The current above which there is no impedance

The step too large to have an impedance asked whether the quantity five rungs of this ladder had been computing exists. An impedance is a ratio of a voltage to a current, and a ratio is a number only if it does not depend on the current; give the amplifier the differential pair’s own tanh and the linear model’s excursion-to-step ratio is flat to under a part per million across five decades while the real one leaves at 10.6 mA.

The slew rate that decides it is $2\,V_T\,\omega_t$ exactly — 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 V/µs because 300 K says so.

There is a second large-signal limit and it is the opposite kind of quantity. An output stage can source and sink only so many milliamps, and how many is a decision somebody made about silicon area. Which of the two binds is not a property of the amplifier.

Four marches, so that the comparison is between measurements #

The stage is the one the load that gets inside the loop opened this ladder with, and it is marched four times with the same netlist and the same step: with neither limit, with the input pair’s tanh alone, with the output stage’s limit alone, and with both. Nothing is estimated and nothing is superposed — each is a march of a netlist that differs from the others by one device.

The output stage is written the way the input pair was, as a tanh, and for the same reason: a real one’s limit is smooth, and the small-signal transconductance at the origin is exactly $1/r_{out}$, so the linear behaviour of the five rungs below is untouched by construction. Setting the limit to infinity returns the netlist to the E-and-resistor pair those rungs used, byte for byte, so no figure below this rung can move.

Each limit is reported as its departure from the linear march rather than as an excursion, because most of an excursion is the linear answer both limits share. Comparing raw excursions reads as the two are within a few per cent at every load, which is true and says nothing.

They change places, and the load decides #

At 0.47 nF the input pair’s departure is 19.4 per cent and the output stage’s is 4.2. At 22 nF it is 0.9 against 2.3. Somewhere near twelve nanofarads they cross, and on either side of the crossing a different physical mechanism is producing the departure.

The input pair’s departure falls with load capacitance, which is the sixth rung’s own result and is worth restating because it runs the wrong way round: a bigger reservoir holds the output node while the loop responds, so the compensation that is worst for the small-signal behaviour is best for this large-signal boundary. The output stage’s departure falls much more slowly, because what it has to supply is the charge the capacitor wants and a bigger capacitor wants more.

The crossing is proportional to the output stage’s rating, and the natural expression for it is $I_{max}/\mathrm{SR}$ — the capacitance the input pair can just about drive at the output stage’s full current. Measured, it sits between one and three times that, because the loop demands more current during the recovery than the step itself asks for.

That expression is worth looking at twice. It is a capacitance, and it is assembled from a number a designer chose — how many milliamps of output stage to pay for — and a slew rate that contains only the thermal voltage and the gain-bandwidth. A boundary between two design decisions has 300 K inside it.

Above the rating there is no impedance at all #

The two limits above are both departures: the ratio is still a number, it has just stopped being the same number at every step. Beyond one point that stops being true.

A load-current step is sustained — the load draws that current until something changes — so if the step exceeds what the output stage can supply, the amplifier can never return the output to where it was. At a 30 mA step into a 20 mA stage the marched ratio is 2,254 Ω where the unlimited model gives 37, and at 100 mA it is 5,374.

Those numbers are not impedances and they are not larger impedances. They are properties of how long the march was watched: the output is falling at $(i_{load} - i_{max})/C_L$ and has not finished. Six rungs of this ladder have been about a quantity that depends on the current; this is the current above which there is no quantity.

Which of the two a designer can change #

The asymmetry between the two limits is the reason this rung exists, and it is worth stating in the form a decision is made in.

The input pair’s limit is not negotiable. Its slew rate is $2\,V_T\,\omega_t$, so for a given gain-bandwidth it is fixed by the thermal voltage — and choosing a faster part raises both the gain-bandwidth and the slew rate together, in proportion, which does not change the ratio between the slew rate and the bandwidth at all. The only way to move it is to leave the bipolar input behind: a field-effect input pair’s transconductance per unit current is lower, so the same tail current buys less gain-bandwidth and more slew rate, which is exactly the trade the exponent that is a square is about.

The output stage’s limit is negotiable and it is bought with area, quiescent current and package. So a designer facing a large capacitive load has one dial that turns and one that does not, and the crossing measured above says which dial is connected: below about twelve nanofarads, buying output current changes nothing.

That is a specific and slightly counterintuitive instruction: what the load sees looking back computed an impedance of 10.00 Ω for one arrangement and 1.2 mΩ for another, and neither of those numbers improves by fitting a bigger output stage while the load is small.

The two limits compound #

Where both mechanisms are present the departure is not the larger of the two. It is between 35 and 60 per cent larger than either alone, at every load capacitance measured.

That is not obvious and it is not a modelling artefact. The input pair’s limit caps the rate at which the compensation capacitor’s voltage can change, which is what tells the output stage what to do; the output stage’s limit caps the current available to act on that instruction. Each makes the other’s recovery start later, so the excursion each is trying to arrest is larger by the time it is arrested.

The reading for a designer is that the two limits cannot be budgeted separately. A stage whose input pair contributes ten per cent and whose output stage contributes ten per cent does not contribute twenty; it contributes about twenty-seven.

What the recovery time does, which is not what the excursion does #

The excursion and the recovery are different quantities and the two limits treat them differently.

Under the input pair’s limit the recovery grows gently: 0.72 µs at a 10 mA step against 0.66 µs linear. Under the output stage’s limit at the same step it is 1.02 µs, and at 30 mA — past the rating — it is 17.6 µs and rising with the step, because the recovery is now $C_L\,\Delta V/I_{max}$ and both factors grow.

A settling specification is therefore the first thing to fail, before any excursion limit is reached, and it fails in a way that a linear model cannot express: the settling time becomes proportional to the step rather than independent of it. The step that is too big measured the same transition for a voltage step at the input, and this is its dual at the output — same amplifier, same two mechanisms, different terminal.

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

Three numbers are usually printed and none of them is the crossing measured above.

Slew rate is printed and is the input pair’s limit, measured as an output slope with the stage driven hard. It is $2\,V_T\,\omega_t$ for a bipolar-input part and is not independently choosable.

Short-circuit current is printed and is not the number wanted here — it is a protection threshold rather than a capability, in the sense a band rather than an edge draws for a switch: it is what the protection circuit allows into a dead short, which is often lower than the linear output current because the protection folds back.

Output voltage against load current is printed as a curve and comes closest — it is the stage’s capability at direct current — but it is a curve of a static quantity and the failure above is dynamic.

What is not printed at all is the interaction. Given the slew rate and the output current, the load capacitance at which they trade places is computable — it is the dashed line above, to within a factor of two or three — and it is the number that says which of the two data-sheet figures a designer should be reading for a given load.

Reading the departure as a specification #

There is a question a designer actually has, and it is not what is the impedance. It is: given this load capacitance and this load step, by how much does the rail move and for how long?

The four marches answer it directly, and the useful form is the pair of numbers rather than either alone. At 10 mA into 2.2 nF the excursion is 0.28 V and the recovery is 0.72 µs; at 10 mA into 22 nF it is 0.12 V and 1.1 µs. The excursion falls with load capacitance and the recovery rises, which is the same trade the second rung of this ladder made with a series resistor and the fourth made with a second feedback path — a quantity moved rather than removed.

Where the seventh rung changes the answer is at the top of the range. The excursion’s fall with capacitance continues indefinitely in the linear model and in the input-limited one; in the output-limited one it stops, because a larger capacitor takes proportionally longer to recharge at a fixed current and the excursion during that time stops falling. There is a load capacitance beyond which more capacitance buys nothing, and it is the same crossing.

What is not in this model #

No supply. The rails here are ideal, so the output stage’s limit is a current and nothing else. A real stage’s current limit and its voltage headroom interact: an output driving a capacitor at the slew rate is dissipating and its rail may move, and what gets through from the rail measures how much of that movement reaches the output. The fifth and sixth rungs both named this and neither has measured it.

No thermal effect. A stage delivering its full rated current into a capacitor for tens of microseconds heats, and both the output stage’s current capability and the input pair’s transconductance are temperature-dependent. On a repetitive load step at a high duty cycle that is a slow drift on top of everything above.

And no protection circuit. Real output stages fold back, and a foldback characteristic is a current limit that falls as the output voltage departs from the rail — which is exactly the condition a capacitive load step creates. That would make the excursion above worse rather than better, and it is not modelled.

The march, and what makes it trustworthy here #

Everything above is a marched netlist rather than a formula, and it is worth saying what the march is checked against, because a large-signal result is only as good as the integration under it.

Two things are checked at every step and both are independent of the answer. Kirchhoff’s current law is rebuilt from the element relations — the capacitor contributes C·dv/dt from its two stored states, the tanh device contributes its own function at its own solved voltage — rather than from the companions the step used, so a sign error in a companion model survives every plot and does not survive the residual. And with the nonlinearity removed the march reproduces the linear integrator to the last digit, which is the same rule applied to the same netlist by different code.

The third check is the one this rung needed and the sixth did not. A linear model’s excursion-to-step ratio must be the same number at every step size, and it is asserted to a part per million — which is what makes the departures above meaningful, because a linear reference that drifted by a per cent would swamp the 0.9 per cent departure measured at 22 nF.

Seven rungs, and what has happened to the quantity #

The first rung of this ladder measured an amplifier’s output impedance with a capacitive load and found the load inside the loop. Rungs two to five bought the margin back, priced what that cost at the floor, and computed what the load sees looking back. The sixth asked whether the resulting number is a number and found it leaving at a current set by the thermal voltage.

This one finds a second current — set by nothing but somebody’s silicon budget — and finds that which of the two decides the answer is a property of the capacitor on the output. Above either of them the ratio is not an impedance; above the output stage’s rating it is not a measurement of the circuit at all, only of how long the march ran.

A ladder that began by computing an impedance has ended by establishing the conditions under which the word applies. That is the shape every model has an edge sets out for the whole collection, arrived at by seven successive measurements of one arrangement rather than by asserting it once.

Two quantities remain named and unmeasured, and they were named by the two rungs below as well: a supply that moves when the load steps, and the interaction between the output stage’s current limit and its voltage headroom. Both are one element away in the same netlist and neither is in these figures.

The first of those is now measured, and the interesting part of the answer is not the number it was expected to produce. The rail the load moves gives the supply an impedance and finds this channel’s own output impedance changing by three parts in ten million, because the loop corrects the supply along with everything else — so the assumption seven rungs made without stating it turns out to have been nearly free. What the same element opens instead is a path to a second amplifier sharing nothing with this one but a wire, and the size of that path is a modelling choice rather than a measurement: 3.84 microvolts per ampere with the compensation capacitor returned to ground and 17.3 millivolts with it returned to the rail, a factor of four and a half thousand from a decision nothing in a block diagram records.

Which is worth reading back into this essay’s own result. The two currents measured here are limits on what one amplifier can do about its own load. The rail is the mechanism by which one amplifier’s load becomes another amplifier’s problem, and it is unaffected by everything this ladder has spent seven rungs measuring — the isolation resistor, the second path, the compensation, the margin. A design that has solved every problem on this ladder has not touched it.

What the boundary is made of #

It is worth naming what kind of boundary each of the two currents is, because they are as different as two numbers in the same units can be.

The input pair’s limit is a physical constant wearing a circuit’s clothes. Twice the thermal voltage times the gain–bandwidth in radians has a temperature in it and nothing else a designer chooses, which puts it in the same family as how small is small signal — one per cent wrong at 7.3 millivolts, 28 per cent of the thermal voltage — and it moves with the room in the way the edges that move with the room collects: proportional to absolute temperature, so a factor of 1.71 across an industrial range, exactly the ratio of the two temperatures.

The output stage’s rating is the opposite kind of number. It is a decision recorded in a data sheet, it has no expression behind it, and it can be bought off with money and die area. The two happen to land within an order of each other for ordinary parts, which is why both appear in this figure at all; on a part built for a different purpose they would be decades apart and only one of them would ever be reached.