Laplace Transform Double Pole

Cuthbert Nyack

Equations

The equations describing the Laplace transform are shown above. For a transient signal, the Laplace transform is a complex function defined over the complex s plane. This contrasts with the Fourier transform which is also a complex function but defined over the real variable w.

Transfer Function of Double pole

The transfer function F(s) of a system with a double pole at -a is shown below.

Surface plot of double pole

The magnitude of the transfer function with a double pole is shown above. In this case the magnitude of F(s) rises much faster as the pole is approached compared with a single pole. Here the pole is cut-off at 75. The frequency response is shown by the red line along the imaginary s axis and reaches a maximum value of ~64. As the order of the pole increases, the peak approaches an almost perfect cylinder of unit radius. The units along the real and imaginary axes are similar to the above plot for a single pole.

Phase plot of Double Pole

The phase plot for a double pole is shown above. Here the discontinuity in phase occurs along a line parallel to the imaginary axis and passing through the double pole. Red line shows the phase associated with the above frequency response. For positive frequencies, the phase changes from 0 to p. A closed path enclosing the double pole will encounter a phase change of 4p while other paths will encounter zero phase change. Vertical and horizontal units are similar to the plot for a single pole.

The impulse and step response for a system with a double pole is shown in the above applet. The impulse response is t ´ e^-at and is shown by the red line. The step response is (1/a²) ´ (1 - e^-at - at ´ e^-at). Orange line in the above applet shows the step response multiplied by (a²). Note that the system becomes unstable as a becomes negative i.e. as double pole moves to the right half of the s plane. vgain can be used to change the impulse response. vertical scale is 0 - +2 when vgain = 1. hgain can be used to see more of the response, horizontal scale is 0 - 4 seconds when hgain = 1.

Frequency Response of a Double Pole

The above Applet shows the Magnitude and Phase response for a system with a double pole at s = -a. When hgain = 1, the horizontal axis is 0 to 8 rad/s. The vertical axis for phase is -p to +p and for magnitude is 0 to +2 when vgain = 1.

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