[078] Loop Bode Plots and Nyquist Diagrams

Why Nyquist diagrams are not a very practical measurement.

Introduction

This article shows the link between loop gain Bode plots and Nyquist diagrams. The Nyquist diagram is a subset of the Bode Plot, omitting crucial design data. The linear scaling of the Nyquist diagram restricts its practicality, and omission of frequency as an explicit variable in the plot is a major drawback. 

Loop Gain Measurements in Modern Control Systems

In control courses in universities around the world, Nyquist plots are used as a tool to assess stability. They work well with prescribed and well behaved finite-gain analytic transfer functions and many theoretical and interesting homework problems can be created. 

However, for working engineers, hardware frequently does not behave as expected. Analytical models are often inaccurate, and pole-zero locations or transfer functions of the power stages are often unknown. In such situations, it is a challenge to predict stability and optimize a closed-loop system. This is where the Bode plot of the loop gain is tremendously useful [1].

But what about Nyquist diagrams in the real world? I have never actually been in a lab where someone wanted to present measurement data as a Nyquist diagram, and there are solid reasons for why this is so. These are: 

1. Nyquist diagrams are concerned with -1 point encirclements, and hence the low gain region of the plot must be clearly shown. To do this, only the region from a gain a little bigger than 1 is shown. Since the plots are always on a linear scale this greatly restricts the available data shown.
2. Nyquist diagrams plot real and imaginary parts of the gain, with frequency as a running parameter. Hence the frequency information is completely missing from the Nyquist plot.
3. Two very different systems with completely different crossover frequencies can have identical Nyquist diagrams. This is no help when trying to improve system performance.
4. When RHP zeros are present in a system (very common for power supplies) the rules for Nyquist become complex.
5. If delays are present in a system the Nyquist criteria are again complicated.
6. Integrators, or even double integrators (seen in PFC circuits), provide another level of complication in completing the Nyquist contour. 

The complexities of rules that go with the Nyquist diagram are difficult to unravel. Study of many online tutorials about the topic are hard to find when dealing with RHP zeros and integrators. Since these comprise the majority of switching power supplies with feedback, application of Nyquist in the lab is very rare. 

Loop Gain Measurement Point

Figure 1 shows a buck converter controlled with a current-mode loop and an outer voltage feedback loop. This is a very common control setup for modern power supplies. The feedback compensation can be implemented in a digital controller, or with analog components as shown. 

Figure 1: Power supply with current-mode control, showing loop measurement point. .