The window the resistors own

What the second path costs at the floor put three noise sources into a capacitively loaded stage — the amplifier’s own voltage noise and the Johnson noise of the two feedback resistors — integrated them over ten hertz to a hundred megahertz, and found the compensation costing nothing at all in the quantity it was suspected of spending. It then did something the rest of this collection ought to copy: it named, in its closing paragraph, the quantities that were missing rather than the ones that were there. The amplifier’s current noise. Its flicker corner. And the impedance the arrangement presents to a load that draws its own current.

The third of those was measured a rung later, by what the load sees looking back, and came out at ten ohms at direct current for the isolation-resistor arrangement and 1.2 milliohms for the two-path one. The other two have stood named and uncounted through four more rungs.

They are not a footnote. An amplifier has two noise generators and this ladder has been carrying one of them, which is a defensible model of a part with no input current and a description of no part that exists. The omission is invisible in the way the worst omissions are: the three-source integration converges, its totals are stable in the fourth digit, every assertion in it holds, and it answers a question about a circuit that has one fewer generator than the circuit on the bench. Nothing in the arithmetic reports a missing source. What reports it is putting the source in.

What is being solved, and the netlist that was already there #

A current generator sits between the amplifier’s inverting input and ground. The non-inverting input’s generator is there too and does nothing, because that input is grounded and a current into a short produces no voltage anywhere.

The transfer from that current to the load needs no new circuit. The input resistor’s own thermal source is a voltage behind a resistor from ground to the summing node; its Norton equivalent is a current of one over that resistor, into the summing node, with the resistor still in place. So the transfer the three-source integration already solved for, multiplied by the resistor, is the transfer a current generator needs — the same operator, the same factorisation, a source transformation rather than a second model. Every frame drawn below checks it at three frequencies against a netlist with the current source explicitly in it, and the two agree to the last bit of the arithmetic — which is what a claim about every netlist rather than about this one is owed.

That leaves four sources in the band, and a fifth when the load node has a real resistor on it: √(4kT/R) as a Norton current driving the impedance the node already presents, which is the quantity what the load sees looking back built and this rung reuses.

The reproduction, which comes before the departure #

A noise integration is exactly the kind of machinery that returns a plausible number for a mis-stated problem, so the first thing asked of it is a question whose answer is already recorded.

Set the current generator to zero and the four-source machinery must return the three-source one. At ten kilohms and twelve picofarads of compensation it returns 28.88 microvolts, against the 28.9 the rung that established it published, and puts 78.0 per cent of the noise power in the two resistors, against the 78 that rung recorded. Nothing is new; that is the point. The instrument reproduces the measurement it is about to disagree with, at the setting where the two must agree.

Now put 0.6 picoamps per root hertz in — an ordinary bipolar-input part, the same generator the floor a circuit has measured beside a four-nanovolt voltage generator. The total goes to 30.13 microvolts and the resistors’ share falls to 71.7 per cent. Four and a third per cent on the total, at the impedance this ladder happens to stand at.

That is a small number and it is the least interesting thing on this page, because the impedance this ladder happens to stand at is a choice nobody defended.

Ten kilohms arrived in the first rung of this argument as the resistor pair that makes an inverting unity gain, and it has been carried through eight rungs without ever being a variable. Every number the ladder has published — a margin of 30.1 degrees, a settling optimum at twelve picofarads, an output impedance of ten ohms, a floor of 28.9 microvolts — was computed at it, and none of the eight essays has a sentence saying what would happen if it were something else. The margin, measured below, moves by under three degrees when it is changed by four decades. The floor moves by a factor of sixty-nine.

Two powers of one quantity #

The feedback network is two ten-kilohm resistors. Scale them both and the closed-loop gain does not move; take the compensation capacitor down in the same proportion and the product Rf·Cf does not move either, so the frequency at which the fast path takes over from the slow one — the handover that the path that buys the error back spent its whole argument placing — is exactly where it was. The loop is the same loop at every point on that axis.

The three noise terms are not.

The amplifier’s voltage noise reaches the load multiplied by the noise gain and by nothing else, so it does not move at all: fitted over four decades of impedance it goes as the 0.005 power of it. Each resistor’s own thermal noise goes as √(4kTR), fitted at the 0.501 power. And the current generator becomes a voltage by flowing in the feedback resistor, fitted at the 0.997 power. A constant, a square root and a straight line — and a square root crosses a straight line twice.

The window is 2.91 kΩ to 85.9 kΩ, and its two edges are the roots of one quadratic whose three coefficients are read off the sweep: a variance that does not move with the impedance, one that grows as its square, and one that grows as the impedance itself. The share measured at each root is a half to within two parts in a hundred, which is the check that the three power laws above are the whole of the behaviour rather than a description of the middle of the range.

Below 2.91 kΩ the amplifier’s voltage noise dominates because the resistors have become quiet. Above 85.9 kΩ its current noise dominates because the resistors have not become noisy fast enough. In between, and only in between, the network is the thing that matters.

Two quantities in that picture are easy to confuse and they are not the same. The impedance at which the part’s share is least is 15.8 kΩ; the impedance at which the noise is least is the bottom of the axis, because the total falls monotonically all the way down. A share is a ratio and a floor is a voltage, and a design that minimises the first has minimised the fraction of its noise that is nobody’s fault while leaving the noise itself sixty-nine times what it could have been. The share is worth knowing because it says which component to change; it is not a thing to optimise.

The impedance at which the part costs least is not eₙ/iₙ #

The noise field’s own result for two generators against a source is that they sum to a minimum at R = eₙ/iₙ, which for this part is 6.67 kΩ, and that the minimum depends only on the product of the two.

Measured inside this loop the part’s share is least at 15.8 kΩ, which is 2.374 times 6.67 kΩ, and the factor decomposes exactly into two quantities that are separately measurable on the same circuit.

The first is the noise gain. The amplifier’s voltage generator arrives at the load multiplied by 1 + Rf/Rin, which here is 2.000, while its current generator arrives multiplied by the feedback resistor and by nothing else. An arrangement with a different closed-loop gain has a different factor here, so the optimum is not a property of the part at all.

The second is bandwidth, and it is the smaller factor and the less obvious one. The two generators do not see the same band. The voltage noise sees an equivalent noise bandwidth of 2.866 MHz and the current noise 2.035 MHz, because they arrive at the load through different transfers — one through the noise gain, which is flat until the loop gives up, and one through the closed-loop response, which is not. The square root of that ratio is 1.187, and 2.000 × 1.187 × 6.67 kΩ is 15.8 kΩ. The noise-bandwidth field’s own quantity turns up again in a circuit designed for something else, exactly as the bandwidth noise sees says it will, and it is worth nineteen per cent of the answer here.

The axis nothing in the loop can see #

Four decades of feedback impedance move the floor from 12.89 microvolts to 892.97 — a factor of sixty-nine — and move the phase margin from 64.72 degrees to 61.79. Under three degrees, across the whole axis, and most of that at the bottom end where the ten-ohm isolation resistor is no longer small against a hundred-ohm feedback resistor.

This matters more than the sixty-nine does, because it says which instrument will not answer the question. The closed-loop gain is exactly minus one at every point on that axis. The handover between the two feedback paths sits at 1.326 megahertz at every point, because the product that sets it was held. And the margin moves by under three degrees. So the load that gets inside the loop would report very nearly the same margin and the resistor that buys the margin back the same repair, whichever end of the axis the network sat at. A designer choosing resistor values from a stability measurement is choosing them from a quantity that is very nearly blind to the choice.

What decides the bottom of the axis, and it is not noise #

The floor falls monotonically as the impedance falls, so nothing in this measurement stops a designer at a hundred ohms and 12.89 microvolts. What stops them is a quantity two rungs below.

A hundred-ohm feedback resistor carries ten milliamps for every volt of output, and it carries them whatever the load is doing. The step too large to have an impedance found this stage’s linear description failing at 10.60 milliamps through the input pair’s own slew rate, and the current above which there is no impedance found a second limit that is nothing but design choice — a twenty-milliamp output stage — binding at a different load from the first. A hundred-ohm network spends half of that rating on its own feedback before the load has drawn anything.

So the impedance scale has a floor set by the output stage and a ceiling set by the current generator, and the quiet region between them is not centred on either. That is the useful form of this rung’s result: the axis is bounded below by something the noise integration cannot see and above by something no stability measurement can.

The two frames either side of the ordinary part are worth taking together, because they say the window belongs to the amplifier rather than to the arrangement.

What this decides, and what it does not #

The useful form of the result is a rule about which component answers a complaint about the floor, and it depends on where the network sits.

Below about three kilohms the answer is the part. Lowering the resistors further buys √R in the two terms that are falling and nothing at all in the one that is not, so the returns collapse: the last decade down, from a kilohm to a hundred ohms, takes the floor from 15.70 microvolts to 12.89 — eighteen per cent, for ten times the current in the feedback network. The only remaining move is a quieter voltage generator. Between three and eighty-six kilohms the answer is the network, and halving both resistors takes their contribution down by √2 without moving the handover frequency at all, since that depends on the product of the feedback resistor and the compensation capacitor and both can be traded. Above eighty-six kilohms the answer is the part again and it is a different generator: a field-effect input, drawn above at a tenth of the current noise and the same four nanovolts. A real one gives some of that back in voltage noise, which the comparison above does not contain — it changes one generator and holds the other — so what the figure shows is the ceiling of the trade rather than the trade.

What none of this decides is the compensation capacitor. The floor at every impedance here was computed with Rf·Cf held at the value the settling optimum set four rungs below, and that value is still the one to use: the noise argument moves the resistors and the capacitor together, and leaves the frequency between them alone.

The corner that is worth a hundredth of a per cent #

The other named quantity is the flicker corner, and the honest report is that in this stage’s own band it is worth nothing.

A corner of a hundred hertz on both of the amplifier’s generators multiplies the total, integrated from a tenth of a hertz to a hundred megahertz, by 1.00009. A corner of ten kilohertz — which no ordinary part has — multiplies it by 1.00927. The reason is arithmetic rather than circuitry: the variance a corner adds is the 1/f term’s own integral, eₙ²·$f_c$·$\ln(f_{hi}/f_{lo})$, which grows as the logarithm of the band, while the white term grows as the band itself. A stage whose loop reaches megahertz is integrating over seven decades of white noise and one logarithm of flicker.

That closed form is drawn against the netlist integration at every point rather than substituted for it, and the two share only the density. They agree to a part in 4074 up to a hundred kilohertz, and above that the closed form runs high — by eighteen per cent at a hundred megahertz — because it assumes flat transfers and the loop has stopped having them. The departure is not an error in either route; it is a measurement of where the assumption behind the shorter one fails, which is the same shape of result as every model has an edge collects.

What the corner does move is the blame rather than the number. In a band that stops at ten hertz the part carries 94 per cent of the power with the corner in and 24 without. A slow instrument reading this stage is measuring the amplifier; a fast one is measuring the resistors. Both are true and they are statements about different measurements of one circuit.

A resistive load, which every rung below set to zero #

The last idealisation this ladder has been carrying is that the load draws no current at direct current. Putting a real resistor there adds the fifth source and it is negligible — under 0.13 per cent of the power at every load drawn. What the resistor does is not its own noise.

It divides the noise against the ten-ohm isolation resistor, and whether it divides the signal with it is the whole difference between the two arrangements. Taken to the amplifier, the feedback regulates a node the load is not on, so the signal falls 16.7 per cent while the noise falls 30.1; taken to the load, the loop regulates the right node and the signal does not move at all while the noise falls twenty per cent. Signal-to-noise improves by 1.193 in the first arrangement and by 1.253 in the second.

A resistive load making a stage quieter for free is the sort of result that should be distrusted, and the reason it is not free is the same one as above: fifty ohms draws twenty milliamps for every volt at the load, which is a whole output stage’s rating spent on a resistor.

What is not here #

The band is 10 Hz to 100 MHz and every total on this page is quoted against it, because a noise number without its band is not a number. Below ten hertz this model is white and no part is; that is what the flicker figure exists to bound, and it bounds it by integrating from a tenth of a hertz instead.

The current generator is white too, and given the same corner as the voltage one. Real parts do not oblige: a bipolar input’s current noise has a corner of its own, usually higher, and a field-effect input’s rises with frequency rather than falling, because it is the gate’s own shot noise multiplied by a capacitance. Neither is in this netlist.

Nothing here contains the diode-like shot noise of the input bias current itself, which the floor a current sets prices against a resistor’s thermal noise and finds them equal when the direct voltage across the thing carrying the current is 2kT/q. The generator used here is a stated density rather than a current times a charge, so the two arguments meet only through the number 0.6 picoamps.

And the resistors are ideal. A hundred-kilohm feedback resistor has excess noise of its own that depends on how it was made and on the direct current through it, and it has a fraction of a picofarad across it that is a real part of the compensation.

The number worth carrying #

2.91 kΩ to 85.9 kΩ, and outside it in both directions the part owns the floor.

The habit that goes with it is about which measurement is entitled to a design choice. The impedance of a feedback network is invisible to the margin, invisible to the settling time, invisible to the closed-loop gain and invisible to the direct-current error — the four quantities eight rungs of this argument have measured — and it moves the noise by a factor of sixty-nine. A quantity that no instrument in the drawer responds to is not thereby a quantity that does not matter; it is a quantity whose instrument has not been picked up yet.