The node that is an inductance

The node that is at ground for a while ended with a list of places its model stops, and the first entry was this one: the amplifier is one pole, and a real part has a second corner, near which the junction’s impedance stops rising at a decade per decade and can peak — the junction goes inductive, which with a capacitance on it is a resonance.

That is a claim about a shape, and a shape is what is drawn here. It turns out to be worth more than a caveat, because it changes what kind of object the summing node is. A resistance that a capacitance loads has a corner. A reactance that a capacitance resonates with has a band, and trouble that arrives over a band with quiet ground on both sides of it is a different diagnostic problem entirely.

Where the overshoot comes from #

The node’s impedance is the open-loop impedance divided by one plus the return ratio, and the essay before it established both halves of that by measuring them separately and finding them to agree to 0.045 per cent over nine decades. The open-loop impedance is two resistors, 909.09 ohms, and it has no frequency in it. So every feature of the curve is a feature of $|1+T|$.

With one pole, $T$ falls from $10^5$ at direct current at twenty decibels a decade and its phase is never more lagging than ninety degrees. So $|1+T|$ falls monotonically from $10^5$ to one, and the node rises monotonically from a tenth of an ohm to 909 and stops. The ceiling is reached because the divisor reached one, and a divisor that only approaches one from above can only produce a curve that approaches its ceiling from below.

With two poles the phase goes further. Near the crossover $T$ is a complex number of about unit magnitude whose angle has passed ninety degrees, so $1+T$ is the sum of one and a vector pointing partly backwards — and its magnitude dips below one. The division is then a multiplication, and the node’s impedance goes above the two resistors in parallel:

Every row is the same circuit with one capacitor changed inside the amplifier, and the one-pole case measured on the same netlist never exceeds 909.09 ohms by two parts in a million. So the peak belongs to the second pole and not to the method — which is the requirement worth making, because a peak in a measured impedance is exactly the shape a numerical artefact takes.

The ordering in that table is also the mechanism. The peak falls monotonically as the second pole moves away from the crossover, because what produces it is the pole’s phase arriving while the loop still has magnitude. A second pole two decades above the gain–bandwidth contributes phase where there is no gain left to divide, and there is nothing to see.

Why “inductive” is the right word and not a metaphor #

A peak in a magnitude is not enough to call something an inductance. The test is the angle, and it is measured rather than inferred.

A third of a decade below the peak the node’s impedance leads its current by more than twenty degrees. A resistance has an angle of zero, a capacitance lags, and an impedance that rises with frequency and leads is an inductance in the only sense that matters to the circuit around it — it stores energy in a way that a capacitance can exchange with.

Which is a shape been drawn before, on a different circuit and from the same cause. The resistance that is below zero measures an emitter follower’s output resistance at 5.5 ohms at direct current and −21.9 ohms at 257 megahertz, and finds the sign not to belong to the transistor at all: with an ideal source at the base there is no negative band, and a hundred nanohenries of wire between the source and the base produces one from 110 to 301 megahertz. The mechanism there is a loop running out with a phase lag still in it, which is the mechanism here, and the consequence there is the consequence here.

That consequence is the part worth carrying. Trouble arrives over a band of load capacitance with quiet ground on both sides of it. That essay finds 4.7 to 100 pF oscillating while 1 pF and 470 pF do not, which is the single most confusing failure mode in this collection: adding capacitance fixes it and removing capacitance fixes it, and a designer who tries one of those and then the other in the wrong order concludes that the fault is intermittent. It is the same class of failure a band rather than an edge is named for, and the only boundary in this collection with two sides to it.

The node’s angle is the loop’s phase, minus a sign #

The identity behind everything above is worth writing down, because it turns a curve a bench can measure into a statement about a quantity a bench cannot.

The node’s impedance is $Z_\mathrm{open}/(1+T)$ with $Z_\mathrm{open}$ two resistors and therefore real. So the node’s angle is minus the angle of $1+T$, exactly, at every frequency. The three regions of the curve are three regions of one complex number:

Where $|T| \gg 1$, $1+T \approx T$, so the node’s angle is minus the loop gain’s phase. A loop lagging ninety degrees gives a node leading ninety — which is why the rising part of the one-pole curve is already an inductance, a fact the essay before it drew and did not name. A node rising at a decade per decade with a ninety-degree lead is an inductance with nothing resistive in it at all.

Where $|T| \ll 1$, $1+T \approx 1$ and the node is the resistors: real, and flat.

And in between, where $|T|$ is about one, $1+T$ is the sum of a unit vector and a rotating one, and both its magnitude and its angle do whatever the loop’s phase there dictates. That is where the peak is, and it is also the only region where the node’s angle passes through zero — twice, on either side of the peak.

So the inductive behaviour is not a feature of the second pole. It is a feature of every loop, present in the one-pole case over five decades, and what the second pole adds is the one thing a one-pole loop cannot produce: a magnitude that goes above the open-loop value. An inductance whose reactance exceeds the resistance it is in parallel with is an inductance a capacitance can resonate with, and that is the threshold the second pole crosses.

What a capacitance does to it, which is not damping #

The slider on the figure is capacitance on the summing node, and what it does is the finding this essay was written for.

The peak grows and moves down in frequency. That is what a capacitance does to an inductance and it is the opposite of what it does to a resistance: a capacitance across a resistance rolls it off, and a capacitance across an inductance resonates with it. With the second pole at thirty kilohertz, 470 picofarads takes the overshoot from 6.25 decibels to 8.37.

So the ordinary intuition about a summing node is exactly inverted in this region. Below the loop’s crossover, capacitance on the node is a load and rolls the impedance down. Above it, capacitance on the node is the other half of a tank. The frequency at which the sense of the effect reverses is the frequency at which the node’s angle crosses zero on its way up, and that is a property of the amplifier’s second pole rather than of anything a layout can reach.

The two failures this separates #

A summing node with capacitance on it is the commonest instability in analogue design, and the essays that measure it are measuring two different things. It is worth being explicit about which is which, because the remedies differ.

The gain the loop closes against is the non-resonant version. There the capacitance at the junction raises the noise gain — the gain the loop closes against is one plus the feedback impedance over everything at the node, and a capacitance makes the second term fall with frequency — so the loop’s crossover moves up the amplifier’s own roll-off and the phase margin falls. Nine picofarads takes that essay’s stage from ninety degrees of margin to forty-five, and a hundred puts twelve decibels of peaking on the response. The mechanism is entirely first-order: more capacitance is always worse, and the cure is a feedback capacitor that cancels the zero.

This essay is the resonant version, and it needs the second pole. Here the capacitance is not raising a crossover; it is resonating with an impedance that has become reactive because the loop’s phase ran out. More capacitance moves the resonance down rather than deepening a monotone trend, and a feedback capacitor does not cancel anything because there is no zero to cancel — the reactance is the amplifier’s, not the network’s.

The diagnostic that separates them is a sweep rather than a value. A first-order problem gets monotonically worse with capacitance; a resonance has a worst capacitance. One measurement at three values of the stray, and the answer is unambiguous.

What the peak is worth in a number a design uses #

An impedance peak of 1.45 decibels is not itself a fault. What makes it worth measuring is that it is the same quantity as the phase margin, read in a different place, and the summing node is where a bench can reach it without cutting the loop.

The relationship is direct. The node’s impedance is $Z_\mathrm{open}/|1+T|$, and the peak of that curve is $Z_\mathrm{open}$ divided by the minimum of $|1+T|$ — which is the closest the Nyquist locus comes to the critical point, and is the quantity a designer would otherwise call the vector margin. So

$$\text{overshoot, in decibels} = -20\log_{10}\min|1+T|$$

and 1.45 decibels of overshoot at the summing node is a vector margin of 0.846, while 6.25 decibels is a vector margin of 0.487.

That is worth having because of what it does not require. Measuring a phase margin means cutting the loop, injecting, and reading what comes back — which is what what is left at crossover does and which needs the loop to be cuttable. Measuring the impedance at the summing junction means driving a current into a node and reading a voltage, with the loop intact and the circuit doing its job. The two are the same number, and the second is available on a finished board.

The caution that goes with it is the one two measurements of one margin makes about the frequency-domain and time-domain routes agreeing: they agree when one pole pair near crossover dominates, and this identity is exact rather than approximate — it is a definition rearranged — but the interpretation of a vector margin as a phase margin is not. A locus that passes close to the critical point at an unusual angle has a good phase margin and a poor vector margin, and the summing node reports the second.

Measuring it, and the instrument that is in the resonance #

The identity above makes the summing node the cheapest place to read a loop’s vector margin, and it is worth being concrete about the measurement, because the instrument is part of the circuit it is measuring and here that is not a caution but the dominant term.

The injection is easy. A signal generator through a large resistor — a megohm, say — into the summing node is a current source to the accuracy of the ratio between that resistor and the node’s impedance, which at a kilohm against a megohm is a part in a thousand. The node’s voltage divided by the injected current is the impedance, and the loop stays closed throughout, which is the whole advantage over cutting it.

The reading is where the trouble is. The node is a kilohm at the peak and the peak is at a couple of hundred kilohertz, so the probe’s own capacitance is in parallel with the thing being measured — and this essay’s table says what that does. Ten picofarads of a low-capacitance probe at 226 kilohertz is seventy kilohms, which is negligible against a kilohm. A hundred picofarads of ordinary coaxial lead is seven kilohms, still large. But the capacitance is not loading a resistance; it is resonating with an inductance, and the table above shows a hundred picofarads raising the peak from 1074 ohms to 1116 and moving it from 226 to 208 kilohertz. The probe has not attenuated the thing it is measuring; it has changed where and how high it is.

That is the instruments field’s standing argument arriving in the one place where it does not merely add an error. The probe is part of the circuit puts the general case — a probe goes in the netlist, and every reading is a reading of the circuit that includes it — and what is unusual here is the sign. A probe on a resistive node reads low, and a reader who knows the probe’s capacitance can correct for it. A probe on an inductive node reads high, at a frequency that is not the unprobed one, and the correction needs the amplifier’s second pole, which is the quantity being measured.

Two ways round it, and they are the two these essays keep arriving at. Measure with two known capacitances and extrapolate to zero, which works because the peak’s dependence on the capacitance is smooth and monotone over this range — the table gives 1074, 1076, 1084, 1116 and 1192 ohms for nothing, 1, 4.7, 22, 100 and 470 picofarads, and the first two are indistinguishable. Or accept the probe as part of the design: if the finished board has thirty picofarads of stray on that node, the useful measurement is the one taken with the probe that brings the total to thirty.

The second is the better answer and it is the one this field prefers, for the reason two terminals measure the leads as well gives: an arrangement that removes an error is worth more than a correction that estimates it, and where neither is available, a measurement made under the conditions the circuit will run in is worth more than one made under conditions it will not.

Where this model stops in its turn #

Two poles, and a real amplifier has more. A third pole adds phase where the second has already taken the locus near the critical point, so the peak is larger and at a lower frequency than two poles predict. The direction is known and the size is not measured here.

The output resistance is fifty ohms and constant. The ceiling is two resistors and one of them terminates at the amplifier’s output, so a real output impedance that rises with frequency adds to the ceiling in the region where the peak is — which makes the overshoot, measured against a constant 909 ohms, an underestimate.

And the excursion on the node is largest exactly here. Every number is a small-signal impedance about a bias point, and the essay before it noted the discomfort: the frequencies at which the node is a poor ground are the frequencies at which its voltage swing is largest, so the linearity assumption is weakest where the measurement is most interesting. With a peak in the curve that is sharper still. How small is small signal puts the amplitude at which linearising an exponential is one per cent wrong at 7.3 millivolts, and a summing node at a peak of 1868 ohms carrying a milliamp is at 1.9 volts.

Still open: the third pole, and the capacitance that is worst #

The worst capacitance, located. The table above sweeps six values and the peak grows at every one of them, so the resonance’s own maximum is outside the range drawn. Finding it — the capacitance at which the peak is highest, and what the peak is there — would turn this essay’s qualitative statement into the band of capacitance a design has to stay out of, which is what the resistance that is below zero has for its follower and this does not have for its summing node.

The vector margin read two ways. The identity between the impedance peak and $\min|1+T|$ is stated above and not measured: the peak comes from a solve with the loop closed and the vector margin from one with it cut, and they should agree to the solver’s precision. That is the two-route check this figure is missing, and it is the one that would make the peak usable as a margin measurement rather than as an argument that it could be.

And the same node driven rather than probed. Everything here drives a current into the node and reads a voltage, which is an impedance. What a circuit does is put a signal current in through an input resistor, so what a reader wants to know is whether the resonance appears in the signal path as peaking in the closed-loop response. It should, at the same frequency and with a height set by the same $\min|1+T|$ — and a figure with both curves on it would say whether the two heights are equal or whether the noise gain’s own shape separates them.

What is checked #

The peak is required to exceed the resistive ceiling, and the one-pole case on the same netlist is required not to — never by more than two parts in a million over two hundred frequencies. That pair is the claim that the overshoot belongs to the second pole rather than to the measurement, and it is the pair that would catch a peak produced by a coarse sweep.

The angle is required, not the shape. A third of a decade below the peak the impedance is required to lead by more than twenty degrees, which is the statement that the word “inductance” is being used literally.

The peak’s height is stated as an ordering across six second-pole frequencies, falling at every step from 6.25 decibels at thirty kilohertz to 0.07 at ten megahertz. Two earlier versions of this were wrong in opposite directions and both are worth recording: requiring a peak unconditionally fails on a fast amplifier where there is none to speak of, and requiring its absence above a cutoff fails just past the cutoff, because 3 MHz still gives 0.21 decibels and nothing switches off. The ordering holds at every setting, and it is also the mechanism rather than a summary of it.

And the high-frequency limit is required differently in the two cases. With nothing on the node it is the two resistors, to five per cent; with a capacitance on it, it is that capacitor’s reactance. The first version required the resistive limit unconditionally and failed at every setting of the slider but the first, which is the standing defect this ledger records most often — a requirement calibrated on the default is not a requirement about the claim.