ELEC 226, Spring 2003
Prof. Rich Kozick
Laplace Transform for Circuit Analysis
The following steps are used to analyze circuits with the Laplace
- 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
- 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
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.