JOURNAL OF TELECOMMUNICATIONS, VOLUME 18, ISSUE 1, JANUARY 2013

7

© 2012 JOT

www.journaloftelecommunications.co.uk

Accurate Analysis of Waveguide Filters by the Coupled Integral Equations

H. Ghorbaninejad and M. Foroughi

Abstract—In this paper the coupled magnetic field integral equation is applied to accurately determine the frequency response

of waveguide band-pass filters in which the irises are closely spaced due to compacting techniques and higher order mode

effects are essential in the filter performance. A three-resonator H plane filter is designed and analyzed in which the thickness of

irises and coupling effect between them have been considered. To take the effect of higher order modes, a coupled set of

magnetic field integral equations (MFIE) is derived and formulated. Finally a set of linear matrix equations are solved using

method of moment (MoM) and entire basis functions which results rapid convergence. The usefulness of the proposed method

and its performance are verified by designing and simulating a three-resonator H plane equal ripple waveguide band-pass filter.

Keywords—Computer-aided deign, Coupled magnetic field integral equations, Method of moment, Waveguide filters.

—————————— ——————————

1 INTRODUCTION

N modern communication systems, microwave filters are widely used components. Wave guide filters are often used as one of the essential component of micro-

wave circuits. The frequency response of such devices must be accurately predicted before actual implementa-tion.

The straightforward approach to construct these filters is using inductive elements such as irises, rods, diaph-ragms and posts as impedance invertors between trans-mission line resonators, which are realized with half wa-velength hollow waveguides [1], [2], [3], [4], [5]. Some other approaches such as using waveguides filled by mul-ti-layer dielectrics [6] and substrate integrated wave-guides (SIWs) [7] have been introduced as band pass fil-ters by now. The long length of waveguide band-pass filters, which is due to hollow waveguide resonators, is a deficiency for them. In [8] the region between two thin irises (diaphragms) is filled fully by dielectrics to reduce the length of resulted filter and the relations between the physical and electrical parameters of the existed asymme-trical impedance invertors are obtained. But the higher order modes and thickness of diaphragms is not taken to account in [8].

Mode matching technique (MMT) is often used to analysis and deign wave guide filters [9]. In this approach the generalized scattering matrix of individual disconti-nuities are calculated separately then cascaded to form final scattering matrix of structure. When two discontinu-ities are strongly closed it is necessary to be considered as a unit block to take higher order modes [10]. To guarantee numerical efficiency new entire domain basis functions has been introduced in [11] to include the non-analytic behavior of y-directed electric field at the edges of the irises.

In this paper we apply the coupled set of magnetic field integral equations for the design and analysis of wa-veguide filters. The derived integral equations are solved by method of moment and using entire basis functions which results rapid convergence. Numerical results ob-tained from the coupled magnetic field integral equations technique are compared to those from the HFSS software [12] to demonstrate the accuracy of the technique.

2 ANALYSIS OF H-PLANE FILTERS BY THE MFIE

Fig. 1 depicts the example of a three resonator H plane filter structure. It consists of a lossless rectangular wave guide of cross section ba and four symmetrical H plane irises of thickness di. The apertures of irises are

ii baa 2 where the width of one side of the ith iris is

ib . This structure is made of nine sections in which the relative permittivity of ith section is ri . We assume that only the fundamental TE10 mode is incident on one side of structure that in most application can be realized. In this structure we have no variation in the y direction and due to symmetrical irises only TEm0 modes, with m, an odd integer, are excited. We expand the magnetic fields in term of magnetic currents. Fig. 2 shows the equivalent magnetic currents for the accurate analysis of structure and obtaining electric and magnetic fields in varies re-gions. In the region I, there are reflected waves due to 1M in addition to the fundamental incident wave. Electric and magnetic fields in the region II is due to magnetic currents of 1M , 2M and so on. In the same way the elec-tric and magnetic field in the region XI is only due to in-duced magnetic current of 4M . The continuity of the magnetic fields across the apertures of irises provides the interaction of all regions that necessitates the formulation of the problem in terms of coupled magnetic field integral equations. ————————————————

H. Ghorbaninejad is with the Electrical Engineering Department of Guilan University.

I

8

One can drive the appropriate spectral domain Green’s functions for these equivalent structures. The enforce-ment of continuity of transverse magnetic fields across the aperture parts of irises provides the interaction of all regions. A coupled set of magnetic field integral equa-tions is obtained by enforcing tangential component of magnetic fields on the aperture part of irises. In the region I, approaching 1M location, for the magnetic field due to 1M which is shown with

1H , we have the following equation:

1

1

1

11 )()(

apm

m dxxMxYH (1)

In region II, approaching 1M location, for the magnet-ic fields due to 1M and 2M which is shown with

1H , we have the following equation:

1

12

1

1

3

1

11

1

1

21

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(2)

In similar way for the magnetic fields on the magnetic

current locations, approaching from left and right are in

the following form:

1

11

1

1

5

1

12

1

1

42

)()(

)()(

apm

m

apm

m

dxxMxsY

dxxMxYH

(3)

2

3

1

7

1

2

1

62

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(4)

1

1

1

9

2

3

1

83

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(5)

2

24

2

1

11

2

23

2

1

103

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(6)

x

a

1d 1

r2

r3

r2

d3

d4

d

4b3

b2

b1

b

10TE

z

I III

IVII

V

VI

VII

VIII

IX

Fig. 1. Top view of a symmetric three H-plane filter.

y

b

1d 1

r2

r3

r2

d3

d4

d

z

1M1M 2M

2M 3M3M 4M4M 5M5M 6M6M 7M7M 8M 8M

1r 2r 3r 4r 5r 6r 7r 8r 9r

Fig. 2. Side view of a symmetric three resonator H-plane filter and equivalent magnetic currents.

9

2

23

2

1

13

2

24

2

1

124

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(7)

3

5

1

15

2

4

1

144

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(8)

2

4

1

17

3

5

1

165

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(9)

3

36

3

1

19

3

35

3

1

185

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(10)

3

35

3

1

21

3

36

3

1

206

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(11)

4

7

1

23

3

6

1

226

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(12)

3

6

1

25

4

7

1

247

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(13)

4

48

4

1

27

4

47

4

1

267

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(14)

4

47

4

1

29

4

48

4

1

288

)()(

)()(

apm

m

apm

m

dxxMxY

dxxMxYH

(15)

4

8

1

308 )()(

apm

m dxxMxYH (16)

In (1) to (16), imY for 30,,1 i , )(x and )(xi

for 4,,1 i are given in Appendix A. After enforcing continuity of the tangential magnetic field across the aperture parts of irises in the spatial domain, we obtain:

11 HHH inc (17)

.8,,2, iHH ii (18)

Where incH is the incident fundamental mode. by ex-

pressing magnetic fields in (17) and (18) in terms of mag-netic currents one can obtained eight coupled magnetic field integral equations. To solve this set of coupled integral equations, we expand magnetic currents in a se-ries of basis functions then apply Galerkin’s method to

each one of them. Let xBij denote the jth element of a

set of basis functions for the functions )(xM i . We have:

.7,5,3,1,)()(

8,,1,)()(

,1

1

ixBxB

ixBCxM

jiij

ij

N

j

iji

i

(19)

If these expansion are used in the integral equations

and Galerkin’s method is applied to each one of them, a

set of coupled linear equations in the expansion coeffi-

cients ijC results. The eight linear equations can be put in

the following form

incUCACA 2211 (20a)

0CACACA 352413 (20b)

0CACACA 483726 (20c)

0CACACA 51141039 (20d)

0CACACA 614513412 (20e)

0CACACA 717616515 (20f)

0CACACA 820719618 (20g)

0CACA 822721 (20h)

Matrices in (20a) to (20h) are given in the Appendix B. It is notable that some of these matrices may be equal in symmetrical conditions, a fact which, is used to reduce the computational effort. Also is worth noting that (20a) to (20h) follow a clear pattern which allows the number of resonators to be varied relatively simple. Here according to the conditions of the structure, the following basis functions are given in [11]

iN

j iii

iiijij

xbabx

abxjCxB

131

)()(

)(sin (21)

The Fourier transform of these basis functions are ex-pressible in terms of Bessel functions of the first kind of order 1/6 and have been given in appendix B.

3 DESIGN PROCEDURE

We consider a band-pass filter consisting of N series re-

sonators and N+1 impedance invertors. The value of im-

pedance invertors and the elements of resonators can be

obtained according to the type of filter transfer function,

10

center frequency and the bandwidth of the filter. This

circuit model can be realized using N+1 shunt induc-

tances and N transmission line sections [3]. In the design

process, the width of irises apertures and the distances

between them are optimized so that the transmission cha-

racteristic of filter can be fitted to that of LC circuit model.

The fitness function is given by (22). In (22) iCIET fS21 and

iLC fS21 are the S-parameters of coupled integral equa-

tion technique model and LC circuit model respectively,

which are evaluated in frequency sample if .

5.0N

1i

2

2121

N

1i

2

2121pS

p

S

NN

1fitness

iLC

iMoM

iLC

iMoM

fSfS

fSfS

(22)

NP and NS are the number of frequency samples in pass-band and each of stop bands respectively. As initial values for optimization, Lengths of resonators sections and the aperture widths of diaphragm can be obtained by [8].

4 DESIGN EXAMPLES AND RESULTS

In this section two 3-order chebyshev type H plane waveguide band pass filter utilizing a WR-90 waveguide (a = 0.9 inches and b = 0.4 inches) with center frequency f0 =10 GHz, the relative bandwidth 9 percent and equal ripples 0.5 have been designed. It is assumes that the sec-

tions II to VIII have electric permittivity 5.3ri for

8,,2 i and sections I and IX have electric permittiv-

ity 191 rr .

In first example the thickness of all irises is assumed to be di =0.1 mm and in the second example it is assumed to be di =1 mm. frequency samples steps is 50 MHz. Opti-mization rang is 8-12 GHz and it is assumed NP=21 fre-quency samples in pass-band and NP=30 frequency sam-ples in each of stop bands. The designed values of irises widths and distances between them, for two design ex-amples are given in table 1. Fig. 3 and Fig. 4 show trans-mission characteristic of designed filters and simulated one for two design examples. Comparison between the calculated and simulated results shows good agreement over the frequency range.

5 CONCLUSION

In this paper analysis of H plane filters using magnetic filed integral equation have been developed. The thick-nesses of irises and the coupling effects between them have been taken into account due to formulation. Two examples of 3-order chebyshev type waveguide band pass filter using different values of iris thickness is de-signed, signifying that the thickness of irises and the coupling effect between them have basic role in the de-sign procedure and filter performance and must be taken into account. Applying this method to design and analy-sis of closely spaced irises is inevitable to obtain filter per-formance before implementation. The numerical and si-mulated verification shows good agreement between coupled magnetic field integral equations method and simulation results.

8 8.5 9 9.5 10 10.5 11 11.5 12-50

-40

-30

-20

-10

0

FREQUECY [GHz]

|S2

1| [

dB

]

MFIE

HFSS

Fig. 3. Transmission response of designed symmetric three resonator H-plane filter with 0.1 mm iris thickness and simulation result.

8 8.5 9 9.5 10 10.5 11 11.5 12-50

-40

-30

-20

-10

0

FREQUECY [GHz]

|S2

1| [

dB

]

MFIE

HFSS

Fig. 4. Transmission response of designed symmetric three resonator H-plane filter with 1 mm iris thickness and simulation result.

TABLE I THE PHYSICAL PARAMETERS OF DESIGNED FILTERS

Lengths and Widths [mm]

Design example 1

Design example 2

di 0.1 1

a1= a4 8.0372 8.6311

a2= a3 4.1699 5.1468

r1= r2 6.6245 6.0379

r3 7.8552 7.5155

11

6 APPENDICES

6.1 Appendix A

In this appendix we give the expressions for imY , )(x

and )(xi

TE

1

1 YYm (A1)

jjjim dYY 2

TE2 coth (A2)

jjjim dYY 2

TE2

1 sinh (A3)

i2

m

i

m YY (A4)

1i3 m

i

m YY (A5)

jjjim rYY 12

TE12

4 coth (A6)

jjjim rYY 12

TE12

5 sinh (A7)

4i6 mim YY (A8)

5i7 mim YY (A9)

Where i = 2, 10, 18, 20 corresponds to j = 1, 2, 3, 4 respec-tively

2628mm YY (A10)

2729mm YY (A11)

TEm YY 930 (A12)

Where

.9,,2,1,0TE ijY ii (A13)

4.,3,2,1,5.0

)2(0

2

0

2

2 iam irii (A14)

9.,7,5,3,1,5.0

0

2

0

2 iam rii (A15)

)/())/(2()( 5.0 axmsinabx (A16)

.4,,1

),/)(())/(2()( 5.0

i

abxmsinbax iiii

(A17)

6.2 Appendix B

1

112

111

1

~~~~

m

IIIIkm

IIkmk

BBYBBYA (B1)

1

11

3

2

~~

m

IIII

kmkBBYA (B2)

1

11

5

3

~~

m

IIII

kmkBBYA (B3)

1

11

6

11

4

4

~~~~

m

II

km

IIII

kmkBBYBBYA (B4)

1

217

5

~~

m

IIkmkBBYA (B5)

1

129

6

~~

m

IIkmkBBYA (B6)

1

2210

228

7

~~~~

m

IIIIkm

IIkmk

BBYBBYA (B7)

1

2211

8

~~

m

IIIIkmkBBYA (B8)

1

2213

9

~~

m

IIIIkmkBBYA (B9)

1

2214

2212

10

~~~~

m

IIkm

IIIIkmk

BBYBBYA (B10)

1

3215

11

~~

m

IIkmkBBYA (B11)

1

2317

12

~~

m

IIkmkBBYA (B12)

1

3318

3316

13

~~~~

m

IIIIkm

IIkmk

BBYBBYA (B13)

1

3319

14

~~

m

IIIIkmkBBYA (B14)

1

3321

15

~~

m

IIIIkmkBBYA (B15)

1

3322

3320

16

~~~~

m

IIkm

IIIIkmk

BBYBBYA (B16)

1

4323

17

~~

m

IIkmkBBYA (B17)

1

3425

18

~~

m

IIkmkBBYA (B18)

1

4426

4424

19

~~~~

m

IIIIkm

IIkmk

BBYBBYA (B19)

1

4427

20

~~

m

IIIIkmkBBYA (B20)

1

4429

21

~~

m

IIIIkmkBBYA (B21)

1

4430

4428

22

~~~~

m

IIkm

IIIIkmk

BBYBBYA (B22)

)1(~

))/(2( 11

15.0 mBYabU I

kk (B23)

Where

12

6/1

6/1

6/1

6/1

3

12

1

2cos

2

2cos

2

)3/2()2/1(2

)(~

0if

ka

am

kmka

amJ

ka

am

kmka

amJ

aa

bdxxBbB

ka

am

i

i

i

i

i

iaib

ib

ikIik

i

(B24)

6/1

6/1

3

12

1

2cos

2

cos)3/4()2/1(

)3/2(

)3/2()2/1(2

)(~

0if

ka

am

kmka

amJ

a

bm

aa

bdxxBbB

ka

am

i

i

i

i

iaib

ib

ikIik

i

(B25)

6/1

6/1

6/1

6/1

6

1

1

2

1

2cos

2

2cos

2

)3/2()2/1(2

)(~

0if

km

kmkmJ

km

kmkmJ

ab

dxxBbB

km

iiaib

ib

ikIIik

(B26)

6/1

6/1

6

1

1

2

1

2cos

2

)3/4()2/1(

)3/2(

)3/2()2/1(2

)(~

0if

km

kmkmJ

ab

dxxBbB

km

iiaib

ib

ikIIik

(B27)

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H. Ghorbaninejad-Foumani was born in guilan, Iran. He received his B. Sc. degree from Guilan University in 2003 and his M. Sc. and Ph. D. degrees from Iran University of Science and Technology (IUST) in 2005 and 2010 respectively, all in telecommunication engi-neering. He is currently assistant professor at department of electric-al engineering of Guilan University. His scientific fields of interest are electromagnetic problems including microwave and spatial filter de-sign, compacting microwave devices and green’s function of micro-wave structures. . M.Foroughi was born in Hamedan,Iran. He received his B.Sc. de-gree from Razi Kermanshah University in 2010 in electronic engi-neering. As well as he’s student M.Sc. degree Guilan University in telecommunication engineering.