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/(a2 + b2)[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 (a2 + b2). 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.