Laplace Transform Single pole.

Cuthbert Nyack

Transfer Function

The transfer function F(s) of a system with 1 pole at -a is shown below. This is usually obtained from first order physical systems or by finding the transform of e^-at.

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 one pole in the complex s plane. 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 -0.125 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. Pole location is shown by the x on the negative real axis. This plot is characterised by a jump in the phase by 2p along the negative real axis from the pole location to minus 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/2.

The above applet shows the impulse and step response of a system whose transfer function has 1 pole at s = -a. The impulse response is e^-at and is shown in red. The step response is (1/a)[1 - e^-at] and is shown in orange. In the above applet the step response is multiplied by a. The vertical scale corresponds to 0 - +2 and the horizontal axis to 0 - +4 seconds when the horizontal gain is one. Purple line corresponds to a unit step. When a becomes negative the system becomes unstable.

Frequency Response of single Pole

The above Applet shows the Magnitude(red) and Phase(green) Response of a system with a single pole. The vertical axis for phase is -p/2 to +p/2 and for magnitude is 0 to +2. In this case, the magnitude is multiplied by a. When hgain = 1, the horizontal scale is 0 to 8rad s^-1.

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