Transmission Lines

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Transmission Lines : 

Transmission Lines

Slide 2: 

At audio frequencies when two pieces of apparatus , say a generator and a load are connected together by a pair of short leads , one expects equal input and output currents. Furthermore because of negligible lead resistance input and output voltages are also expected to be equal. However, when the lead length is appreciable and when the frequencies belong to R.F. Spectrum , neither of these equalities of current and voltages hold. The input and output voltages differ not only in magnitude but in phase.

Slide 3: 

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 .

Slide 4: 

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.




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 jωdl. The current also changes by small amount dI as a result of current flow through the capacitance Cdl and conductance Gdl.

figure : 


Slide 31: 

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 impedance only 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.

Slide 32: 

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 travelling back towards the generator without a change in phase , but the voltage is reflected with a 180 phase reversal.


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

Slide 34: 

We can express the reflection coefficient in terms of standing wave ratio ; the exact relation between the two is given below.


IMPEDANCE INVERSION For a quarter wavelength line or an odd multiple of λ/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 λ/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 λ/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

Slide 36: 

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 the characteristic 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 λ/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.


STUBS A quarter wave transformer may be used for matching the transmission line to the load Z L if Z L is purely resistive. If however a load is complex ,impedance, matching can be done by tuning out the resistive portion by an inductor or a capacitor and then match with a λ/4 transformer. Transmission lines with a short circuit at the far is more often used than the lumped components at higher frequencies. Another method of matching is by use of stubs with short circuits at the far end.


IMPEDANCE MATCHING BY USE OF STUBS A small section of short circuited transmission line is connected in shunt with the main transmission line. The distance l from the load and the length l’ of the stub are so chosen that the reflected wave produced by shunting impedance of the stub is equal and opposite to the reflected wave 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.

Matching : 

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 l from the receiving end impedance ZR as shown in figure.

Slide 44: 

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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