Transmission Line

Field and Wave Electromagnetic

Chapter9

Theory and Applications of Transmission Lines

Transmission Line

TEM (Transverse electromagnetic) waves guided by transmission lines.

( along the guiding line )

The three most common types of guiding structures that support TEM waves.

⒜ Parallel-plate transmission line ⇒ striplines

⒝ Two wire transmission line

⒞ Coaxial cable : No stray fields

E ⊥ H ⊥ k

Electromagnetic Theory 2 3

① y polarized

② Propagating in the +z direction

cf) Fringe fields at the edges of the plates are neglected.

TEM Wave along a Parallel-Plate Transmission Line (1)

0 z

E

=

yEy

=

yE e^−γ

0 z

x

H xH x E e

^γ

η

= = −

− _η^γ : propagating constant : intrinsic impedance

TEM Wave along a Parallel-Plate Transmission Line (2)

③ Assuming perfect conductor and a lossless dielectric

④ Boundary conditions At and

ε η μ

με ω β γ

=

=

= j j

= 0

y

y = d

0 ,

0 0

,

0 = ⇒ = = =

=

_{n x z y}

t H E E H

E

⎜⎜⎝

⎛

=

−

×

=

s t

t

J H H n

E E

) ˆ₂ ( _{1 2}

2

1 ⎜⎜

⎝

⎛

=

=

−

⋅

n n

s

B B

D D n

2 1

2 1

2 ( )

ˆ

ρ

Electromagnetic Theory 2 5

TEM Wave along a Parallel-Plate Transmission Line (3)

At

At

0 j z

sl sl y

y D ⋅ = ρ ⇒ ρ = ε E = ε E e

^{− β}

n = y

y n y

= 0 , ˆ = ˆ

ˆ ˆ

0 ^{j z}

sl sl x

y H J J zH z E e

^β

η

× = ⇒ = − =

−

y n d

y

= , ˆ = − ˆ

0 j z

su sl y

y D ρ ρ ε E ε E e

^{− β}

− ⋅ = ⇒ = − = −

ˆ ˆ

0 ^{j z}

su su x

y H J J zH z E e

^β

η

− × = ⇒ = = −

−

TEM Wave along a Parallel-Plate Transmission Line (4)

⑤

E

& satisfy Maxwell’s equation

E j H

H j E

ωε ωμ

=

×

∇

−

=

×

∇ H

x

y

H x H

E y

E = ˆ , = ˆ

y

x

E j H dE j H

ωμ dz ωμ

∇× = − ⇒ =

x

y

H j E dH j E

ωε dz ωε

∇× = ⇒ =

①

②

Electromagnetic Theory 2 7

TEM Wave along a Parallel-Plate Transmission Line (5)

Integrating ① over y from 0 to d,

cf)

∫

∫

^d

^E

^y

^{dy = j}

^d

^H

^x

^dy

dz d

0

0

ωμ

0

( )

0^d

( )

d y y

V z = − ∫ E dy = − E z d

; Potential difference from the lower plate to the upper plate

(z ) J H

_x

=

_su

w J z

I ( ) =

_su

assuming

J

_su

= z ˆ J

_su

( z )

Where is the width of the plate The total current flowing in the +z direction

w

TEM Wave along a Parallel-Plate Transmission Line (6)

Then i.e.

where

Flux linkage per unit current Integrating ② over x from 0 to w,

[ ]

( )

_su

( )

_su

( )

dV z d

j J z d j J z w

dz ωμ ω μ ^⎛ w ^⎞

− = = ⎜ ⎝ ⎟ ⎠

) ) (

( j LI z dz

z

dV = ω

−

) ( H m w

L = μ d

: inductance per unit length of the parallel-plate transmission line

①’

∫

∫

^w

^H

^x

^{dx = j}

^w

^E

^y

^dx

dz d

0

0

ωε

1m d

x y

z

1

BS

I I

Hd d

Hw w

μ μ

= Φ =

= =

Electromagnetic Theory 2 9

TEM Wave along a Parallel-Plate Transmission Line (7)

i.e.

where

①’ & ②’ : Time-harmonic transmission line equations.

( )

_y

( )

d I z j E z w

dz ωε

∴ =

( ) ( )

y

w w

j E z d j V z

d d

ω ε ^{⎛ ⎞} ω ε ^{⎛ ⎞}

= ⎜ ⎟ = − ⎜ ⎟

⎝ ⎠ ⎝ ⎠

( ) ( )

d I z j CV z

dz ω

− =

( F m )

d

C = ε w

: capacitance per unit length of the parallel-plate transmission lines

②’

0 0

)

^w _x

( ),

^w _{y y}

( ) cf ∫ H dx = I z ∫ E dx = E z w

) ( )

( z d V z E

_y

= −

TEM Wave along a Parallel-Plate Transmission Line (8)

Combining ①’ & ②’

The solutions of the above wave equations are waves propagating in the +z direction.

) ) (

(

) ) (

(

2 2

2

2 2

2

z dz LCI

z I d

z dz LCV

z V d

ω ω

−

=

−

=

Wave equations

⎢ ⎢

⎣

⎡

=

=

−

−

z j

z j

e I z I

e V z V

β β

0 0

) (

)

(

Electromagnetic Theory 2 11

TEM Wave along a Parallel-Plate Transmission Line (9)

The impedance at any location that looks toward an infinitely long transmission line

⇒ Characteristic impedance of the line

The propagating velocity

0 0

0

( ) ( )

( ) V

V z L

Z = I z = I = C Ω

ε η μ

w d w

Z

₀

= d =

1 1

( / )

u

p

m s

LC ω

β με

= = =

C L LC

L L

I V

LI j V j

LI dz j

dV

=

=

=

=

=

−

ω ω β

ω ω β

ω

0 0

0 0

Lossy Parallel-plate Transmission Lines (1)

Two loss mechanism dielectric loss

① Dielectric loss : dielectric medium have a non-vanishing loss tangent

i.e. permitivity

(cf) Reminding

σ

ohmic loss

conductivity of the dielectric medium

ε

∫ ∫

⋅

−

= ⋅

=

L

E d l s d D V

C Q

Line integration from the lower

potential to the higher potential Integration over a surface enclosing

the positive conductor

∫ ∫

⋅

−

= ⋅

L

E d l

s

d

ε E

Electromagnetic Theory 2 13

Lossy Parallel-plate Transmission Lines (2)

∫ ∫

∫ ∫

⋅

⋅

= −

⋅

⋅

= −

=

s d E

l d E s

d J

l d E I

R V

^{L L}

σ

σ

= ε

=

∴ G

RC C

d w d

C w

G ε σ

ε σ ε

σ _{= ⋅ =}

=

∴

: Conductance per unit length

(dielectric medium)

Lossy Parallel-plate Pransmission Lines (3)

② Ohmic loss

If the parallel-plate conductors have a very large but finite conductivity , ohmic power will be dissipated in the plates.

⇒Nonvanishing axial electric field at the plate surfaces (conduction current)

σc

Ez

zˆ

1

*

ˆ Re( ˆ ˆ )

av

2

z x

P = yp

_σ

= zE × xH

The average power per unit area dissipated in each of the conducting plates

: y component (loss)

Electromagnetic Theory 2 15

Lossy Parallel-plate Pransmission Lines (4)

Consider the upper plate

Surface impedance of an imperfect conductor :

For upper plate

cf) ⇒ only surface current flows

x

su

H

J =

Z

s

( )

t s

s

Z E

= J Ω

: The ratio of the tangential component of the electric field to the surface current density at

the conductor surface

c x z su

z

s

H

E J

Z = E = = η

: Intrinsic impedance of the plate conductor

1 ,

1 >>

>> f σ

c

Lossy Parallel-plate Pransmission Lines (5)

Intrinsic impedance of good conductor

∴ The ohmic power dissipated in a unit length of the plate having a width w is

c c s

s s

j f jX

R

Z σ

μ ) π 1 ( +

= +

= 1

2

Re(| | )

2

^{su s}

P

_σ

= J Z

(

²

)

|

2

2 |

1 J

_su

R

_s

W m

=

wP

σ

su x

s su z

J H

Z J E

=

=

Electromagnetic Theory 2 17

Lossy Parallel-plate Pransmission Lines (6)

( m )

f w w

R R

c c

s

⎟ = Ω

⎠

⎜ ⎞

⎝

= ⎛

σ μ π 2 2

Effective series resistance per unit length for both plates of a parallel-plate

transmission line of width w

( )

1

2

2

R

s

P wp I W m

σ σ

w

⎛ ⎞

= = ⎜ ⎟

⎝ ⎠

The power dissipated when a sinusioidal current of amplitude I flows through a

resistance R_s/w

wJ

su

I where =

⇒ Power loss in upper plate only

General Transmission Line Equations (1)

cf) Difference between transmission lines and ordinary electric networks

Electric Network T.L.

Physical dimensions ≪ λ Discrete circuit elements

(lumped parameters) No standing wave

Physical dimension ~ λ Distributed-parameter

Standing wave except under matched conditions

Electromagnetic Theory 2 19

General Transmission Line Equations (2)

Distributed parameters For differential length Δz

R : resistance per unit length(for both conductors ) (Ω/m) L : inductance per unit length ( for both conductors) (H/m) G : conductance per unit length (S/m)

C : capacitance per unit length (F/m) Series

element Shunt element

⎢⎣

⎡

⎢⎣

⎡

General Transmission Line Equations (3)

Kirchhoff’s voltage law

let

Kirchhoff’s current law at node N

let

①, ② : General transmission line equations.

0 ) , ) (

, ) (

, ( )

,

( − + Δ =

∂ Δ ∂

− Δ

− v z z t

t t z z i L t z zi R t z v

t t z L i t z z Ri

t z v t z z v

∂ + ∂

Δ =

− Δ

− + ( , )

) , ) (

, ( ) , (

( , ) ( , )

0 v z t ( , ) i z t

z Ri z t L

z t

∂ ∂

Δ → − = +

∂ ∂

^①

0 ) , ) (

, ) (

, (

) ,

( − + Δ =

∂ Δ + Δ ∂

− Δ + Δ

− i z z t

t t z z z v C t z z zv G t z i

t t z C v t z z Gv

t z z i

∂ + ∂

∂ =

− ∂

→

Δ ( , )

) , ) (

,

0 (

^②

Electromagnetic Theory 2 21

General Transmission Line Equations (4)

For time harmonic,

cf) cosine reference

V(z), I(z) : functions for the space coordinate z only, both may be complex

then,

[ ]

[

^{j t}

]

t j

e z I t

z i

e z V t

z v

ω ω

) ( Re ) , (

) ( Re ) , (

=

=

⎥ ⎥

⎥ ⎥

⎦

⎤

+

=

−

+

=

−

) ( ) ) (

(

) ( ) ) (

(

z V C j dz G

z dI

z I L j dz R

z dV

ω ω

⇒ Time-harmonic transmission line equations

Wave Characteristics on an Infinite T.L. (1)

From the coupled time-harmonic T.L. equations

where

) ) (

(

) ) (

(

2 2

2

2 2

2

z dz I

z I d

z dz V

z V d

γ γ

=

=

: propagation constant

: attenuation constant (Np/m)

) ( ) )(

( + +

⁻¹

= +

= α j β R j ω L G j ω C m γ

γ α

①’

②’

Electromagnetic Theory 2 23

Wave Characteristics on an Infinite T.L. (2)

The solutions of ①’ and ②’

wave amplitudes

For an infinite line (semi-infinite line with the source at the left end )

z z

z z

e I e

I z I z I z I

e V e

V z V z V z V

γ γ

γ γ

−

− +

− +

−

− +

− +

+

= +

=

+

= +

=

0 0

0 0

) ( )

( )

(

) ( )

( )

(

) , )(

,

(V₀⁺ I₀⁺ V₀⁻ I₀⁻

γ ω L j R I

V I

V +

=

−

=

₋⁻

+ +

0 0 0

0

→vanishes. (no reflected waves) only waves traveling in the +z direction

e^γz

Wave Characteristics on an Infinite T.L. (3)

cf) uniform plane waves in a lossy medium

C j G

L j R C

j G L j R I

V z I

z Z V

ω ω ω

γ γ

ω

+

= +

= +

= +

=

=

₊⁺

0 0

0

( )

) (

Characteristic impedance

; independent of z

z z

e I z I z I

e V z V z V

γ γ

− + +

− + +

=

=

=

=

0 0

) ( )

(

) ( )

(

( )( ^' )

j j j

γ α = + β = ωμ ^′′ + ωμ ωε ^{′ ′′} + ωε

ε ε

μ η μ

+ ′

′′

+ ′

= ′′

j j

c

Electromagnetic Theory 2 25

Wave Characteristics on an Infinite T.L. (4)

1. Lossless line

a. Propagation constant

b. Phase velocity

LC j

j β ω α

γ = + = 0

LC α

β ω

=

=

(A linear function of ω)

u

_p

= = LC 1 β

ω

(Non-dispersive)

Wave Characteristics on an Infinite T.L. (5)

c. Characteristic impedance

C jX L

R

Z

₀

=

₀

+

₀

=

C R

₀

= L

0

= 0 X

(constant)

(Non-reactive line)

Electromagnetic Theory 2 27

Wave Characteristics on an Infinite T.L. (6)

2. Low-loss line

a. Propagation constant _{12 12}

1

1 ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ +

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ +

= +

= j C

G L

j LC R

j

j β ω ω ω

α γ

1 1

2 2

R G

j LC

j L j C

ω ω ω

⎛ ⎞⎛ ⎞

≅ ⎜ + ⎟⎜ + ⎟

⎝ ⎠⎝ ⎠

1 1 2

R G j LC

j L C

ω ω

⎡ ⎛ ⎞ ⎤

≅ ⎢ ⎣ + ⎜ ⎝ + ⎟ ⎠ ⎥ ⎦

1 ,

2

C L

R G LC

L C

α ^{⎛ ⎞} β ω

∴ ≅ ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠ ≅

(Approximately a linear function of ω)

Wave Characteristics on an Infinite T.L. (7)

b. Phase velocity

c. Characteristic impedance

u

_p

= ≅ LC 1 β

ω

(Non-dispersive)

2 1 2

1 0

0

0

1 1

−

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ +

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ +

= +

= j C

G L

j R C

jX L R

Z ω ω

⎥ ⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝ ⎛ − +

≅ C

G L R j C

L

ω 2 1 1

2 0 ,

₀

1

0

⎟ ≅

⎠

⎜ ⎞

⎝ ⎛ −

−

≅

≅ C

G L R C

X L C R L

ω

Capacitive reactance

Electromagnetic Theory 2 29

Wave Characteristics on an Infinite T.L. (8)

3. Distortionless line

a. Propagation constant

b. Phase velocity

⎟⎠

⎜ ⎞

⎝⎛ = C G L R

( ) ⎟

⎠

⎜ ⎞

⎝

⎛ +

+

= +

= j C

L L RC j R

j β ω ω

α γ

( ^{R j L} )

L

C + ω

= C ,

R LC

α L β ω

∴ ≡ =

(A linear function of ω)

u

_p

= = LC 1 β

ω

(constant)

Wave Characteristics on an Infinite T.L. (9)

c. Characteristic impedance

C L C

L j RC

L j jX R

R

Z =

+

= + +

=

ω ω

0 0

0

0

0

0

R L

C X

=

=

(constant)

Electromagnetic Theory 2 31

Wave Characteristics on Finite Transmission Line (1)

The general case of a finite transmission line (Z₀) terminated in an arbitrary load impedance Z_L.

A sinusoidal voltage source with internal impedance Z_g is connected to the line at z=0.

⇒ Reflected waves exist on unmatched lines

°

g∠0 V

L L L

l z

I Z V I

V ⎟ = =

⎠

⎜ ⎞

⎝

⎛

= : Cannot be satisfied without term unless Ze^γz _L=Z₀

Wave Characteristics on Finite Transmission Line (2)

General solutions for the time-harmonic one-dimensional Helmholtz equations

(cf) - circuit theory

matched condition (Z_g=Z_L^*) →maximum transfer of power

- T.L.

line is matched when Z_L=Z₀. → no term

z z

z z

e I e

I z I

e V e

V z V

γ γ

γ γ

−

− +

−

− +

+

=

+

=

0 0

0 0

) (

) (

0 0

0 0

0

Z

I V I

where V

₊⁺

= −

₋⁻

=

e^γz

Electromagnetic Theory 2 33

Wave Characteristics on Finite Transmission Line (3)

Four unknowns

cf) not independent because of the constraint by the relations at z=0 and z=l

Let z=l

and

−

− + +

0 0 0

0

, I , V , I

V

: from the wave equation solutions

( )

( )

0 0 0 0

0 0

0 0

0 0

1 2 1 2

l l l

L L L

l l

L l

L L

V V e V e V V I Z e

V V

I e e

V V I Z e

Z Z

γ γ γ

γ γ

γ

+ − − +

+ −

− − −

⎛ = + ⎛⎜ = +

⎜ ⇒ ⎜

⎜ = − ⎜

⎜ ⎜ = −

⎝ ⎝

L L

L Z

V =I

Wave Characteristics on Finite Transmission Line (4)

New variable : distance measured backward from the load

( ) ( )

[ ]

( ) ( )

[

₀ ^{( )} ₀ ^{( )}

]

0

) ( 0 )

( 0

) 2 (

) 2 (

z l L

z l L

L

z l L

z l L

L

e Z Z e

Z Z Z

z I I

e Z Z e

Z I Z

z V

−

−

−

−

−

−

−

− +

=

− + +

=

∴

γ γ

γ γ

z l z′= −

( ) ( )

[ ]

( ) ( )

[

L ^z

]

z L

L

z L

z L

L

e Z Z e

Z Z Z

z I I

e Z Z e

Z I Z

z V

− ′

′

− ′

′

−

− +

′ =

− + +

′ =

γ γ

γ γ

0 0

0

0 0

) 2 (

) 2 (

Electromagnetic Theory 2 35

Wave Characteristics on Finite Transmission Line (5)

In order to simplify the above equations, using hyperbolic functions

2cosh

z z

e

^{γ ′}

+ e

^{−γ ′}

= γ z ^′ e

^γz′

− e

^−γz′

= 2sinh γ z ^′

( )

( )

0

0 0

( ) cosh sinh

( ) sinh cosh

L L

L L

V z I Z z Z z

I z I Z z Z z

Z

γ γ

γ γ

′ ′ ′

∴ = +

′ = ′ + ′

Two equations can provide the voltage and current at any point along a transmission line in terms of and .I_L,Z_L,γ Z0

Wave Characteristics on Finite Transmission Line (6)

z Z

z Z

z Z

z Z Z

z I

z z V

Z

L L

+ ′

′ + ′

= ′

′

= ′

′ γ γ

γ γ

cosh sinh

sinh cosh

) (

) ) (

(

0 0 0

z Z

Z

z Z

Z Z

L L

+ ′ + ′

= γ

γ tanh tanh

0 0 0

Impedance when look toward the load end of the line at a distance z’ from the load

Electromagnetic Theory 2 37

Wave Characteristics on Finite Transmission Line (7)

( at the source end of the line)

the generator looking into the line sees an input impedance Z_i

l Z

Z

l Z

Z Z Z

Z

L L l

z z

i

γ

γ tanh ) tanh

(

0 0 0

0

+

= +

=

′==

(cf) Impedance transformation

_i ⁱ _g

g i

g i

g i

V Z V

Z Z

I V

Z Z

⎡ =

⎢ +

⎢ ⎢

⎢ =

⎢⎣ +

(

=⁰

)

′=l z z

Wave Characteristics on Finite Transmission Line (8)

The average power delivered to the input terminals of the line

The average power delivered to the load

For a lossless line

( ) P

av i

= Re [ ] V

i

I

i^* _z=0,z′=l

2 1

( ) [ ]

L L L

L L z

l L z L L

av R I R

Z I V

V

P ²

2

0 ,

*

2 1 2

Re 1 2

1 = =

= _{= ′=}

( ) ( )

Pav i = Pav L

Electromagnetic Theory 2 39

Wave Characteristics on Finite Transmission Line (9)

If

⇒ No reflected waves

0, Z

Z_L = Z(z′)=Z₀

⎪⎭

⎪ ⎬

⎫

=

=

=

=

∴

−

−

−

−

z i z l L

z i z l L

e I e

e I z I

e V e

e Z I z V

γ γ

γ

γ γ

γ

) (

) (

) (

)

(

₀ Waves traveling in +z

direction

Transmission Line as Circuit Elements (1)

Transmission line having inductive or capacitive impedance

⇒ impedance matching between a generator and a load.

Frequency band : 300 MHz ~ 3GHz

cf) f < 300MHz : line’s physical dimension is too long f > 3GHz : waveguide is preferred

For lossless T.L.

Input impedance at distance from the load(Z_L) end

( ) ^{j l j l}

l R

Z

j β γ β β

γ = ,

₀

=

₀

, tanh = tanh = tan

l jZ

R

l jR

R Z Z

L L

i

β

β tan tan

0

0

0

+

= +

Impedance transformations by

lossless transmission line

l

Electromagnetic Theory 2 41

Transmission Line as Circuit Elements (2)

Special cases

1. Open-circuit termination

(

Z_L →∞

) l l jR

j R jX

Z

_{i i}

β

β ^cot

tan

⁰

0 0

0

= = − = − cf l l

λ β ² π

) =

Transmission Line as Circuit Elements (3)

X_i0 can be either capacitive or inductive depending on If

In practice, it is impossible to have an infinite load impedance at the end of a transmission line.

⇒ At high freq. ⇒ coupling and radiation

β .l

1, tan

l l l

β

<<

β

≅

β

0

0 0

1

i i

R L C

Z jX j j j

l LCl Cl

β ω ω

∴ = ≅ − = − = −

; Impedance of a capacitance of Cl farads

Electromagnetic Theory 2 43

Transmission Line as Circuit Elements (4)

2. Short circuit termination

(

^Z_L ⁼⁰

) l

jR jX

Z

_is

=

_is

=

₀

tan β

<< 1 β l

Ll j

Z

_is

= ω

: Impedance of inductance

Transmission Line as Circuit Elements (5)

3. Quarter-wave section :

⎟

⎠

⎜ ⎞

⎝

⎛ = =

, 2 4

β π λ l l

) , 3 , 2 , 1 4 (

) 1 2

( − =

= n n

l λ

) 2 1 2 4 ( ) 1 2

2 ( λ π

λ

β l = π n − = n −

±∞

= β l tan

2 0 i

L

Z R

= Z

Quarter wave line ⇒ impedance inverter.

quarter wave transformer.

Electromagnetic Theory 2 45

Transmission Line as Circuit Elements (6)

4. Half-wave section

cf) The characteristic impedance and the propagation constant Open-circuited line,

Short-circuited line,

⎟ ⎠

⎜ ⎞

⎝

⎛ = λ β = π l

l ,

π 2 λ β

n l n

l = ⋅ , = 2

L

i

Z

Z =

∴

(Half-wave line)

Only for lossless.

For lossy case, this properties are valid only for Z_L=Z₀

0 tan β l =

l Z

Z

Z_L →∞ : _io = ₀coth

γ

l Z

Z

Z_L →0 : _is = ₀ tanh

γ

Transmission Line as Circuit Elements (7)

5. Lossy line with a short-circuit termination

is io

Z Z Z =

∴

₀

1 1

1 tanh

^is

( )

io

Z m

l Z

γ =

^{− −}

0 0

sinh( )

tanh cosh( )

is

j l

Z Z l Z

j l

α β

γ α β

= = +

+

l l

j l l

l l

j l Z l

β α

β α

β α

β α

sin sinh

cos cosh

sin cosh

cos sinh

0

+

= +

Electromagnetic Theory 2 47

Transmission Line as Circuit Elements (8)

For

For

l n

l n

l n

l , sin 0, cos ( 1)

2 ⇒ = = = −

⋅

= λ β π β β

0 tanh 0

( )

Zis Z

α

l Z

α

l

∴ = ≅ assuming 1

tanh l

l l α

α α

<<

≅ : Series resonant circuit condition

(

n oddnumber

)

l n n

l = ⋅ ⇒ = , =

2 4

β π λ

0 cosβl=

0 0

is tanh

Z Z

Z αl αl

∴ = ≅ : Very large

: Parallel-resonant circuit condition

Reference & Homework

Ref. Microwave engineering by David M.

Pozar page 330-336

Electromagnetic Theory 2 49

Lines with Resistive Termination (1)

both incident and reflected wave exist Z0

Z_L ≠

( ) ( )

( ) ( )

( ) ( )

0 0

( ) ( )

0 0

0

( ) 2 ( ) 2

l z l z

L

L L

l z l z

L

L L

V z I Z Z e Z Z e

I z I Z Z e Z Z e

Z

γ γ

γ γ

− − −

− − −

⎡ ⎤

= ⎣ + + − ⎦

⎡ ⎤

= ⎣ + − − ⎦

( ) ( )

[ ]

( ) ( )

[

L ^z

]

z L

L

z L

z L

L

e Z Z e

Z Z Z

z I I

e Z Z e

Z I Z

z V

− ′

′

− ′

′

−

− +

′ =

− +

+

′ =

⇒

γ γ

γ γ

0 0

0

0 0

) 2 (

) 2 (

⇒

−

′ = l z z

z z

e e

− ′

′ γ

γ : right traveling wave (incident wave) : left traveling wave (reflected wave) where

Lines with Resistive Termination(2)

cf) current reflection coefficient of the load impedance Z

⎥ ⎦

⎢ ⎤

⎣

⎡

+ + − +

′ =

^{′ − z′}

L z L

L

L

e

Z Z

Z e Z

Z I Z

z

V

^{γ 2γ}

0 0

0

) 1

2 ( ) (

[

^z

]

z L

L

Z Z e e

I +

′

+ Γ

− ′

= (

₀

)

^γ

1

^2γ

2

Γ

Γ

+ =

= −

Γ

^jθ

L

L

e

Z Z

Z Z

0

0 : Voltage reflection coefficient of the load impedance Z_L where

[

^z

]

z L

L

Z Z e e

Z z I

I ′ = +

₀ ^γ′

− Γ

^−2γ′

0

1 ) 2 (

)

(

Electromagnetic Theory 2 51

Lines with Resistive Termination(3)

2 0

( 2 )

0

( 2 )

0 0

( ) ( ) 1

2

( ) 1

2

( ) ( ) 1

2

j z j z

L L

j z

L j z L

j z

L j z L

j

V z I Z R e e

I Z R e e

I z I Z R e e

R

β β

θ β

β

θ β

β

γ β

Γ

Γ

′ − ′

′

′ −

′

′ −

=

′ = + ⎡⎣ + Γ ⎤⎦

⎡ ⎤

= + ⎣ + Γ ⎦

⎡ ⎤

′ = + ⎣ − Γ ⎦

•

•

For a lossless transmission line,

0

0 0

( ) ( cosh sinh )

( ) ( sinh cosh )

L L

L L

V z I Z z Z z

I z I Z z Z z

Z

γ γ

γ γ

′ = ′+ ′

′ = ′+ ′

From the expression using hyperbolic functions

Lines with Resistive Termination(4)

0

0

, , cosh cos , sinh sin

( ) cos sin

( ) cos sin

,

L L L

L L

L L

L L L L L

j V I Z j j j

V z V z jI R z

I z I z jV z

R

Z R V I R

V

γ β θ θ θ θ

β β

β β

= = = =

′ = ′+ ′

′ = ′+ ′

=

•

=

•

For lossless l

ine

If

2

2 0 2

2

2 2

0 0

( ) cos sin

( ) cos sin ,

L

L

L L

z V z R z

R

R L

I z I z z where R

R C

β β

β β

⎛ ⎞

′ = ′+ ⎜ ⎟ ′

⎝ ⎠

⎛ ⎞

′ = ′+⎜ ⎟ ′ =

⎝ ⎠

Electromagnetic Theory 2 53

Lines with Resistive Termination(5)

max min

0

1 1

1 1

0, 1 ( )

1,

L

V s

s V s

s Z Z

s

+ Γ −

= = Γ =

− Γ +

Γ = = =

Γ = −

•

→ ∞

•

Standing-wave ratio (SWR)

For a lossless transmission line

when Matched load

whe

max min

min max

0 ( )

1, ( )

2 2 , 0, 1, 2, ...

L

L

M

Z

s Z

V I

z n n

V I

θ_Γ β π

=

Γ = + → ∞ → ∞

− ′ = − =

n Short circuit

when Open circuit

and occur when

cf)

and occur together wh

2 z_m (2n 1) , n 0, 1, 2, ...

θ_Γ − β ′ = − + π = en

Lines with Resistive Termination(6)

0

0 0

0

0

0

, ,

, 0 ( 0)

, 0 ( )

L

L L

L

L

L

R R

Z R Z R

R R

R R

R R

θ

θ π

Γ Γ

= = Γ = −

+

> Γ > =

< Γ < = −

①

②

For resistive terminations on a lossless line,

and real

and real

cf )

Electromagnetic Theory 2 55

Lines with Resistive Termination(7)

0 max min 0

0

0 0

0 max min

: ,

: ,

L

L L L

L

L L L

L L

R R

R R V V V V s

R R

R R

R R V V V V s

R R

> = = ∴ =

< = = ∴ =

cf)

Lines With Arbitrary Termination(1)

Lines With Arbitrary Termination

Let

Neither a voltage maximum nor a voltage minimum appears at the load ( ' 0)

If we let the standing wave continue by an xtra e d

L L L

Z R jX

at z

= +

−

=

− istance,

it will reach a minimum

Electromagnetic Theory 2 57

Lines With Arbitrary Termination(2)

0 0 0

0

'

tan tan 1. Find from s. use 1

1

2. Find from z . use 2 for 0.

3. Find , which is the ratio of ( ) at ( )

m m

L i at z i i

onto the right

m m

m m

L

R jR l

Z Z R jX R

R jR l

s s

z n

Z V z z

I z

β β

θ θ β π

′=

Γ Γ

= = + = +

+

Γ Γ = −

+

= ′ − =

′ ′

′

0

0

0.

1

1

j

L L L j

m

Z R jX R e

e R R

s

θ θ

Γ Γ

=

= + = + Γ

− Γ

=

zi

m m 2

Z l

λ

+ =

Transmission Line Circuits (1)

Constraint at the load side (Boundary condition) 0

Constraint at the generator end where 0

Transmission Line Circu its

L L L

V I Z at z l or z

z and z l

−

= = ′ =

− = ′ =

Voltage generator : Internal impedance:

0

and from the condition of load impedance

g

g

i g i g

V Z

V V I Z at z and z l

V

∴ = − − ① = ′ =

( )

^l

1

2 ^l

IL

Z Z e^γ

⎡

e^{− γ}

⎤

= + ⎣ + Γ ⎦ − ②