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 z1 (at t ¼ 0).

Fig. 8-1