Chapter 11
Transmission Lines
•1
Transmission Line Theory
• High frequency circuit analysis
• At microwave frequencies, the dimensions of circuit
components and lines are comparable with the
wavelength – signal phase is critical
• Need to take line lengths into consideration in analysis –
include phase delays – treat currents and voltages as
waves
• Current and voltage wave propagation depends on the
inductance and capacitance of the lines
• Apparent input impedances vary with line length
•2
1
Transmission Line Theory
Review of KVL and KCL
I1
R
I2
L
G
V1
C
V2
Applying KVL:
𝐕𝟐 = 𝐕𝟏 − 𝐈𝟏 × 𝐑 − 𝐋
𝝏𝐈𝟏
= 𝐕𝟏 − 𝐈𝟏 × (𝐑 + 𝒋𝝎𝐋)
𝝏𝒕
Applying KCL:
Time Harmonic Current/Voltages
𝝏𝐕𝟐
𝐈𝟐 = 𝐈𝟏 − 𝐕𝟐 × 𝐆 − 𝐂
= 𝐈𝟏 − 𝐕𝟐 × (𝐆 + 𝒋𝝎𝐂)
𝝏𝒕
•3
Transmission Line Theory
• Lumped-Element Circuit Model
-
Consider incremental length of T-line
I(z, t)
I(z+Δz, t)
RΔz
V(z, t)
LΔz
GΔz
CΔz
V(z+Δz, t)
Δz
R – series resistance per unit length
L – series inductance per unit length
G – shunt conductance per unit length
C – shunt capacitance per unit length
•4
2
Transmission Line Theory
I(z, t)
I(z+Δz, t)
RΔz
LΔz
GΔz
V(z, t)
CΔz
V(z+Δz, t)
Δz
Kirchhoff voltage law (KVL)
V ( z , t )  Rz  I ( z , t )  Lz 
Kirchhoff current law (KCL)
I ( z , t )
 V ( z  z , t )
t
I ( z , t )  Gz  V ( z  z , t )  Cz 
V ( z   z , t )
 I ( z  z , t )
t
In differential form for time-harmonic signals
dI ( z )
dV ( z )
  (G  jC )V ( z )
  ( R  j L ) I ( z )
dz
dz
•5
Transmission Line Theory
• Wave Propagation on T-line
d 2V ( z )
d 2 I (z)
  2V ( z )  0
  2I (z)  0
2
2
dz
dz
Helmholtz Equation
Line current and voltage are given by
V ( z )  V0 exp(z )  V0 exp(z )
I ( z )  I 0 exp(z )  I 0 exp(z )


I ( z )  V0 exp(z )  V0 exp(z ) / Z 0
Propagation Constant:     j  ( R  jL)(G  jC )
Characteristic Impedance:
Z0 
( R  j L )
(G  j  C )
•6
3
Transmission Line Theory
• Lossless Line (R = 0 and G = 0)
    j  j LC
   LC

 0
Propagation constant is purely real and no decay in signal amplitude
Wavelength:

2


Characteristic Impedance:
2
 LC
L
C
Z0 
•7
Lossless Transmission Line
• Terminated Line
IL
VL
Z0 β
ZL
l
V ( z )  V0 exp( j z )  V0 exp( j z )


I ( z )  V0 exp( j z )  V0 exp( j z ) / Z 0
At the load,
ZL 
V ( 0) V0  V0

Z0
I ( 0) V0  V0
Reflection Coefficient:
V0  V0 

Z L  Z0 
V0
Z L  Z0
Z L  Z0
Z L  Z0
•8
4
Lossless Transmission Line
• Current and voltage standing waves patterns
V ( z )  V0 exp( j z )   exp( j z )
I (z) 
V0
exp( j z )   exp( j z )
Z0
Vmax  V0 1   
Vmin  V0 1   
• Standing Wave Ratio (SWR)
SWR 
1 
1 
•9
Standing Waves
• Voltage on Lossless T-line
V ( z )  V0 exp(  j  z )  V0 exp( j  z )
 V0 exp(  j  z )  V0 exp( j  z )
 V0  V0 exp(2 j z )  exp(  j z )
V ( z )  V0 1   cos(2 z )  j  sin(2 z ) exp(  j  z )
• Note the amplitude of the line voltage, for a purely real
 is
2
2
V ( z )  V0  1   cos 2 z     sin 2 z 
=V0 1   2  2 cos 2 z
•10
5
Standing Waves
2
||=1
||=0.5
||=0.2
1.8
|V(z)|/ V+0
1.6
L=|  |e j0
1.4
L=|  |e j//2
1.2
L=|  |e j
1
L=|  |e j3/2
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
Distance, 
•11
Lossless Transmission Line
• Impedance Transformation
l
Z0
Zin
(  l ) 
Z in 
ZL
V0 exp( j l )
  exp(2 j l )
V0 exp( j l )
V (  l ) V0 exp( j l )   exp( j l )

Z0
I (  l ) V0 exp( j l )   exp( j l )
Z in 
1   exp(2 j l ) Z
1   exp(2 j l ) 0
•12
6
Lossless Transmission Line
• Impedance Transformation – Lossless Line
Z0 
l
Z0
Zin
ZL
L
C
𝜷 = 𝝎 𝑳𝑪
Input Impedance:
Z in  Z 0
Z L  jZ0 tan l 
Z0  jZ L tan l 
Input impedance
depends on length
of T-line
•13
Lossless Transmission Line
• Impedance Transformation - Special Cases
(a) Short (𝒁𝑳 = 𝟎)
Z in  jZ0 tan l
(b) Open (𝒁𝑳 = ∞)
Z in   jZ0 cot l
(c) l   / 2
Z in  Z L
(d) l   / 4
Z02
Z in 
ZL
•14
7
Lossless Transmission Line
• Quarter-wave Transformer
Z1
Z0
RL
λ/4
Zin
Match input impedance to
Z in 
Z12
 Z0
RL
Z1  RL Z0
•15
Lossless Transmission Line
• Input Impedance
l
IIN
IIN
ZS
ZS
Z0
ZIN
ZL
ZIN
VS
VS
Actual T-Line Circuit
I IN 
VS
( Z S  ZIN )
Equivalent Circuit
PL 
1
2
I IN RIN
2
Power dissipated in load, since line is lossless
•16
8
Lossless Transmission Line
• Input Impedance – Application
l
IIN
IIN
Z0
ZS
Z1
ZS
ZL
ZIN
Z1
VS
VS
ZIN
VS
ZS
ZIN
VS
ZIN
Equivalent
Circuit I
Actual Circuit
I IN 
IIN
Equivalent
Circuit II

ZIN  Z1 || ZIN
 Z  Z  
IN
 S

•17
Lossless Transmission Line
• Time Averaged Power

1
Pav  Re V ( z ) I ( z )*
2
 2



1 V0
2

1   P  P
2 Z0
• Return Loss
RL  20 log10 
•18
9
Lossless Transmission Line
• Power Delivered to Load


1
1
RIN
2
PL  Re V ( l ) I ( l )*  VS
2
2
2
( RIN  RS )  ( X IN  X S )2
Load Matched to Line,
PL 
Z 0  ZIN
IIN
Z0
1
2
VS
2
2
( Z0  RS )2  X S
ZS
Generator Matched to Loaded Line RS  ZIN
2
ZIN
VS
V
1
RS
2
PL  VS
 S
2
2
( RS  RS )
8 RS
•19
Lossless Transmission Line
• Available Power
Maximize power delivered to load
P
0
RIN
yields
RIN  RS
X IN   X S
Z IN  Z S* - conjugate matching
Power delivered to load
2
1 VS
Pmax 
 Pa
8 RS
- available power
•20
10
Lossy Transmission Line
• Low-Loss Line
    j
 ( R  jL)(G  jC )

R 
G 
 1 

 ( jL)( jC ) 1 
j  L 
jC 


j R
G 
 j LC 1  


2

L

C  



1 R
 GZ 0 
2  Z0

  
   LC
•21
Lossy Transmission Line
• Characteristic Impedance:
Z0 
( R  j L )
L

(G  jC )
C
• Reflection Coefficient:
(  l )  e 2 jl e 2l
• Input Impedance:
Zin  Z0
Z L  Z0 tanhl 
Z0  Z L tanhl 
• Input Power and Load Power:
 2


1 V0
2
PIN 
1  ( l ) e 2l
2 Z0
 2

1 V0
2
PL 
1 
2 Z0

•22
11
Distributed Line Parameters
Parameter
R (/m)
Coaxial Line
Two-Wire Line
Planar Line
1 1 1

2 c  a b 
1
a c
𝟐
𝒘𝜹𝝈𝒄
L (H/m)

2
ln
G (S/m)
2
b
ln
a
C (F/m)
2
b
ln
a
b
a
d

d
cosh 1

2a
w

cosh 1
w
d
2a
d

cosh 1
w
d
2a
d
•23
Smith Chart
• Reflection Coefficient

Z L  Z0 zL  1

  e j
Z L  Z0 zL  1
zL 
j
ZL 1   e

Z 0 1   e j
rL  jx L 
1  r2  i2
rL 
(1  r ) 2  i2
(1  r )  ji
(1  r )  ji
xL 
2i
(1  r ) 2  i2
•24
12
Smith Chart
2

 1 
r 
 r  L   i2  

1

r
1

r
L 
L 


2
r  U 2  i2  a 2
2

 

r  1   i  1    1 
xL   xL 

2
2
r  12  i  V 2  b 2
• Equations represent circles in complex Γ domain Compare
with
 x  p 2   y  q 2  r 2
•25
Smith Chart
Constant rl Circle
Constant xl Circle
Constant  Circle
•26
13
Smith Chart
*
* zL = 1 + j
* zL = 2 - j
* zL = 0.5 – 0.2 j
*
*
2.0
•27
Smith Chart
• Radial distance represents
magnitude of Γ
• Angle represents phase of Γ
• Constant rL circles are centered
on Γr axis (Γi = 0)
• Constant xL circles are centered
along Γr = 1
• A given load ( zL = rL+ jxL ) is
represented by a point in the
complex Γ domain
|Γ|
*
1.0
2.0
•28
14
Smith Chart
• Since (  l )   exp( 2 j l )
adding a length, l , of T-line is
equivalent to rotating the load
point clockwise along a constant
radius circle (constant |Γ| ) by an
angle 2βl radians
*
• Half a wavelength ( λ/2 ) line is
equivalent to a full circle,
yielding the same impedance as
the load.
*
2.0
•29
Smith Chart
• Note zL = 1 is at the origin, where
|Γ| = 0. This represents the
load which is matched to Z0 .
• The outer radius |Γ| = 1,
represents complete mismatch,
i.e. reactive loads, with the
upper half being inductive and
the lower half being capacitive
• The normalized admittance can
be obtained by rotating the load
point by 180.
*
o
*
2.0
•30
15
Analytical vs. Smith Chart
• Analytical
• Smith Chart
V ( z )  V0 exp( j z )  V0 exp( j z )
 V0 exp(  j z )1   exp(2 j z )

(  l ) 
Negative 
Positive 
*
*
V0 Z L  Z 0

V0 Z L  Z 0
V0 exp(  j l )
  exp( 2 j l )
V0 exp( j l )
Vmax  V0 1   
Vmin  V0 1   
All ZL values map on to  plane
Adding a length of line maps to a
rotation along the constant ||
circle
•31
Analytical vs. Smith Chart
• Analytical
• Smith Chart
Purely reactive
load, || = 1
(  l )   exp(2 j l )
Z in (  l ) 
*
V ( l )
I (l )
 Z0
Z L  jZ 0 tan  l 
Z 0  jZ L tan  l 
l /2
Z in  Z L
l   /4
Z in 
Z02
ZL
*
The impedance value at any point
on the circle is the input impedance
•32
16
Impedance Matching
• Analytical
• Smith Chart
Purely real loads can be
matched by using ¼-wave
transformers – 1 variable
parameter, Z0
B = - j cotan l2
l1
*
B
/4
*
ZIN
Z1  Z L Z IN
ZL
Complex loads can be
matched by using single stub
tuners – 2 variable
parameters, l and B
l2
Z0=50
ZIN
Z0=50
ZL
l1
•33
Transients
• The impact of propagation delay on currents and
voltages along a transmission line in the transient state
Current at 𝑧 = 0, immediately after the switch is closed,
𝑽𝒈
𝑰 𝒛 = 𝟎, 𝒕 = 𝟎 = 𝑰𝟎 =
𝒁𝒈 + 𝒁𝟎
The voltage at input to the transmission line,
𝒁𝟎
𝑽 𝟎, 𝟎 = 𝑽𝟎 = 𝑰𝟎 𝒁𝟎 =
𝑽
𝒁𝒈 + 𝒁𝟎 𝒈
•34
17
Transients
• After the switch is closed waves, 𝐼 = 𝐼 and 𝑉 = 𝑉 will
propagate along the line with a velocity,
𝟏
𝒖=
𝑳𝑪
• The transit time to travel the length of the line,
𝒕𝟏 = 𝒍/𝒖
• 𝐼 and 𝑉 will be incident on the load at 𝑡 = 𝑡 and this
will generate reflected waves, 𝐼 and 𝑉 .
• The voltage at the load at 𝑡 = 𝑡 is given by the sum of
the incident and reflected waves as
𝑽 𝒍, 𝒕𝟏 = 𝑽 + 𝑽 = 𝑽𝟎 + 𝚪𝐋 𝑽𝟎 = (𝟏 + 𝚪𝐋 )𝑽𝟎
where the load reflection coefficient is
𝚪𝐋 =
𝒁𝑳 𝒁𝟎
𝒁𝑳 𝒁𝟎
•35
Transients
• Similarly, the current at the load at 𝑡 = 𝑡 is given by the
sum of the incident and reflected waves as
𝑰 𝒍, 𝒕𝟏 = 𝑰 + 𝑰 = 𝑰𝟎 − 𝚪𝐋 𝑰𝟎 = (𝟏 − 𝚪𝐋 )𝑰𝟎
• At 𝑡 = 2𝑡 , the reflected wave will reach the input to the
transmission line (𝑧 = 0), and there will be a reflection at
the input.
• The current and voltages at 𝑧 = 0 and 𝑡 = 2𝑡 are
𝑽 𝟎, 𝟐𝒕𝟏 = 𝚪𝐆 𝚪𝐋 𝑽𝟎 + (𝟏 + 𝚪𝐋 )𝑽𝟎
𝑰 𝟎, 𝟐𝒕𝟏 = −𝚪𝐆 − 𝚪𝐋 𝑽𝟎 /𝒁𝟎 + (𝟏 − 𝚪𝐋 )𝑽𝟎 /𝒁𝟎
where the generator reflection coefficient is given by
𝒁𝑮 − 𝒁𝟎
𝚪𝐆 =
𝒁𝑮 + 𝒁𝟎
•36
18
Transients
• The current and voltage waves will
continue to bounce back and forth
between the load and the generator.
• The bounce diagram on the right
illustrates these reflections
• The reflections will decrease in amplitude
and will reach a steady state
• As 𝑡 → ∞, the transmission line will look
like any wire and the voltage and current
are given by
𝒁𝑳
𝑽 𝟎, ∞ = 𝑽 𝒍, ∞ =
𝑽
𝒁𝑳 + 𝒁𝑮 𝒈
𝑽𝒈
𝑰 𝟎, ∞ = 𝑰 𝒍, ∞ =
𝒁𝑳 + 𝒁𝑮
Voltage
𝑡=0
Current
𝑡→∞
•37
Transients
• Example 11.8
•38
19