7/31/2019 Transmission Lines Upload
1/47
Transmission Lines
7/31/2019 Transmission Lines Upload
2/47
7/31/2019 Transmission Lines Upload
3/47
An equivalent circuit of a transmission line can be developed by
considering a pair of straight wires of equal size; this line is
known as the parallel wire line.
Since the wires are of uniform size , the resistance of the
conducting material of which the wires are made may be
assumed to be uniformly distributed along their lengths.
The magnetic field links the wires and hence an inductance is
said to be present .This again is distributed uniformly along the
length of the line. Since this inductance impedes the current
flow , it is effectively in series with the uniformly distributed
resistance .
7/31/2019 Transmission Lines Upload
4/47
The fact that the input and output currents are different suggest
the possibility of an admittance between the wires . This shunt
admittance consists of a conductance and a capacitance in parallel.
The presence of capacitance is because the line consists of two
conductors separated by air-dielectric.
Because the dielectric is not perfect , a conduction current will
flow between the wires. This leakage path may be represented by
a conductance between the wires.
7/31/2019 Transmission Lines Upload
5/47
FIGURE OF EQUIVALENT CIRCUIT OF A
TRANSMISSION LINE
7/31/2019 Transmission Lines Upload
6/47
BASIC TRANSMISSION LINE EQN
On one side of the transmission line is a generator and on the
other side is a load shown by a purely resistive element.
Consider length dl on the transmission line and the voltages
and currents on two sides of dl .
The voltage changes by an amount dE as a result of the drop
produced by line current I flowing through the resistance Rdl
and reactance jdl.
The current also changes by small amount dI as a result of
current flow through the capacitance Cdl and conductance
Gdl.
7/31/2019 Transmission Lines Upload
7/47
figure
7/31/2019 Transmission Lines Upload
8/47
7/31/2019 Transmission Lines Upload
9/47
7/31/2019 Transmission Lines Upload
10/47
7/31/2019 Transmission Lines Upload
11/47
7/31/2019 Transmission Lines Upload
12/47
7/31/2019 Transmission Lines Upload
13/47
7/31/2019 Transmission Lines Upload
14/47
7/31/2019 Transmission Lines Upload
15/47
7/31/2019 Transmission Lines Upload
16/47
7/31/2019 Transmission Lines Upload
17/47
7/31/2019 Transmission Lines Upload
18/47
7/31/2019 Transmission Lines Upload
19/47
7/31/2019 Transmission Lines Upload
20/47
7/31/2019 Transmission Lines Upload
21/47
7/31/2019 Transmission Lines Upload
22/47
7/31/2019 Transmission Lines Upload
23/47
7/31/2019 Transmission Lines Upload
24/47
7/31/2019 Transmission Lines Upload
25/47
7/31/2019 Transmission Lines Upload
26/47
7/31/2019 Transmission Lines Upload
27/47
7/31/2019 Transmission Lines Upload
28/47
7/31/2019 Transmission Lines Upload
29/47
7/31/2019 Transmission Lines Upload
30/47
7/31/2019 Transmission Lines Upload
31/47
STANDING WAVES
When the load impedance is equal to characteristic impedance the
load absorbs all the power , and the only waves that are present
there are the travelling waves of voltage and current travelling from
the generator to the load.
If the load impedance differs from the characteristic impedanceonly some power is absorbed and the rest reflected back.
We have two sets of V and I , one travelling towards the load and
the other travelling back to the generator. These two sets of
travelling waves are travelling in opposite directions and then the
interference between the two results in an interference pattern
known as standing waves.
7/31/2019 Transmission Lines Upload
32/47
Consider a short circuited lossless line , at the load end a
voltage minimum and a current maximum occurs , because
the load is a short circuit here and the current will, therefore,
have a finite value, since the line has finite impedance. Again a
half wavelength from the load the voltage is maximum and the
current is zero.
After reflection from short circuit, the current starts travellingback towards the generator without a change in phase , but the
voltage is reflected with a 180 phase reversal.
7/31/2019 Transmission Lines Upload
33/47
STANDING WAVE RATIO
To describe the character of voltage distribution on a transmission
line a quantity termed as SWR is defined. SWR is expressed in
terms of the ratio of maximum to minimum amplitudes
The SWR is a measure of the mismatch between the load and the line ;
and in all practical measurements this quantity is determined first.
If the incident wave amplitude is E1 and the reflected wave is E2 , SWR
can be expressed as
7/31/2019 Transmission Lines Upload
34/47
We can express the reflection coefficient in terms of standing wave
ratio ; the exact relation between the two is given below.
7/31/2019 Transmission Lines Upload
35/47
IMPEDANCE INVERSION
For a quarter wavelength line or an odd multiple of /4, the
impedance at the source , the impedance at the source , seen when
looking towards the load is given by the relation:
When the load is mismatched standing waves of voltage and current
are set up along the line with a node and antinode being repeated
after each /2. If the load is not a short circuit the voltage and current
minima are not zero , thereby resulting in an SWR other than infinity.
Moreover the current nodes and voltage nodes are separated by a
distance of /4. At points of voltage nodes or current antinodes ,
the line impedance is low while at points of voltage antinode or
current node , the line impedance is high
7/31/2019 Transmission Lines Upload
36/47
This amounts to saying that the impedance at points of voltage
nodes is inversely proportional to the impedance at points of
voltage antinodes. The above equation states this mathematically
and the proportionality constant happens to be the square of thecharacteristic impedance of the line.
The quarter wave line provides impedance transformation upto
the highest frequency at which the transmission lines are used.
Eq. 1.47 shows that impedance at the input of a quarter wave
line depends upon load impedance and the characteristic
impedance of the interconnecting transmission lines.
If z0 can be varied ,the impedance at the input of/4 transformer
will be varied accordingly ,and the thus may be matched thus be
matched to the main line. This is particularly important in nearly
all transmission lines ,since for maximum power transfer the load
must be matched to the line itself.
7/31/2019 Transmission Lines Upload
37/47
7/31/2019 Transmission Lines Upload
38/47
IMPEDANCE MATCHING BY USE OF STUBS
A small section of short circuited transmission line isconnected in shunt with the main transmission line.
The distance lfrom the load and the length lof the stub are
so chosen that the reflected wave produced by shunting
impedance of the stub is equal and opposite to the reflectedwave existing on the line at that point because of mismatched
load Z L .
Thus the stub cancels out the two reflected waves .The
formulae for calculation of stub lengths and position lengths
are derived below.
7/31/2019 Transmission Lines Upload
39/47
7/31/2019 Transmission Lines Upload
40/47
7/31/2019 Transmission Lines Upload
41/47
7/31/2019 Transmission Lines Upload
42/47
7/31/2019 Transmission Lines Upload
43/47
Matching
(a) single stub matching : Matching a transmission line by a short circuited stub is
commonly employed to correct mismatch. The length of the stub is l and it is
placed at a distance lfrom the receiving end impedance ZR as shown in figure.
7/31/2019 Transmission Lines Upload
44/47
At R.F. ,Z0 is a pure resistance and at a length l the impedance
R1+jX1 is such that R1=R0.We now proceed to find analytically
the length and position of the stub required for matching .In
accordance with equation ( 1.39a) we have ,
7/31/2019 Transmission Lines Upload
45/47
7/31/2019 Transmission Lines Upload
46/47
7/31/2019 Transmission Lines Upload
47/47
Top Related