Analog Insydes

Plotting

To show the plotting functions in Analog Insydes, let's define a transfer function. This describes what comes out of a circuit if you put in s:

H1[s_] := 1000. / (1. + 10.1*s + s^2)

Bode plot

The Bode plot is perhaps the most commonly used graphing scheme for visualizing frequency responses of linear analog systems. It consists of two separate charts that display magnitude and phase of a transfer function on a logarithmic and a linear scale vs. frequency, the latter being scaled logarithmically. The magnitude values are usually given in decibels (dB) and the phase values in degrees.

BodePlot[H1[I omega], {omega, 0.001, 1000.}];

[Graphics:../Images/AnalogInsydes_gr_19.gif]

Nyquist plot

The Nyquist plot is a parametric plot of the real and imaginary part of a transfer function in the complex plane as the frequency parameter sweeps through a given interval. Nyquist plots are particularly useful for stability analysis in control system design because one can immediately check whether a negative feedback loop meets Nyquist's stability criterion: If the Nyquist curve of the open-loop system wraps around the point -1 on the real axis, then the corresponding closed-loop system is unstable.

NyquistPlot[H1[I omega], {omega, 0.001, 1000.},
PlotStyle -> RGBColor[1, 0, 0], PlotPoints -> 100]

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We determine whether the corresponding closed-loop system is stable or unstable by taking a closer look at the region around the point (-1, 0) using the PlotRange option.

NyquistPlot[H1[I omega], {omega, 0.1, 1000.},
PlotStyle -> RGBColor[1, 0, 0], PlotPoints -> 100,
PlotRange -> {{-3, 1}, {-1, 1}}]

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(-1, 0) is left of the curve, so the system is stable.

Nichol plot

A Nichol plot is similar to a Nyquist plot, but it shows gain on a logarithmic scale (dB) vs. phase on a linear scale (degrees) with an axis origin at the point (0 dB, -180°). The advantage of Nichol's chart is the ease by which gain and phase margins can be determined graphically. The gain margin is simply the negative value of the gain axis intersect. The phase margin is equal to the distance between the axis origin and the phase axis intersect.

H3[s_] := 20*(3 + s) / (s * (5 + s) * (20 + 5*s + 2*s^2))

NicholPlot[H3[I omega], {omega, 0.1, 5.},
PlotStyle -> RGBColor[1, 0, 0], PlotPoints -> 100,
PlotRange -> {{-200, -80}, {-15, 2}}]

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The gain margin is 12.5 dB; the phase margin is about 80 degrees.

The root locus plot

Root locus can be plotted for normal transfer functions.

RootLocusPlot[H3[s]]

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But usually a root locus plot shows the locus of the poles and zeros of H(s, k) in the complex plane as it varies within an interval {k\_0..k\_1}. By default, poles are red, and zeros are green.

H4[s_, a_] := (a + 2*s + s^2) / (10 + 3*a*s + 4*s^2 + s^3)
RootLocusPlot[H4[s, a], {a, 3, 5}]

-Graphics-

Simplification functions

ApproximateMatrixEquation ApproximateTransferFunction

Utilities

DXFGraphics CSDFRead CSDFInterpolatingFunctionList
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