Nyquist plot

The Nyquist plot for a linear system with frequency response function G(s) (s being the complex frequency in the Laplace domain) consists of a parametric plot.

In Cartesian coordinates, the real part of the response function Re(G(s=jω)) is plotted on the x-axis. The imaginary part Im(G(s=jω)) is plotted on the y-axis. The frequency is swept as a parameter, resulting in a plot per frequency.

The same plot can be described using polar coordinates, where gain of the response function |G(s=jω)| is the radial coordinate, and the phase of the response function arg(G(s=jω)) is the corresponding angular coordinate.

The frequency is annotated on the curve with frequency values in Hz or rad/s using a logarithmic scale. Powers of 10 are indicated by empty circles and intermediate values are indicated by filled circles.

You can refer to the frequency diagram settings to generate a Nyquist plot.

Figure 52: Example of Nyquist plot

Closed-loop stability

In most cases, control engineers utilize feedback when designing control systems. This is often accomplished using a PID controller system. Where there is regular feedback, control theory can be used to determine how the system responds to such feedback. In practically all such systems stability is important and control theory can help ensure stability is achieved.

We consider a system whose response function is G(s); when placed in a closed loop control with an unit gain in the feedback loop, the closed loop response function then becomes

Assessment of the stability of a closed-loop negative feedback system is done by applying the gain margin and phase margin stability criterion to the Bode plot of the open-loop system (i.e. the same system without its feedback loop). Stability is determined by looking at: the number of encirclements of the point (−1, 0). The range of gains over which the system is stable can be determined by looking at crossings of the real axis

In the example below, for the frequency where the phase reaches -180 degrees (crossing the real axis), the Nyquist diagram shows that the magnitude of the red curve is greater than 0 dB and that stability is therefore not respected; while the magnitude of the blue curve is lower than 0 dB and that stability is therefore respected with a gain margin of around 5 dB.

Figure 53: Nyquist plot - Stability