Circuit Response
every model has an edge, and the figure carries it
Field

Before the steady state

Everything on the frequency axis assumes a settled circuit. These are the figures about getting there — and about the one limit no transfer function contains, where a step becomes large enough that the response stops being a scaled copy of a smaller one.
00.50011.50024output (volts), for a 1 V step inthe final valuesolid: from the poles · dashed: stepped forwardtime (milliseconds) above · the same span as a fraction, belowgap between the two routes (volts)1e-71e-61e-51e-41.0m10m1.0e+2m1solved, then checked — residues against 500 trapezoidal stepsthe numerical route is out by 1.7e-3 V

One step, computed twice

A step response from the poles is exact. The same step walked forward in time is not, and the difference between them is the trapezoidal rule's own error rather than anything about the circuit. It falls by a factor of four every time the step is halved, which is a claim about a method and can be watched.

 the step this produces00.50011.5001234σjωζ = 0.3000ω₀ = 1592 Hzsolved, then checked — poles by rooting the determinantnatural frequency recovered to 6 digits

Where the behaviour is written down

Two numbers in the complex plane contain everything a second-order circuit will ever do. Their distance from the origin is the natural frequency, the cosine of their angle is the damping — and the fastest-settling circuit is not the critically damped one, which is the case the textbooks name.

 00.2500.5000.750105101520time (microseconds)output, divided by the size of its own step20 mV step1.0e+2 mV step5.0e+2 mV step2 V step8 V steplinear below 79.6 mVsolved, then checked — integrated with the rate limitscaling fails above a 79.6 mV step

The step that is too big

A linear circuit scales — double the input and the output doubles, exactly. A real amplifier does not, because its output can only move at a fixed rate, and the amplitude at which the two stop agreeing is about eighty millivolts for an ordinary part. No transfer function contains that number, because no transfer function can.

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