# Laplace Transform
1 Zero 2 CC Poles

Cuthbert Nyack

## Equations

The equations describing the Laplace transform are shown above.
For a transient signal, the Laplace transform is a complex
function defined over the complex s plane. This contrasts with
the Fourier transform which is also a complex function but
defined over the real variable w.

## Transfer Function, 1 zero, 2 poles

Transfer function below has poles at -a + jb, -a - jb and a zero
at -1/z. This way of writing the transfer function means that
it becomes equal to the one for 2 poles when z = 0,

## Surface plot of magnitude

Magnitude plot differs from the one above by having a dip at
the zero where the magnitude of the transfer function goes to
zero.

## Surface plot of phase

The phase plot with a zero differs significantly from that
without a zero. As before the phase jump start off at the poles
but this time they move towards the real axis. When they reach
the real axis both travel along the axis in different directions.
One moves towards the zero and ends there. The other moves
towards minus infinity. This behaviour is similar to that
observed in root locus plots used for the analysis of control
systems.

Above applet shows how the impulse and step responses are affected
by adding a zero to the transfer function. For step(orange) the
vertical scale is 0 - +2. Step response is multiplied by
(a^{2} + b^{2}).
For Impulse response(red) the vertical scale is -1 - +1 when
vgain = 1. The horizontal scale is 0 - 8p
when hgain = 1.

## Magnitude and Phase response
for 1 zero and 2 conplex conjugate poles

Above Applet shows magnitude and phase response for a system
with 1 zero and 2 complex conjugate poles.

*Return to main page*

*Return to page index*

Copyright 1996 © Cuthbert A. Nyack.