Question

The phasor voltage on a lossless 50Ω transmission line connected to a purely resistive load is given by

V~(z) = 10(e⁻ʲ⁵z - 0.1eʲ⁵t)

i. What is the wave number (k) of this voltage?

Answer 1

The wave number (k) of the phasor voltage V~(z) = 10(e⁻ʘ⁵z - 0.1e⁸⁵t)i is 5, which is the coefficient of z in the exponent of the spatial part of the phasor.

Step-by-step explanation:

The wave number (k) in a phasor expression like V~(z) = 10(e⁻ʘ⁵z - 0.1e⁸⁵t)i, associated with a voltage on a transmission line, is the coefficient of the position variable z in the exponent of the complex exponential term that represents the spatial variation of the wave. In this case, it is given directly by the numerical coefficient adjacent to z in the exponent. Thus, for the phasor voltage V~(z), the wave number (k) is 5 per unit length (assuming z is in meters if the phase is in radians).

The wave number (k) of a voltage is related to its wavelength by the equation:

k = 2π/λ

Given that the phasor voltage is V~(z) = 10(e⁻ʲ⁵z - 0.1eʲ⁵t)i, we can see that the voltage is sinusoidal with a spatial component e⁻ʲ⁵z and a temporal component eʲ⁵t.

The wave number (k) is therefore 5.