Page 13 - Circuit Analysis II with MATLAB Applications

Chapter 1

Second Order Circuits

T his chapter discusses the natural, forced and total responses in circuits containing resistors,

inductors and capacitors. These circuits are characterized by linear second-order differential
equations whose solutions consist of the natural and the forced responses. We will consider
both DC (constant) and AC (sinusoidal) excitations.

1.1 The Response of a Second Order Circuit

n
A circuit containing energy storage devices (inductors and capacitors) is said to be an nth-order cir-
cuit, and the differential equation describing the circuit is an nth-order differential equation. For exam-
ple, if a circuit contains an inductor and a capacitor, or two capacitors or two inductors, along with
other devices such as resistors, it is said to be a second-order circuit and the differential equation that
describes it is a second order differential equation. It is possible, however, to describe a circuit having
two energy storage devices with a set of two first-order differential equations, a circuit which has
three energy storage devices with a set of three first-order differential equations and so on. These are
*
called state equations but these will not be discussed here.
The response is found from the differential equation describing the circuit, and its solution is
obtained as follows:

1. We write the differential or integrodifferential (nodal or mesh) equation describing the circuit. We
differentiate, if necessary, to eliminate the integral.
2. We obtain the forced (steady-state) response. Since the excitation in our work here will be either a
constant (DC) or sinusoidal (AC) in nature, we expect the forced response to have the same form
as the excitation. We evaluate the constants of the forced response by substitution of the assumed
forced response into the differential equation and equate terms of the left side with the right side.
Refer to Appendix B for the general expression of the forced response (particular solution).

3. We obtain the general form of the natural response by setting the right side of the differential
equation equal to zero, in other words, solve the homogeneous differential equation using the
characteristic equation.

4. Add the forced and natural responses to form the complete response.
5. We evaluate the constants of the complete response from the initial conditions.

* State variables and state equations are discussed in Signals and Systems with MATLAB Applications, ISBN 0-
9709511-3-2 by this author.

Circuit Analysis II with MATLAB Applications 1-1
Orchard Publications