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

2. 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

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

4. 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.

5. Summary of the s-Domain Equivalent Circuits

6. 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

7. 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.

8. 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

9. 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.

10. 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.

11. 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)

12. 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.

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

16. 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.

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

18. 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.

19. THE END