3.1 SINUSOIDAL STEADY-STATE PHASOR ANALYSIS 103
a systematic algebraic approach for determining both the forced and the natural components
of a network response, is then taken up in Section 3.3. The concept of atransfer functionis
also introduced along with its application to solve circuit problems. Then, in Section 3.4 the
network response to sinusoidal signals of variable frequency is investigated. Also,two-port
networksand block diagrams, in terms of their input–output characteristics, are dealt with in
this chapter. Finally, computer-aided circuit simulations using PSpice and PROBE, as well as
MATLAB are illustrated in Sections 3.5 and 3.6. The chapter ends with a case study of practical
application.
3.1 SINUSOIDAL STEADY-STATE PHASOR ANALYSIS
The kind of response of a physical system in an applied excitation depends in general on the
type of excitation, the elements in the system and their interconnection, and also on the past
history of the system. The total response generally consists of aforced responsedetermined by
the particular excitation and its effects on the system elements, and anatural responsedictated
by the system elements and their interaction. The natural response caused by the energy storage
elements in circuits with nonzero resistance is alwaystransient; but the forced response caused
by the sources can have atransientand asteady-statecomponent. Theboundary conditions
(usuallyinitial conditions), representing the effect of past history in the total response, decide
the amplitude of the natural response and reflect the degree of mismatch between the original
state and the steady-state response. However, when excitations are periodic or when they are
applied for lengthy durations, as in the case of many applications, the solution for the forced
response is all that is needed, whereas that for the natural response becomes unnecessary. When
a linear circuit is driven by a sinusoidal voltage or current source, all steady-state voltages
and currents in the circuit are sinusoids with the same frequency as that of the source. This
condition is known as thesinusoidal steady state. Sinusoidal excitation refers to excitation
whose waveform is sinusoidal (or cosinusoidal). Circuits excited by constant currents or voltages
are called dc circuits, whereas those excited by sinusoidal currents or voltages are known as
ac circuits.
Sinusoids can be expressed in terms of exponential functions with the use of Euler’s identity,
ejθ=cosθ+jsinθ (3.1.1)
e−jθ=cosθ−jsinθ (3.1.2)
cosθ=
ejθ+e−jθ
2
(3.1.3)
sinθ=
ejθ−e−jθ
2 j
(3.1.4)
wherejrepresents the imaginary number
√
−1. The reader is expected to be conversant with
complex numbers.
If we are able to find the response to exponential excitations,ejθore−jθ, we can use the
principle of superposition in order to evaluate the sinusoidal steady-state response. With this in
mind let us now study the response to exponential excitations.
Responses to Exponential Excitations
Let us considerAestas a typical exponential excitation in whichAis a constant andsis acomplex-
frequency variablewith a dimension of 1/second such that the exponentstbecomes dimensionless.
