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EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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Page 1: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

EMLAB

Microwave Engineering

Chap. 3.7 ~ Chap. 3.11

Page 2: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

2

EMLAB

3.7 Stripline

Stripline ≒ “Flattened-out” coax

TEM mode phase velocity, propagation constant

Characteristic impedance

𝑣𝑝=1

√𝜇0𝜖0𝜖𝑟=𝑐

√𝜖𝑟

𝑍 0=√ 𝐿𝐶=√𝐿𝐶𝐶

=1𝑣𝑝𝐶

Page 3: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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EMLAB

3.7 Stripline

Z0 구하기 (1) – Conformal mapping 을 통해 구한 exact solution 을 curve fitting 하여

구함 .

Z0 는 정해져 있고 stripline 의 너비 W 를 구하려는 경우𝑊𝑏

={ 𝑥 𝑓𝑜𝑟 √𝜖𝑟 𝑍 0<120Ω0.85−√0.6−𝑥 𝑓𝑜𝑟 √𝜖𝑟 𝑍0>120Ω

𝑍 0=30𝜋

√𝜖𝑟𝑏

𝑊 𝑒+0.441𝑏

𝑊𝑒

𝑏=𝑊𝑏− { 0 𝑓𝑜𝑟

𝑊𝑏

>0.35

(0.35−𝑊𝑏 )2

𝑓𝑜𝑟𝑊𝑏

<0.35

Page 4: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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EMLAB

3.7 Stripline (cont’d)

Z0 구하기 (2) – approximate electrostatic solution

Φ (𝑥 , 𝑦 )={ ∑𝑛=1𝑜𝑑𝑑

𝐴𝑛cos𝑛𝜋 𝑥𝑎

sinh𝑛𝜋 𝑦𝑎

𝑓𝑜𝑟 0≤ 𝑦 ≤ 𝑏2

∑𝑛=1𝑜𝑑𝑑

𝐵𝑛cos𝑛𝜋 𝑥𝑎

sinh𝑛𝜋𝑎

(𝑏− 𝑦 ) 𝑓𝑜𝑟 𝑏2≤ 𝑦 ≤𝑏

필드가 내부 도체 근처에만발생한다는 가정 → |a|/2 에 도체벽a>>b

𝛻𝑡2Φ (𝑥 , 𝑦 )=0 𝑓𝑜𝑟|𝑥|≤ 𝑎

2,0≤ 𝑦 ≤𝑏

𝑘𝑥2+𝑘𝑦

2=0

Page 5: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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EMLAB

3.7 Stripline (cont’d)

at y=b/2 An=Bn.

surface charge density

𝐸𝑦=−𝜕Φ𝜕 𝑦

={ −∑𝑛=1𝑜𝑑𝑑

𝐴𝑛(𝑛𝜋𝑎 ) cos𝑛𝜋 𝑥𝑎 cosh𝑛𝜋 𝑦𝑎

𝑓𝑜𝑟 0≤ 𝑦 ≤ 𝑏2

∑𝑛=1𝑜𝑑𝑑

𝐵𝑛 (𝑛𝜋𝑎 )cos 𝑛𝜋 𝑥𝑎 cosh𝑛𝜋𝑎

(𝑏−𝑦 ) 𝑓𝑜𝑟 𝑏2≤ 𝑦 ≤𝑏

𝜌𝑆=𝐷𝑦 ¿

Page 6: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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EMLAB

3.7 Stripline (cont’d)

surface charge density 가 상수임을 가정 , 미지수 An 을 구함 .

strip(center conductor) (-W/2 ~ W/2) 전압 :

𝑉 avg=1𝑊 ∫

−𝑊2

𝑊2

∫0

𝑏2

𝐸𝑦 (𝑥 , 𝑦 )𝑑𝑦𝑑𝑥=∑𝑛=1𝑜𝑑𝑑

𝐴𝑛 ( 2𝑎𝑛𝜋𝑊 )sin 𝑛𝜋𝑊2𝑎 sinh

𝑛𝜋𝑏2𝑎

∴𝐴𝑛=2𝑎 sin(𝑛𝜋𝑊 /2𝑎)

(𝑛𝜋 )2𝜖0𝜖𝑟 cosh (𝑛𝜋 𝑏/2𝑎)

𝜌𝑆 (𝑥 )={1 𝑓𝑜𝑟|𝑥|<𝑊 /20 𝑓𝑜𝑟|𝑥|>𝑊 /2

𝜌𝑆=2𝜖0𝜖𝑟∑𝑛=1𝑜𝑑𝑑

𝐴𝑛 (𝑛𝜋𝑎 )cos 𝑛𝜋 𝑥𝑎 cosh𝑛𝜋𝑏2𝑎 양변에 곱하고

-a/2 ~ a/2 구간에서 x 에 대해 적분

cos𝑛𝜋 𝑥𝑎

Page 7: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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EMLAB

3.7 Stripline (cont’d)

strip(center conductor) 의 단위 길이당 총 전하

stripline 의 단위 길이당 capacitance

𝐶=𝑄𝑉 avg

=𝑊

∑𝑛=1𝑜𝑑𝑑

𝐴𝑛( 2𝑎𝑛𝜋𝑊 )sin 𝑛𝜋𝑊2𝑎 sinh

𝑛𝜋 𝑏2𝑎

F /m

𝑄= ∫−𝑊 /2

𝑊 /2

𝜌𝑆 (𝑥 )𝑑𝑥=𝑊 Coul /m

Page 8: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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EMLAB

3.8 Microstrip Line

Quasi-TEM mode phase velocity, propagation constant

εe: effective dielectric constant.

𝑣𝑝=𝑐

√𝜖𝑒 단 ,1<𝜖𝑒<𝜖𝑟

𝜖𝑒=𝜖𝑟+12

+𝜖𝑟 −12

1√1+12𝑑 /𝑊

Page 9: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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EMLAB

3.8 Microstrip Line (cont’d)

W,d 등을 알고 있을 때 Z0 구하기

Z0 가 알려진 경우 W/d 비율 구하기

𝑍 0={60

√𝜖𝑒ln( 8𝑑𝑊 +𝑊

4𝑑 ) 𝑓𝑜𝑟 𝑊𝑑 ≤1120𝜋

√𝜖𝑒[𝑊𝑑 +1.393+0.667 ln(𝑊𝑑 +1.444)]𝑓𝑜𝑟 𝑊

𝑑≥1

𝑊𝑑

={ 8𝑒𝐴

𝑒2 𝐴−2𝑓𝑜𝑟 𝑊

𝑑<2

2𝜋 [𝐵−1−ln (2𝐵−1 )+

𝜖𝑟−12𝜖𝑟 {ln (𝐵−1 )+0.39− 0.61

𝜖𝑟 }] 𝑓𝑜𝑟 𝑊𝑑 >2

Page 10: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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EMLAB

3.9 The Transverse Resonance Technique

도파관의 transverse cross section ( 직사각형 혹은 원형 등 ) 에 전송선로 모델을 도입 ,

차단 주파수를 구함 .

차단주파수에서 도파관의 정재파 = 공진주파수에서의 전송선로

TE0n Modes of a Partially Loaded Rectangular Waveguide

Characteristic impedance for TE mode

𝛽=√𝜖𝑟 𝑘02−𝑘𝑦𝑑2 =√𝑘02−𝑘𝑦𝑎2

𝑍 𝑖𝑛𝑟 (𝑥 )+𝑍 𝑖𝑛

𝑙 (𝑥 )=0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 r:right, l: left

𝑍 𝑑=𝑘𝜂𝑘𝑦𝑑

=𝑘0𝜂 0𝑘𝑦𝑑

,𝑍𝑎=𝑘0𝜂0𝑘𝑦𝑎

𝑘𝑦𝑎 tan𝑘𝑦𝑑𝑡+𝑘𝑦𝑑 tan𝑘𝑦𝑎 (𝑏−𝑡)=0

Page 11: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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3.10 Wave Velocities and Dispersion

The speed of light in a medium

The phase velocity

Dispersion:

Group velocity: 협대역 신호의 전파 속도

신호 f(t)

𝐹 (𝜔 )=∫−∞

𝑓 (𝑡 )𝑒− 𝑗𝜔𝑡 𝑑𝑡↔ 𝑓 (𝑡 )= 12𝜋 ∫

−∞

𝐹 (𝜔 )𝑒 𝑗𝜔𝑡 𝑑𝜔

lossless matchedTL or WG

𝐹 𝑜 (𝜔 )=𝑍 (𝜔 )𝐹 (𝜔 )

Page 12: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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3.10 Wave Velocities and Dispersion (cont’d)

lossless TEM mode 출력 신호 (propagation constant 가 주파수의 1 차 함수인 경우 )

시간지연 O, distortion X.

lossless TEM propagation constant

Narrow input signal

f(t) 의 높은 주파수 성분을 , (carrier frequency)

Output signal spectrum (freq. domain, time domain)

𝑓 𝑜 (𝑡 )= 12𝜋 ∫

−∞

𝐹 (𝜔 )|𝑍 (𝜔)|𝑒 𝑗 (𝜔𝑡 −𝜓 )𝑑𝜔

𝑆 (𝜔 )=∫−∞

𝑓 (𝑡 ) cos𝜔𝑜 𝑡 𝑒𝑗𝜔𝑡 𝑑𝑡=𝐹 (𝜔−𝜔𝑜 )

𝑆𝑜 (𝜔 )= 𝐴2 [𝐹 (𝜔−𝜔𝑜 )+𝐹 (𝜔+𝜔𝑜) ]𝑒− 𝑗 𝛽 𝑧

Page 13: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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EMLAB

3.10 Wave Velocities and Dispersion (cont’d)

F(ω) 가 협대역 신호일 경우 (ωm≪ωo) propagation constant β 를 테일러 급수 전개

라 하면

𝑣𝑔=1𝛽𝑜′ =( 𝑑 𝛽𝑑𝜔 )

−1|𝜔=𝜔𝑜

𝛽 (𝜔 )=𝛽 (𝜔𝑜 )+ 𝑑 𝛽𝑑𝜔|

𝜔=𝜔 𝑜

(𝜔−𝜔𝑜 )+ 12𝑑2 𝛽𝑑𝜔2|

𝜔=𝜔𝑜

(𝜔−𝜔𝑜 )2+….

𝑠𝑜 (𝑡 )= 𝐴2𝜋

ℜ{𝑒 𝑗 (𝜔𝑜 𝑡− 𝛽𝑜 𝑧 ) ∫−𝜔𝑚

𝜔𝑚

𝐹 ( 𝑦 )𝑒 𝑗 (𝑡 −𝛽𝑜′ 𝑧 ) 𝑦𝑑𝑦 }

Page 14: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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EMLAB

3.11 Summary of Transmission Lines and Waveguides (cont’d)

Page 15: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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EMLAB

3.11 Summary of Transmission Lines and Waveguides (cont’d)

Ridge Waveguide (BW ↑, Power handling capacity ↓)

ridge – cutoff freq. 를 낮춰주는 역할

Dielectric Waveguide

εr2>εr1.

lossy at bends or

junctions.

Slotline

quasi-TEM mode

Page 16: EMLAB Microwave Engineering Chap. 3.7 ~ Chap. 3.11

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3.11 Summary of Transmission Lines and Waveguides (cont’d)

Coplanar waveguide

quasi-TEM mode

Covered microstrip