The Laplace Transform in Circuit Analysis CHAPTER 13 The Laplace Transform in Circuit Analysis

CHAPTER 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 13.6 The Transfer Function and the Convolution Integral

CHAPTER CONTENTS 13.7 The Transfer Function and the Steady- State Sinusoidal Response 13.8 The Impulse Function in Circuit Analysis

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 In these equations, I0 is the initial current through the inductor, and V0 is the initial voltage across the capacitor.

Summary of the s-Domain Equivalent Circuits

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 V = Z I

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 know. 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.

13.4 The Transfer Function The transfer function is the s-domain ration of a circuit’s output to its input. It is represented as Where Y(s) is the Laplace transform of the output signal, and X(s) is the Laplace transform of the input signal

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.

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, h(t) 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.

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)

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.

The approximation of y(t). The impulse response Summing the impulse responses

13.7 The Transfer Function and the Steady-State Sinusoidal Response To 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.

If x(t) = A cos(ωt + ø), H(jω) = |H(jω)|e jθ(ω) Then STEADY-STATE SINUSOIDAL RESPONSE COMPUTED USING A TRANSFER FUNCTION

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.

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