Z Plane and Stability.

Cuthbert Nyack

The above diagram shows the relation of poles on the Laplace s plane and on the z plane. Pole 1a in the s plane maps to 1b in the z plane, 2a to 2b etc.

Stability of a Single Pole

The time behaviour resulting from a single pole in the s and z planes is shown by the 2 equations above. In the s plane a pole located on the real axis in the left half plane where a < 0 produces an exponentially damped stable response, e.g. pole at 1a is stable. In the z plane a pole on the positive real z axis and within the unit circle (a < 1) produces a converging series and a stable response. e.g. pole 1b in the z plane. On the other hand pole 2a to the right of the imaginary axis in the s plane and 2b outside the unit circle in the z plane produce unstable responses.

Stability of a Pair of CC Poles

The stability of complex conjugate poles in the s and z plane can be investigated using the above 2 equations. In the s plane stable responses result when a < 0 and in the z plane stable responses result when d < 1. Pole pairs 3a, 3b, 5a and 5b are stable while pairs 4a and 4b are unstable.

In summary, poles within the unit circle in the z plane are stable while poles outside the unit circle produce unstable responses.

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