J Infrared Milli Terahz Waves (2009) 30:1161–1169
DOI 10.1007/s10762-009-9541-3
The Analysis of the Parameters and the Functions of Rectangular Waveguide E-plane Filter
Yuyuan Yang & Xinghui Yin
Received: 16 March 2009 / Accepted: 21 June 2009 / Published online: 8 July 2009
# Springer Science + Business Media, LLC 2009
Abstract In this paper, to analyze the discontinuity structure of E-plane metal insert by mode matching method, and metal insert thickness and resonator length impact on filter design.
Keywords Mode matching method . E-plane . Filter . Metal insert thickness
1 Introduction
E-plane waveguide filter was found out by Konishi in 1974[1]. This type of filters combine the advantages of simple structure, low passband insertion losses, easily produced, so that it is widely applied in ku-band and ka-band. To this structure, the analysis methods mainly are Variation method and Mode matching method. Taking account into the high-order mode nearby the discontinuity and the metal thickness impact on scattering parameters, mode matching method analyzes waveguide discontinuity structure more precisely. So it is a high-precision numerical method. In this paper, mode matching method is applied to analyze the metal thickness impact on the performance of the filter. To sum up the filter design method relying the diversification of metal thickness by experiments and simulations. Combined with simulation results, the impact caused by the resonator length and metal thickness is analyzed.
2 Basic theory
The basic unit structure of rectangular waveguide E-plane filter as shown in Fig. 1.
This work was supported by the Natural Science Foundation of Jiangsu Province of China under Grant No. BK2008357.
Y. Yang (*) : X. Yin
Hohai University, Nanjing, China e-mail: yangyuyuan1@yahoo.cn
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J Infrared Milli Terahz Waves (2009) 30:1161–1169 |
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Fig. 1 The basic unit structure of E-plane filter.
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FIII |
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a |
FI |
II |
Fi II |
F III |
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2 B II |
B iII |
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III |
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t |
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a1 |
FI |
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F I |
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B |
III |
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i |
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BI |
BiI |
BiIII |
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y |
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For each subregion v=I, II, III (Fig. 1), the electrical fields are: |
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M |
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GmIII sin a |
x FmIII e jkzm z þ BmIII ejkzm z |
ð1aÞ |
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EyIII ¼ m¼1 |
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X |
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mp |
III |
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III |
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N |
GnII sin a a2 |
ðx a2Þ FnII e jkznz þ BnII ejkznz |
ð1bÞ |
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EyII ¼ n¼1 |
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X |
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np |
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EyI |
¼ i |
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GiI sin a1 |
x FiI e jkziz |
þ BiI ejkziz |
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ð1cÞ |
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¼ |
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ip |
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X |
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The corresponding equations of Hx are obtained by multiplying equations (1a) to (1c)
by YmIII, YnII and YiI .
Where
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Kzm |
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2 |
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2 |
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2a |
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III |
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w2m0"0 ðmp=aÞ2 |
w2m0"0 ðmp=aÞ2 0 |
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ð Þ |
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j ðmp=aÞ w2m0"0 |
w m0"0 ðmp=aÞ h0 |
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Kzn |
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8 q2 |
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ð ð |
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w2m0"0 ðnp=ða a2ÞÞ2 |
"0 |
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w2m0"0 |
np= |
a a2 2 |
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0 |
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ð Þ |
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j |
ðnp=ða a2ÞÞ w2m0 |
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w m0"0 ðnp=ða a2ÞÞ h0 |
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Kzi |
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8 q2 |
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2c |
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I |
¼ |
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w2m0"0 ðip=a1Þ2 |
w2m0 |
"0 ðip=a1Þ2 0 |
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ð Þ |
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j ðip=a1Þ w2m0"0 |
w m0 |
"0 ip=a1 h0 |
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With |
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: q |
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III |
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KIII |
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zm |
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ð3aÞ |
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Ym |
¼ |
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wm0 |
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J Infrared Milli Terahz Waves (2009) 30:1161–1169 |
1163 |
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II |
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KII |
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zn |
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Yn |
¼ |
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ð3bÞ |
wm0 |
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KI |
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zi |
ð3cÞ |
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Yi |
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wm0 |
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Normalization (G) can be obtained:
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III |
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wm0 |
Gm |
¼ 2 |
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abKzmIII |
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wm |
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rð Þ |
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GnII ¼ 2 |
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a a2 bKII |
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zn |
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wm0 |
Gi |
¼ 2 |
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a1bKziI |
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The continuity conditions of electric and magnetic field at z=0:
ð4aÞ
ð4bÞ
ð4cÞ
III |
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EyII |
a2hxha |
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Ey |
¼ |
0 I |
a1hxha2 |
ð |
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Þ |
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< Ey |
0 x a1 |
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h h |
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Hx |
0hxha1 |
ð Þ |
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H |
III |
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HxII |
a2 x |
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5b |
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h h |
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For each subregion (V=I, II, II), FmV ; BVm can be obtained by complex mathematical calculations. Available to the following three matrix equations:
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FIII þ BIII |
¼ LEI FI |
þ BI þ LEII FII þ BII |
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ð6aÞ |
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LHI FIII |
BIII ¼ FI |
BI |
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ð6bÞ |
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LHII |
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FIII BIII |
¼ FII |
BII |
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ð6cÞ |
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Where coupling integral matrix elements are: |
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s |
a1 |
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mp |
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np |
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I |
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KIII |
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I |
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zm |
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x |
sin |
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LE |
mn ¼ 2 |
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aa1KI |
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sin |
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a1 |
x dx ¼ |
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LH |
nm |
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ð7aÞ |
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zi 0 |
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q |
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mp |
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np |
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II |
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KIII |
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Z |
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II |
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ð Þ |
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zm |
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x |
sin |
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LE mn |
¼ 2 |
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a a a2 |
KII |
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sin |
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a2 |
ðx a2Þ |
dx ¼ |
LH |
nm |
ð7bÞ |
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zn a2 |
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1164 J Infrared Milli Terahz Waves (2009) 30:1161–1169
Rearrange
S |
¼ |
2 |
LEI |
LEII |
II |
3 |
1 |
2 |
LEI |
LEII |
II |
3 |
ð Þ |
I |
0 |
II |
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0 |
II |
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LH |
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4 |
LH |
5 |
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0 |
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LH |
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0 |
I |
LH |
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Assumed that the metal length is w, the features of discontinuity plane at z=w and at z= 0 are identical, but in the opposite direction, so S z¼w ¼ ðSjz¼0ÞT .
According to the microwave network theory, the length of waveguide is w, considering N modes, then the scattering matrix elements as follows:
S11w ¼ S22W ¼ 0 ; S12W ¼ S21W ¼ D
Where
n o
D ¼ diag e jKznV w
Using network cascade method, the two-port scattering matrix of discontinuity district formed by E-plane metal insert can be calculated[2]:
ðSÞ11 ¼ ðSÞ22 ¼ nI þ U W ðI þ UÞ 1W o 1 nI U W ðI þ UÞ 1W o |
ð9aÞ |
ðSÞ12 ¼ ðSÞ21 ¼ nI þ U W ðI þ UÞ 1W o 1 W nU ðI þ UÞ 1ðI UÞo |
ð9bÞ |
Where |
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II
I ¼ X LVE nU þ 2DV U DV DV 1DV oLVH
V ¼I
II
W ¼ X 2LVE DV U DV DV 1LVH
V ¼I
With U is unit matrix
Then T-equivalent circuit parameters can be calculated:
jxs ¼ 1 s12 þ s11
1 s11 þ s12
2s12
jxp ¼ ð1 s11Þ2 s212
ð10aÞ
ð10bÞ
According to equations (10a) and (10b), K-impedance converter can be calculated:
Φ ¼ tan 1 2xp þ xs tan 1xs |
ð11aÞ |
K ¼ tan Φ=2 þ tan 1xs |
ð11bÞ |