Nichols plot

The Nichols plot for a linear system with frequency response function G(s) (s being the complex frequency in the Laplace domain) consists of displaying the magnitude 20log10|G(s=jω)| as a function of the phase arg(G(s=jω)).

This plot may be compared to the Bode plot in which the two inter-related graphs, the magnitude 20log10|G(s=jω)| as a function of the frequency *log10(ω)*and the phase arg(G(s=jω)) as a function the frequency log10(ω) are plotted.

The frequency is the parameter along the curve that is annotated 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 Nichols plot.

Figure 49: Example of Nichols 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

Loci of constant 20log10|M(s=jω)| and arg(M(s=jω)) are overlaid to allow the designer to obtain the values of the closed loop response function from the Nichols plot of the open loop response function, they are called Hall circles.

Figure 50: Hall circles

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 Nichols plot of the open-loop system (i.e. the same system without its feedback loop). Stability is determined by looking at the position of the critical point (−180 degree, 0 dB) as you progress along the curve with increasing frequency:

• Gain margin

    The gain margin is the distance on the vertical axis (magnitude) between the nominal curve and the critical point when the phase *arg(G(s=jω))* reaches -180 degree.
  

• Phase margin

    The phase margin is the distance on the horizontal axis (phase) between the nominal curve and the critical point when the gain *20log10|G(s=jω)|* reaches 0 dB.
  

In the example below, for the frequency where the phase reaches -180 degrees, the Nichols 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 51: Nichols plot - Stability