Nyquist plot

A Nyquist diagram, also referred to as the Nyquist graph or Nyquist plot, the locus of the output of a control circuit with the frequency as a parameter dar. It is used in the control art amplifier design and signal processing to describe the stability of a system with feedback,. It is named after the Swedish-American physicist Harry Nyquist.

The Nyquist diagram is a parametric function graph of a complex function in the normal case of a transfer function of a Fourier - LTI system in the complex plane. It serves a similar purpose as the Bode diagram, namely the representation of functions with complex-valued output values ​​:

In contrast to the Bode diagram is shown in the Nyquist plot the magnitude and phase in a single diagram, namely by drawing the real and imaginary parts of the output value directly in the complex plane. A line is formed by using all the possible values ​​for the operating parameters. Alternatively, the magnitude and phase of the output value can be entered, the reference to frequency and phase response of the Bode plot is near. A major difference from the Bode diagram is that often no values ​​of the function parameter be registered with the Nyquist diagram, which is why like any statement about kink frequencies from the graph can be made.

The benefits of the Nyquist diagrams is that the stability of the feedback system can be easily predicted by representing this curve. In this case, stability and other properties can be improved by changing the plot graph. See: Nyquist stability criterion of

Nyquist and similar diagrams are classic methods for predicting the stability of a circuit. They have been supplemented or indeed supplanted by computer-based mathematical tools in recent years, but they are particularly likely to give the developer an intuitive feel for the circuit behavior.

Experimental determining a Nyquistdiagramms

One can imagine the following experimental setup: A circuit consisting of the series connection of a resistor and a capacitor (low pass / RC element) is acted upon by a function generator with a sinusoidal voltage. On an oscilloscope the input voltage and the output voltage ( voltage across the capacitor ) can be measured. For the input of the following applies:

The output has a different amplitude and a phase shift:

They are:

If you determined for each parameter and the complex frequency response given by:

In the second picture the Nyquistdiagramm an RC circuit ( PT1 element ) is shown. The xa / xe labeled line corresponds to a value of the dependent function in the complex plane. The locus moves, starting from 1 increasing to the original, thereby forming a semi-circle from. The amplitude decreases with increasing, so it is a low-pass.

Calculating a Nyquistdiagramms

As an example for the calculation of the Nyqistdiagramms It is a simple PT1 element. To return to the example of the resistor and the capacitor, Kp = 1 and T1 = RC.

The complex number in the denominator can be cut out by complex conjugate Expand:

We obtain the real and imaginary parts:

Thus, the magnitude and phase is calculated

The extreme values ​​are obtained as follows:

This results in a semi-circle as shown in the chart above under " Experimental determination of the Nyqistdiagramms ".

Drawing a Nyquistdiagramms

To plot the transfer function ( Fourier frequency range):

The drawing of the function is now carried out by merely substituting values ​​for parameters, which results in complex numbers, which are then entered into the diagram and connected. In order to cover a wide range, are logarithmically increasing values ​​of omega and limit considerations for benefit. Furthermore, it is useful to calculate the axis interface by the real and imaginary parts is equal to zero and by transforms.

For example:

Simplified sketch of the Nyquist plot

A quick sketch of the locus can be done in certain cases, with a simplified procedure. The transfer function is given in the following form:

In addition, the following conditions must be met: m < q n, and the poles and zeros may not be the right of the imaginary axis. The beginning of the locus for ω = 0 (measured from the real axis ) at an angle of -Q * 90 ° rotated, and the locus to rotate clockwise, until - (q n -m) * 90 ° for ω → ∞. When m = 0, the locus rotates monotonically and there are no changes in the curvature of the locus. Because m < q n is the locus ends at the origin.