A system defined by linear differential equation needs an excitation signal. This page is an introduction to the commonly used excitation signals in circuit analysis. The signals can be classified as random and deterministic signal. A random signal is stochastic in nature and the study of such signals is beyond the scope of this page. A deterministic signal can be expressed as a function of time and its value is known explicitly at any instant. The response of the first-order and second-order systems to some of the commonly used deterministic signals is within the scope of this text. The commonly used signal is either a singularity function or a complex frequency function.
The term singularity function refers to a class of excitations which can be expressed as a polynomial in t. A function f(t) is called a singularity function if f(t) or any of its derivatives is discontinuous. The commonly used singularity functions are:
The set of singularity functions used in circuit analysis are shown in Fig. 2.
An impulse function is infinite in value when t = 0. When t is not equal to zero, the value of impulse function is zero. The unit step function changes abruptly at t = 0. When t > 0, the value of unit step function is unity and it is zero when t < 0. The ramp function increases linearly with time, when t > 0 and it is zero when t < 0. The impulse function is the derivative of the unit step function, and the unit step function is the derivative of the ramp function. In other words, the unit step function is the integral of the impulse function and the ramp function is the integral of the unit step function. The properties of these signals are described more in details in the chapter on signals.
Equation (119) 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 (120). When a is greater than zero and and w is zero, the resultant function is an exponentially decaying function, as defined by equation (121).
When a is zero and and w is greater than zero, the resultant function is a sinusoidal function, as defined by equation (122). 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 plot of an exponentially decaying with a time constant of 1 second and an initial value of 1 is shown above.
A cosine function with an amplitude of 1 and w = 10 radians per second is shown above.
The plot of a complex frequency function is shown above. It is decaying with a time constant of 1 second and has an initial value of 1. The value of w is 10 radians per second.
The next page describes how the particular solution of a linear differential equation can be obtained.