Pole–zero plot

The pole-zero diagram short PN diagram, the poles and zeros of the transfer function of a system in the complex plane Represents the system can be an electric system, such as a filter, but it may also to be controlled, a its mechanical system, for example a vehicle with a vehicle dynamics control. The most common application found pole-zero diagrams in communication engineering and control engineering.

From a pole - zero plot can be concluded among other things, the magnitude and phase response of the frequency response of a system as well as its impulse and step response. Thus it forms a valuable basis for analysis, synthesis and stability considerations of circuits, filters, and other transmission systems.

The creation and use of a pole-zero diagram requires specific knowledge of mathematics and systems theory. In the transfer function, the poles and zeros can be represented, is the Laplace transform of the impulse response or the z-transform of a system. The display is in a complex plane. Usually Simple poles are marked by a cross, multiple poles by a double cross and zeros by a small circle. Is practically supports the creation of pole-zero diagrams or the derivation of transfer functions or other system properties from pole-zero diagrams today often by software.

Significance of poles and zeros locations

From the position of the poles can be seen, among other things, whether a system is causal and stable. Pole determine the timing of the system. The system is stable if all the poles of the transfer function in the open left half-plane ( LHE) of the diagram lie. Realizable ( causal) systems have at least as many poles as zeros. From the spacing of all the poles and zeros for a frequency in the diagram one can estimate the frequency transfer properties. Transients are shown by two complex conjugate poles. Complex poles in the open left half-plane point to decaying oscillations. All these illustrative chart interpretations and many other interpretations of this kind can be obtained with the system theory of communications.

If you move on the frequency axis from to, rotates each pole in the LHE, and every zero in the right half-plane (RHE ) the phase. Each zero in the LHE causes a phase rotation.

Example

High-pass 2nd order

As can be seen, has a high-pass 2nd order a complex conjugate pair of pole and a double zero at the origin. The system is therefore stable.

Minimum phase systems have no zeros in the RHE.