IEEE Solid-States Circuits Magazine - Spring 2022 - 25

and outside can be answered with a
simulation. The Nichols chart's purpose
then is not necessarily to determine
whether your circuit is stable
or not but to 1) give you a conceptual
framework for what's going on and 2)
inform you intuitively and quantitatively
how things like gain or delay
changes could cause an instability to
come or go.
A dramatic example of the Nichols
chart's usefulness can be envisio ned
using Figure 5. By inspection, it should
be clear that any change in loop-gain
magnitude would shift the Nichols
PRACTICAL TIPS FOR TRYING THIS YOURSELF
■ People are often caught off guard when SPICE returns a perfectly
normal-looking ac response for a loop that later turns out
to be unstable. As long as it can find a dc operating point-even
an unstable one-SPICE is happy. Do not be lulled into complacency
by a benign-looking frequency response.
■ Many circuit simulators allow you to define blocks based on
S-domain polynomials. Although these usually work fine for
a small-signal analysis, they can produce unexpected results
when you try to confirm stability with a transient simulation.
Painful as it is, you may have to build your polynomial with
real circuit poles and zeros to generate a transient.
■ It is tempting to simulate closed-loop performance by using
H(s) G(s)/[1G(s)].=+ This is mathematically correct of
course, but simulators can have trouble with it. Imperfect cancellation
of numerator and denominator terms leads to unnecessarily
high-order equations and weird numerical errors. Instead,
use something like the MATLAB feedback() function, which
knows what you're trying to do and returns the correct result.
The syntax is sysH = feedback(sysG,1), where the " 1 " indicates
unity-gain feedback.
■ Here's some MATLAB code to get you started. This will reproduce
the 2D and 3D plots of Figures 1 and 4.
This code is available at https://github.com/CMangelsdorf/SSCM/releases/tag/examples
s=tf('s');
sysG=60*(s/2+1)/((s/10−1)*(10*s+1)*(s/100 +1)*(s/1000 +1)*(s/10000+1));
sysH1=feedback(sysG,1);
sysH2=feedback(0.1*sysG,1);
StabilityStr= {'unstable','stable'};
fprintf(' H1 is %s\n',StabilityStr{isstable(sysH1)+1});
fprintf(' H2 is %s\n',StabilityStr{isstable(sysH2)+1});
figure('Name','Bode Plot','NumberTitle','off','WindowStyle','Docked');
margin(sysG); hold; margin(0.1*sysG);
figure('Name', '2-D Nichols Plot','NumberTitle','off','WindowStyle','Docked');
nichols(sysG,0.1*sysG); ngrid;
figure('Name', '3-D Nichols Plot','NumberTitle','off','WindowStyle','Docked');
y=linspace(−30,60); x=linspace(−360,0);
[X,Y]=meshgrid(x,y);
G=(10.^(Y/20)).*exp(j*X*pi/180);
H=G./(1+G);
surfh=surf(X,Y,20*log10(abs(H)),'FaceColor','white');
axis([−360,0,−30,40,−30,40]); hold;
[magG,phaseG,wG]=bode(sysG);
magG1=squeeze(magG); phaseG1=squeeze(phaseG);
[magG,phaseG,wG]=bode(0.1*sysG);
magG2=squeeze(magG); phaseG2 = squeeze(phaseG);
[magH1,phaseH1]=bode(feedback(sysG,1),wG);
[magH2,phaseH2]=bode(feedback(0.1*sysG,1),wG);
magH1=squeeze(magH1); magH2=squeeze(magH2);
plot3(phaseG1,20*log10(magG1),20*log10(magH1),'Color','b','LineWidth',3);
plot3(phaseG2,20*log10(magG2),20*log10(magH2),'Color','r','LineWidth',3);
xlabel('Open Loop Phase (deg)');
ylabel('Open Loop Gain (dB)');
zlabel('Closed Loop Gain (dB)');
rotate3d on;
IEEE SOLID-STATE CIRCUITS MAGAZINE
SPRING 2022
25
https://www.github.com/CMangelsdorf/SSCM/releases/tag/examples
IEEE Solid-States Circuits Magazine - Spring 2022

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