That is, nonlinearities are approximated as various linear gains between points. When you do AC analysis, it's a small signal steady state analysis. Which is precisely what SPICE does in the transient analysis mode: implicit integration, to solve the differential equation, given stimuli (arbitrary sources) and some system (the circuit and model you've entered). But, that can be done one small step at a time, if nothing else. Well, with the small cost of still having to solve the differential equation.
#Laplace transforms an dtransfer functions plus#
Incidentally, if you start with the differential form - some function for input, plus derivatives of the output - you can plug in a function and get a result directly. Then you have transient analysis, were you look at the frequency and the dampening factor, this is used for controls/stability.
So for linear circuits you have steady state analysis, where you look at the system by the frequency, this used for filter. Hence, the fourier transform is a special case of the laplace transform, notably when there are no transients. sigma = 0) then you are now in the frequency domain, which is what you would get if you took the fourier transform of the transfer function to begin with. If the laplace transform of the transfer function has an ROC (region of convergence) that contains the imaginary axis (j*omega axis) then assuming no transients (dampening factor i.e. Note that in the laplace transform, s = sigma + j*omega, where omega is the frequency and sigma is some dampening factor. If you wait for the transients to go away, you are now dealing with frequency analysis which is dealt with by fourier transform. how does your system act when turned on (an impulse), how does it react to a certain input, does it overshoot its target, undershoot? How fast does it respond, is that too fast or too slow? These are all questions that are answered with control theory, and can be tuned by adding the appropriate compensators using OP amps, this is known as PID control. This is the transient side of things i.e. Taking the laplace transform of that function opens up an entire field of analysis called control theory, which deals with the stability of the system. So you take your linear circuit and define the input and the output through some differential equations, you take the ratio of the two (input over output) and you have the transfer function.
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