ELECTRIC CIRCUITS EIGHTH EDITION JAMES W. NILSSON & SUSAN A. RIEDEL ELECTRIC CIRCUITS EIGHTH EDITION

THE LAPLACE TRANSFORM IN CIRCUIT ANALYSIS CHAPTER 13 THE LAPLACE TRANSFORM IN CIRCUIT ANALYSIS © 2008 Pearson Education

CONTENTS 13.1 Circuit Elements in the s Domain 13.2 Circuit Analysis in the s Domain 13.3 Applications 13.4 The Transfer Function 13.5 The Transfer Function in Partial Fraction Expansions © 2008 Pearson Education

CONTENTS 13.6 The Transfer Function and the Convolution Integral 13.7 The Transfer Function and the Steady- State Sinusoidal Response 13.8 The Impulse Function in Circuit Analysis © 2008 Pearson Education

13.1 Circuit Elements in the s Domain We can represent each of the circuit elements as an s-domain equivalent circuit by Laplace-transforming the voltage-current equation for each elements: Resistor: V = RI Inductor: V = s LI – LI0 Capacitor: V = (1/s C)I + V0 /s © 2008 Pearson Education

13.1 Circuit Elements in the s Domain In these equations, I0 = initial current through the inductor, V0 = initial voltage across the capacitor. V = L {v}, I = L {i) © 2008 Pearson Education

13.1 Circuit Elements in the s Domain The resistance element. Time domain Frequency domain © 2008 Pearson Education

13.1 Circuit Elements in the s Domain An inductor of L henrys carrying an initial current of I0 amperes © 2008 Pearson Education

13.1 Circuit Elements in the s Domain The series equivalent circuit for an inductor of L henrys carrying an initial current of I0 amperes © 2008 Pearson Education

13.1 Circuit Elements in the s Domain The parallel equivalent circuit for an inductor of L henrys carrying an initial current of I0 amperes © 2008 Pearson Education

13.1 Circuit Elements in the s Domain The s-domain circuit for an inductor when the initial current is zero © 2008 Pearson Education

13.1 Circuit Elements in the s Domain A capacitor of C farads initially charged to V0 volts © 2008 Pearson Education

13.1 Circuit Elements in the s Domain The parallel equivalent circuit for a capacitor initially charged to V0 volts © 2008 Pearson Education

13.1 Circuit Elements in the s Domain The series equivalent circuit for a capacitor initially charged to V0 volts © 2008 Pearson Education

13.1 Circuit Elements in the s Domain The s-domain circuit for a capacitor when the initial voltage is zero © 2008 Pearson Education

13.1 Circuit Elements in the s Domain We can perform circuit analysis in the s- domain by replacing each circuit element with its s-domain equivalent circuit. The resulting equivalent circuit is solved by writing algebraic equations using the circuit analysis techniques from resistive circuits. © 2008 Pearson Education

13.1 Circuit Elements in the s Domain Summary of the s-domain equivalent circuits © 2008 Pearson Education

13.2 Circuit Analysis in the s Domain Circuit analysis can be performed in the s domain by replacing each circuit element with its s-domain equivalent circuit. Ohm’s Law in the s-domain © 2008 Pearson Education

13.3 Applications Circuit analysis in the s domain is particularly advantageous for solving transient response problems in linear lumped parameter circuits when initial conditions are known. © 2008 Pearson Education

13.3 Applications It is also useful for problems involving multiple simultaneous mesh-current or node-voltage equations, because it reduces problems to algebraic rather than differential equations. © 2008 Pearson Education

13.3 Applications The Natural Response of an RC Circuit The capacitor discharge circuit An s-domain equivalent circuit An s-domain equivalent circuit © 2008 Pearson Education

13.3 Applications The Step Response of a Parallel Circuit The step response of a parallel RLC circuit An s-domain equivalent circuit © 2008 Pearson Education

13.3 Applications The Step Response of a Multiple Mesh Circuit The multiple-mesh RL circuit An s-domain equivalent circuit © 2008 Pearson Education

13.3 Applications The Use of Thévenin’s Equivalent A circuit to be analyzed using Thévenin’s equivalent in the s domain An s-domain model of the circuit A simplified version of the circuit, using a Thévenin’s equivalent © 2008 Pearson Education

13.3 Applications The Use of Superposition A circuit showing the use of superposition in s-domain analysis The s-domain equivalent for the above circuit © 2008 Pearson Education

13.3 Applications The Use of Superposition The circuit with Vg acting alone The circuit with Ig acting alone © 2008 Pearson Education

13.3 Applications The Use of Superposition The circuit with energized inductor acting alone The circuit with energized capacitor acting alone © 2008 Pearson Education

13.4 The Transfer Function The transfer function is the s-domain ratio of a circuit’s output to its input. It is represented as Y(s) is the Laplace transform of the output signal, X(s) is the Laplace transform of the input signal. © 2008 Pearson Education

13.5 The Transfer Function in Partial Fraction Expansions The partial fraction expansion of the product H(s)X(s) yields a term for each pole of H(s) and X(s). The H(s) terms correspond to the transient component of the total response; the X(s) terms correspond to the steady-state component. © 2008 Pearson Education

13.5 The Transfer Function in Partial Fraction Expansions If a circuit is driven by a unit impulse, x(t) = δ(t), then the response of the circuit equals the inverse Laplace transform of the transfer function, y(t) = L -1{H(s)} = h(t) © 2008 Pearson Education

13.5 The Transfer Function in Partial Fraction Expansions A time-invariant circuit is one for which, if the input is delayed by a seconds, the response function is also delayed by a seconds. © 2008 Pearson Education

13.6 The Transfer Function and the Convolution Integral The output of a circuit, y(t), can be computed by convolving the input, x(t), with the impulse response of the circuit, h(t): © 2008 Pearson Education

13.6 The Transfer Function and the Convolution Integral The excitation signal of x(t) A general excitation signal Approximating x(t) with series of pulses Approximating x(t) with a series of impulses © 2008 Pearson Education

13.6 The Transfer Function and the Convolution Integral The approximation of y(t) The impulse response Summing the impulse responses © 2008 Pearson Education

13.6 The Transfer Function and the Convolution Integral © 2008 Pearson Education

13.6 The Transfer Function and the Convolution Integral © 2008 Pearson Education

13.7 The Transfer Function and the Steady-State Sinusoidal Response We can use the transfer function of a circuit to compute its steady-state response to a sinusoidal source. To make the substitution s = jω in H(s) and represent the resulting complex number as a magnitude and phase angle. © 2008 Pearson Education

13.7 The Transfer Function and the Steady-State Sinusoidal Response If x(t) = A cos(ωt + ø), H(jω) = |H(jω)|e jθ(ω) then Steady-state sinusoidal response computed using a transfer function © 2008 Pearson Education

13.8 The Impulse Function in Circuit Analysis Laplace transform analysis correctly predicts impulsive currents and voltages arising from switching and impulsive sources. The s-domain equivalent circuits are based on initial conditions at t = 0-, that is, prior to the switching. © 2008 Pearson Education

13.8 The Impulse Function in Circuit Analysis A circuit showing the creation of an impulsive current The s-domain equivalent circuit © 2008 Pearson Education

13.8 The Impulse Function in Circuit Analysis The plot of i(t) versus t for two different values of R © 2008 Pearson Education

13.8 The Impulse Function in Circuit Analysis A circuit showing the creation of an impulsive voltage The s-domain equivalent circuit © 2008 Pearson Education

THE END © 2008 Pearson Education