An inverting unity gain, and the 100 pF that only the loop can see
Drawn above at its default parameters, which is almost never how an essay calls it. A placement states the numbers that essay is arguing about, so the figure a reader meets is about that argument rather than about the generator — 98% of the placements on this site pass one, and the phase that raised that number from 12% found eight captions describing a figure the page was not showing.
At those defaults the edge it states is 45° at 9.00 pF — the right-hand slot
of the caption strip, which on this site is never used for anything else, and which is read
back out of the drawing above rather than out of the code that wrote it.
It belongs to Feedback, and the margin, which is to say a change to it is a
change to lib/figures/feedback.js.
It takes a slider on capacitance at the summing junction (picofarads) with 7 settings,
and every one of them has passed the same assertions as the frame above — a figure whose
circuit stops doing what its caption says at any setting stops the build.
Called by 14 essays
which is the blast radius of changing it
The gain the loop closes against
An inverting amplifier with two equal resistors has a gain of one and a loop that closes against two, so it has half the bandwidth of a follower built from the same part — 4.99 megahertz against ten. 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 buffer that is not a buffer
An emitter follower is reached for when something has to be driven without being loaded: unity gain in, high impedance seen, low impedance presented. The last of those is a number with a range, and the range is narrow. At a kilohm of source the emitter presents 19.08 ohms at low frequency and 67 ohms at 29 megahertz, because the current gain that made it small is falling — and the peak is worst in the middle of the slider, so it cannot be avoided by making the source stiffer or softer.
The load that gets inside the loop
Hanging a capacitor on an amplifier's output changes nothing a reader can find in any expression for its gain, and takes the phase margin of a unity-gain inverter from ninety degrees to thirty. The mechanism is fifty ohms of output resistance that no data sheet page puts next to the stability page: the load works against it, the pole that results is in the forward path, and forty-five degrees arrives at 905 picofarads — which is a metre of coaxial cable.
The resistor that buys the margin back
Two point two nanofarads takes a unity-gain inverter's phase margin from ninety degrees to thirty. Ten ohms between the amplifier and the load restores forty-five, twenty-three restores sixty, and it works for a reason that reads as a cheat: the feedback is taken from the wrong side of the resistor, so its pole is outside the loop. Take the feedback from the load instead — which is what anyone controlling the load would do — and the same resistor makes every value worse. What it costs is that the loop no longer regulates the load's node at all: ten ohms is one per cent of error into a kilohm, at direct current, uncorrected.
What the second path costs at the floor
The arrangement that repaired a capacitively loaded amplifier was suspected of paying for itself in noise, because that is how compensations usually pay. It does not: it has no peak in its noise gain at all, and the total at the load falls from 54.9 microvolts to 28.9 as the capacitor is added. What it costs is settling, and the capacitor that is quietest is eighteen times the capacitor that settles fastest — a trade the phase margin says nothing about, because the margin is comfortable at both.
Where the trouble is at the input
Every other arrangement in this field has its difficulty at the output — a capacitive load, an isolation resistor, a load that draws current. A photodiode amplifier has it at the input, and the capacitance causing it is not a parasitic: it is the diode's junction, which is the price of its area, and area is what a photodiode is bought for. The feedback resistor's own noise is 127 nV/√Hz against the amplifier's 4, and the amplifier is still ninety-five per cent of the noise at a large diode.
The factor the expression leaves out
The classical compensation for a photodiode amplifier is quoted both as the forty-five degree choice and as the maximally flat one, and it is neither: it leaves 1.18 dB of peaking and 52.4° of margin. Flat is at exactly √2 times it — fitted at 1.4186 against 1.4142, at every detector from 3 pF to 1 nF. And the third quantity the capacitor is supposed to trade, the noise, does not move at all inside the signal band: three compensations spanning a factor of fourteen give 8.37 against 8.36 µV in a 4.36 kHz measurement and 230 against 65 µV over the whole plane.
The error a bigger resistor cannot help
Every other quantity in a photodiode amplifier improves with a larger feedback resistor: the signal grows as R and the resistor's own noise as √R, so the ratio goes as √R. The bias current does not behave like either — it flows in the same resistor the signal does, so its contribution is the ratio of the two currents with the resistance cancelled, 0.1 per cent at 25 °C and the same 0.1 per cent at a megohm and at a gigohm. The offset behaves the other way, growing as one plus the resistor over the diode's own leakage, and at 85 °C the two are 6.40 and 0.74 per cent against 0.10 and 0.11 at room temperature.
The tee that charges for its own compensation
A feedback tee makes a large transimpedance out of small resistors, and R₁(1 + R₂/R₃) + R₂ is the whole of what is usually said about it. The expression has no term for the noise gain, and the tap sets that to 1 + R₁/R₃ — nine, where a single feedback resistor of any value gives exactly one, because at direct current the source is a capacitor. One quantity then does everything: the signal and R₁'s own noise are multiplied together so the tee buys no signal-to-noise ratio at all, and the phase margin RISES from 1.8° to 15.9°.
The node that does not care how many
Put thirty-two inputs on a summing junction instead of one and its ceiling falls from 909 ohms to 31.2 — the input resistors in parallel with the feedback resistor, twenty-nine times stiffer. Every other part of the curve is unchanged to five per cent, because the open-loop impedance and the loop gain both scale as one over the count and cancel. So the stiffer node buys nothing: the per-channel leakage saturates instead of reaching unity, and the leakage into every other channel together tends to exactly one, whatever the count.
The compensation that costs nothing
A photodiode amplifier can be compensated with a capacitor across its feedback resistor, or with a feedback tee, whose raised noise gain also raises the phase margin. At one transimpedance, 500 kΩ on 33 pF of detector, each design's margin is drawn against its output noise over its own bandwidth. The capacitor dominates at every margin the tee can reach: at 50° it carries 85 µV where the tee carries 347, in the same 400 kHz. The two keep the same band to within 1.3 per cent at equal margin, so the tee's noise buys no bandwidth. The conjecture that the answer would turn on which noise dominates does not survive either. With the amplifier's voltage noise cut tenfold, the tee is still 5.6 times worse.
The drift a tee multiplies
A feedback tee reaches a large transimpedance with small resistors, and on a photodiode its price has been counted in noise: at direct current its noise gain is its own factor, where a single resistor's is one. The same noise gain multiplies the amplifier's offset. Calibrated at 25 °C and read at −40, a 10 MΩ amplifier with 1 µV/K of offset drift is 6.56 pA wrong through one resistor and 65.1 pA through a tee of factor ten — ten times, exactly, while the input bias current's error is identical through both. So the tee's drift penalty is hidden wherever the bias current dominates, which for a field-effect input doubling every 10 K is the hot end, and fully exposed at the cold end. The worst error over −40 to 85 °C comes within a factor of two of the single resistor's only above 10 MΩ for a tee of ten, and above 100 MΩ for a tee of a hundred.
The gain added to buy a margin
An amplifier sold as stable only at gains of ten — 100 MHz of gain–bandwidth, its second pole at 20 MHz — has 35.0° of margin and 4.4 dB of peaking as a gain of −1, because that gain closes its loop against two. A 1.25 kΩ resistor from the summing junction to ground raises the noise gain to ten without touching the signal gain, which stays one to a part in ten thousand, and the margin becomes 65.6°: still 14.1 MHz of bandwidth, 2.19 times what a unity-stable part from the same die gives at its natural gain. The price is everything the amplifier contributes, multiplied by ten instead of two — 64.0 nV/√Hz at the output against 20.5 — and a capacitor in series with the resistor buys back the low-frequency half of it and none of the wideband total.
The input that costs every other input
A summing amplifier gives each of its inputs a gain of −1 and closes its loop against one more than the number of inputs, so every input added divides every channel's bandwidth and multiplies the amplifier's own noise — 6.43 MHz with one input, 1.17 MHz with eight, 603 kHz with sixteen, on a 10 MHz part. Above 5.4 inputs the amplifier's 5 nV/√Hz, multiplied by n + 1, outgrows the resistors' Johnson noise, which rises only as its square root. A muted channel whose source is grounded still costs: a sixteen-input mixer is 11.5 dB louder with two channels in use than one whose unused resistors are disconnected. A channel of gain 0.1 beside one of gain 10 is 13.7 dB noisier than alone. A tree of smaller summers doubles the bandwidth and does not make the sum any quieter.