XIX International Symposium on Theoretical Electrical EngineeringJuly 16 − 19, 2017, Ilmenau, Germany TO3-3

Abstract— A semi-analytical method to analyze post-wall waveguides and circuits is presented. The propagation constant of the

fundamental TE mode, the attenuation constant due to transversal leakage loss and the effective width of an equivalent

rectangular waveguide are calculated. The S-parameters of post-wall passive circuits such as e.g. bandpass filters are analyzed

using the image theory combined with the lattice sums technique.

Index Terms— compact bandpass filters, modal analysis, periodic structures, post-wall waveguide.

I. INTRODUCTION

The post-wall waveguide is formed by periodically

distributed (usually metallic) posts or posts with a high

dielectric permittivity [1]. It allows the “planarization” of

non-planar structures such as conventional rectangular

waveguides and they can completely be integrated together

with planar structures onto the same substrate with the same

processing or fabrication techniques. Functional post-wall

circuits are designed by inserting additional metallic or

dielectric posts into the post-wall waveguide.

II. FORMULATION AND DISCUSSIONS

The post-wall waveguide is confined by two N-layered

square lattice of conducting circular posts (each forming a

planar lattice) embedded in a dielectric substrate that is

vertically bounded by two parallel conducting plates

(Fig.1). The electromagnetic fields are uniform in the y-

direction ( 0y ) and the dominant mode 10TE is

excited.

If we assume a longitudinal field variation of ei z, the

guided wave propagating within the waveguide channel

/ 2 ax is expressed as follows:

+( , ) ( , ) ( , ) U Uy x z x z x zE c c (1)

with

( , ) exp ( / 2) exp( ) mx z i x a i z U (2)

( 0, 1, 2, )c c (3)

where 2 2

02 / , , ,s s sh k k

s is the permittivity of the dielectric substrate, stands

for the propagation constant, and c are the column

vectors whose elements represent the amplitudes of the

transversally up-going and down-going -th Floquet

mode.

The dispersion equation to determine the mode

propagation constant is expressed as follows [2]:

det[ ( , ) ( , )] 0N I W R (4)

where the signs stand for even (–) and odd (+) modes,

respectively, ,( )N R denotes the generalized reflection

matrix of the confining N-layered periodic arrays of

circular rods and ( , ) W denotes the phase shift along x-

axis. Solving dispersion equation based on a perturbation

analysis, the attenuation constant (due to the leakage loss)

and the effective width of the equivalent rectangular

waveguide are calculated. To validate the proposed

formalism, numerical examples for post-wall waveguides

and post-wall waveguide-based passive circuit, such as

bandpass filter, are presented and compared to those

reported in the references yielding a perfect agreement

between the S-parameters over a wide frequency range. It

is important to mention that the proposed semi-analytical

formulation is not time-consuming, which could be

considered as a major advantage of the proposed

formulation. The proposed method is a very apt tool for

designing advanced multi-functional, ultra-compact

submillimeter passive circuit devices.

REFERENCES

[1] D. Deslandes and K. Wu, “Accurate modeling, wave mechanisms,

and design considerations of a substrate integrated waveguide,”

IEEE Trans. Microwave Theory Tech., vol. 54, no. 6, pp. 2516-

2526, 2006.

[2] K. Yasumoto, H. Maeda and V. Jandieri, “Analysis of Post-Wall

Waveguides Using a Model of Two-Dimensional Photonic Crystal

Waveguides,“ Proceedings of the IEEE International Conference

on Signal Processing and Communication (ICSC-2015), Noida,

India, pp. 74-79, April, 2015.

Modal Analysis of Post-Wall Waveguides and

Compact Circuits Vakhtang Jandieri1, Hiroshi Maeda2, Kiyotoshi Yasumoto2 and Daniel Erni1

1General and Theoretical Electrical Engineering (ATE), Faculty of Engineering, University of Duisburg-Essen, and

CENIDE – Center for Nanointegratin Duisburg-Essen, D-47048 Duisburg, Germany 2 Department of Information and Communication Engineering, Fukuoka Institute of Technology, Fukuoka, Japan

E-mail: vakhtang.jandieri@uni-due.de

Fig. 1. Schematic of a post-wall waveguide bounded

by two adjacent N-layered post-wall arrays.

− 63 −

Modal Analysis of Post-Wall Waveguides and Compact Circuits

Vakhtang Jandieri1, Hiroshi Maeda2, Kiyotoshi Yasumoto2 and Daniel Erni1

1 General and Theoretical Electrical Engineering (ATE) and CENIDE – Center Nanointegration Duisburg-Essen, Duisburg, Germany

2 Department of Information and Communcation Engineering, Fukuoka Institute of Technology, Fukuoka, Japan

ISTET 2017Ilmenau, Germany

18th July, 2017

1. Background

2. Model of 2D-Photonic Crystal Waveguide

3. Analysis of Post-wall Waveguides

4. Attenuation Constant and Leakage Loss

5. Numerical Examples for Post-Wall Waveguides

6. Numerical Examples for Compact Circuits

7. Conclusions

Content

Two-dimensional model fields The posts may be assumed to be infinitely long. Mode fields like modes in rectangular waveguide

Planar microwave (millimeter wave) circuitsb λ�

( / 0)y∂ ∂ =

0TEm

passive element

rectangularwaveguide

rectangularwaveguideconducting plate

conducting platedielectric substrate

metallic post

z x

y

bh

2r

aµ 0ε s

1. Background

Hirokawa and M. Ando, “ Single-layer feed waveguide consisting of posts for plane TEM wave Excitation in parallel plates,” IEEE Trans. Antennas Propagat., vol. 46, no. 5, pp.625-630, 1998.

A very good alternative of the conventional non-planar guiding devices is a Post-Wall Waveguide also calledSubstrate Integrated Waveguide. It allows the “planarization” of non-planar structures such as rectangularwaveguides and they can completely be integrated together with planar structures onto the same planarsubstrate with the same processing or fabrication techniques.

2D-Photonic Crystals

Photonic crystal

Unit cell

zx

x

z

Photonic crystal waveguide

2. Model of Photonic Crystal Waveguides

x

z

hx

N

N

h

( , )N β ωR

( , )N β ωRa

Dielectric or conductor circular rods of infinite length

conducting plate

conducting platedielectric substrate

metallic post

z x

y

bh

a

2r

µ 0ε s

1-layer structure

a

h2r

ε µ0,s

z

x

Consider TE modes. If we assume a longitudinal field variation of exp(iβz), the two-dimensional guided wave propagating along the guiding region |x| < a/2, is expressed as follows:

[ ]

+

2 20

( , ) ( , ) ( , )

( , )

2 / , , ,

exp ( ( / 2)) exp( )

y

lm

s s s

x z x z x z

x z

h k k

E

i x a i z

β β π κ β ω ε µ

κ β δ

+ − −

±

= ⋅ + ⋅

=

= + = − =

U U

U � �

� � ��

∓

c c

[ ], ,( ) ( )

,( ) exp( )

N

mi aω β ω β

ω β κ δ

± ⋅=

=

W R

W

∓

� �

c c

2 2

0

det[ ( , ) ( , )] 0 (1)

( , ) [e ], ( 2 / ) (2)

( , ) : Generalized r

l

N

lm l s

s s

N

i a k l h

k

κ

β ω β ω

β ω δ κ β π

ω ε µ

β ω

− =

= = − +

=

I W R

W

R

Rigorous Formulation

000

eflection matrix of -layered post walls

1 e ( , ) 0

Only the f

(3)

unda

i a N

N

Rκ β ω− =

Long Wavelengths Approxination

mental Floquet mode with 0 is propagating and all other Floquet modes are evanescent.

l =

3. Analysis of Post-wall Waveguides

0 00

00 00

00

1 e ( , ) 0 ( ) (4)

( , ) 1.0 and ( , ) 1.0 (5)

( )

( , )

Ni a

N N

N

R

R R

R

κ β ω β ω

β ω β ω

β ω

β ω

− ⇒=

≤

Mode propagation constant

Properties of

�

0 000

200

( )

1 ( , ) small perturbation related to leakage

( , ) 1 e e 1

N e

N

i a a i a

e

R

a a

R

a

κ κ

β ω

β ω − +∆ −

−

= +

= − = = −

⇒

⇒

∆ Effective width:

Equivalent rectangular waveguide

0 (for and satisfying Eq.(3)) (6) ea π ω β

κ=

4. Attenuation Constant

0 0(

Let us introduce a small attenuation constant of the guided wave due to the leakage and express the propagation constant as:

)the attenuation constant due to the leakage is given as fol

a)

iβ β α α β= + <<

000 0

0

2

0 00

0 00

ln ( , )

0.5

lows:

(7)

The attenuation constant also can be calculated using a perturbation analysis taking into account the principle of power conservation.

[sin(

)

b)

N

Nx

z

Ra

SP

F

a

κα β ω

β

α κ

β κ κκ

=−

< >= = ×

+

1

2

0

( 0)

] e [sinh( ) ] (8)M

a N

Ma a a Rγβ γ γ

γ

−

−

=−≠

+ +

∑ ��� � �

���

5. Numerical Examples for Post-Wall Waveguides

0

/ 2.33 2.0 0.4 7.2 Posts: perfect conductor

s

h mmr mma mm

ε ε ====

Parameters

a

h2r

ε µ0,s

z

x

V. Jandieri, H. Maeda, K. Yasumoto and D. Erni, “Analysis of Post-Wall Waveguides and Circuits Using a Model of Two-Dimensional Photonic Crystals,” Progress in Electromagnetics Research M (PIER M), vol.56 pp. 91-100, 2017.

K. Yasumoto, H. Maeda and V. Jandieri, “Analysis of Post-Wall Waveguides Using a Model of Two-Dimensional Photonic Crystal Waveguides,“ Proceedings of the IEEE International Conference on Signal Processing and Communication (ICSC-2015), Noida, India, pp. 74-79, April, 2015 (Invited Paper)

0 00

10

det[ ( , ) ( , )] 0 (rigorous) 1 e ( , ) 0 (long wavelength approximation)

/ 2 TE

N

Ni a R

h

κ

β ω β ωβ ω

β π

− =

− =

Eq.(1): I W REq.(3):

Propagation constant of mode

.(3) .(1)For 1- layer structure, the difference between is less than0.09%.

Eq Eqβ β−

100 : transmission coefficient, 2.0 , 0.4 , 7.2 F h mm r mm a mm= = =

2 2 2 20

0( ( / ))e s s e

e

a k k a

aπ κ β β πκ

⇒= − = −=

Effective width of equivalent rectangular waveguide

V. Jandieri, H. Maeda, K. Yasumoto and D. Erni, “Analysis of Post-Wall Waveguides and Circuits Using a Model of Two-Dimensional Photonic Crystals,”Progress in Electromagnetics Research M (PIER M), vol.56 pp. 91-100, 2017.

K. Yasumoto, H. Maeda and V. Jandieri, “Analysis of Post-Wall Waveguides Using a Model of Two-Dimensional Photonic Crystal Waveguides,“ Proceedings of the IEEE International Conference on Signal Processing and Communication (ICSC-2015), Noida, India, pp. 74-79, April, 2015 (Invited Paper)

[6] D. Deslandes and K. Wu, “Accurate modeling, wave mechanisms, and design considerations of a substrate integrated waveguide,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 6, pp. 2516-2526, 2006.

000 0

0

2

0 00

1

2

0 0 00

( 0)

ln ( , )

0.5

(7)

[sin( ) ] e [sinh( ) ] (8)

N

Nx

z

Ma N

M

Ra

SP

F

a a a a Rγ

κα β ω

β

α κ

ββ κ κ γ γκ γ

−

−

=−≠

=−

< >= = ×

+ + +

∑ ��� � �

���

Attenuation constant calculated using (7) and (8)plotted by the dashed-dotted line and dashed line,respectively, is compared with Ref. [6] representedby the solid line.

0

/ 2.2 5.165 0.3875 11.759 Posts: perfect conductor

ε ε ====

s

h mmr mma mm

Parameters

a

h2r

ε µ0,s

z

x

βh/2πEq.(1) Eq.(3)

f [GHz] N=1 N=2 N=3 N=∞ N=18.0 0.04152 0.04056 0.04056 0.04060 0.04246

10.0 0.15980 0.15956 0.15956 0.15956 0.15998

12.0 0.23368 0.23352 0.23352 0.23353 0.23394

14.0 0.29844 0.29836 0.29836 0.29853 0.29878

V. Jandieri, H. Maeda, K. Yasumoto and D. Erni, “Analysis of Post-Wall Waveguides and Circuits Using a Model of Two-Dimensional Photonic Crystals,” Progress in Electromagnetics Research M (PIER M), vol.56 pp. 91-100, 2017.

K. Yasumoto, H. Maeda and V. Jandieri, “Analysis of Post-Wall Waveguides Using a Model of Two-Dimensional Photonic Crystal Waveguides,“ Proceedings of the IEEE International Conference on Signal Processing and Communication (ICSC-2015), Noida, India, pp. 74-79, April, 2015 (Invited Paper)

6. Numerical Examples for Compact Circuits

Bandpass filter with three circular posts of conductor: (a) Original post-wall waveguidestructure; (b) Equivalent rectangular waveguide structure, and (c) S-parameters of the circuit (a).

Theory of Images

Bandpass filter with three circular posts of conductor: (a) Original post-wall waveguidestructure; (b) Equivalent rectangular waveguide structure, and (c) S-parameters of the circuit (a).

Bandpass filter with three circular posts of conductor: (a) Original post-wall waveguidestructure; (b) Equivalent rectangular waveguide structure, and (c) S-parameters of the circuit (a).

Bandpass filter with three circular posts of conductor: (a) Original post-wall waveguidestructure; (b) Equivalent rectangular waveguide structure, and (c) S-parameters of the circuit (a).

Remarks on PWWs Formed by Dielectric Posts

Which is preferable as the material of posts, a metal (PEC) or a dielectric?

Can the equivalent rectangular waveguide model with the effective width ae be defined for a PWW formed by dielectric posts?

When designing PWW devices, we should bear in mind the difference in wave propagation between PWWs of a metal (PEC) and PWWs of a dielectric near the cutoffs.

Reflection

Leakage

Leakage

Cutoff in a rectangular waveguideCutoff in a post-wall waveguide

with metallic posts

••

Cutoff in a dielectric waveguideCutoff in a post-wall waveguide

with dielectric posts

••

Two different mechanisms of waveguide cutoff

7. Conclusions

A photonic crystal waveguide model for post-wall waveguides has been proposed.A simplified dispersion equation of post-wall waveguides based

on the long wavelengths approximation has been presente

•

•d.

Using the dispersion equation, the propagation constants and effective width of post-wall waveguides can be easily and accurately calculated.The long wavelengths approximation, which is valid w

•

• hen only the fundametal Floquet mode is propagating, can be applicable to various post-wall waveguide geometries reported in literatures.Compact circuits - bandpass filters - have been accurately a• nalyzed.

Thank you very much