Download Transmission Line Eq..

_____ Notes _____
Lossless Transmission Line
Signal loss occurs by two basic mechanisms: signal power can be
dissipated in a resistor [or conductance] or signal currents may be
shunted to an AC ground via a reactance. In transmission line
theory, a lossless transmission line does not dissipate power.
Signals, will still gradually diminish however, as shunt
reactances return the current to the source via the ground path.
For the power loss to equal zero, R = G = 0. This condition
occurs when the transmission line is very short. An oscilloscope
probe is an example of a very short transmission line. The
transmission line equation reduces to:
2 v
 2v

LC
x 2
t 2
15 – 6
2 i
2 i
2  LC
x
t 2
To determine how sinusoidal signals are affected by this type of
line, we simply substitute a sinusoidal voltage or current into the
above expressions and solve as before, or we could take a much
simpler approach. We could start with the solution for the
general case:
R  jLG  jC
  j   
Let R = G = 0, and simplify:
  j 
 jL jC
  LC j
2
Equating the real and imaginary parts:
0
   LC
15 – 7
This expression tells us that a signal travelling down a lossless
transmission line, experiences a phase shift directly proportional
to its frequency.
Wireless Communications Systems
15 - 1
Transmission Line Theory
_____ Notes _____
PHASE VELOCITY
A new parameter, known as phase velocity, can be extracted
from these variables:
Vp 
1


meters/second
LC 
15 – 8
Phase velocity is the speed at which a fixed point on a wavefront,
appears to move. In the case of wire transmission lines, it is also
the velocity of propagation., typically: 0.24c  Vp  0.9c .
The distance between two identical points on a wavefront is its
wavelength () and since one cycle is defined as 2 radians:

2

and   2f
therefore: Vp  f
In free space, the phase velocity is 3 x 108 meters/sec, the speed
of light. In a cable, the phase velocity is somewhat lower because
the signal is carried by electrons. In a waveguide transmission
line, the phase velocity exceeds the speed of light.
Substituting 13 – 8 into 13 – 5, we obtain:

v  vo cos  t  LC x
Or

15 – 9
Distortionless Transmission Line
A distortionless transmission line is not the same as a lossless
one. A lossless line modifies the phase characteristics of the
signal but does not consume power. A distortionless line does not
distort the signal phase, but does introduce a signal loss. Since
common transmission lines are not super conductors, a
distortionless line does produce attenuation distortion.
Phase distortion does not occur if the phase velocity Vp is
constant at all frequencies.
By definition, a phase shift of 2 radians occurs over one
wavelength .
2
Transmission Line Theory
_____ Notes _____
Since:
Vp  f
Then:

Vp 
2

and
f 

2
2  


 2 
This tells us that in order for phase velocity Vp to be constant, the
phase shift coefficient  must vary directly with frequency  .
Recall:   R  jLG  jC    j
The problem now is to find . This can be done as follows:
R  jL 
G  jC
  
 jL 
 jC
 jL 
 jC 
 j LC 1
R
G
1
jL
jC
The 2nd and 3rd roots can be expanded by means of the
Binomial Expansion. Recall:
1 x n  1  nx 
nn  1 2 nn  1n  2  3
x 
x  ...
2!
3!
In this instance n = 1/2. Since the contribution of successive
terms diminishes rapidly,  is expanded to only 3 terms:
2
2 
 1 R

1  R   1 G
1  G  
  j LC 1
 


1




 


 2 jL 8  jL  

 2 jC 8  jC  


2
 1 G
1  G  1 R
1 RG 
1
 


 
2
 2 jC 8  jC  2 jL 4 LC 


 1  R  G 2 1  R 2

 j LC  

  


 16  jL  jC  8  jL 


2
2
2 
R   G  1  R   G  
 1 
 
 

 

 16 
jL

  jC  64  jL   jC  


Since j equate the imaginary terms to find .
3
Transmission Line Theory
_____ Notes _____


 1  G 2 1 RG
1  R 2 1  R 2  G 2 
   LC 1  


 
  


 
 
 




  
2
L 


L
 
C
 
8 
64
 8 C 4 LC


Difference of Squares
Very Small
 1 R
G 2 
LC 1  




 

 8 L C 
Note that if
R
G

L C
   LC
then:
From this we observe that  is directly proportional to . This
means that the requirement for distortionless transmission is:
RC  LG
If we equate the real terms, we obtain:

RG
The Frequency Domain
Signal analysis is often performed the frequency domain. This
tells us how the transmission line affects the spectral content of
the signals they are carrying.
To determine this, it is necessary to find the Fourier Transform of
the transmission line equation.
2 v
v
2 v

RGv


RC

LG


LC
2
t
t 2
Recall: x
and recall the Fourier Transform:
Ff t F  

e
 jt
f t dt

To prevent this analysis from ‘blowing up’, we must put a
stipulation on the voltage function namely, that it vanishes to
zero at an infinite distance down the line. This comprises a basic
boundary condition.
let
4
v  0 as x  
Transmission Line Theory
_____ Notes _____
This stipulation is in agreement with actual laboratory
experiments. It is well known that the signal magnitude
diminishes as the path lengthens.
Likewise, a time boundary condition, that the signal was zero at
some time in the distant past and will be zero at some time in the
distant future, must be imposed.
let v  0 as t  
Although engineers have no difficulty imposing these
restrictions, mathematical purists, are somewhat offended. For
this and other reasons, other less restrictive transforms have been
developed. The most notable in this context, is the Laplace
transform, which does not have the same boundary conditions.
Having made the necessary concessions in order to continue our
analysis, we must find the Fourier Transform corresponding to
the following terms:
v 
 and
t 
Fv , F

Let:

2v 


2

t 

F
Fv  V
Then applying the transform on the derivative, we obtain:
v 

t 
F


e
 jt

v
dt
t
This equation can be solved by using integration by parts:
 u dv  uv   v du
u  e  j t  du   je  jt dt
v
and dv 
v  v
t
let

v 
 jt
 F   e  jt v |
dt
  v  je
t 



Applying the boundary conditions when t goes to infinity makes
the 1st term disappear.
5
Transmission Line Theory
_____ Notes _____

v 
 F   j e  jwt v dt
t 


Note that the resulting integral is simply the Fourier Transform.
In other words:
v 
 j Fv  jV
t 
F

 2 v 
2
2   j 
t
 
F
Similarly:
Fv   j2 V
We can now write the transmission line equation in the frequency
domain:
 2V
2
2  RGV  RC  LG j V  LC j V
x
Where: V  V   Fvt
Rearranging the terms, we obtain:
 2V
 RG   RC  LGj   jL jCV
x 2
Or
Since
 2V
 R  jLG  jCV
x 2
R  jLG  jC    j  
Then
Or
 2V
  2V
x 2
 2V
  2V  0
x 2
This represents the most general form of the transmission line
equation in the frequency domain. This equation must now be
solved for V to observe how voltage (or current) varies with
distance and frequency. This can be done by assuming a solution
of the form:
V  Ae  x  Be x
forward
wave
6
reverse
wave
Transmission Line Theory
_____ Notes _____
These terms represent an exponential decay as the signal travels
down the transmission line. If we ignore any reflections,
assuming that the cable is infinitely long or properly terminated,
this simplifies to:
V  Vo e x
To verify whether this assumption is correct, substitute it into the
equation, and see if a contradiction occurs. If there is no
contradiction, then our assumption constitutes a valid solution.
2
V e x   2 Vo e x  0
x 2 o

Vo e  x   2 Vo e x  0
x
 2 Vo e x   2 Vo e x  0


00
Thus we validate the assumed solution. This tells us that in the
frequency domain, the voltage or current on a transmission line
decays exponentially:
V  Vo e x
where

R  jL G  jC 
     j
  propagation constant
 = attenuation coefficient
 = phase coefficient
Where:
 
R2  2 L2 G 2   2 C2

1  1 L 
C 
tan    tan 1  

 R 
 G 
2 
 R  jLG  jC  cos
  Im R  jL G  jC  sin
  Re
and:
In exponential notation, a sinusoid may be represented by a
rotating unity vector, of some frequency:
e jt  cost   j sint
Note that the magnitude of this function is 1, but the phase angle
is changing as a function of t.
7
Transmission Line Theory
_____ Notes _____
Vo  e jt
If we let:
Then


t 
x 
j 
V  e jt e x  e jt e   jx  e x 
e
attenuation
phase

vs. x
vs. t and x
This result is quite interesting because it is the same solution for
the transmission line equation in the time domain. The term
x
e represents an exponential decay. The signal is attenuated as
length x increases. The amount of attenuation is defined as:
Attenuation in Nepers = lne x
Attenuation in dB
 x
= 20loge x  8.68589x
This allows us to determine the attenuation at any frequency at
any point in a transmission line, if we are given the basic line
parameters of R, L, G, & C.
jt x 
The term e
represents a rotating unity vector since:
e
jt x 
 cost  x   j sint  x 
The phase angle of this vector is x radians.
Characteristic Impedance
The characteristic impedance of a transmission line is also
known as its surge impedance, and should not be confused with
its resistance. If a line is infinitely long, electrical signals will
still propagate down it, even though the resistance approaches
infinity. The characteristic impedance is determined from its AC
attributes, not its DC ones.
Recall from our earlier analysis:
v
i
 Ri  L
x
t
and
i
v
 Gv  C
x
t
Taking the Fourier Transform of these expressions, we obtain:

V  RI  jLI and
x
8

I  GV  jCV
x
Transmission Line Theory
_____ Notes _____
If voltage and current are forcing functions of frequency, they are
not functions of distance. i.e. the frequency of a signal remains
constant regardless of how long the transmission line is, therefor:

V  V and
x

I I
x
Consequently, we may write the above equations as:
V  RI  jLI  R  jLI
I  GV  jCV  G  jCV
Taking the ratio of these two attributes in order to obtain
impedance:
V R  jL I

I G  jC V
V 2 R  jL

 
 
 I  G  jC
Zo 
R  jL
G  jC
This can be simplified for the low and high frequency case:
Z olow freq  
R
G
Zo high freq  
L
C
Skin Effect
Under DC conditions, electrons are uniformly spread throughout
the cross-section of the conductor. However, as frequency
increases, electrons have a tendency to redistribute themselves,
and migrate towards the outer surface.
The skin depth is the distance from the surface, where the current
density has dropped to 1/e of its surface value. [The conductor is
assumed to have a thickness of at least three times the skin
depth]. Under such conditions, the solid conductor can be
replaced by a hollow one.
9
Transmission Line Theory
_____ Notes _____
skin depth =  
1
meters
 f 
where
  permeability
  conductivity
Conductivity of copper: copper = 5.81 x 107 mhos per meter.
Permeability of copper: copper = 4  x 10-7 henries per meter.
-2
10
Skin Depth vs. Frequency
-3
Depth in Meters
10
-4
10
Copper
Aluminium
-5
10
-6
10
3
10
4
10
5
10
6
10
Frequency
7
10
8
9
10
10
RELATIVE PERMEABILITY
Material
Copper
Silver
Gold
Bismuth
Plastics
Aluminum
Titanium
Palladium
Nickel
Cobalt
  r o
Comments
r
0.99999
Diamagnetic
0.99998
“
0.99996
“
0.99983
“
~1.0
“
1.000021 Paramagnetic
1.00018
“
1.00082
“
250
Ferromagnetic
600
“
where :
o  4  107 henries/meter
For most transmission line dielectrics:   o
and since
10
o 
o
o
then  =



o
 r o

377
r
Transmission Line Theory
_____ Notes _____
RELATIVE DIELECTRIC CONSTANTS
r
1.0
1.0006
2.0
2.25
2.5
3.0
5.0
7.5
Material
Vacuum
Air
Teflon
Polyethylene
Paraffin paper
Rubber
Mica
Glass
   r o
where :
o 
1
9
 10
farads/meter
36
RELATIVE CONDUCTIVITY AND RESISTIVITY
Material
Aluminum
Brass
Copper
(annealed)
Gallium
Gold
Iron
Lead
Mercury
Nichrome I
Nickel
Silver
Steel
Tantalum
Tin
Titanium
Tungsten
Zinc
r
0.610
0.256
1.0
0.0176
0.7062
0.178
0.0782
0.01789
0.01538
0.198
1.05
0.0189
0.111
0.149
0.0209
0.307
0.294
r
1.64
3.9
1.0
56.8
1.416
5.6
12.78
55.6
65.0
5.05
0.95
52.8
9.0
6.7
47.8
3.25
3.4
   rc
 c  1.7241 108 ohms/meter
   r c
c  5.81 10 7 mhos/meter

1

L
ohms
A
L  length
R
A  cross -sectional area
11
Transmission Line Theory
_____ Notes _____
resistance per square = Rsq 
1
ohms

Twin Lead Cable
Twin lead cables are used to connect television receivers to set or
roof mounted antennas. The typical characteristic impedance of
these cables is 300Ω.
d
D
2
D

 
  1
 d 
for:
L
 2D 
ln  h/m
  d 
1 .2 1 0
In d u ctan ce
1 .1 1 0
1 10
9 10
8 10
7 10

2D 
ln 
 d 
C
-6
-6
Twin Lead
In d u ctan ce/m v s. D/d
-6
-7
-7
-7
3
4
5
6
7
8
9
10
9
10
D/d
3 .5 1 0
-11
Cap acitan ce p er Meter
Twin Lead
Cap acitan ce/m v s. D/d
3 10
2 .5 1 0
2 10
-11
-11
-11
3
12
4
5
6
D/d
7
8
f/m
Transmission Line Theory
_____ Notes _____
TWIN LEAD FIELDS
Coaxial Cable
Coax is used primarily in cable TV applications. The typical
characteristic impedance of these cables is 75Ω.
D
d
2
for:
L
 D 
ln  h/m
2  d 
C
D

 
  1
 d 
2
f/m
D 
ln 
 d 
G
2
mhos/m
D
ln
d
13
Transmission Line Theory
_____ Notes _____
5 10
In d u ctan ce p er Meter
4 .5 1 0
4 10
3 .5 1 0
3 10
2 .5 1 0
2 10
-7
-7
Co ax
In d u ctan ce/m v s. D/d
-7
-7
-7
-7
-7
3
1 .2 1 0
Cap acitan ce p er Meter
1 .1 1 0
1 10
9 10
8 10
7 10
6 10
5 10
4
5
6
D/d
7
8
9
10
8
9
10
-10
-10
Co ax
Cap acitan ce/m v s. D/d
-10
-11
-11
-11
-11
-11
3
4
5
6
D/d
7
Transient Analysis
When a DC source is attached to a transmission line, a voltage
pulse travels down the line the far end towards the load. If the
voltage surge meets any discontinuity or change in impedance, a
portion of the signal will be reflected back to the source.
Equilibrium will eventually be achieved as the resulting
bouncing signals diminish to zero.
Assuming a lossless transmission line, the fraction of the voltage
reflected back to its origin, is known as the reflection coefficient,
and is given by:
  o
D  D
 D  o
where
D  reflection at the discontinuity
 D  impedance at the discontinuity
 o  characteristic impedance of the transmission line
The instantaneous voltage on a transmission line can be thought
of as being composed of three components:
• The initial condition
14
Transmission Line Theory
_____ Notes _____
• The incoming signal
• The outgoing reflection
EXAMPLE
In the circuit illustrated below, let:
Vs = 10 volts
Rs = source resistance of 150 Ω
Zo = transmission line characteristic impedance of 50 Ω
ZL = load impedance of 25 Ω
Vs = source voltage of 10 volts
T = propagation time down the transmission line
Rs
Vs
Zo
Vin
VL
RL
If the switch is closed at t = 0, the instantaneous voltage Vin is
given by:
Vin t  0  VS 
o
50
 10 
 2.5v
RS   o
150  50
The reflection coefficient at the load is given by:
L 
Z L  Zo 25  50
1


Z L  Zo 25  50
3
The reflection coefficient at the source is given by:
S 
ZS  Zo 150  50 1


ZS  Zo 150  50 2
Since it takes T seconds for the input signal to travel down the
line, the initial voltage at VL = 0.
When t = T, the input signal arrives and a voltage appears at VL.
and the total result is:
15
Transmission Line Theory
_____ Notes _____
1




VL  
V
V
V
2.5

 


 1.6667v
i
in
in  L  0  2.5  
t T
3

initial condition incoming signal reflected signal
-.83333
Notice that in this example, the reflected signal is negative and
subtracts from the incoming signal. Had RL been greater than Zo,
the reflected signal would have been positive. The -0.8333 v
reflected signal component is sent back to the source where it is
reflected again. At t = 2T, the voltage at Vin is given by:
1





VL  
V
Vin

 V

 
 1.2501v
i
in  S  2.5  .8333  .8333
t 2T
2

initial condition incoming signal reflected signal
-.4167
The -0.4167 volt reflection is sent back down to the load:
VL t  3T  1.6667  .4167  .4167 
1
 1.3889v
3
This process continues until equilibrium is reached. Both ends of
this lossless loop will converge at a final value of:
V final  Vin  VL  VS 
Time
0
T
2T
3T
4T
5T
6T
7T
RL
25
 10 
 1.429v
RS  RL
150  25
Vin
2.5
2.5
1.25
1.25
1.4583
1.4583
1.4236
1.4236
VL
0
1.6667
1.6667
1.3889
1.3889
1.4352
1.4352
1.4275
The derivation of these numbers is often more readily apparent
from a ‘bounce diagram’. This type of sketch shows the size of
the signal that is reflected, back and forth between the two ends
of the transmission line.
16
Transmission Line Theory
_____ Notes _____
Time
Vin
2.5
0
VL
0
2.5
1.6666
T
-.8333
1.25
2T
Reflected Signal
-.4166
1.3889
3T
.1389
1.4583
4T
.0694
1.4352
5T
-.0231
1.4236
6T
-.0116
1.4274
7T
.0039
2.5
2
Vin
1.458
1.4236
1.25
1
T
2T
3T
4T
5T
6T
7T
8T
t
2
1.6667
VL
1.4352
1.3889
1.4275
1
0
T
2T
3T
4T
5T
6T
7T
8T
t
From the forgoing analysis, we can conclude that:
Termination
Open circuit
ZL > Zo
ZL = Zo
ZL < Zo
Short circuit
Voltage Reflection
Total, in phase, positive
Current Reflection
Total, out of phase,
negative
Partial, in phase positive Partial, out of phase,
negative
None
None
Partial, out of phase, Partial, in phase positive
negative
Total, out of phase, Total, in phase, positive
negative
17
Transmission Line Theory
_____ Notes _____
Reflections are undesirable. They send power back to the source,
which may ultimately cause damage, and they can lead to the
development of standing waves when the transmission line is
continuously excited.
Standing Waves
If waves are allowed to reflect back and forth on a transmission
line, the incident and reflected waves will interact to create a
standing wave. This is somewhat similar to the vibration of a
stringed instrument. The ratio of maximum to minimum voltage
of a standing wave is known as the VSWR† .
VSWR 
Emax I max 1 


Emin
Imin 1 
The VSWR can take on any value between 1 and ∞.
If the load is purely resistive, this expression may be simplified
to:
Zo
RL
VSWR  
or 


RL
Z
L

whichever is larger
If VSWR = ∞, total reflection occurs. Ideally for a matched load,
VSWR = 1 and there are no reflections. The energy contained in
the reflected wave may be dissipated in the cable itself as I2R
losses or absorbed by the generator.
A VSWR ≠ 1may be acceptable if the transmission line is used as
a tuned circuit or a reactive load.
Quarter Wavelength Impedance Transformer
To terminate a transmission line with a resistive load RL not
equal to its characteristic impedance, a /4 section with a
characteristic impedance of Z  Zo RL can be placed between
the two as an impedance transformer.
†
18
Voltage Standing Wave Ratio
Transmission Line Theory
_____ Notes _____
Two reflections occur; one at the input of the matching section,
and one at the load, but they are equal and anti-phase and
therefore cancel out.
The impedance of any point along a transmission line is given
by:
Z  jZo tanh x 
Z  Z o  L

Z o  jZ L tanh x 
Z  jZ o tanx 
Z  Z o  L

Z o  jZ L tanx 
This expression simplifies to
lossless transmission line [R = G = 0].
for the
Smith Chart
All values on the Smith chart are normalized to the characteristic
impedance of the line.
Z  
Z
Zo
The location of all impedances along the transmission line are
described by circles on a Smith chart.
19
Transmission Line Theory
_____ Notes _____
Inductance Loci
[j]
Resistance Loci
Capacitance Loci
[-j]
Smith Chart
20
Transmission Line Theory
_____ Notes _____
Assignment Questions
Quick Quiz
1. Signals are attenuated [linearly, exponentially] as they
propagate down a transmission line.
Analytical Questions
5. Show that the attenuation coefficient for a distortionless
transmission line is given by:

RG
5. Given the following circuit:
RS  50
10 volts
Zo  91
RL  50
Transmission Line
a) Calculate the reflection coefficient
b) Sketch the voltage waveform going into the transmission
line and at the load for 5 propagation time constants, when
the switch is closed
6. A coaxial cable with a .75 mm center conductor diameter,
using Teflon as the dielectric (r = 2), has a characteristic
impedance of 75 . Stating all assumptions, find:
a) The diameter of the outer conductor
b) The inductance per meter
c) The capacitance per meter
d) The phase velocity
7. A 150 MHz, 1 Vrms signal is injected into a transmission
line with the following characteristic values:
R = .025 /m
21
Transmission Line Theory
_____ Notes _____
L = .02 nH/m
C = .15 pfd/m
G = 1 S/m
a) Find the signal amplitude and phase angle after 1000
meters
b) Calculate the low and high characteristic impedance
c) Find the magnitude and phase angle of the cable
impedance at 250 MHz
d) Find the magnitude of the impedance half way along the 1
Km cable, if it is terminated in 50 Ω.
8. Explain why a lumped model of a transmission line does not
give the same results as a distributed model.
9. What is another name for characteristic impedance?
10. Fill in the reflection polarity in the following table:
Voltage Reflection
Current Reflection
Open Circuit
Short Circuit
11. Given the following circuit:
RS  50
10 volts
Zo  75
RL  100 
Transmission Line
a) Sketch the approximate waveforms going into the
transmission line and at the load for 3 propagation time
constants, when the switch is closed.
b) Find the final quiescent voltage on the transmission line.
c) Calculate the VSWR if the circuit is modified as follows:
22
Transmission Line Theory
_____ Notes _____
RS  75 
1 Vp-p
100 M Hz
Z o  75
RL  100 
20 meter Transmission Line
12. A coaxial cable with a 2 mm center conductor, uses Teflon as
the dielectric, and has a characteristic impedance of 91 .
Stating all assumptions, find:
a) the diameter of the outer conductor
b) The inductance per meter
c) The capacitance per meter
Composition Questions
1. What is a lossless transmission line?
2. Sketch and label the field patterns for a twin lead cable.
3. What conditions must be met for distortionless transmission?
23
Transmission Line Theory
_____ Notes _____
For Further Research
Anderson, Edwin M., Electric Transmission Line Fundamentals,
Reston Publishing, Reston, 1985
Chipman, Robert A., Transmission Lines,
Kreyszig, Erwin, Advanced Engineering Mathematics
Magnusson,
Philip C., Transmission Lines and Wave
Propagation, Allyn and Bacon, Boston, 1970
Poularikas & Seely, Signals & Systems
Reed & Reed, Mathematical Methods in Electrical Engineering
Sinnema,
William, Electronic Transmission Technology,
Prentice-Hall, Englewood Cliffs, 1979
Reference Data for Radio Engineers, IT&T
24