導波管スロットアンテナの数値固有モード基底関数 …...july 15, 2002 tokyo...

July 15, 2002 Tokyo Institute of Technology

No. 1東工大移動通信グループ合同輪講

導波管スロットアンテナの数値固有モード基底関数を用いた

モーメント法解析

Takuichi Hirano

Tokyo Institute of TechnologyAndo and Hirokawa Lab.

Takada Lab.



Outline

なぜMoM/FEM解析法が必要になったのか？MoM/FEM解析法の説明

導波管クロススロットアレーの設計



Objective

導波管クロススロットアレーの設計複雑（任意）な形状を有するスロットの解析

クロススロットノッチ付リングスロット楕円スロットなど

円偏波放射素子

円偏波放射素子

I字型スロットT字型スロット



Waveguide Crossed Slot Array



Past Work

MotorFan 1994. 112本の直線スロットの合成として解析 → 正確ではない



(b) A waveguide crossed-slot linear array

(c) A waveguide matching crossed-slot

(a) A waveguide crossed-slot array

Short circuited

Beam direction

Short circuited

Infinitely-long waveguide

Infinite ground plane Infinite ground plane

A Waveguide Crossed Slot Array

Circularly polarized waveRadiates 100%



クロススロットの円偏波放射原理（直感的説明）

T1 T2 T3 T4



Principle of Leaky-wave Operation

0λ

gλ

tθ

tθ

Incident wave

0sin λθλ =tg

gt λ

λθ 0sin =

This relation is satisfied when the slot spacing isnot equal to one guide wavelength.

Beam direction is independent of slot spacing



Feeding Waveguide

π-Junction

Short Circuit

Input Port

gλ

Inductive Post

Coupling Window



Analysis of One Slot

解析法の確立＆確認



III （半無限空間）

Integral Equations

intM−extM

extM−

intM

→ 連立積分方程式

開口面上の磁界の連続条件

I （導波管）

II （壁厚）

dSdSdSS mmS mmS mm )()( ext)III(ext)II(int)II(

intextint

MGMGMG −⋅=⋅+−⋅ ∫∫∫∫∫∫dSdSdS

S mmS mmS mmext)II(int)II(int)I(

extintint

)( MGMGMG ⋅+−⋅=⋅ ∫∫∫∫∫∫

)I(mmG

)II(mmG

)III(mmG

領域 I のグリーン関数

領域 II のグリーン関数

領域 III のグリーン関数



int1M

int1M−

exi1M

exi1M−

Mode1 Mode2

MoM/FEM Analysis

Basis functions (by FEM)

L+++= 332211 mmmM AAA

III （Half free space）

I （Waveguide）

II （Wall thickness）

Weighting coefficients (by MoM)

X-shaped waveguideEigenmode functions



)coth(

)ˆ()ˆ()coth(11

t

dSdSty

i

S suiu

S ouiuii

γ

γ

−=

⋅×⋅×−= ∫∫∑ ∫∫ hnehne

)( iuδ=

Effect of the Wall Thickness

Magnetic field mode functionElectric field mode function

Orthogonality of mode functions

No mode summation

Self-reaction

mi



Edge-based FEM

xx λ=A

固有ベクトル→固有モード関数の界分布

固有値問題

固有値→カットオフ周波数fcu

{ } zuzutu

ue γ−+= )()()( rererE



Eigenmode Basis Functions

n̂×= EM

Mode1 Mode2 Mode3



ルーフトップ関数

法線成分連続

発散が０

Edge-based FEM 基底関数=MoM ルーフトップ関数→ MoM のリアクションを従来の方法で計算できる



Frequency vs. S21

-4.5-4

-3.5-3

-2.5-2

-1.5-1

-0.50

10 11 12 13 14 15

Frequency [GHz]

Am

plitude

[dB

]

Cal. (t=1.27 [mm]) Exp. (t=1.27 [mm]) Cal. (t=0)

-20

-15

-10

-5

0

5

10

15

10 11 12 13 14 15

Frequency [GHz]P

has

e [

Deg]

Cal. (t=1.27 [mm]) Exp. (t=1.27 [mm]) Cal. (t=0)

11S 21S

31S



Radiation Pattern

Radiation Pattern (Observation Angleφ=90°,12.0 GHz)

-30

-25

-20

-15

-10

-5

0

5

-90 -60 -30 0 30 60 90

Observation Angleθ [Deg]

Am

plitude

[dB

]

Spin Linear (Exp.) Emax (Cal.) Emin (Cal.)

θ



解析法の信頼性を確認できた



Analysis of a Matching Crossed Slot

Short circuited

Beam direction

Short circuited

Infinitely-long waveguide

Infinite ground plane Infinite ground plane



X字型導波管の固有モード関数（磁流基底関数）

214

11

Mode1 Mode2 Mode3Cutoff Wavenumbers 1:1.3:1.24, Excitation coefs 1.1:1.18:0.00079

105 deg



0

1

2

3

4

5

6

0.7 0.75 0.8 0.85 0.9 0.95 1

Mode1

Mode2

Mode3

12 / ll

1l

×cuk

カットオフ波数の変化

w

l1 l2

105 deg

w/l1=2/14

l

w

l

π=cuk

c.f. 方形導波管のカットオフ波数

Mode1 Mode2 Mode3

設計モデル



θ

φ

105.9deg

-97.4dB

-97.7dB

遠方界（ビームチルト角）

位相差: 103.5 deg

円偏波放射原理

実数分布→直線偏波!

105 deg

Mode1 Mode2

ビームチルト方向

1.1 1.18

スロット長（共振周波数）の差による



円偏波放射原理アニメーション

θ

)exp( 11 θjA

)exp( 22 θjA

°= 30θ の例

++±=−=

ππθθθ nAA

212

21

円偏波放射条件



Array Design

導波管1本

導波管が並んだ２次元アレーの設計と基本的に変わらない。



int1M

int1M−

exi1M

exi1M−

int2M

int2M−

exi2M

exi2M−slot #1 slot #2

Mode1 Mode2

MoM/FEM Analysis

Basis functions (by FEM)

L+++= 332211 mmmM AAA

III （Half free space）

I （Waveguide）

II （Wall thickness）

Weighting coefficients (by MoM)

X-shaped waveguideEigenmode functions



Setting of the Problem

One waveguide

Uniform illuminationCircularly polarized waveat a beam direction

Frequency: 11.85GHzBeam-tilting angle : 50deg

Purpose

Satellite

DBS

tθ



Design Procedure

1. Determinethe waveguide size andthe number of elements

2. Determine initial slot lengthsusing infinite periodic model

3. Finite array analysis

4. Modify slot parameters

Simulate mutual couplings

tθ

ab tθ

Beam widthmma 0.16=

Number of slots=8Leaky-wave

MoM/FEM

t=0.5mmp= -3.5mmw=2.0mmrot=111deg



Power Relation and Slot Coupling

12.5% of incident power per each slot87.5% (7 slots), 100% (8 slots)

Weak coupling(short slot)

Strong coupling(long slot)

Uniform Aperture Illumination

#1 #2 #3 #4 #5 #6 #7

1/8 1/7 1/6 1/5 1/4 1/3 1/2Slot coupling:12.5% per each slotRadiated power:

1

Matchingcrossed slot

Controlled by slot length



0

10

20

30

40

50

60

10 10.5 11 11.5 12

Slot length (mm)

Rad

iation p

ow

er

(%)

0

0.5

1

1.5

2

2.5

Axi

al r

atio

at a

beam

angl

e (dB

)

xpyp

b t

Periodic walls

Periodic walls

Infinitely-long waveguidea

TE10 wave

Ground plane

xp

yp

a

xθ

w

l

p

Determine Initial Slot Parameters

Large in this case

)47.0(0.12 0λmm

)20(0.500 0λmm

Infinite periodic model

slot no. coupling (%) slot length (mm)1 12.5 10.212 14.3 10.373 16.7 10.534 20.0 10.755 25.0 11.016 33.3 11.357 50.0 11.868 100.0



-50

-40

-30

-20

-10

0

-90 -60 -30 0 30 60 90

Angle (deg)

Rela

tive a

mplit

ude (

dB

)

Radiation Pattern (Init.)

50

Beam-tiltingangle

1.07dB

θ

-10.89dB

Semi-majorSemi-minor

θ



tθ

0.00

2.00

4.00

6.00

8.00

10.00

1 2 3 4 5 6 7

Slot no.

Axi

al ra

tio a

t th

e b

eam

dire

ction (

dB

)

0.00

20.00

40.00

60.00

80.00

100.00

Phase

of

R.H

.C.P

wave

(deg)

E-field Radiated by Each Slot (Init.)

0.00

5.00

10.00

15.00

20.00

1 2 3 4 5 6 7

Slot no.

Rad

iation p

ow

er

(%)

-20.00

-15.00

-10.00

-5.00

0.00

Rel.

pow

. at

the b

eam

dire

ction (

dB)

Not uniform enough



Correction of the Power

0

10

20

30

40

50

60

10 10.5 11 11.5 12

Slot length (mm)

Rad

iation p

ow

er

(%)

At the present slot length (1),differential coefficient (2) is calculated numericallySlot length is corrected by the 1st orderapproximation using the difference betweenthe present (3) and desired power (4)and the differential coefficient.Above procedure is applied to all the slots simultaneously

(1)(2)(3)

(4)

Corrected slot length

Rep.

Error of the radiationpower

Init. length



-50

-40

-30

-20

-10

0

-90 -60 -30 0 30 60 90

Angle (deg)

Rel

ativ

e am

plitu

de (d

B)

Radiation Pattern (Corrected, 3 Iterations)

50

Beam-tiltingangle

0.91dB

θ

-11.55dB

θ

Semi-majorSemi-minor



-40

-35

-30

-25

-20

-15

-10

-5

0

10 11 12 13 14 15

Frequency (GHz)

¦S11¦ (d

B)

MoM/FEM

HFSS

-800

-600

-400

-200

0

200

400

600

800

10 11 12 13 14 15

Frequency (GHz)

arg(

S11)

(deg)

MoM/FEM

HFSS

-50

-40

-30

-20

-10

0

10 11 12 13 14 15

Frequency (GHz)

¦S21¦ (d

B)

MoM/FEM

HFSS

-300

-240

-180

-120

-60

0

60

120

180

240

10 11 12 13 14 15

Frequency (GHz)

arg(

S21)

(deg)

MoM/FEM

HFSS

Freq. vs. S11 and S21 (Corrected, 3 Iterations)

S11(amplitude) S11(phase)




Freq. vs. Axial Ratio (Corrected, 3 Iterations)

0

1

2

3

4

5

6

7

8

9

10

10 11 12 13 14 15

Frequency (GHz)

Axi

al r

atio

(dB

)

15.8%

11.85GHzDesigned

θ



0.00

5.00

10.00

15.00

20.00

1 2 3 4 5 6 7

Slot no.

Rad

iation p

ow

er

(%)

-20.00

-15.00

-10.00

-5.00

0.00

Rel.

pow

. at

the b

eam

dire

ction (

dB)

0.00

2.00

4.00

6.00

8.00

10.00

1 2 3 4 5 6 7

Slot no.

Axi

al ra

tio a

t th

e b

eam

dire

ction (

dB)

0.00

20.00

40.00

60.00

80.00

100.00

Pha

se o

f R

.H.C

.P w

ave

(deg)

E-field Radiated by Each Slot (Corrected, 3 Iterations)

Uniform enoughrθ

xθ

tθ



Correction (Method 2)

rθ

xθ



-50

-40

-30

-20

-10

0

-90 -60 -30 0 30 60 90

Angle θ (deg)

Rela

tive

am

plitude

(dB

)

Radiation Pattern

50

Beam-tiltingangle

0.1dB

θ



-50

-40

-30

-20

-10

0

10 11 12 13 14 15

Frequency (GHz)

¦S11¦ (d

B)

MoM/FEM

HFSS

-800

-600

-400

-200

0

200

400

600

800

10 11 12 13 14 15

Frequency (GHz)

arg(

S11)

(deg)

MoM/FEM

HFSS

-50

-40

-30

-20

-10

0

10 11 12 13 14 15

Frequency (GHz)

¦S21¦ (d

B)

MoM/FEM

HFSS

-300

-240

-180

-120

-60

0

60

120

180

240

10 11 12 13 14 15

Frequency (GHz)

arg(

S21)

(deg)

MoM/FEM

HFSS

S11 and S21

S11(amplitude)

S11(phase)




0

1

2

3

4

5

6

7

8

9

10

10 11 12 13 14 15

Frequency (GHz)

Axi

al r

atio

(dB

)

Axial Ratio

29.5%

11.85GHzDesigned



0.00

5.00

10.00

15.00

20.00

1 2 3 4 5 6 7

Slot no.

Rad

iation p

ow

er

(%)

-20.00

-15.00

-10.00

-5.00

0.00

Rel.

pow

. at

the b

eam

dire

ction (

dB)

0.00

2.00

4.00

6.00

8.00

10.00

1 2 3 4 5 6 7

Slot no.

Axi

al ra

tio a

t th

e b

eam

dire

ction (

dB

)

0.00

20.00

40.00

60.00

80.00

100.00

Phase

of

R.H

.C.P

wave

(deg)

E-field Radiated by Each Slot



Array Design 2



W.G. Crossed Slot Array

-50

-40

-30

-20

-10

0

-90 -60 -30 0 30 60 90

Angle (deg)

Rel

ative

am

plitude

(dB

)

99.8% Radiates

0.6dB



Design Model for a Matching Crossed Slotxpyp

ab

Short circuited

ab t

Short circuited

Short circuitedSame magnetic currents



Frequency Characteristics

-50

-40

-30

-20

-10

0

10 11 12 13 14 15

Frequency (GHz)

¦S11

¦ (dB

)

0

1

2

3

4

5

6

7

8

9

10

10 11 12 13 14 15

Frequency (GHz)

Axi

al r

atio

(dB

)



Conclusion

○ MoM/FEM解析法を１素子で確認した

　　（→任意形状スロットの解析も可能）○ MoM/FEM解析法で終端の整合クロススロットを設計した○ MoM/FEM解析法を用いて一次元導波管

　　クロススロットアレーの設計を行った○ リングスロット、楕円スロットの解析を行った

◎ 速い！◎ 壁厚の影響の評価が簡単！◎ 正確！◎ 円偏波の動作原理（物理現象）が理解できる！

MoM/FEM解析法の利点

導波管スロットアンテナの 数値固有モード基底関数 …...july 15, 2002 tokyo...

Documents

導波管スロットアンテナの数値固有モード基底関数 …...july 15, 2002 tokyo...