Page 172 - Schaum's Outline of Theory and Problems of Electric Circuits

Higher-Order Circuits

and Complex Frequency

8.1 INTRODUCTION

In Chapter 7, RL and RC circuits with initial currents or charge on the capacitor were examined and
first-order differential equations were solved to obtain the transient voltages and currents. When two
or more storage elements are present, the network equations will result in second-order differential
equations. In this chapter, several examples of second-order circuits will be presented. This will
then be followed by more direct methods of analysis, including complex frequency and pole-zero plots.

8.2 SERIES RLC CIRCUIT
The second-order differential equation, which will be examined shortly, has a solution that can take
three different forms, each form depending on the circuit elements. In order to visualize the three
possibilities, a second-order mechanical system is shown in Fig. 8-1. The mass M is suspended by a
spring with a constant k. A damping device D is attached to the mass M. If the mass is displaced
from its rest position and then released at t ¼ 0, its resulting motion will be overdamped, critically
damped,or underdamped (oscillatory). Figure 8-2 shows the graph of the resulting motions of the
mass after its release from the displaced position z 1 (at t ¼ 0).

Fig. 8-1

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