# Laplace Transform
1 Real 2 CC Poles

Cuthbert Nyack

## Transfer Function

The transfer function F(s) of a system with a pole at -a and
complex conjugate poles at b - jc and b + jc.is
shown below.

## s Plane

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

## Surface Plot of Magnitude

The magnitude of the transfer function with a single pole on
the negative real axis and two complex conjugate poles is
shown above. The poles are at -1.725, -0.525 + 0.5j and
-0.525 - 0.5j.

## Surface Plot of Phase

The phase plot for a 3 pole system is shown above. The phase
starts at each pole and goes to infinity at ~120 degrees to
one another.

If you have Netscape 3.01 or other browser with a VRML viewer, then a 3-d
view of the magnitude, phase, real and imaginary parts of the Laplace
Transform of a system with 3 poles can be seen by clicking on one of
the links below. The values at the poles do go to infinity, but have been
truncated for convenience. The +ve imag axis coincides with the +ve z-axis
in the usual vrml coordinate system.
*MAGNITUDE*
*PHASE*
*REAL*
*IMAGINARY*

The impulse (red line) and step (orange line)
response for a system with 3 poles
is shown in the above applet. Horizontal axis is 0 - 8pi when
hgain = 1. For step vertical is 0 - 2, for impulse vertical
is -1 - +1 when vgain = 1.

## Frequency Response for a
system with 1 real pole and 2 complex conjugate poles

Above Applet shows the Magnitude (red line)
and Phase response (green line) for a system with
1 real and 2 complex conjugate poles. Horizontal axis is 0 to
4rad/s when hgain = 1. For Phase vertical axis is
-3p/2
to +3p/2 and for magnitude it is 0 to 2
when vgain = 1.

## Surface Plot of Magnitude of 4 poles

## Impulse and Step Response for a
system with 2 pairs of complex conjugate poles

Poles are at a + jb, a - jb, c + jd and c - jd. Horizontal axis
is 0 to 8p when hgain = 1. Impulse response is in red
and step response in orange.

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Copyright 1996 © Cuthbert A. Nyack.