The step too large to have an impedance

The rung below this one drove the load node of three capacitive-load compensations with a step of current and reported an impedance — ten ohms at direct current for the amplifier-feedback arrangement, 1.2 milliohms for the two-path one, and a recovery whose length moves with the compensation capacitor.

An impedance is a voltage divided by a current. It is a number only if the ratio does not depend on the current, and every model in the five rungs below this one is linear, so every one of them answers a step of any size with the same waveform scaled. That is the assumption this rung is about, and the five rungs below named it in their closing note: a load step large enough to slew the amplifier, at which point none of it is linear and the impedance stops being a quantity.

The slew rate is not a free parameter #

The amplifier here is three parts and only the first is new: an input transconductor with a tanh characteristic, a compensation capacitor it charges, and a unity buffer with an output resistance.

That tanh is not a convenient saturating function chosen for the purpose. It is the differential pair’s own law, measured in the semiconductors field, and it is why the numbers below contain the thermal voltage. A pair with a tail current $I$ carries $I\tanh(v/2V_T)$, so its transconductance at zero is $I/2V_T$ and its output current saturates at $I$ — which is the whole of the slew mechanism.

Two consequences fall out before anything is marched and both are checked rather than assumed.

The small-signal parameters are unchanged. $g_m = I/2V_T$ and $\omega_t = g_m/C_c$, so the same gain-bandwidth comes out of the same numbers, and the linear behaviour of the stage is the linear behaviour the five rungs below measured.

The slew rate is not a free parameter. It is $I/C_c$, and substituting the two expressions above gives

$$\mathrm{SR} = 2\,V_T\,\omega_t$$

— the thermal voltage times the gain-bandwidth, with no design choice in it at all. A ten-megahertz bipolar-input part slews at 3.25 volts per microsecond because 300 kelvin says so, and the only ways out of it are to make the pair slower per amp of tail current (degeneration, which the semiconductors field prices), to use a device with a softer law (a field-effect pair, whose square law saturates later), or to add a current boost that operates only on large signals.

That is a striking constraint and it is the same shape as several others in this collection: a quantity a designer thinks of as a specification turning out to be a physical constant times another specification. The 60 mV per decade of a diode drop and the 26 mV that decides how small a small signal is are the same constant appearing in the same way.

Two marches of one netlist #

Both traces come from the same netlist marched from the same initial state, so what is drawn is a difference between two marches rather than between a march and a formula. Below a few milliamps they lie on top of each other and cannot be told apart.

At thirty milliamps they can. The excursion is 1,109 millivolts against the linear model’s 709, and the recovery takes 1,025 nanoseconds against 660. The input error the pair saw peaked at 408 millivolts — 15.8 thermal voltages — while the tanh is already a per cent from linear at 9.0 millivolts.

The shape of the departure is the informative part. The slewing trace is not a scaled version of the linear one and it is not a delayed one either: it has a straight segment in it, where the output is moving at a rate the circuit has no say in, and the loop is open for the duration. During that segment the amplifier is not slower than the model says. It is out of current, which is a different failure, and the two are distinguishable by the fact that a slower amplifier’s response scales with the step and this one’s does not.

The impedance stops being a number at 10.6 milliamps #

The linear model’s ratio is flat to under a part per million across five decades of step size. That is not a measurement of the circuit; it is a measurement of what linearity means, and it is drawn because the other curve has to be compared against something whose flatness is exact.

The real one leaves at 10.6 milliamps, defined as the step at which the recovery takes a tenth longer than the linear model’s. The excursion-to-step ratio there has risen from 23.6 to 27.1 ohms, fifteen per cent, by thirty milliamps it is 37.0 — fifty-seven per cent above the linear value — and by three hundred it is 54.5, more than twice it, and no longer a property of the circuit at all.

The number worth carrying away is not 10.6 milliamps. It is that the input error at the departure is 3.63 thermal voltages — 93.9 millivolts — because that is the quantity that names the mechanism. Move the load capacitance over two decades and the current at which the impedance departs moves from 7.4 to 29 milliamps, and the input error at which it happens moves from 5.3 thermal voltages to 1.3: both change, and only one of them stays in single figures. A departure that always happens within a factor of a few of the pair’s own scale is the pair running out of current. A departure that happened at a fixed number of milliamps would be something else — an output stage’s limit, a protection circuit, a supply — and the two are told apart by exactly this.

Why a larger capacitor departs at a larger current #

The slider’s own result deserves a paragraph, because it runs against the obvious expectation.

A bigger load capacitor makes the stage worse by every measure the four rungs below use: less phase margin, a longer settling, a lower crossover. It might reasonably be expected to slew sooner as well. It does not — the departure moves from 7.4 milliamps at 220 pF to 29 at 47 nF, four times further out.

The reason is what a load-current step does to the input. The step is absorbed initially by the load capacitance, which holds the node while the loop responds, so the rate the amplifier is asked to produce falls as the capacitance rises. A large capacitor is a reservoir that buys time, and buying time is precisely what a slew-limited amplifier is short of. What it costs is the thing the rungs below measured: the excursion is larger and lasts longer, and the ringing that a marginal phase margin produces is more of it.

So the compensation that is worst for the small-signal behaviour is best for the large-signal boundary, and a designer optimising one has moved the other in the direction they were not looking. That is the same shape as the finding the fifth rung closed on — the capacitor that is best for the noise is twenty-four times slower to recover — and it is the third quantity that capacitor decides.

Why the boundary is far above the small-signal one #

There is an apparent contradiction here worth resolving, because two essays in this collection give very different amplitude boundaries for the same device.

The semiconductors field measured a differential pair’s own amplitude limits and found 1.03 millivolts for one per cent distortion and 7.30 for one per cent gain error. This essay’s departure is at 93.9 millivolts of input error, which is thirteen times the larger of those.

Both are right and they are answering different questions. Those boundaries are where the pair stops being linear — where its output current departs from $g_m v$ by a per cent. This one is where the pair stops being able to supply the current the loop is asking for, which is a much later event: tanh is 76 per cent of the way to its limit at 3.63 units of argument, so most of the tail current is available and the loop is merely slow rather than open.

The intermediate region is real and is not drawn as a boundary by either essay. Between about ten millivolts and a hundred, the amplifier is neither linear nor slewing: its transconductance is falling smoothly, so the loop gain is falling, so the settling is lengthening — continuously, with no edge anywhere. That is why the departure has to be defined by a criterion (a tenth of the recovery) rather than found: there is no kink in the curve to bisect on.

What this does to the five rungs below #

Each of the five is a statement about a linear circuit and each keeps a bound.

The margin and the settling are small-signal quantities measured on a cut loop, and they describe the recovery only after the slewing has finished. A stage that slews for six hundred nanoseconds and then settles in three hundred has a settling time of nine hundred, and its phase margin says nothing about the first two thirds of that.

The direct-current error survives untouched: it is a statement about the steady state, and a steady state has no rate in it.

The noise survives too, for the opposite reason — it is a small-signal quantity by construction, since the noise is microvolts and nothing in the tanh has departed at microvolts.

The output impedance is the one that does not survive, and it is the rung immediately below. Its number — the ten ohms, the 1.2 milliohms — is a limit as the step goes to zero, which is the right definition and is not the quantity a load step measures. For a load that steps by a milliamp, the limit is the answer. For one that steps by fifty, it is not an answer at all.

And the comparison between the three compensations, which is what the whole anchor exists for, is worth a sentence of its own. The departure measured here is a property of the amplifier rather than of the compensation, so all three arrangements leave at about the same input error — but the input error a given load step produces depends on the arrangement’s own loop gain, so they leave at different currents. The arrangement with the best small-signal impedance drives its input hardest for a given load step, and therefore departs soonest.

What a specification sheet says about this, and what it does not #

The two numbers a data sheet gives — an output impedance and a slew rate — are on different pages and are not usually read together. This essay is the arithmetic that joins them, and the joining is short.

The output impedance is a small-signal quantity, quoted against frequency, and it is exactly right in the limit of a vanishing step. The slew rate is a large-signal quantity, quoted as one number, and it is exactly right in the limit of an infinite one. Between them there is a load-step size at which the first stops applying, and no data sheet states it, because it is not a property of the amplifier: it depends on the loop the amplifier is in and on what is hung on its output.

Computing it takes one solve and one march. The input error a load step $\Delta I$ produces is roughly $\Delta I\,Z_{\text{out}}(0)/A$, with $A$ the loop gain at the frequency the step’s own spectrum lives at, and the departure is where that error reaches a couple of thermal voltages. That is a one-line estimate and it is within a factor of two of every number in this essay — which is enough to know whether a design is near the boundary or three decades from it, and knowing which is usually the whole question.

The estimate also says why the problem is getting worse rather than better. Loads that need capacitive-load compensation are loads that draw current, and the currents drawn by modern digital loads step by amps in nanoseconds. The tail current of an input pair has not changed.

What is not modelled #

The output stage cannot run out of current. The buffer here is ideal apart from its output resistance, so the only limit in the circuit is the input pair’s tail. A real amplifier has an output current limit too — often a few tens of milliamps, often with a foldback that reduces it further as the output voltage departs from the supply — and for a load step it is usually the binding one. That would give a second departure, at a current rather than at a number of thermal voltages, and the two would be distinguishable by exactly that.

The supply does not move. The load current comes from somewhere, and in a real board it comes from a rail with an impedance of its own, so a thirty-milliamp step is also a rail step. That is the quantity the five rungs below also named as unmeasured and it stays unmeasured here.

There is no recovery mechanism. A real amplifier that has slewed may have saturated an internal node, and coming out of that takes a further time that no model of this kind contains — which is why a measured settling curve sometimes has a flat portion after the slewing and before the exponential.

And the compensation capacitor is one. The tanh saturates the current into a single capacitor, which is the classical Miller-compensated topology. A part with feedforward, a class-AB input stage, or a slew-boost circuit is a different nonlinearity, and the whole point of those arrangements is that $2V_T\omega_t$ is not a law they obey.

What the gate checks #

The linear model’s impedance is asserted to be the same number at every step on the sweep, to within a part per million, which is what makes the comparison meaningful and would fail if the two marches had drifted apart for a numerical reason rather than a physical one.

The nonlinear excursion is asserted never to be below the linear one, at any step, which is the claim that the tanh can only limit and never help.

The departure is asserted to be at a few thermal voltages of input error rather than at a current — between one and twelve — so a version whose boundary had come out at a millivolt or at a volt would fail rather than be reported.

The slew rate is asserted against $2V_T\omega_t$ computed independently from the tail current and the compensation capacitor, which is the check that the macro-model’s small-signal parameters and its large-signal ones came from the same device.

And the current-law residual is checked at every step of both marches, which is what caught the sign error in the linear transconductor: written the other way round the linear march is a positive feedback loop and diverges to 10¹¹² volts, and it was reported as an output impedance before that check was read.

The sixth rung, and what a rung is for #

This anchor now has six essays on it and it is worth saying what the sequence has been.

The first four asked what the load does to the signal: the margin, the settling, the direct-current error, the noise. The fifth turned round and asked what the load sees, and found a quantity none of the four could — an impedance that is ten ohms where the loop gain suggested milliohms, because the isolation resistor is outside the loop.

This one asks whether that quantity exists. It does, over a range, and the range is stated in the input pair’s own units — which is the same answer this collection gives about an ideal amplifier, a lumped element and a small-signal model, and is the reason the premise of the site is written the way it is. Nothing has been refuted. A number has been given the amplitude at which it stops being one.