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 + 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.

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.