ELEC 226: Laplace transform for circuit analysis

ELEC 226
Prof. Rich Kozick

Laplace Transform for Circuit Analysis

The following steps are used to analyze circuits with the Laplace transform.

Find the initial voltage across all capacitors and the initial current through all inductors. Be sure to clearly define the voltage polarities and current directions.
Draw the circuit in the s-domain for t > 0:
- Replace all voltage, current, and source waveforms by their Laplace transforms.
- Replace resistors, capacitors, and inductors by their s-domain equivalent circuit elements using Table 13.1. Note that these equivalent circuit elements include the s-domain impedance as well as the initial conditions.
Analyze the circuit in the s-domain to find expressions for the Laplace transform of the voltages and/or currents of interest (e.g., I₁(s), I₂(s), ..., V₁(s), V₂(s), ...).
For this circuit analysis, you can use all of the tools that we studied in Chapters 1-5, including Ohm's law (generalized to impedances, V(s) = I(s) Z), KCL, KVL, series/parallel equivalent impedances, voltage/current dividers, delta-Y transformations, node voltage analysis, mesh current analysis, source transformation, Thevenin/Norton equivalent circuits, superposition, and op amps (ideal model and realistic models).
Find the inverse Laplace transform to obtain the waveforms for the voltages and/or currents of interest (e.g., i₁(t), i₂(t), ..., v₁(t), v₂(t), ...).

Transfer Function, Poles, Zeros, and Frequency Response

Recall the definition of frequency response in sinusoidal steady-state analysis:

H(j w) = (Phasor of output) / (Phasor of input)

What is the meaning of frequency response? How do you measure it in the lab?

The transfer function H(s) is defined similarly in the s-domain (assuming all initial conditions are zero):

H(s) = (Laplace transform of output) / (Laplace transform of input)

The frequency response is equal to the transfer function evaluated along the s = jw (imaginary) axis. Why??

The transfer function is a ratio of polynomials: H(s) = N(s) / D(s)

The roots of the numerator N(s) are called the zeros.
The roots of the denominator D(s) are called the poles.

Why are the terms poles and zeros used? What is their significance? How are they used to analyze and design frequency-selective filters? Let's look at some familiar RC, RL, and RLC circuits from the point of view of transfer functions, poles, and zeros.