The node that is at ground for a while

“Virtual earth” is the most useful three-word abstraction in analogue design, and like every model on this site it has a frequency in it that nobody writes down. The node is at ground because a loop is holding it there, so it is at ground exactly as well as the loop is strong — and the loop’s strength falls at twenty decibels a decade from the amplifier’s first pole, which for an ordinary part is ten hertz.

The measurement #

An inverting amplifier: a kilohm from the input to the summing junction, ten kilohms from the junction to the output, an amplifier with $10^5$ of gain and a megahertz of gain–bandwidth between the junction and the output. The input source is set to zero, which is what an impedance at a node means — everything else left as it is and the independent sources zeroed — so the kilohm terminates at ground.

Then one ampere is driven into the junction and the voltage is read. That is the definition of an impedance rather than a formula for one, and it is the same routine the networks field uses on a one-port.

Three regions, and each has a name.

Below ten hertz the loop gain is flat at its direct-current value and the junction is 0.101 Ω. Ten hertz is the amplifier’s own pole — its gain–bandwidth over its direct-current gain — and the impedance there is $\sqrt2$ times the floor, which is what a first-order corner does.

Between there and about a megahertz the impedance rises a decade per decade, measured at 1.00 decades per decade, because the loop gain is falling at exactly that rate and the junction is the open-loop impedance divided by it.

Above the crossover the impedance is 909.5 Ω, which is 1 kΩ in parallel with 10 kΩ. That is what is left when the amplifier has stopped participating: two resistors meeting at a node.

Two routes #

The second route is the return ratio. The loop is cut at the amplifier’s input, injected, and what comes back round is the loop gain $T$; the junction’s impedance is then predicted as the open-loop impedance — the plain 909.09 Ω of the two resistors — divided by $|1 + T|$.

The two agree to 0.045% across thirty frequencies spanning nine decades. They share the netlist and nothing else: one inverts a matrix with a current source in it, the other inverts a different matrix with the loop opened and reads a node voltage.

The rule the slider produces #

Move the gain–bandwidth and the impedance curve slides bodily sideways. The floor does not move, because the floor is the direct-current loop gain and that is set by $A_0$ rather than by the bandwidth; the ceiling does not move, because it is two resistors. What moves is the whole of the sloped middle.

Pick a threshold — one per cent of the input resistor, which is ten ohms here — and bisect for the frequency at which the junction reaches it:

Every row is the gain–bandwidth divided by a thousand. A summing junction stops being a virtual earth at a thousandth of the amplifier’s gain–bandwidth product, and the factor of a thousand is not a coincidence of these component values: it is the hundred that is $R_f/R_\text{in} + 1$ divided into the reciprocal of the one per cent, near enough, and it moves with those two choices and with nothing else.

For a garden-variety part running audio, that is a kilohertz. The node a designer is treating as ground for the whole of the audio band is ten ohms at the top of it and a hundred ohms an octave and a half above.

Where the 909 ohms comes from #

The ceiling is worth deriving because it is the one number in the essay that a reader can check without a solver, and because it is not the number most people would guess.

With the amplifier removed entirely, the junction has two resistors on it: the kilohm going to ground, because the input source was zeroed, and the ten kilohms going to the amplifier’s output, which with no amplifier is also at ground through its own output impedance of fifty ohms. So the junction sees 1 kΩ in parallel with 10.05 kΩ, which is 909.5 Ω, and the measured ceiling is 909.5 Ω.

The guess most people would make is the ten kilohms, on the reasoning that the feedback resistor is the one connected to the thing that stopped working. It is the smaller of the two that dominates, and the smaller of the two is the input resistor — so the ceiling of a summing junction’s impedance is its input resistor, near enough, whatever the feedback resistor is.

That has a consequence for the sloped middle as well. The whole curve is the ceiling divided by $|1+T|$, so a design with a smaller input resistor has a proportionally lower impedance at every frequency, and the frequency at which the junction reaches a stated fraction of the input resistor is unchanged. The rule of a thousandth of the gain–bandwidth survives changing both resistors together, which is why it is worth stating as a rule.

A number for the crosstalk #

Two signals summed into one junction through a kilohm each. A signal on the second develops a voltage on the junction of its own current times the junction’s impedance, and that voltage pushes a current back down the first input’s resistor into whatever is driving it.

The leakage, as a fraction of the second channel’s own current, is the junction impedance over the input resistance — which is the table at the top of this essay divided by a thousand:

Twenty decibels per decade of separation, starting from eighty. A summing mixer built on a megahertz part has forty decibels of channel separation at a kilohertz and twenty at ten, which is audible, and the fix is not better resistors — it is a faster amplifier, because the whole curve moves with the gain–bandwidth and with nothing else in the circuit.

This is the mechanism behind the usual advice to sum into a virtual earth only from sources that can drive it, and behind the observation that a passive summing network followed by one gain stage often measures better than an active summing junction, which is otherwise a puzzle.

What goes wrong when it is not ground #

Three consequences, and they are the reason the number is worth having.

Two inputs summed into one junction talk to each other. A current from the second input develops a voltage on the junction, and that voltage drives a current back down the first input’s resistor into whatever is driving it. The crosstalk is the junction’s impedance over the input resistance: a part in ten thousand at direct current, a per cent at a kilohertz, and unity somewhere past the crossover. A summing mixer’s channel separation is this curve.

A capacitance on the junction becomes a gain. The noise gain of an inverting stage is $1 + Z_f/Z_\text{in}$ where $Z_\text{in}$ includes everything at the junction, so a photodiode’s capacitance or a long input cable’s puts a zero in the noise gain at the frequency where its reactance equals the input resistance — and what is amplified by it is the amplifier’s own input noise, which was not in the signal path at all.

And the input impedance is not the input resistor. An inverting amplifier’s input impedance is usually quoted as $R_\text{in}$, on the reasoning that the far end of it is at ground. It is $R_\text{in}$ plus the junction’s impedance, which is a per cent high by a kilohertz and twice the quoted value where the junction reaches 1 kΩ.

The third of those is the one that reaches a specification sheet. An inverting stage whose input impedance is quoted as “1 kΩ” has an input impedance of 1.010 kΩ at a hundred hertz, 1.10 kΩ at ten kilohertz and 1.91 kΩ at a megahertz — which for a stage driven from a source with any impedance of its own is a gain that falls with frequency for a reason that is nowhere in the gain expression. The usual symptom is a high-frequency roll-off a decade below where the gain–bandwidth product says it should be, blamed on the amplifier, and cured by lowering the source impedance rather than by changing the part.

The same failure as the essay before it #

The quantity this essay measures and the sensitivity the previous one measures are the same loop gain read twice, and they run out together.

At the default part, the sensitivity reaches one — feedback no longer helping — at 6.33 kHz, and the junction passes one per cent of the input resistor at 995 Hz. Those are different thresholds on different quantities, so the numbers differ; what is identical is the slope between them and the frequency at which both saturate.

That is the honest general statement about feedback and it is worth putting in one sentence: everything a loop does well, it does in proportion to its loop gain, and every one of those things stops at the same place. The gain accuracy, the low output impedance, the virtual earth, the distortion reduction and the disturbance rejection are five names for one division.

The practical form of that is a single question to ask of any circuit with a loop round it: what is the loop gain at the frequency of interest? Everything else follows, and it follows by a division rather than by a rule of thumb. A designer who knows the loop gain is 100 at the working frequency knows that the gain error is a per cent, the summing junction is a hundredth of its ceiling, the distortion is a hundredth of the open-loop figure and the disturbance rejection is forty decibels — all of it, from one number, without a second measurement. What is left at crossover is where that one number is obtained the way a bench obtains it: cut the loop, drive one side of the cut, and measure what comes back to the other.

The sentence is worth stating strongly and then bounding, because it has two real exceptions in this collection and both are instructive.

The first is that near crossover the division is by a complex number smaller than one, so the loop makes things worse rather than merely stopping. This essay’s own curve shows it and how much of the amplifier gets through measures it directly: the fraction of a device change reaching the answer rises above one near crossover, so feedback there makes the gain more device-dependent than no feedback at all, and below fifty degrees of margin the fraction changes sign on the way.

The second is larger and it is a genuine exception to the sentence rather than a qualification of it. What gets through from the rail measures a disturbance rejection on two circuits with identical loop gain, identical crossover and phase margins agreeing to a millionth of a degree — and finds them sixty decibels apart, because the disturbance reaches the output through a forward path the loop divides rather than through the loop itself, and the size of that path is the pass device’s own intrinsic gain. So “the disturbance rejection is forty decibels” does not follow from the loop gain alone; it follows from the loop gain and the open-loop value of the quantity being rejected, and the second of those is a property of the topology that a block diagram does not show.

Which sharpens the sentence rather than weakening it. Everything a loop does well it does by dividing an open-loop quantity by $1+T$, and knowing $T$ tells a designer the divisor and nothing about the dividend.

Where the model stops #

The amplifier is one pole. A real part has a second corner, and near it 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, and that is where an inverting stage with a long input lead rings.

That failure is measured elsewhere on this site, on a different circuit and with the numbers in it. The resistance that is below zero finds an emitter follower’s output resistance at 5.5 Ω at direct current and −21.9 Ω at 257 MHz, with the sign not belonging to the transistor: with an ideal source at the base there is no negative band at all, and a hundred nanohenries of wire between source and base produces one from 110 to 301 MHz — so 4.7 to 100 pF on the emitter oscillates while 1 pF and 470 pF do not, a band of load capacitance with quiet ground on both sides of it. An inverting stage’s summing junction is the same object: an impedance that rises with frequency because a loop is running out, meeting a stray capacitance, with the oscillation confined to a band rather than starting above a threshold. And the gain the loop closes against puts the ordinary, non-resonant version of it in numbers on this exact node — nine picofarads at the summing junction, less than a scope probe, takes the phase margin from ninety degrees to forty-five, and a hundred picofarads puts twelve decibels of peaking on a response whose designed gain is nought decibels and whose measured gain at a kilohertz has not moved by three parts in a million.

The output impedance is fifty ohms and constant. A real amplifier’s output impedance rises with frequency too, which adds to the 909 Ω ceiling above the crossover rather than being negligible against it.

And nothing here is nonlinear. Every number is a small-signal impedance about a bias point. What happens when the junction is driven far enough to take the amplifier out of its linear range is not a larger impedance; it is a different circuit, and the loop is not holding anything.

How far is far enough is measured twice in this collection, and the two answers are usefully different. The step that is too big puts the boundary at about eighty millivolts for an ordinary part, arriving as a rate limit — the output can only move so fast, so the response stops being a scaled copy of a smaller one — and no transfer function contains that number because no transfer function can. How small is small signal puts a much smaller one under it, 7.3 millivolts, for the amplitude at which linearising an exponential is one per cent wrong, which is 28 per cent of the thermal voltage rather than a small fraction of it. The junction measured here is nominally at ground and the excursion on it is the input divided by the loop gain, so at low frequency it is microvolts and neither boundary is near; above crossover it is the whole of what the divider passes, and both are.

That is worth stating because it inverts the usual reading of the curve. The frequencies at which the junction is a poor ground are exactly the frequencies at which its excursion is largest, so the small-signal assumption underlying the measurement is weakest precisely where the measurement is most interesting.

What the summing node decides #

A node at ground for a thousandth of the amplifier’s gain-bandwidth decides three later results. The gain the loop closes against is where a capacitance at that node separates the signal gain from the noise gain. Where the trouble is at the input is the transimpedance case, where the detector’s capacitance is the largest thing at the node. The ideal amplifier, and where it stops being one is the finite product all of it comes from, and What is left at crossover is the loop gain the node’s impedance is one over.

The gate #

The high-frequency limit is asserted to be the two resistors in parallel, to two parts in a thousand. That is the claim that the amplifier has entirely stopped participating, and it is the one that would fail first if the model were wrong.

The direct-current value is asserted to be below a thousandth of it, which is the loop gain doing its job.

The slope is measured over a decade that follows the amplifier, not over a fixed one. The first version measured between one and ten kilohertz, which at 100 kHz of gain–bandwidth is already past the top of the rise and returned 0.83 decades per decade — a real flattening, measured in the wrong place and reported as a defect.

The two routes are asserted to agree to two per cent across nine decades, which is a loose bound for a reason: the return-ratio route uses the open-loop impedance as a constant, and near the ceiling that is no longer a good decomposition. Where it is a good decomposition the two agree to 0.045%.