# CHAPTER 9

*The Complete Response of Circuits with Two Energy Storage Elements*

**IN THIS CHAPTER**

**9.1** Introduction

**9.2** Differential Equation for Circuits with Two Energy Storage Elements

**9.3** Solution of the Second-Order Differential Equation—The Natural Response

**9.4** Natural Response of the Unforced Parallel *RLC* Circuit

**9.5** Natural Response of the Critically Damped Unforced Parallel *RLC* Circuit

**9.6** Natural Response of an Underdamped Unforced Parallel *RLC* Circuit

**9.7** Forced Response of an *RLC* Circuit

**9.8** Complete Response of an *RLC* Circuit

**9.9** State Variable Approach to Circuit Analysis

**9.10** Roots in the Complex Plane

**9.11** How Can We Check …?

**9.12** **DESIGN EXAMPLE**—Auto Airbag Igniter

**9.13** Summary

Problems

PSpice Problems

Design Problems

**9.1** *Introduction*

In this chapter, we consider second-order circuits. A second-order circuit is a circuit that is represented by a second-order differential equation. As a rule of thumb, the order of the differential equation that represents a circuit is equal to the number of capacitors in the circuit plus the number of inductors. For example, a second-order circuit might contain one capacitor and one inductor, or it might contain two capacitors and no inductors.

For example, a second-order circuit could be represented by the equation

where *x*(*t*) is the output ...