Electric Circuits, 9th Edition P30. Designed for use in a one or two-semester Introductory Circuit Analysis or Circuit Theory Course taught in Electrical or Computer Engineering Departments. Electric Circuits 9/e is the most widely used introductory circuits textbook of the past 25 years. As this book has evolved over the years to meet the changing learning styles of students, importantly, the underlying teaching approaches and philosophies remain unchanged. | 266 Natural and Step Responses of RLC Circuits Figure A circuit used to illustrate the natural response of a parallel RLC circuit. introducing these three forms we show that the same forms apply to the step response of a parallel RLC circuit as well as to the natural and step responses of series RLC circuits. Figure A circuit used to illustrate the step response of a parallel RLC circuit. Figure A circuit used to illustrate the natural response of a series RLC circuit. Introduction to the Natural Response of a Parallel RLC Circuit The first step in finding the natural response of the circuit shown in Fig. is to derive the differential equation that the voltage v must satisfy. We choose to find the voltage first because it is the same for each component. After that a branch current can be found by using the current-voltage relationship for the branch component. We easily obtain the differential equation for the voltage by summing the currents away from the top node where each current is expressed as a function of the unknown voltage v v 1 Z T dv p y vdr 0 C 0. K L Jq at We eliminate the integral in Eq. by differentiating once with respect to t and because 1 is a constant we get Figure A circuit used to illustrate the step response of a series RLC circuit. 1 dv v d2v - - C - 0. R dt L dt2 8-2 We now divide through Eq. by the capacitance C and arrange the derivatives in descending order d2v 1 dv v d RC dt C Comparing Eq. with the differential equations derived in Chapter 7 reveals that they differ by the presence of the term involving the second derivative. Equation is an ordinary second-order differential equation with constant coefficients. Circuits in this chapter contain both inductors and capacitors so the differential equation describing these circuits is of the second order. Therefore we sometimes call such circuits second-order circuits. The General Solution of the Second-Order Differential Equation We can t .
