Commonly used practical excitation function for a network are:
1. a dc signal,
2. an exponentially decaying signal,
3. a sinusoidal signal and
4. a sinusoidal signal with an exponentially decaying envelope.
The use of complex frequency enables these signals to be described in a general manner. Let f(t) be defined by equation (3.1).
Equation (3.1) describes the complex frequency function. It is essentially an exponential function with a complex exponent. The advantage of using this function is that it describes four functions, depending on the values of a and w. When both of them are of zero value, the resultant function is a unit step function, as defined by equation (3.2). When a is greater than zero and and w is zero, the resultant function is an exponentially decaying function, as defined by equation (3.3). Since the power of the natural logarithm base e has no dimension, the unit of a is designated to be neper per second, where the neper is a unit with no dimension. Normally a is called the neper frequency.
When a is zero and and w is greater than zero, the resultant function is a sinusoidal function, as defined by equation (3.4). When both of them are greater than zero, the resultant function is a complex frequency function. It is an oscillating function, but the magnitude of oscillation decays with time. Hence the complex frequency function has an exponentially decaying envelope. The damped sinusoid can be defined as shown by equation (3.6). Equation (3.7) is the Laplace transform of the damped sinusoidal function.
The plot of poles of the damped sinusoid function is shown in Fig. 29. For a decaying sinusoid, the real part of the pole is negative. The plot of a damped sinusoid is shown below. The plot is generated using a MathCad program. The complex frequency signal is defined as shown below.
In this function, a = 0.5 and w = 4p . The plots are shown for a duration of 2 s.
We can extract the real and the imaginary parts and plot these functions.
Plot of the Imaginary part of the damped sinusoid signal
Plot of the Real part of the damped sinusoid signal
Polar Plot of the damped sinusoid signal
As shown by the polar plot, the magnitude of the damped sinusoid signal is 10 at t = 0, and it spirals inwards in the counterclockwise direction as t increases.
This page has described the complex frequency function. The next page is on energy and power signals.