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Transmission Lines


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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 .


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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.


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FIGURE OF EQUIVALENT CIRCUIT OF A

TRANSMISSION LINE


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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.


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figure


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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.


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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.


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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


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We can express the reflection coefficient in terms of standing wave

ratio ; the exact relation between the two is given below.


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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


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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.


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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.


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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.


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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 ,


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