*s*-Domain Analysis: Understanding Poles and Zeros of F(s)

Laplace transforms can be used to predict a circuit’s behavior. The Laplace transform takes a time-domain function *f(t)*, and transforms it into the function *F(s)* in the *s-*domain. You can view the Laplace transforms *F(s)* as ratios of polynomials in the *s*-domain. If you find the real and complex roots (poles) of these polynomials, you can get a general idea of what the waveform *f(t)* will look like.

For example, as shown in this table, if the roots are real, then the waveform is exponential. If they’re imaginary, then it’s a combination of sines and cosines. And if they’re complex, then it’s a damping sinusoid.

The roots of the polynomial in the numerator of *F(s)* are *zeros*, and the roots of the polynomial in the denominator are *poles*. The poles result in *F(s)* blowing up to infinity or being undefined — they’re the vertical asymptotes and holes in your graph.

Usually, you create a *pole-zero* diagram by plotting the roots in the *s*-plane (real and imaginary axes). The pole-zero diagram provides a geometric view and general interpretation of the circuit behavior.

For example, consider the following Laplace transform *F(s)*:

This expression is a ratio of two polynomials in *s*. Factoring the numerator and denominator gives you the following Laplace description *F(s)*:

The *zeros*, or roots of the numerator, are *s* = –1, –2. The *poles*, or roots of the denominator, are *s* = –4, –5, –8.

Both poles and zeros are collectively called *critical frequencies* because crazy output behavior occurs when *F(s)* goes to zero or blows up. By combining the poles and zeros, you have the following set of critical frequencies: {–1, –2, –4, –5, –8}.

This pole-zero diagram plots these critical frequencies in the *s*-plane, providing a geometric view of circuit behavior. In this pole-zero diagram, X denotes poles and O denotes the zeros.

Here are some examples of the poles and zeros of the Laplace transforms, *F(s**)*. For example, the Laplace transform *F*_{1}*(s)* for a damping exponential has a transform pair as follows:

The exponential transform *F*_{1}*(s)* has one pole at *s = –**α* and no zeros. Here, you see the pole of *F*_{1}*(s)* plotted on the negative real axis in the left half plane.

The sine function has the following Laplace transform pair:

The preceding equation has no zeros and two imaginary poles — at *s = +j**β** and s = –j**β*. Imaginary poles always come in pairs. These two poles are *undamped*, because whenever poles lie on the imaginary axis *j**ω*, the function *f(t)* will oscillate forever, with nothing to damp it out. Here, you see a plot of the pole-zero diagram for a sine function.

A ramp function has the following Laplace transform pair:

The ramp function has double poles at the origin (*s* = 0) and has no zeros.

Here’s a transform pair for a damped cosine signal:

The preceding equation has two complex poles at *s = **α** + j**β* and *s = **α** – j**β* and one zero at *s = –**α*.

Complex poles, like imaginary poles, always come in pairs. Whenever you have a complex pair of poles, the function has oscillations that will be damped out to zero in time — they won’t go on forever. The damped sinusoidal behavior consists of a combination of an exponential (due to the real part *α* of the complex number) and sinusoidal oscillator (due to the imaginary part *β* of the complex number).

Here, you see depicted the pole-zero diagram for a damped cosine.