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Analysis of reflections on a transmission line

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A transmission line of length l, characteristic impedance Z0 = 100 Ohm, and one-way time of flight T = l/c is connected at z=0 to a 100 volt DC battery through a series source resistance Rs = 100 Ohm and a switch. The z=l end is loaded by a 300 Ohm resistor.

a) The switch at the z=0 end has been closed for a very long time so that the system is in the DC steady state. What are the values of the positive and negative traveling wave voltage amplitudes V{subscript}+ (z-ct) and V{subscript}_ (z+ct)?

Please see the attachment for circuit diagram.

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Solution Preview

See the attached file.

In general we can describe a voltage propagating down the line in the forward direction by

V+ = V(+)*exp(-gamma*x) (1)

Where gamma is the propagation constant of the line in question and is equal to alpha + jbeta, alpha being the attenuation per unit length on the transmission line, beta being the phase change per unit length of the line. Here we denote V(+) as the magnitude of the voltage at any point.

We will in turn consider the transmission line to be lossless so we can say α = 0 in (1) then we get an expression for V+ described by (2).

V+ = V(+)*exp(-j*beta*x) (2)

Now we can deduce beta as the number of radians change per unit length (phase change per unit length on the line). This is equal to the source angular frequency (omega) divided by the phase velocity of propagation along the line given as (c). Thus we can say that

beta = omega/c (3)

Applying (3) in (2) we get

V+ = V(+)*exp(-j*omega*x/c) (4)

As we are given a DC source and in the steady state omega = 0 ...

Solution Summary

The solution determines what are the values of the positive and negative traveling wave voltage amplitudes.