Laplace Transform 1 Zero 2 CC Poles

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, 1 zero, 2 poles

Transfer function below has poles at -a + jb, -a - jb and a zero at -1/z. This way of writing the transfer function means that it becomes equal to the one for 2 poles when z = 0,

Surface plot of magnitude

Magnitude plot differs from the one above by having a dip at the zero where the magnitude of the transfer function goes to zero.

Surface plot of phase

The phase plot with a zero differs significantly from that without a zero. As before the phase jump start off at the poles but this time they move towards the real axis. When they reach the real axis both travel along the axis in different directions. One moves towards the zero and ends there. The other moves towards minus infinity. This behaviour is similar to that observed in root locus plots used for the analysis of control systems.

Above applet shows how the impulse and step responses are affected by adding a zero to the transfer function. For step(orange) the vertical scale is 0 - +2. Step response is multiplied by (a² + b²). For Impulse response(red) the vertical scale is -1 - +1 when vgain = 1. The horizontal scale is 0 - 8p when hgain = 1.

Magnitude and Phase response for 1 zero and 2 conplex conjugate poles

Above Applet shows magnitude and phase response for a system with 1 zero and 2 complex conjugate poles.

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