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TRANSMISSION LINE THEORY
(TEM Line)
A uniform transmission line is defined as the one
whose dimensions and electrical properties are
identical at all planes transverse to the direction of
propagation.
Circuit Representation of TL’s
A uniform TL may be modeled by the following circuit
representation:
R: Series resistance per unit length of line (for both
conductors (ohm/m)).
L: Series inductance per unit length of line (Henry/m).
G: Shunt conductance per unit length of line (mho/m).
C: Shunt capacitance per unit length of line (Farad/m).
The line is pictured as a cascade of identical sections,
each of z long.
Since z can always be chosen small compared to the
operating wavelength, an individual section of line may
be analyzed using ordinary ac circuit theory. In the
following analysis, we let 0z , so the results are
valid at all frequencies (hence for any physical time
variation).
Applying the Kirchhoff’s voltage law to the line section
gives:
( , )( , ) ( ) ( , ) ( ) ( , )
i z tv z t R z i z t L z v z z t
t
Rearranging yields:
( , ) ( , ) ( , )( , )
v z z t v z t i z tR i z t L
z t
Letting 0z , we get,
( , ) ( , )( , )
v z t i z tR i z t L
z t
Now applying Kirchhoff’s current law to the line
section gives:
( , )
( , ) , ( , )v z z t
i z t G z v z z t C z i z z tt
Rearranging yields:
( , ) ( , ) ( , )( , )
i z z t i z t v z z tG v z z t C
z t
Letting 0z
( , ) ( , )( , )
i z t v z tG v z t C
z t
Then the time domain TL or telegrapher equations are:
( , ) ( , )( , )
v z t i z tR i z t L
z t
( , ) ( , )( , )
i z t v z tG v z t C
z t
The solution of these equations, together with the
electrical properties of the generator and load, allow
us to determine the instantaneous voltage and current
at any time t and any place z along the uniform TL.
Lossless Line: For the case of perfect conductors (R=0)
and insulators (G=0), the telegrapher equations reduce
to the following form:
( , ) ( , )v z t i z tL
z t
( , ) ( , )i z t v z tC
z t
( , ) ( , )v z t i z tL
z z z t
( , )i z t vL L C
t z t t
Or,
2 2
2 2
2 2
2
( , ) ( , )0
( , ) ( , )0
v z t i z tLC
z t
i z t i z tLC
z t
Wave equation’s for voltage and current on a lossless
TL.
Although real lines are never lossless, lolessness
approximation for practical TL’s is very usefull.
TRANSMISSION LINES WITH SINUSOIDAL EXCITATION
We will only consider the sinusoidal steady-state
solutions.
Transmission-Line Equations:
Under sinusoidal steady state conditions, the TL
equations take the form:
( )( ) ( )
( )( ) ( )
dV zR j L I z
dz
dI zG j C V z
dz
Where ( )V z and ( )I z are voltage and current phasors.
The real sinusoidal voltage and current waveforms are
obtained from:
( , ) Re ( )
( , ) Re ( )
j t
j t
v z t V z e
i z t I z e
Wave Propagation on a TL
The second order differential equations for ( )V z and
( )I z are:
22
2
22
2
( )( ) 0
( )( ) 0
d V zV z
dz
d I zI z
dz
where 1/2
j R j L G j C
complex propagation constant.
attenuation constant (Np/m).
phase constant (rad/m).
The solution for ( )V z is:
0 0( )
( ) ( ) ( )
z zV z V e V e
V z V z V z
Where 0V
and 0V
are constants independent of z, and
0
0
( )
( )
z
z
V z V e
V z V e
Represent voltage waves traveling on the line in the
positive and negative z directions respectively.
WAVE PROPAGATION ON A TL
The second order equations for V(z) and I(z) are:
22
2
22
2
( )( ) 0
( )( ) 0
d V zV z
dz
d I zI z
dz
The solution for V(z) is:
0 0( ) z zV z V e V e
Where 0V
is a constant of voltage waves traveling in the
forward direction, 0V
is a constant of voltage waves
traveling in the reverse direction, independent of z.
Now consider the equation:
( )
( ) ( )dV z
R j L I zdz
If we substitude the solution for V(z) into the above
equation we get,
0 0 ( ) ( )z zV e V e R j L I z or,
0 0( )( ) ( )
z zI z V e V eR j L R j L
1/2
( )
( )
R j L G j C
R j L R j L
1/2
( )
( )
G j C
R j L R j L
Define
1/2
0( )
R j LZ
G j C
the characteristic impedance of the TL. Then,
0
1
( )R j L Z
So,
0 00 0
0 0
( )
( ) ( )
z z z zV VI z e e I e I e
Z Z
I z I z
Where,
0 00 0
0 0
V VI I
Z Z
Constants independent of z.
0 0
0 0
z zV VI z e I z e
Z Z
Lossless Line:
For the lossless line R=0, G=0.
i) 0
LZ
C
ii)
R j L G j C
j L j C j LC
j
LC Propagation constant
0 (no loss)
iii)
0 0( ) j z j zV z V e V e
0 0
0
1( ) ( )j z j zI z V e V e
Z
iv) If 0 0 0 0
j jV V e V V e then
0 0( , ) cos cosv z t V t z V t z
0 0
0
1( , ) cos cosi z t V t z V t z
Z
v) 2 2 1
2 f LC f LC
vi) 1
puLC LC
So,
1 p
p
p
vu
fLC
u f
TERMINATED TRANSMISSION LINE
Infinitely Long TL
For the infinite line, only forward traveling waves exist and
therefore at z=0,
00 0
0
0in
VV V I I
Z
VZ Z
I
Suppose now that the infinite line is broken at z . Since the
line to the right of z is still infinite, its input impedance is 0Z
and therefore replacing it by a load impedance of the same value
does not change any of the conditions to the left of z . This
means that a finite line terminated in its characteristic impedance
is equivalent to an infinitely long line. Like the infinite line, a finite
length line, terminated in 0Z has no reflections. Also, its input
impedance is equal to 0Z and independent of the line length.
00
0
in g
g
ZV V V
Z Z
0
0
g
in
g
VI I
Z Z
0inV V
00
0
in
VI I
Z
So,
0
0
0
g
g
ZV V
Z Z
And
0
0
( ) z j z
g
g
ZV z V e e
Z Z
0
( )g z j z
g
VI z e e
Z Z
The time-averaged power absorbed by the load is:
*
* 0
0 0
1 1Re Re
2 2
gl j l l j l
L L L g
g g
VZP V I V e e e e
Z Z Z Z
2 2
2 2002
00
1
2 2
g gl l
L
gg
V VZP e Z e
Z ZZ Z
If 0 (lossless line)
2
0
0
1
2
g
L in
g
VP Z P
Z Z
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