPLAR


288 J. Bornemann, F. Taringou and Z. Kordiboroujeni

Figure 1. Substrate-integrated waveguide with microstripinput and all-dielectric waveguide output ports.

As pointed out in Section 1, the MMT is especially effi-cient if a reduced set of modes can be used. Due to the smallthickness of the substrate, only TEm0 modes are consideredin all-dielectric waveguides. In all microstrip sections, thequasi TEM as well as all applicable TEm0 modes are con-sidered. Since the TEM mode is the lowest possible TMmode, the overall electromagnetic field can be calculatedfrom two axial vector potential components, Aez and Ahz ,as

EE D1

j!"r � r � .Aez Eez/ � r � .Ahz Eez/;

EH D1

j!�r � r � .Ahz Eez/Cr � .Aez Eez/;

(1)

where the TE modes are represented by

A�hz DXm

qZ�hm

a�

m�

s2

a�bcos°m�a�.x � x�/

±„ ƒ‚ …

T �hm.x/

� ŒF �m exp.�jk�zmz/C B�m exp.Cjk�zmz/�

(2)

and the TEM mode by

A�ez DqY�`

ypa�b„ƒ‚…

T �e`.y/

ŒF �` exp.�jk�z`z/ � B�` exp.Cjk�z`z/�:

(3)

In (2) and (3), � identifies the section of equivalent wave-guide width a� with x� as its lower coordinate, b is the sub-strate height, F and B are the forward and backward travel-ling wave amplitudes, respectively, kz’s are the propagationconstants

k�zi D

s�!c

�2"r eff �

� i�a�

�2; (4)

where i D 0 for the TEM mode, c is the speed of light infree space, and the wave impedances are given by

Z�hi D!�0

k�ziD

1

Y �hi

; Y �` D!"0"r eff

k�z`

D1

Z�`

: (5)

Note that the vector potential components in (2) an (3) arepower normalized so that the coupling integrals, e.g., for thecase shown in Figure 2 can be computed from

.Jl l/0;0 D

ZA1

.rT Ie`/ � .rT

IIe`/da;

.Jlh/0;k D

ZA1

.rT Ie`/ � .rT

IIhk � Eez/da;

.Jhl /m;0 D

ZA1

.rT Ihm � Eez/ � .rT

IIe`/da;

.Jhh/m;k D

ZA1

.rT Ihm/ � .rT

IIhk/da;

(6)

where the TEM-to-TEM coupling occupies the upper leftmatrix element in J.

Once the coupling integrals J are obtained, the couplingmatrix M follows from

M D Diag®qY I`m

¯.J/Diag

®qZII`k

¯: (7)

This technique will now be applied to the individual discon-tinuities involved in Figure 1. For individual combinationsof modal scattering matrices and adding lengths of certainsections, the reader is referred to [16].

2.1 Microstrip-to-microstrip Discontinuity

Figure 2 shows the discontinuity and the boundary condi-tions involved. Note that widths a� represent the equiva-lent and frequency-dependent waveguide widths of the ac-tual microstrip lines with finite conductor thickness andfrequency-dependent permittivity. Moreover, this discon-tinuity includes models to shift the reference plane due tofringing fields. For details on microstrip transmission lineparameters, the reader is referred to [20].

For this case, the specific coupling integrals according to(2), (3) and (6) are

.Jl l/0;0 D

ra1

a2; .Jhl /m;0 D 0;

.Jlh/0;k D�p2

pa1a2

Z Ca1=2�a1=2

cos°k�a2

�x C

a2

2

�±dx;

.Jh;h/m;k D2

pa1a2

Z Ca1=2�a1=2

cos°m�a1

�x C

a1

2

�±� cos

°k�a2

�x C

a2

2

�±dx:

(8)



Mode Matching for Substrate-integrated Waveguide Components 289

Figure 2. Microstrip-to-microstrip discontinuity.

Note that Th in (2) depends only on x, whereas Te in (3)depends only on y.

The modal scattering matrix for Figure 2 is defined as"BI

FII

#D

"S11 S12S21 S22

#"FI

BII

#(9)

and once the coupling matrix M is computed from (7) and(8), the scattering submatrices follow as

S11 D ŒMMT C U��1�ŒMMT � U�;

S12 D 2ŒMMT C U��1M D ST21;

S21 DMT ŒU � S11� D ST12;

S22 D U �MT S12;

(10)

where U denotes the unit matrix.

2.2 Microstrip-to-dielectric-waveguide Discontinuity

Figure 3 shows the boundary conditions of the microstrip todielectric waveguide discontinuity. Width a1 is the equiva-lent waveguide width of the actual microstrip line [20].

For this case the coupling integrals are

.Jlh/0;k D

p2

pa1a2

Z Ca1=2�a1=2

sin°k�a2

�x C

a2

2

�±dx

.Jh;h/m;k D�2pa1a2

Z Ca1=2�a1=2

cos°m�a1

�x C

a1

2

�±� sin

°k�a2

�x C

a2

2

�±dx

(11)

and the remaining procedure follows (7) and (10) to de-termine the modal scattering matrix. Note that the all-dielectric waveguide supports no TEM mode; thus only twodifferent coupling integrals have to be considered comparedto four in (8).

Figure 3. Microstrip-to-all-dielectric-waveguide disconti-nuity.

2.3 Discontinuity Between All-dielectric Waveguides

Figure 4 shows the boundary conditions of a discontinuitybetween two all-dielectric waveguides as required, e.g., atthe output of the circuit in Figure 1. Since the practicalrealization of this circuit would be on a single rectangularsubstrate, the interface between the two waveguides is se-lected as a magnetic wall as in the two previous discontinu-ities involving microstrip lines. The coupling integrals forthis case are simply those of the TEm0 modes on both sidesof the discontinuity

.Jh;h/m;k D2

pa1a2

Z Ca1=2�a1=2

sin°m�a1

�x C

a1

2

�±� sin

°k�a2

�x C

a2

2

�±dx;

(12)

Figure 4. Discontinuity between two all-dielctric wave-guides.




because neither sections support TEM-mode propagation.The modal scattering matrix is again computed with (7) and(10).

2.4 Via Holes Within All-dielectric Waveguide

This discontinuity is different from the previous ones as thedielectric waveguide with N � 1 via holes represents an N -furcated waveguide according to Figure 5. Note that thelength of the via holes is exaggerated in order to fit in therelated wave amplitudes.

Coupling integrals for this discontinuity involve cou-pling from the dielectric waveguide (section 0) to each ofthe N -furcated waveguides (sections n/. Thus N couplingintegrals are obtained.

.J nhh/mk D2

pa0an

Z xnCan

xn

sin°m�a0

�x C

a0

2

�±� sin

°k�an

�x � xn

�±dx:

(13)

Applying (7) to each of the N coupling integrals, the elec-tric field matrix equation, e.g. for N D 3, can be arrangedas follows

�F0 C B0

�D�

M1 M2 M3�„ ƒ‚ …

M

264F1 C B1

F2 C B2

F3 C B3

375; (14)

from which the scattering matrix of the discontinuity atz D 0 and z D L can be obtained according to (10). Con-sidering the diagonal matrix between the two discontinu-ities, which is comprised of individual matrices

Dn D Diag¹exp.�jknzkL/º; (15)

the overall modal scattering matrix of a via-hole section iscomputed according to [16].

p

Figure 5. Discontinuities between all-dielectric waveguideand N -furcated waveguide formed by N � 1 via holes.

This completes the presentation of individual disconti-nuities. The only issue that has been left out is a possibleoff-centre connection between two sections. Such an offsetis easily incorporated as a shift in the sin and cos functionsinvolved in the coupling integrals.

3 Results

Figure 6 shows an analysis of 10 equally spaced via-hole pairs with microstrip input and all-dielectric wave-guide output according to Figure 1. The substrate is cho-sen as RT/duroid 5880 with "r D 2:2, substrate heightb D 508 µm and metallization thickness t D 17 µm. Inthe mode-matching approach, the microstrip transformerat the input is approximated by five individual sections,whereas in CST and HFSS, it is modelled as shown. Verygood agreement is observed between this approach (MMT),HFSS and CST, with the MMT minimum located betweenthose of CST and HFSS.

A four-pole dual-mode filter in SIW technology is shownin Figure 7(a). The original waveguide design is from [21]and has been redesigned in all-dielectric waveguide for the20 GHz frequency range and transferred to SIW technologyusing RT/duroid 6002 substrate. The SIW filter is then anal-ysed and fine-optimized using the method presented here.Figure 7(b) shows a direct comparison between this method(MMT) and the resonant-mode field solver in CST. Excel-lent agreement is observed but the MMT code is faster thanCST by a factor of seven. An analysis including all dielec-tric and metallic losses (not shown here) reveals a midbandinsertion loss of 0.93 dB which, according to approxima-tions in [22], determines the unloaded Q factor to about 550.

The same filter, but with microstrip ports, is shown inFigure 8(a). Figure 8(b) compares the results obtained with

Figure 6. Comparison between results obtained with thismethod (MMT), HFSS and CST at the example of Fig-ure 1.



Mode Matching for Substrate-integrated Waveguide Components 291

(a)

(b)

Figure 7. Four-pole dual-mode filter in SIW technologywith all-dielectric waveguide ports (a) and performancecomparison between this method (MMT) and CST’sresonant-mode analysis (b).

(a)

( ) y ( )

(b)

Figure 8. Four-pole dual-mode filter in SIW technologywith microstrip ports (a) and performance comparison be-tween this method (MMT) and CST’s frequency domainsolver (b).

this method (MMT) and the frequency domain solver ofCST. Some differences are observed in the return loss per-formance. However, the discrepancies are small and occurbelow the �20 dB level. Therefore, the modelling proce-dure involving microstrip ports in MMT is verified by theCST results.

4 Conclusions

Substrate-integrated waveguide components with rectangu-lar/square via holes are effectively and accurately modelledby a mode-matching approach. Two different feeds are con-sidered, all-dielectric waveguide feeds and microstrip trans-formers, which allow the design to be carried out for inte-gration with other SIW circuits or as a stand-alone com-ponent. This is demonstrated for a four-pole dual-modeSIW filter example. The results obtained with the mode-matching approach are verified by comparison with data ofcommercially available field solvers.

Acknowledgments

The authors wish to acknowledge support for this workfrom the Natural Sciences and Engineering Research Coun-cil of Canada.

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