# 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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© 2006 Cuthbert A. Nyack.