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MODE-MATCHING ANALYSIS OF SUBSTRATE-INTEGRATED WAVEGUIDE CIRCUITS

Jens Bornemann and Farzaneh Taringou

Department of Electrical and Computer Engineering, University of Victoria,

Victoria, BC, V8W 3P6, Canada

ABSTRACT

A mode-matching approach is presented for the analysis of

substrate-integrated waveguide

numerical technique takes advantage of recently developed

fabrication techniques employing rectangular-shaped via

holes. Discontinuity models

waveguides and sections with arbitrary numbers of vias are

presented and combined into a powerful analysis tool which

can be used straightforwardly for the design of SIW

components. The influence of the overall substrate width on

the circuit performance is investigated. It is found that the

computational domain can be significantly reduced without

impacting on the computed performances. A design example

involving a back-to-back impedance transformer is

presented. The results are verified by comparison with the

commercially available field solver CST Microwave Studio.

Index Terms— Substrate-integrated

technology, numerical modeling,

techniques, computer-aided design.

1. INTRODUCTION

Substrate-integrated waveguide (SIW) circuits form a

reasonable compromise between microstrip and waveguide

technologies [1]. Their advantages made an impact in the

lower millimeter-wave frequency range [2], [3] where

microstrip components are

waveguides too bulky.

Although a large variety of different SIW components

have been presented in the recent literature, actual circuit

designs involve commercial field solvers such as Ansoft

HFSS or CST Microwave Studio. While the accuracy of

such packages is undisputed, their numerical effort in circuit

optimization is tremendous and time-consuming. Thus other

techniques, which are more tailored to waveguide-type

modeling, have emerged. The Mode-Matching Technique

(MMT) is used for dispersion analysis of SIWs [4], and the

Boundary Integral − Resonant Mode Expansion (BI-RME)

method for equivalent-circuit extraction [5]. One of the

reasons for not using such techniques in the full-wave

synthesis and optimization of SIW circuits is the circular

shape of the via holes with respect to the Cartesian

coordinate system of the principle SIW discontinuities

(SIW) circuits. The

involving all-dielectric

waveguide

mode-matching

increasingly lossy and

which form the SIW component. With the emergence of

new fabrication techniques, however, more arbitrary via

shapes become feasible [2], [6], and rectangular and square

via shapes have successfully been implemented [7].

Therefore, this paper presents a MMT approach for the

analysis and design of SIW components with square via

holes. The numerical procedure follows fundamental MMT

principles [8] and uses an all-dielectric waveguide as a low-

reflective SIW feed [9].

2. THEORY

Fig. 1 shows a SIW formed by ten equally spaced square via

hole pairs with equivalent-width all-dielectric waveguide

ports at both ends.

Fig. 1. Substrate-integrated waveguide with square via holes and

all-dielectric waveguide ports.

In order to analyze such a circuit with a MMT

procedure, two basic discontinuities need to be solved: first,

the transition between two all-dielectric waveguides of

different width (Fig. 2a) and, secondly, a number of

rectangular vias embedded in an all-dielectric waveguide

(Fig. 2b).

Since the substrate height is usually much smaller than

the waveguide widths, a modal analysis based on TEm0

modes is sufficient. Following procedures in [8], the vector

potentials in regions ν=-1, 0, n (n=1to N) (c.f. Fig. 2) can be

written as

a

A Z Acosx

a

−

−−

⎡

⋅−

⎣

()()

111

1

hz

1

11

?

11

?

zz

2

F expjk zB expjk z

−−

?

−

?

−

−

?

−

⎧

⎨

⎩

⎫

⎬

⎭

π ⎛⎞

⎟

⎠

=+

⎜

⎝

⎤

⎦

++

∑

?

?

?

(1a)

IEEE CCECE 2011 - 000579

CCECE 2011 Niagara Falls, Canada

978-1-4244-9789-8/11/$26 ©2011 IEEE

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()()

0

hz

0

m

0

m

0

m

0

0

m

0

zm

0

m

0

zm

a

m

a

A Z A cosx

2

F expjkz B expjkz

⎧

⎨

⎩

⎫

⎬

⎭

π⎛⎞

⎟

⎠

=+

⎜

⎝

⎡

⎣

⎤

⎦

⋅−++

∑

(1b)

()

()()

n

hz

n

k

n

kn

k

n

n

k

n

zk

n

k

n

zk

k

a

A Z A cosxx

F expjk zB expjk z

⎧

⎨

⎩

⎫

⎬

⎭

π

=−

⎡

⎣

⎤

⎦

⋅−++

∑

(1c)

where F and B are the forward and backward travelling

wave amplitudes, respectively; kz are the propagation

constants, A the normalization coefficients and Z the wave

impedances, given by [8]

2

2

zir

i

0

i

zii

i

a

k

c

a

i

ωμ

2

A

a h

ν

1

Z

kY

ν

ν

ν

ν

π

ν

νν

⎛⎞

⎟

⎠

ωπ

⎛

⎜

⎝

⎞

⎟

⎠

= ε −⎜

⎝

=

==

(2);

c is the speed of light and h the substrate height.

(a)

(b)

Fig. 2. Discontinuities involved in the MMT process.

Considering the magnetic-wall boundary conditions for

the field-matching process in Fig. 2a, the coupling matrix

for this discontinuity is

M DiagY J Diag

=

?

{}( )

{}

10

m

Z

−

(3a)

with

( )

J

1

1

a2

0

1

m

a2

10

1 0

−

a

a2m

a

sinx sinx dx

a22

a a

−

−

+

−

−

−

⎧

⎨

⎩

⎫

⎬

⎭

(3b)

⎧

⎨

⎩

⎫

⎬

⎭

ππ⎛⎞

⎟

⎠

⎛

⎜

⎝

⎞

⎟

⎠

=++

⎜

⎝

∫

?

?

from which the modal scattering matrix follows

straightforwardly

1

11

0

21

S

F

⎢⎥

⎣

⎣⎦

1

12

0

22

SS

S

BF

B

−−

⎡

⎢

⎤

⎥

⎡

⎢

⎢

⎣

⎤

⎥

⎥

⎡

⎢

⎤

⎥

⎦

=

⎦ (4a)

[]

1

TT

11

1

TT

21 12

TT

122111

T

2212

S MMU MMU

S 2 MM

⎡

⎣

UMS

SM U S

−

S

SU M S

−

−

⎡

⎣

⎤

⎦

⎡

⎣

⎤

⎦

=+−

⎤

⎦

=+=

==

=−

(4b)

where U is the unit matrix, and T means transposed.

The discontinuity in Fig. 2b involves N coupling

matrices, capturing the coupling from region 0 to each

region n (n=1 to N). However, since the number of modes in

region 0 is higher than that in region n, the column-type

coupling matrices are arranged in one large matrix with the

single right side vector holding all forward and backward

traveling wave amplitudes Fn and Bn. The individual

coupling matrices are

m

M DiagYJ

=

{}( )

{}

n0nn

k

DiagZ

(5a)

with

( )

()

nn

n

xa

n

0

n

mk

x

0n

0 n

a a

a

2m

a

k

a

J sinx sinxx dx

2

+

⎧

⎨

⎩

⎫

⎬

⎭

⎧

⎨

⎩

⎫

⎬

⎭

ππ

⎛

⎜

⎝

⎞

⎟

⎠

=+−

∫

(5b)

With all N output-port amplitudes combined in a single-port

vector, the scattering matrix of the N-furcated waveguide

discontinuity is as in (4b). The final modal scattering matrix

of N via holes in an all-dielectric waveguide (Fig. 2b)

0

11v

0L

21v

S

F

⎢⎥

⎣

⎣⎦

is obtained by cascading a single diagonal matrix containing

the transmission parameters Dn of every single region n=1 to

N up to the center location (z=L/2) of the via holes (Fig. 2b)

⎧

=

⎨

⎩

and flipping the resulting scattering matrix, i.e., cascading

the same scattering matrix but with input and output ports

interchanged.

The final overall modal scattering matrix of the

structure in Fig. 1 is then obtained by following general

procedures of cascading scattering matrices, e.g. [8].

0

12v

S

S

0L

22v

S

BF

B

⎡

⎢

⎤

⎥

⎡

⎢

⎢

⎣

⎤

⎥

⎥

⎦

⎡

⎢

⎤

⎥

⎦

=

(6)

nn

zk

L

2

D Diag expjk

⎫

⎬

⎭ (7)

⎛

⎜

⎝

⎞

⎟

⎠

−

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3. RESULTS

The first step in designing a substrate-integrated waveguide

is to specify the substrate and the frequency range. For this

work, the substrate is chosen as RT Duroid 5880 with a

relative permittivity of εr=2.2 and a substrate height of

h=508μm. For the frequency range of 18 GHz to 28 GHz, a

SIW cutoff frequency of approximately 15 GHz was chosen

which, following guidelines in [10], results in circular vias

of diameters of 0.72mm and center-to-center (longitudinal)

spacings of 1.02mm. The (transverse) width of the SIW

measured between the centers of the via holes is

avia=7.28mm. The equivalent waveguide width of such an

arrangement is aequ=6.71mm according to [11]. In order to

employ the MMT procedure outlined in the previous

section, the circular vias are translated to square ones based

on the assumption that the area of the via hole be preserved.

Thus all via holes used in this work have a cross section of

0.64mm x 0.64 mm. Moreover, all examples presented

below have been obtained using all modes up to 300 GHz

for the modeling of discontinuities and modes up to 150

GHz for the combination of modal scattering matrices.

The quality of a SIW circuit and its transition to an all-

dielectric waveguide of width a-1=aequ (Fig. 2a) is measured

by the reflection coefficient seen into the ports. Fig. 3

depicts the performance of the circuit shown in Fig. 1.

Fig. 3. Performance of the circuit shown in Fig. 1 and comparison

between this method (solid lines) and CST (dotted lines).

The reflection coefficient is -50 dB or better, and the

results are confirmed by the commercial field solver CST

Microwave Studio. For the CST simulations, the accuracy is

set to the minimum limit of -80 dB. The agreement between

the MMT approach presented here and the CST results is

excellent and remarkable, especially considering the fact

that the two curves that are compared are both at such a low

level. This result instills confidence in the design of future

SIW components. It is worthwhile to point out that a

different computation of the equivalent waveguide width as

recently published in [12] results in a reflection level that is

10 dB higher than that shown in Fig. 3 and thus is not as

well matched to the SIW as the results obtained from [11].

Another critical point involving the computational

domain is associated with the substrate width of the via-

holed section (a0 in Fig. 2). Therefore, the performance of

the circuit in Fig. 3 has been re-evaluated using different

substrate widths normalized to the center-to-center SIW

width avia. The resulting performances are shown in Fig. 4. It

is observed that the outer substrate walls can be located very

closely to the via holes and that the performance is nearly

independent of their actual location. Thus the computational

domain for SIW circuit modeling can be reduced. As the

substate wall distance is increased, a limit is reached

(a0/avia=3 in Fig. 4) when fields are propagating outside of

the SIW and recombine with those in the actual SIW. Then

resonance peaks as the one observed at 24.6 GHz in Fig. 4

occur and will affect the accuracy of the MMT modeling

process. As the substrate width is further increased (not

shown here), more resonance peaks appear and make it

difficult to obtain a reliable performance evaluation of a

SIW circuit.

Fig. 4. Influence of the substrate width on the performance of the

circuit in Fig. 1.

As a design example, Fig. 5 shows the performance of a

back-to-back connection of a two-section impedance

transformer in SIW technology. The transformer has been

designed using a standard H-plane waveguide transformer

synthesis routine [8]. As the exact corners and straight walls

of the waveguide model cannot exactly be reproduced by

the via holes in the SIW component, fine optimization of

transformer lengths and width becomes necessary. For this

purpose, additional vias are added at the transitions between

different waveguide widths so that the width can be

optimized without creating large spaces between via holes.

As shown in Fig. 5b, excellent agreement is observed

between the results of the MMT approach and those of CST,

thus verifying the correctness of the MMT modeling process

presented in this paper. Note that all S11 levels are below -40

dB.

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(a)

(b)

Fig. 5. A back-to-back two-section impedance transformer (a) and

performance comparison (b) between this method (solid lines) and

CST (dotted lines).

4. CONCLUSIONS

A modeling approach for the analysis and design of SIW

circuits is presented. The mode-matching procedure takes

advantage of via holes of square cross sections and captures

the modal interactions of all individual discontinuities

involved. An advantage of this technique is that the actual

substrate width can be reduced to extend just a fraction

beyond the vias which gives rise to a reduction in the overall

computational domain.

Examples include a SIW section fed by all-dielectric

waveguide ports and a back-to-back connection of a two-

section impedance transformer. Excellent agreement with

results obtained from CST Microwave Studio verifies the

modeling process presented in this paper.

5. REFERENCES

[1] K. Wu, D. Deslandes, and Y. Cassivi, “The substrate integrated

circuits - A new concept for high-frequency electronics and

optoelectronics,” Proc. 6th Int. Conf. Telecommunications in

Modern Satellite, Cable and Broadcasting Service (TELSIKS

2003), vol. 1, pp. PIII-PX, Oct. 2003.

[2] E. Moldovan, R.G. Bosisio, and K. Wu, “W-band multiport

substrate-integrated waveguide circuits,” IEEE Trans. Microwave

Theory & Tech., vol. 54, pp. 625-632, February 2006.

[3] D. Stephens, P.R. Young, and I.D. Robertson, “Millimeter-

wave substrate integrated

photoimageable thick-film technology,” IEEE Trans. Microwave

Theory Tech., Vol. 53, pp. 3832-3838, Dec. 2005.

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waveguides and filters in

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