7.7 DIFFERENTIAL EQUATIONS

Many systems in engineering are modeled by linear DEs with constant coefficients to describe how various quantities (signals, voltages, currents, and so on) change with time. For convenience in our discussion, we provide a simple example for a second-order RLC circuit.

FIGURE 7.17 Parallel RLC circuit. Current source is is switched on at t = 0 as indicated by the unit step-function u(t).

nc07f017.eps

Example 7.22. A parallel RLC circuit (with resistor R, inductor L, and capacitor C) has an independent current source is that is switched on at time t = 0. The current through the inductor can be modeled by the following second-order DE:

(7.156) Numbered Display Equation

where the coefficients {a0, a1, a2} are functions of {R, L, C} and u(t) is the unit-step function. A circuit diagram is provided in Figure 7.17 for which it can be shown

(7.157) Numbered Display Equation

Depending on the coefficient values, the solution of (7.156) can take on one of three forms:

(7.158) Numbered Display Equation

(7.159) Numbered Display Equation

(7.160)

where and are functions of { ...

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