# Laplace Transform
Single pole.

Cuthbert Nyack

## Transfer Function

The transfer function F(s) of a system with 1 pole at -a is
shown below. This is usually obtained from first order physical
systems or by finding the transform of e^{-at}.

## s plane

The location of the pole in the complex s plane is shown below.

## Surface plot of Magnitude

The above diagram shows a surface plot of the magnitude of the
Laplace transform function F(s) when F(s) has one pole in the
complex s plane. The plot is characterised by its "flatness"
over most of the s plane except at the pole where it rises
to infinity. The height of the peak has been cut-off at 15.
The red line shows the magnitude of F(s) along the imaginary
s axis and corresponds to the frequency response of a system
which has Laplace Transform F(s). Here the pole is at -0.125
and the maximum in the frequency response is ~8.

## Surface plot of Phase

The phase plot corresponding to the above magnitude plot is shown
above. Pole location is shown by the x on the negative real
axis. This plot is characterised by a jump in the phase by 2p
along the negative real axis from the pole location to minus
infinity. A closed path surrounding the pole will encounter
a phase change of 2p, while any other path will not. The red line
is the phase along the imaginary axis and corresponds to the
phase associated with the frequency response. Along the positive
imaginary axis, the phase changes from 0 to -p/2.

The above applet shows the impulse and step response of a system
whose transfer function has 1 pole at s = -a. The impulse response
is e^{-at} and is shown in red. The step response is
(1/a)[1 - e^{-at}] and is shown in orange. In the
above applet the step response is multiplied by a. The
vertical scale corresponds to 0 - +2 and the horizontal
axis to 0 - +4 seconds when the horizontal gain is one.
Purple line corresponds to a unit step. When a becomes negative
the system becomes unstable.

## Frequency Response of single Pole

The above Applet shows the Magnitude(red) and Phase(green) Response of a
system with a single pole. The vertical axis for phase is
-p/2 to +p/2
and for magnitude is 0 to +2. In this case, the magnitude is
multiplied by a. When hgain = 1, the horizontal scale is 0 to
8rad s^{-1}.

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Copyright 1996 © Cuthbert A. Nyack.