Before the steady state

Two numbers without solving for the waveform

Where a step response starts and where it ends are two limits of the transfer function, and neither needs the waveform. Both are exact here — 1.000000000 volts at the end and the whole step at the first instant — and one of them is a lie waiting to happen: take the damping to zero and the final-value theorem still returns 1.000000 for a response that swings between 0 and 2 for ever. Its condition is not on the transfer function but on where the poles are, and the practical condition is narrower still: at five ohms the poles are safely in the left half-plane and sixty cycles is not enough time.

17 min read 8 figures Two routes to a number Refused, not extrapolated Every model has an edge

Assumes: One step, computed twice · Where the behaviour is written down

A step response is a curve, and computing it costs something: a residue expansion, or a march through time, or a laboratory. Two points of that curve cost nothing at all.

Where it starts is $H(\infty)$ , because at the instant the step arrives every capacitor is a short circuit and every inductor an open one. Where it ends is $H(0)$ , because after long enough the reverse is true. Both are single evaluations of a function that has to be built anyway, and neither requires the waveform between them.

Those are the initial and final value theorems, and they are among the most useful things in the subject. One of them is also the most quietly wrong.

Fig. 1 A series RLC at 50.3 kHz driven by a one-volt step, with the capacitor voltage and the inductor voltage drawn together — the same two limits read at opposite ends of the same network. The line is the final value the theorem gives, the panel carries both predictions against both measurements, and the slider takes the damping to zero.

One network, two outputs, two limits

A series RLC gives both theorems something non-trivial to say, provided the two outputs are taken separately.

Across the capacitor, the response starts at nothing and ends at the source. At $t = 0^+$ the capacitor is uncharged and holds the node at zero; after everything has settled the inductor is a piece of wire and the resistor carries no current, so the capacitor has the whole supply.

Across the inductor, it is the other way round. At $t = 0^+$ the inductor’s current cannot change instantaneously, so it must take the entire step; after everything has settled it has none of it.

Both predictions come from solves rather than from the algebra above. The direct-current solve returns 1.000000000 for the capacitor, and a solve at ten million times the natural frequency returns $1.4\times10^{-14}$ — which is a numerical zero, and the honest version of “the capacitor is a short circuit at $t = 0$ ”.

The route that can see an initial value, and the route that cannot

The marched route walks the network forward in time by the trapezoidal rule, which is what a circuit simulator does and what this site’s transients field is largely about. It reproduces the capacitor voltage beautifully. It cannot read the initial value at all.

The reason is worth having, because it looks like a bug and is a property of the method. The node between the resistor and the inductor is not a state variable: its voltage is whatever the source has left after the inductor’s current has been decided, and at $t = 0$ that jumps discontinuously from zero to one. A trapezoidal step across a discontinuity does not land on the answer, it straddles it. The first two marched samples are 1.995 and −0.010 volts, alternating about the truth, and their mean is the exact voltage at the instant between them.

So the initial value is exactly the quantity a march is worst at, and this figure computes it from the residue expansion instead — poles from the matrix, residues from the numerator, and the derivative taken analytically so the inductor’s voltage comes out of the same expansion without a second march. That route gives 1.000000 at $t = 0$ , against $H(\infty) = 1.000000$ from the solve.

The two routes are then compared where both are trustworthy. Over sixty cycles at a damping ratio of 0.079 they agree to $3.8\times10^{-4}$ of a volt.

One step response, computed twice: from the poles, and by walking the network forward. A damping ratio of 0.22, so the overshoot is 49.2%. The two curves are drawn on top of each other; the panel below is the difference between them, which is the trapezoidal rule's error at 500 steps and reaches 1.70e-3 V. — Fig. 2 The two routes drawn against each other in the field’s own essay about them. The residue expansion is exact and the march is not, and the difference is the trapezoidal rule’s own error, which falls with the square of the step. In the figure above that error is what sets the tolerance on the comparison rather than anything about the circuit.

What the final-value theorem’s condition actually is

It is usually stated as a footnote: the theorem holds provided the system is stable. That is not quite the condition and the difference is the whole of this essay.

The condition is that $sY(s)$ must have all of its poles strictly inside the left half-plane. For a step response $Y(s) = H(s)/s$ , so the $s$ cancels and the condition is on the poles of $H$ — but the statement is about the poles, not about the function’s value anywhere. Nothing about $H(0)$ says whether $H(0)$ is the final value.

The figure demonstrates that by walking the poles towards the imaginary axis. Every frame has the same $H(0)$ , to nine decimals:

resistance	damping ratio	worst pole, real part	inside ±2% after
200 Ω	0.316	−0.316 ω₀	1.8 cycles
100 Ω	0.158	−0.158 ω₀	3.7 cycles
50 Ω	0.0791	−0.0791 ω₀	7.6 cycles
20 Ω	0.0316	−0.0316 ω₀	19.6 cycles
5 Ω	0.00791	−0.00791 ω₀	more than 60
1 Ω	0.00158	−0.00158 ω₀	more than 60
none	0	4×10⁻¹⁷ ω₀	never

At the last row the poles are on the axis, the theorem’s condition fails, and the theorem still returns a number. It returns 1.000000, confidently, for a waveform that swings between zero and two volts for ever and is within two per cent of the answer for 1.1 per cent of the time.

The same network with the resistor taken out, and a final value that never arrives. computed by solving, not by drawing. A 50.3 kHz series RLC driven by a one-volt step, with the capacitor voltage and the inductor voltage drawn together. Two limits of the transfer function are two points of the waveform and neither needs the waveform: H(0) = 1.000000 is where the capacitor ends up, and H(∞) across the inductor is 1.0000, which is what it does at the first instant — the expansion gives 1.000000 for it at t = 0. With no resistor in the netlist the poles are on the imaginary axis — real part 4.44e-17 of ω₀ — and the theorem returns 1.000000 for a waveform that swings between 0 and 2 for ever. Only 1.1% of the last twenty-four cycles are within ±2% of that answer, and the mean of them is 0.99987: the number is the waveform's average rather than its limit, because there is no limit for it to be. — Fig. 3 The same network with the resistor taken out of the netlist. The final value the theorem gives is unchanged and the response never arrives at it. What the number turns out to be is the waveform’s *mean* — 0.99987 of it, measured over the last twenty-four cycles — which is a true and quite different statement from the one the theorem is usually read as making.

The zero-ohm resistor that does not exist

The undamped frame is built by leaving the resistor out rather than by setting it to zero, and that is not a stylistic choice. The assembler refuses a resistance of zero by name:

R1 is given a resistance of 0, which is not a resistance. A perfect short circuit is a constraint between two nodes rather than an element with a law of its own, and stamping it as an admittance of infinity is how a matrix acquires a row of nonsense.

Which is the right refusal, and it forces the honest arrangement. The lossless case is a different netlist with one fewer element in it, not the same netlist at the end of a parameter sweep, and the figure’s last slider position says so.

Four networks the solver refuses. Each has no answer, for a reason that is a fact about the circuit rather than about the arithmetic. The solver names the reason; it does not return a number. — Fig. 4 The refusals the machinery makes, which are the reason the frame above had to be built differently. A network that does not determine its own node voltages is declined and named; an element given a value that is not a value of that kind is declined before assembly.

The condition that actually bites

The pole condition is a clean mathematical statement and it is not the one an engineer needs, because it is satisfied by circuits whose final value nobody will ever see.

At five ohms the poles are at −0.0079 of the natural frequency: strictly inside the left half-plane, so the theorem holds and its answer is correct. The response reaches ±2% of that answer after about seventy-eight cycles, which is 1.6 milliseconds, and inside a sixty-cycle observation window it is inside the band for fifteen per cent of the time.

So there is a second boundary, and it is the one worth computing. The envelope decays as $e^{-\zeta\omega_0 t}$ , so reaching two per cent takes $\ln 50 / (\zeta\omega_0)$ , which is $0.6226/\zeta$ cycles. Setting that to the sixty cycles the figure marches gives a damping ratio of 0.0104, and a resistance of

R = 2\zeta\sqrt{L/C} = 6.56\ \Omega

Bisecting on the marched response instead gives 6.50 Ω, and the two agree to one per cent. Below that resistance the theorem is true and useless: it returns the right number for a waveform that will not have got there before the measurement stops.

That is the site’s usual shape arriving in an unusual place. The model — “the final value is $H(0)$ ” — has an edge, and the edge is not a frequency or an amplitude but a resistance, given an observation window. Change the window and the resistance moves with it.

A step on a series RLC at ζ = 0.32, and the two numbers read off H(s). computed by solving, not by drawing. A 50.3 kHz series RLC driven by a one-volt step, with the capacitor voltage and the inductor voltage drawn together. Two limits of the transfer function are two points of the waveform and neither needs the waveform: H(0) = 1.000000 is where the capacitor ends up, and H(∞) across the inductor is 1.0000, which is what it does at the first instant — the expansion gives 1.000000 for it at t = 0. Here the damping ratio is 0.316, the response is inside ±2% after 1.8 cycles, and 100.0% of the last twenty-four cycles sit there. The poles are at a real part of -3.16e-1 of ω₀, which is the condition the final-value theorem actually has — not a property of H but of where sY(s) has its poles. — Fig. 5 The well-damped end of the same slider, where the theorem is both true and useful: inside ±2% after 1.8 cycles and flat thereafter. Everything the theorem promises is delivered here, which is why the promise is easy to over-generalise.

A step on a series RLC at ζ = 0.16, and the two numbers read off H(s). computed by solving, not by drawing. A 50.3 kHz series RLC driven by a one-volt step, with the capacitor voltage and the inductor voltage drawn together. Two limits of the transfer function are two points of the waveform and neither needs the waveform: H(0) = 1.000000 is where the capacitor ends up, and H(∞) across the inductor is 1.0000, which is what it does at the first instant — the expansion gives 1.000000 for it at t = 0. Here the damping ratio is 0.158, the response is inside ±2% after 3.7 cycles, and 100.0% of the last twenty-four cycles sit there. The poles are at a real part of -1.58e-1 of ω₀, which is the condition the final-value theorem actually has — not a property of H but of where sY(s) has its poles. — Fig. 6 A hundred ohms, where the damping ratio is 0.158 and the response is inside ±2% only after 3.7 cycles. The condition that actually bites is not the final-value theorem’s — that is exact here, H(0) = 1.000000 — it is that the theorem says nothing about *when*, and at this damping the answer it gives is not the answer for several cycles.

Why only one of the two lies

The asymmetry between the two theorems is not an accident of how they are usually presented, and it is worth stating because it decides which one needs checking.

The initial-value theorem asks about $t \to 0^+$ , and the limit it takes is $s \to \infty$ . Every network here is proper — its numerator degree never exceeds its denominator’s — so that limit exists and is finite, and there is nothing further to verify. The one caveat is the plus sign: it is the value just after the step, not the value at it, and for the inductor voltage above those two are zero and one respectively. A theorem about a one-sided limit cannot be checked against a two-sided measurement, which is the only way to get it wrong.

The final-value theorem asks about $t \to \infty$ , and a limit at infinity may simply not exist. That is the whole difference: $H(\infty)$ is a number about an instant that certainly happens, and $H(0)$ is a number about an instant that may never arrive. The first is a value; the second is a promise, and the poles are the terms.

Where the poles being on the axis is the design

A network whose poles sit exactly on the imaginary axis sounds like a pathology. It is the point of an oscillator.

The applied field’s Wien bridge is built so that its pole pair crosses the axis at exactly one gain, and the crossing is found by bisection on the netlist rather than quoted. On one side of it the amplitude decays, on the other it grows, and at the crossing the linear network has a final value that the circuit never reaches — which is precisely the failure drawn above, arranged on purpose.

The resolution there is that a real oscillator is not linear: a limiter sets the amplitude, and the final value that exists is a limit cycle rather than a point. Nothing linear predicts it, and the final-value theorem is one of the things that cannot.

A step on a series RLC at ζ = 0.79, and the two numbers read off H(s). computed by solving, not by drawing. A 50.3 kHz series RLC driven by a one-volt step, with the capacitor voltage and the inductor voltage drawn together. Two limits of the transfer function are two points of the waveform and neither needs the waveform: H(0) = 1.000000 is where the capacitor ends up, and H(∞) across the inductor is 1.0000, which is what it does at the first instant — the expansion gives 1.000000 for it at t = 0. Here the damping ratio is 0.791, the response is inside ±2% after 0.6 cycles, and 100.0% of the last twenty-four cycles sit there. The poles are at a real part of -7.91e-1 of ω₀, which is the condition the final-value theorem actually has — not a property of H but of where sY(s) has its poles. — Fig. 7 Five hundred ohms: ζ = 0.791 and inside ±2% after 0.6 cycles. Where the poles being on the axis is the design is the oscillator two fields over, and it is exactly the case this theorem refuses — with poles on the imaginary axis there is no final value, and the theorem’s own condition is what excludes it.

The initial value, and what it is used for

The other theorem is less treacherous and gets less attention, which is a pity because it answers a question that comes up constantly: what does this circuit do the instant something happens?

A probe’s compensation is an initial-value question. A ten-times probe’s divider has one ratio at direct current, set by the resistors, and another at the first instant, set by the capacitors. The whole business of compensating it is making $H(0)$ and $H(\infty)$ equal.

A regulator’s load step is an initial-value question. The instant a load current appears, the output capacitor is still charged and the loop has not moved, so the step lands on the capacitor’s equivalent series resistance. That first number is $H(\infty)$ for the output impedance, and it is the one that decides whether the specification is met.

A charge-sharing calculation is an initial-value question. Two capacitors connected together redistribute charge in a time set by whatever resistance is between them, and the value they redistribute to is $H(0)$ while the peak current is $H(\infty)$ .

In each case the useful number is available from two evaluations and the waveform is not needed at all — which is what makes the pair of theorems worth the page.

A step on a series RLC at ζ = 3.2, and the two numbers read off H(s). computed by solving, not by drawing. A 50.3 kHz series RLC driven by a one-volt step, with the capacitor voltage and the inductor voltage drawn together. Two limits of the transfer function are two points of the waveform and neither needs the waveform: H(0) = 1.000000 is where the capacitor ends up, and H(∞) across the inductor is 1.0000, which is what it does at the first instant — the expansion gives 1.000000 for it at t = 0. Here the damping ratio is 3.16, the response is inside ±2% after 3.9 cycles, and 100.0% of the last twenty-four cycles sit there. The poles are at a real part of -1.62e-1 of ω₀, which is the condition the final-value theorem actually has — not a property of H but of where sY(s) has its poles. — Fig. 8 Two kilohms: ζ = 3.16, overdamped, inside ±2% after 3.9 cycles — slower than the lightly damped case rather than faster. The initial value, and what it is used for, is the other half of the pair: it is read off the same transform without solving anything, and it is the number a designer checks a simulation against in its first nanosecond.

What is checked, and what would break it

The figure holds four things at once, and each one fails differently.

The direct-current solve and the marched final value agree to five decimals wherever the response has settled; if the netlist were assembled wrongly they would not, and no amount of correctness elsewhere would hide it.

The residue expansion and the march agree on the capacitor’s whole waveform to $3.8\times10^{-4}$ of a volt over sixty cycles at the default damping. That tolerance is set by the trapezoidal rule, not chosen: at two hundred steps a cycle the march warps the frequency slightly, so a response that is still ringing sixty cycles later has accumulated a phase error, and the disagreement grows as the damping falls — 9.6×10⁻⁵ at ζ = 0.316 and 3.1×10⁻² with no resistor at all. It is reported rather than tuned away, because it is the same finding this field’s first essay is about, arriving in a figure that is not about it.

The poles are recovered from the matrix and their real parts are asserted to be strictly negative where a resistor is present and zero — $4\times10^{-17}$ of the natural frequency — where one is not.

And the arrival is asserted in opposite directions on the two sides of the window boundary: where the response has had time, at least 99% of the last twenty-four cycles must be inside the band; where it has not, fewer than half of them, with the mean still landing on the theorem’s answer. A single assertion covering both would have been a claim that is true in the way that says nothing.

What is left

Two evaluations of a transfer function give two points of a waveform, exactly, for nothing. That is a genuinely good bargain and it is worth taking.

What comes with it is a condition that is stated about poles and used as though it were about stability, and a second condition — that the observation outlast the settling — that is not stated at all. The first is satisfied by every circuit in this collection that has a resistor in it. The second is violated by one of them at five ohms, which is a perfectly ordinary value for a perfectly ordinary resonant circuit.

The number to carry away is 6.50 ohms, and the sentence to carry with it is that the number depends on how long the measurement lasts.

A limit theorem with a condition about time

A final-value theorem returning 1.000000 for a response that swings between 0 and 2 for ever is an unusually pure example of a result that is exactly right and useless, and it is worth setting beside the collection’s other two.

An exact answer to a different question is the closest: a reconstruction returns an alias above half the sample rate to 1.6 parts in a thousand, which is the same accuracy it returns the input with below the boundary, and it is wrong about the input by twice the amplitude. Its error is not a degradation but exactness about something else.

The capacitor that was right once is the practical one: a correction capacitor sized by one division, exact for the load it was computed from at the frequency it was computed at, and capable of making an installation worse than it was before anything was fitted.

What this essay adds to the pair is a condition that is about duration rather than about frequency or load. The theorem’s stated condition is on where the poles are; the practical condition is that the measurement has to outlast the settling, and at five ohms sixty cycles does not. That is the same practical form the amplitude nothing linear predicts arrives at from the other direction — a run stopped in the middle of a growth reporting 51 millivolts against a settled 703 — and the remedy is the same in both: compute the run length from the circuit rather than fixing it, and report whether the answer had stopped moving.

What the two theorems are actually for

Given that one of them is a lie waiting to happen, it is worth being clear about why both are worth having, and the answer is that they are the only quantities in this collection available before a response is computed.

An initial value and a final value come from two limits of a transfer function, so they cost two evaluations and no waveform at all. That makes them a check on everything computed afterwards — one step, computed twice compares a marched response against a residue expansion and needs both routes to agree, and a march whose first sample disagrees with the initial-value theorem is wrong about its initial conditions rather than about its integration. And where the behaviour is written down is where the condition on the second theorem lives: the poles, which are the same object the response is built from.

Which is the practical division. Use the theorems as endpoints and as checks, and use them for design only where the poles have already been looked at — because their condition is a statement about the poles rather than about the transfer function, and it is the one part a formula does not carry.

Which is the ordinary shape of a theorem’s condition in this collection. The statement travels, the condition does not, and the condition is usually about something the statement does not mention — a pole position here, a load angle in the power field, a waveform in the metering one. Reuniting the two is the whole of what these pages do. What is unusual here is that the theorem states its own condition and the condition is still lost, because it is stated in a variable — pole position — that the object being handed around is not written in.

Which suggests the form a statement of it should take. Not “the final-value theorem requires the poles to be in the left half-plane”, which is true and travels badly, but “the final value is what the response settles to, and this circuit settles in n cycles” — a duration, attached to the same number, in the units the measurement is made in. The 6.50 ohms on this page is that sentence for one circuit.

Part 1 on value theorems

One argument about Value theorems, and one of 2 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

$A step through r sections starts as (t/τ)^r: it reaches 1% at 10.1 µs, 105 µs, 243 µs, 380 µs, 508 µs for r = 1 to 5. Solved, and expanded two ways. The step response of buffered RC sections of time constants τ, τ/2, … τ/r, with τ = 1 ms, on logarithmic axes. The relative degree of the recovered transfer function is r, so the first r − 1 derivatives of the step are zero at the start and the r-th is lim s^r·H(s) = r!/τ^r, read off the network solved far above its poles and off the expansion of H about infinity; the step therefore starts as (t/τ)^r, a straight line of slope r. The expansion about infinity and the residue expansion agree to a part in a million where both are well conditioned. The output reaches 1% at 10.1 µs (r = 1), 105 µs (r = 2), 243 µs (r = 3), 380 µs (r = 4), 508 µs (r = 5), and half its final value at 693 µs, 1.23 ms, 1.58 ms, 1.84 ms, 2.04 ms. For these time constants the whole step is (1 − e^(−t/τ))^r, checked against both expansions, so the time to a fraction ε is −τ·ln(1 − ε^(1/r)).$ The start a step takes from infinity Part 2 — The initial-value theorem reads where a step starts off H at infinite frequency. Apply it again to s·H, s²·H and on, and it reads how the step starts: the first r − 1 derivatives are zero for a network r degrees more poles than zeros, and the r-th is the ratio of the leading coefficients. So a step through r sections begins as a power of time — for sections of τ, τ/2, … τ/r, exactly (t/τ) to the r — and reaches one per cent at 10.1 µs through one section, 105 µs through two and 508 µs through five. Put a zero anywhere, even a thousand times above every pole, and the step starts linearly instead, with a slope of twice the zero's time constant over τ² that is the larger term for the first two of them.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third axis, after the field and the idea: the things themselves, and every essay that touches each one.

Final value theorem Initial value theorem Lossless network Poles Residues Settling time Transfer function Trapezoidal rule

The cancellation that leaves a tail poles, residues, settling time
Three cliffs, and where they are poles, residues, settling time
The best damping is not the one to build poles, settling time
The gap a derivative needs poles, residues
Two exponentials, and where they meet poles, settling time

One network, two outputs, two limits #

The route that can see an initial value, and the route that cannot #

What the final-value theorem’s condition actually is #

The zero-ohm resistor that does not exist #

The condition that actually bites #

Why only one of the two lies #

Where the poles being on the axis is the design #

The initial value, and what it is used for #

What is checked, and what would break it #

What is left #

A limit theorem with a condition about time #

What the two theorems are actually for #