Where the trouble is at the input

A photodiode produces a current, and a current is not a thing anybody can read. The standard way to turn it into a voltage is one resistor and one amplifier: the diode drives the inverting input, a feedback resistor carries the current, and the output is the current times that resistor. A megohm turns a microamp into a volt.

It is the simplest arrangement in this field and it is the one most likely to oscillate, and the reason is a component nobody chose. A photodiode has a junction capacitance, thirty picofarads for an ordinary small one and hundreds for a large one, and it is not a parasitic to be minimised — capacitance is proportional to junction area, and area is what the diode was bought for. The bigger the signal, the worse the stability.

The noise gain, which is where the field’s own machinery applies #

The gain the loop closes against is not the gain the signal gets, and this field measured that two rungs back. Here they could hardly be more different.

The signal gain is the transimpedance: a current in, a voltage out, and the resistor decides it. The noise gain — the gain from the amplifier’s own input to its output — is set by the feedback fraction, which here is a divider between the feedback resistor and the input capacitance. At direct current the capacitance is an open circuit and the fraction is one; above $1/2\pi R_f(C_d + C_f)$ the capacitance is taking the divider over and the fraction falls, so the noise gain rises, at twenty decibels a decade.

The amplifier’s own gain falls at twenty decibels a decade. A rising curve and a falling curve close at forty, and forty decibels a decade of closure is a phase margin of nothing at all.

That is the same argument this field makes about a capacitive load on the output, arriving at the other terminal, and it is why an arrangement with no gain in it in the ordinary sense — one resistor, unity feedback fraction at direct current — is unstable.

The cure is a small capacitor across the feedback resistor. It puts a pole in the noise gain at $1/2\pi R_fC_f$, flattening it above there, and if that pole is placed near the closure the two curves meet at twenty decibels a decade again.

The two expressions, and how conservative they are #

The classical choice makes the noise-gain pole coincide with the closure and gives

$$C_f = \sqrt{\frac{C_d}{2\pi R_f\,\mathrm{GBW}}}, \qquad \mathrm{BW} = \sqrt{\frac{\mathrm{GBW}}{2\pi R_f C_d}}$$

Both are drawn here against a measurement rather than used as definitions. At each diode capacitance the feedback capacitor is bisected on the solved loop until the phase margin is exactly 45°, and the bandwidth is bisected on the solved magnitude of the transimpedance itself.

The expression asks for more capacitance than 45° needs, by a factor between 0.81 and 0.83 at every point, and correspondingly promises less bandwidth than the stage has, by a factor of about 1.34.

That is worth more than either exact agreement or disagreement would be. A fixed factor across two and a half decades says the expression has the right shape and a conservative constant — it is placing the pole at the closure rather than at the margin, which is a slightly different requirement — so a designer using it gets a stage that works and leaves a third of the bandwidth on the table.

The square-root law itself holds exactly. Fitted over the whole sweep the bandwidth goes as the −0.50 power of the capacitance: a diode of four times the area costs half the bandwidth, not three quarters of it.

Uncompensated, and the shape of the failure #

Leaving the feedback capacitor out is worth a measurement rather than a warning, because the failure is not oscillation and that is what makes it dangerous.

At a tenth of a picofarad — which is stray, not a component — the margin is 9.2° and the transimpedance peaks 15.9 dB above its direct-current value at 220 kHz. The stage does not oscillate. It works, its bandwidth measures 339 kHz, which is better than the compensated 295, and its step response rings for a dozen cycles.

A designer measuring the frequency response and reading off a −3 dB point gets a better number than the compensated stage gives. The defect is in a quantity nobody plots.

What a photodiode is, in a netlist #

Everything above treats the diode as a current source with a capacitance across it, and it is worth saying where that model stops, because two of its boundaries are design quantities rather than fine print.

The capacitance depends on the reverse bias. A junction’s depletion width grows as the square root of the voltage across it, so its capacitance falls as the inverse square root — biasing a photodiode at five volts instead of zero roughly halves its capacitance and, by the square-root law above, buys about forty per cent more bandwidth for nothing but a supply. What it costs is dark current, which rises with bias and is a signal-like error rather than a noise one.

There is a shunt resistance across it too, of hundreds of megohms, and at zero bias it is what converts the amplifier’s own input offset voltage into an input current. It is left out here because a megohm of feedback against a hundred megohms of shunt is a per cent, and because including it would make the noise gain’s low-frequency end a divider rather than unity — which is a real effect and belongs to the essay about offsets rather than to this one.

And the current source is only a current source while the diode is reverse biased or at zero volts. The virtual earth is what keeps it there: an amplifier whose loop has given up — saturated, slewing, or oscillating — lets the diode’s own junction voltage rise, at which point it stops being a current source and starts being a diode. That is a boundary this collection is built to state and it has an amplitude: the photocurrent at which the summing node departs from zero by a thermal voltage.

Sixteen decibels that nothing plots #

The uncompensated case is worth its own measurement rather than a warning, because of how it fails.

At a tenth of a picofarad of feedback capacitance — stray, not a component — the margin is 9.2° and the transimpedance peaks 15.9 dB above its direct-current value at 220 kHz. Nothing oscillates. The stage has a bandwidth of 339 kHz, which is higher than the compensated stage’s 295, and a signal-to-noise ratio that is worse for the same reason.

A designer who measures the frequency response and reads off the −3 dB point gets a better number from the broken circuit. The peak is at a frequency the specification does not mention, the ringing is in a step response nobody took, and the noise penalty is in an integral nobody computed. Three quantities have to be looked at before the defect is visible, and the one everybody looks at points the wrong way.

That is why the compensation here is bisected on the phase margin rather than chosen to maximise a bandwidth. Maximising the bandwidth of this arrangement is a procedure that converges on the broken circuit.

The noise, and the part that is thirty times quieter #

Now the quantity that decides whether the arrangement is any good, and the answer that nobody guesses.

There are two noise sources. The feedback resistor has its own Johnson noise, $\sqrt{4kTR_f}$, which at a megohm is 126.6 nV/√Hz. The amplifier has its own input-referred voltage noise, 4 nV/√Hz for an ordinary part. The resistor is more than thirty times noisier.

Each is solved from the netlist separately — one unit source in the circuit at a time, the other turned off — and the transfer from it to the output is integrated over frequency. Neither is a bandwidth-times-density estimate.

At an ordinary diode the amplifier is two thirds of the noise power and at a large one it is nineteen twentieths — the part whose density is thirty times lower.

The reason is the noise gain again, and it is the same curve that caused the stability trouble. The amplifier’s own noise is multiplied by a factor that rises with frequency, so its contribution grows towards the top of the band and the total is dominated by the highest frequencies the stage passes. The resistor’s noise is not: it goes through the same low-pass the signal does, flat and then falling.

One curve decides both the stability and the noise, which is why a transimpedance stage is a single design problem rather than two, and why “use a quieter amplifier” and “use a smaller resistor” are advice that points in opposite directions.

What the resistor buys, and what it costs #

The feedback resistor is the one free parameter left, and every quantity here depends on it.

The signal is proportional to $R_f$. The resistor’s noise is proportional to $\sqrt{R_f}$. So the signal-to-noise ratio against the resistor’s own noise improves as $\sqrt{R_f}$, and a larger resistor is unambiguously better on that count — which is the standard argument for using the largest one the application allows.

Against the amplifier’s noise it does the opposite. A larger $R_f$ moves the noise-gain zero down in frequency, so the rising region starts earlier and there is more of it inside the band. And the bandwidth falls as $1/\sqrt{R_f}$ from the same expression that gives the compensation.

So the resistor trades signal-to-noise against bandwidth at a rate of $\sqrt{R_f}$ in one and $1/\sqrt{R_f}$ in the other, which is the cleanest statement of what the arrangement is: the product of the transimpedance and the square of the bandwidth is roughly a constant of the amplifier and the diode, and everything a designer can choose moves along that curve rather than off it.

The same shape, three fields apart #

This arrangement is the third place in the collection where a capacitance at an amplifier’s input decides something that gets attributed to the amplifier, and putting the three beside each other says what the pattern is.

In the semiconductor field, the frequency a device sets for itself draws a common-emitter stage’s two picofarads of base-collector capacitance into its input multiplied by one plus the gain, and the bandwidth that follows is nothing to do with the device’s own transition frequency. The Miller approximation predicts 643 kHz where the solved network gives 504 — twenty-two per cent optimistic, with no room in it for the second pole or for the right-half-plane zero the network also has. The cure is a cascode, and the device that never sees the swing prices it: the bandwidth really is fourteen times better, and it costs two volts of a five-volt supply.

In the instruments field, the corner the instrument has no part in puts a kilohm of source imbalance and ten picofarads at each input of a 95 dB instrumentation amplifier and finds the rejection falling twenty decibels a decade above 290 hertz, reaching 84 dB at a kilohertz on a part that is still doing 95. What converts common mode into differential is the difference of two time constants, the instrument’s own rejection is untouched and irrelevant, and the cure is a capacitor on the quiet input.

Here, the source’s capacitance and the feedback resistor set a noise gain that rises, and the cure is a pole in it — a capacitor across the feedback resistor, chosen to put that pole where the noise gain would otherwise have kept climbing, which is the one repair on this list that costs bandwidth directly rather than costing accuracy or headroom. The gain the loop closes against is where that quantity is isolated on the simplest possible circuit: nine picofarads at the summing junction of a unity-gain inverter — less than a scope probe — takes the phase margin from ninety degrees to forty-five, while the measured gain at a kilohertz has not moved by three parts in a million.

Three different symptoms — a bandwidth, a rejection, a margin — from one arrangement: a capacitance at a node whose impedance the designer was thinking of as something else. In each case the specification that gets blamed belongs to the amplifier and the quantity that decides the answer belongs to what is connected to it. And in each case the capacitance is small enough to be invisible on a schematic: two picofarads, ten picofarads, and a diode’s junction, none of which anyone draws.

The fourth member of the family is at the output rather than the input and is this field’s own: the load that gets inside the loop, where fifty ohms of output resistance and the load’s capacitance make a pole inside the loop, and forty-five degrees arrives at 905 picofarads — a metre of coaxial cable. The cures there — an isolation resistor, a second feedback path — have no analogue here, because a photodiode cannot be isolated from the summing junction by a resistor without the resistor’s own noise arriving in its place.

That asymmetry is worth one more sentence, because it is the reason this arrangement is harder than the other three rather than merely different. In all three of the others the offending capacitance is a parasitic: a device’s junction, a wiring stray, a probe. Something can be done about a parasitic, even if it is expensive. Here the capacitance is the specification — it is proportional to the diode’s area, and area is the sensitivity being bought — so the only repairs available are ones that accept it, which is why every design decision on this page is about the feedback network and none of them is about the source.

What is checked #

Every point on the compensation sweep is asserted to be bisected to within half a degree of 45° on the solved loop, so the curve is a measurement of a stated condition rather than a plot of an expression. The classical expression is then asserted to be conservative and consistently so — between 0.6 and 0.95 of the measured capacitance, and varying by less than fifteen per cent across the sweep — which is two assertions where one would have been ambiguous: a wrong expression and a conservative one both disagree, and only the second disagrees by a constant factor.

The square-root law is fitted over the whole sweep rather than asserted at one point, and required to land between −0.42 and −0.58.

The noise argument carries the surprise as an assertion rather than as prose: the resistor’s density is asserted to be more than twenty times the amplifier’s, and the amplifier is asserted to be more than nine tenths of the noise power at the largest diode. That the dominant source changes hands inside the range is asserted too, so the claim is about the diode rather than about a particular pair of parts.

What is not modelled: the amplifier’s own current noise, which flows in the feedback resistor and is a third source this essay does not have — the floor a circuit has is where that generator is measured, at 0.6 pA/√Hz beside a 4 nV/√Hz voltage generator, with the ratio of the two setting a source resistance of 6.67 kΩ at which the sum is least; the diode’s shot noise, which is proportional to the photocurrent and is therefore signal-dependent, and which the floor a current sets prices against the resistor’s — the two are equal when the direct voltage across the thing carrying the current is $2kT/q$, 50.0 millivolts at 290 kelvin, whatever the resistance and whatever the current; the resistor’s own parasitic capacitance, which for a megohm is a fraction of a picofarad and is a real part of $C_f$ — often most of it; and the input’s second capacitance, the amplifier’s own, which is in the netlist at three picofarads and is lumped with the diode’s rather than being a separate design quantity.