Transmission line exam problem 1

Figure 1

1.   Obtain expressions for the voltage and current standing wave patterns on a lossless open circuited transmission line.

Sketch these patterns.

Answer

The forward traveling wave is incident on the open circuit at z = 0.

V_f = A cos(wt - ß z)

Phasor representation of the forward traveling voltage wave is

V⁺ = A e^{-jß z}

At the open circuit (z = 0), the wave is reflected.

Phasor representation of the reflected voltage wave is

V^- = A e^{jß z}

The total voltage is the sum of the incident and reflected waves.

V = V⁺ + V^- = A(e^{-jß z} + e^{jß z}) = 2A( e^{jß z} + e^{jß z})/2 = 2A cos(ß z)

The magnitude of the voltage on the line is the magnitude of 2A cos(ß z) The magnitude is plotted in Figure 1.

Since for the forward traveling wave, V⁺ = Zo I⁺
And for the reflected wave V^- = -Zo I^-
The total current is

I = (A/Zo)(e^{-jß z} - e^{jß z}) = 2j(A/Zo)( e^{-jß z} - e^{jß z})/2j = -2j(A/Zo) sin(ß z)

The magnitude of the curent on the line is the magnitude of -2(A/Zo)j sin(ß z)

The magnitude is plotted in Figure 1.