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A pole-zero plot shows the location in the complex plane of the poles and zeros of the transfer function of a dynamic system, such as a controller, compensator, sensor, equalizer, filter, or communications channel. By convention, the poles of the system are indicated in the plot by an X while the zeros are indicated by a circle or O.
In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0. A meromorphic function may have infinitely many zeros and poles. This is the case for the gamma function (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer.
that is, the sum of the angles from the open-loop zeros to the point (measured per zero w.r.t. a horizontal running through that zero) minus the angles from the open-loop poles to the point (measured per pole w.r.t. a horizontal running through that pole) has to be equal to , or 180 degrees.
the numerator is equal to zero whenever z K = −α. This has K solutions, equally spaced around a circle in the complex plane; these are the zeros of the transfer function. The denominator is zero at z K = 0, giving K poles at z = 0. This leads to a pole–zero plot like the ones shown.
The block diagram on the right shows the second-order moving-average filter discussed below. The transfer function is: = + + = + +. The next figure shows the corresponding pole–zero diagram. Zero frequency (DC) corresponds to (1, 0), positive frequencies advancing counterclockwise around the circle to the Nyquist frequency at (−1, 0).
The potential analogue method was proposed by Darlington [12] as a simple way to choose pole-zero positions for delay networks. The method allows the designer to implement a delay characteristic by locating poles and zero on the complex frequency plane intuitively, without the need for complicated mathematics or the recourse to reference tables.
The open-loop transfer function is equal to the product of all transfer function blocks in the forward path in the block diagram. The closed-loop transfer function is obtained by dividing the open-loop transfer function by the sum of one and the product of all transfer function blocks throughout the negative feedback loop. The closed-loop ...
The second Figure 3 does the same for the phase. The phase plots are horizontal up to a frequency factor of ten below the pole (zero) location and then drop (rise) at 45°/decade until the frequency is ten times higher than the pole (zero) location. The plots then are again horizontal at higher frequencies at a final, total phase change of 90°.