Small Signal Simulation in the s-Plane
Frédéric Poullet, Gilles Depeyrot, Florian Espalieu
In analog circuit design, small-signal analysis is one of the first steps for studying the stability of the circuit or obtaining the transfer function (voltage gain, trans-impedance, power rejection, etc.). This analysis linearizes all of the non-linear devices in the circuit; the resultant linear circuit is then analyzed over a range of frequencies.
Historically, small-signal analysis originated from sinusoidal steady-state analysis and then from the usage of phasors (functions in the form of ejw), and not from usage of Laplace (functions in the form of es). This is certainly why small-signal analysis in SPICE simulators is done versus frequencies and not in the s-plane, despite the fact that Walter R. Evans  has shown earlier the importance of determining the phase of the loop transfer function L(s) at arbitrary points on the s-plane for studying control subjects.
Another reason can be that visualizing a function over complex numbers can be much more difficult than with a function over real numbers. Indeed, for complex functions, maps between two-dimensional spaces are involved whose graph would be in four-dimensional space.
This drawback has been overcome in SMASH by the use of a technique known as domain coloring, and introduced by Frank Farris and Bernd Thaller, and thus allows visualizing small-signal analysis in the s-plane.
Keywords: Small-signal analysis, Evans Root Locus, Domain coloring, Control system, Stability, Transfer function
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