Laplace Transform 3 Real Poles

Cuthbert Nyack

Transfer Function

The transfer function F(s) of a system with 3 poles at -a, -b and -c is shown below.

s Plane

The location of the poles 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 three poles in the complex s plane on the negative real axis. 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 -0.225, -0.925 and -1.725.

Surface plot of Phase

The phase plot corresponding to the above magnitude plot is shown above. The phase jump now has a more complicated behaviour. One path goes from the leftmost pole to minus infinity along the negative real axis. Path 2 starts from the middle pole and moves to the right. Path 3 starts from the pole to the right and moves to the left. When 2 and 3 meet they go off the real axis. 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 -3p/2.
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.

The above applet shows the impulse(red line) and step response of a system whose transfer function has 3 pole at s = -a, -b and -c on the negative real axis. The impulse response is shown in red. vgain affects only the impulse response gain and the vertical scale is 0 - 2 when vgain = 1. Orange line shows the step response which is multiplied by (a*b*c). Horizontal scale is from 0 to 12 seconds when hgain = 1.

Frequency Response for a system with 3 real poles

Above Applet shows the Magnitude (red line) and Phase (green line) response for a system with 3 real poles. Horizontal axis is 0 to 8rad/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.
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Copyright 1996 Cuthbert A. Nyack.