# Laplace Transform
2 CC Poles

Cuthbert Nyack

## Transfer Function

The transfer function F(s) of a system with two complex
conjugate poles at -a + jb and -a - jb is shown below.
This is usually obtained from second order physical systems.
Second order systems are those described by second order
differential equations e.g. resonant systems with one
resonant frequency.

## 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 two complex
conjugate poles in the
complex s plane. The plot is characterised by its "flatness"
over most of the s plane except at the poles 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 poles are at
-0.125 + 0.5j and -0.125 - 0.5j
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. This plot is characterised by a jump in the phase by
2p
along lines parallel to the imaginary axis from the pole
location to
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.

The above applet shows the impulse and step response of a system
whose transfer function has two complex conjugate poles at
s = -a -jb and s = -a + jb. The impulse response
is (1/b) ´ e^{-at}
´ sin(bt) and is shown in red.
The step response is
(1/(a^{2} + b^{2})[1 - e^{-at}
´ cos(bt) - (a/b)
´ e^{-at} ´ sin(bt)] and is
shown in orange. In the
above applet the step response is multiplied by (a^{2} + b^{2}).
The
vertical scale corresponds to 0 - +2 and the horizontal
axis to 0 - +8p seconds when the
horizontal gain is one.
Purple line corresponds to a unit step. Both the impulse and step
responses have the characteristic oscillatory behaviour of
second order systems. Reducing a produces more oscillations while
changing b changes the frequency of oscillation.

## Magnitude and Phase response
for 2 complex conjugate poles

Above Applet shows the frequency response for a system with
2 complex conjugate poles. Magnitude in red and Phase in
green. Horizontal axis is 0 to 8p when hgain = 1.
For Phase vertical axis is -p
to +p and for magnitude,
it is 0 to 2 when vgain = 1.

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Copyright 1996 © Cuthbert A. Nyack.