2. TRANSMISSION LINES 7e Applied EM by Ulaby and Ravaioli

Types of Transmission Lines

Fields of Transmission Lines

Transmission Lines = Roads for Wave

A transmission line connects a generator to a load

Transmission lines include: • Two parallel wires • Coaxial cable • Microstrip line • Optical fiber • Waveguide • etc.

- Dispersion (분산 = 주파수에 따른 불균일 특성)

- 분산이 크면 신호왜곡 증가

- Distortion (신호왜곡)

Modes in Transmission Line Mode: 전기장의 분포형태

- TEM (transverse electromagnetic) mode: Ez = 0, H

z = 0, # of conductors > 1

fc = 0 (차단 주파수), Example = coaxial cable TEM mode

- Quasi-TEM: TEM with mixed dielectric, fc (cutoff frequency) = 0

Example = microstrip line

- TE (transverse electric) mode: Ez = 0, f

c > 0, Example = rectangular waveguide

- TM (transverse magnetic) mode: Hz = 0, f

c > 0, Example = coaxial TM01 mode

- Hybrid mode: TE+TM, fc > 0, Example = optical fiber HE11 mode

Electric and Magnetic Fields in A Transmission Line

,V d I d= − ⋅ = ⋅∫ ∫E L H L

Voltage and Current: obtained from E and H

- 전송선에는 전기장 파동, 자기장 파동이 진행

- 회로이론 적용을 위해 전송선의 전압, 전류 개념 적용

Transmission Line Circuit Model

Expressions will be derived in later chapters

Transmission Line Parameters: R, L, G, C R ', G ', C ' from electrostabics; L ' from magnetostatics

Transmission-Line Equations

ac signals: use phasors

Telegrapher’s equations

( , ) Re ( )

( , ) Re ( )

j t

j t

v z t V z e

i z t I z e

ω

ω

=

=

( )

( )

d VR j L I

d z

d IG j C V

d z

ω

ω

′ ′− = +

′ ′− = +

2 22

2 2

2

( )( ) 0 (wave equation)

( )( )

d V d VR j L G j L V V

d z d z

R j L G j L

ω ω γ

γ ω ω

′ ′ ′ ′− + + = − =

′ ′ ′ ′= + +

0 0( ) (voltage phasor)z zV z V e V e

γ γ+ − − += +

0 00 0

0 0( ) (current phasor)z z z z

V VI z I e I e e e

Z Z

γ γ γ γ+ −

+ − − + − += + = −

0

0

: voltage wave traveling in + direction

: voltage wave traveling in direction

z

z

V e z

V e z

γ

γ

+ −

− + −

0 0( ) ;z j z j z jV z V e A e e e V A e

γ ϕ α β ϕ+ + − − − += = =

( )( ) (propagation constant)R j L G j C jγ ω ω α β′ ′ ′ ′= + + = +

Phase constant

complex propagation constant

attenuation constant

(전파상수)

(감쇠정수)

(위상정수)

0 0

1

10 10

( ) ; 1 m,

: Np/m, 1 Np/m = 1 / 2.718 0.37

1(dB/m) 20log ( ) (20log ) 8.68

zV z V e z V V e

e

e e

α α

α

α

α α

+ − −

−

−

= = =

= =

= − = =

: phase constant (rad/m)

22 ; wavelength

: phase velocity (m/s)p

v

β

πβλ π β λλ

ωβ

= → = =

=

( , ) Re( ) cos( )j t zv z t V e A e t z

ω α ω ϕ β−= = + −

0 00 0

0 0( ) (current phasor)z z z z

V VI z I e I e e e

Z Z

γ γ γ γ+ −

+ − − + − += + = −

0 00 0 0

0 0

( ) ( ) (characteristic impedance)( )

V VR j LZ R jX

G j C I I

ωω

+ −

+ −

′ ′+= = + = = − Ω

′ ′+(특성임피던스)

0

0

R j L Z

G j C

Z

ω γ

γω

′ ′+ =

′ ′+ =

Example 2-1: Air Line

Lossless Transmission Line

2

0

00

0, 0 (Lossless)

=

0,

1,

1p

R G

L C

L C

LZ

C

L Z C

Z

v

L C

λ ω

α β ω

β βω ω

′ ′= =

′ ′−

′ ′= =

′=

′

′ ′= =

=′ ′

Lossless Line: Lossless TEM Line:

0 0( )( )

1

r r

p

L C

v

µε µ µ ε ε

β ω µε

µε

′ ′ = =

=

=

Material Parameters in Transmission Lines

0

0

(1 tan )(1 tan )

tan

r m

r e

j j

j j

µ µ µ µ µ δε ε ε ε ε δσ ωε δ

′ ′′= − = −′ ′′= − = −′=

0

1 2

: conductor's conductance

;

c

s s

s

c

R R fR R

W W

σ

π µσ

= + =

- Conductor (도체, 금속)

- Dielectric (유전체,플라스틱)

Low-Loss Transmission Lines ,R L G Cω ω′ ′ ′ ′<< <<

1/2 1/2

0 1 1 1 12 2

12 2 2 2

R j L L R G L R Gj j

G j C C j L j C C L C

L R G R

Z

Gj

C L C L C

L

C

ωω ω ω ω ω

ω ω ω ω

−′ ′ ′ ′ ′ ′ ′ ′ + = = + + − + ′ ′ ′ ′ ′ ′ ′ ′+

′ ′ ′ ′ ′ = + − − ′ ′ ′ ′ ′

′′

1/2 1/22

0 0

( )( ) 1 1

1 12 2

12 2 2 2

2 212

R GR j L G j C L C

j L j C

R Gj L C j j

L C

R G R Gj L C j

L C L C

R R G

Z

GL C

L YC

γ ω ω ωω ω

ωω ω

ωω ω ω ω

ωω

αω

′ ′ ′ ′ ′ ′ ′ ′= + + = − + + ′ ′

′ ′ ′ ′ − − ′ ′

′ ′ ′ ′ ′ ′= − − + ′ ′ ′ ′

′ ′ ′ ′= + = ′ ′

′ ′+

00

1, Y

Z

L Cβ ω

=

=

′ ′

Distortionless Transmission Line

19

Distortionless Transmission Line: Heaviside Condition ( )(1)( 1) (0) ( ) ( , : constants)

(0)a j bV

V z V e H f e e a b

V

γ γ ω− − − += = → = = =

1/2 1/22( )( ) 1 1

1

if

(Heaviside

, , and should be frequency-independent.

condition)

Note:

R GR j L G j C L C

j L j C

Rj L C

Rj L C R G j L C

L C

R G

L C

R L G C

j

L

ω ω

ω ω

γ ω ω ωω ω

ωω

′′ ′ ′ ′ ′ ′+ = +

′ ′

′ ′=′ ′

′ ′ ′ ′ ′ ′ ′ ′= + + = − + + ′ ′

′ ′ ′

′ ′ ′ ′

− = ′

1/2

0

1/2

1 1

if

R j L L R G

G j C C j L j C

Z

L

C

R G

L C

ωω ω

ω

ω

ω

−′ ′ ′ ′ ′ += = + +

′′

′

′ ′ ′ ′ ′+

=

′=′

′

Reflected Wave

In general:

wave along +z because coefficients of t and z have opposite signs

wave along –z because coefficients of t and z have the same sign

Microstrip Line - A Quasi-TEM line

Phase velocity in dielectric:

Phase velocity for microstrip:

Quasi-TEM : Ez & Hz very small

0 2= (wavelength on microstrip)re

λ πλβε

=

: effective dielectric constant

(1+ ) / 2

re

r re r

ε

ε ε ε≤ ≤

p

r

cv

ε=

p

re

cv

ε=

Microstrip (cont.)

s = w/h

1 1 12 2 1 12 / ( / )

r r

re

w h

ε εε + −= +

+

22

030 4 8 8ln 1

/ / /re

Z

w h w h w h

πε

= + + +

Coaxial Transmissioni Line (TEM Line)

7 700

1 2; ; 4 10 H/m, 5.8 10 S/m (copper)s s

s

R R fR R

W W

π µµ π σ

σ−′ = + = = ⋅ = ⋅

120 0 0 0; 8.854 10 F/m, 2.25 (PE = polyethylene)

r rL C µ ε µ ε ε ε ε−′ ′ = = = ⋅ =

tan ( )CG Cσ δ ω

ε′

′ ′= =

Skin Depth (AC Resistance)

Table: Copper (Cu) skin depth