The capacitor across the upper resistor

What gets through from the rail ended on the one input a regulator’s loop does not reject. The output is held at the reference times one plus the upper divider resistor over the lower, four here, so whatever the reference does the output does four times over, and no amount of loop gain changes that because it is the loop working rather than failing. The essay then named the standard repair: a capacitor across the divider, so that at high frequency the reference is passed at a gain of one instead of four.

It named the lower resistor. That is the wrong one, and measuring the three possibilities — no capacitor, a capacitor across the lower resistor, a capacitor across the upper — turns up a second thing the recommendation assumed: that four is the gain the reference sees above a kilohertz at all.

Four is the gain below a kilohertz #

With no capacitor anywhere the reference’s gain is four to three decimals at low frequency and then rises, to 5.681 at 7.72 kilohertz, before falling away. The rise is the loop’s own. A closed loop reproduces its input times the ideal gain times T/(1 + T), and near the crossover, where the loop gain’s magnitude is close to one and its phase is 46.5 degrees from the edge, 1 + T is smaller than T and the ratio exceeds one. A reference is amplified by 1.42 times more than the divider says, at the frequency where the loop is least in control.

This is the same denominator the two essays below this one on the ladder found in other places. A source below a frequency found the output impedance peaking at 1.95 ohms at ten kilohertz, and the rejection essay found the rail passing at +5.79 decibels there. The reference’s gain peaks within an octave of both, because all three are something divided by 1 + T. So the reference’s noise near seven or eight kilohertz arrives at the output at 5.7 times its size, not four, and the repair has more than the divider to undo.

Which resistor the capacitor goes across #

The fraction of the output fed back to the amplifier is the divider’s ratio, a quarter here, and the reference’s gain is its reciprocal. A capacitor in the divider changes that fraction at high frequency, and the two positions change it in opposite directions.

Across the upper resistor, from the output to the feedback node, the capacitor bypasses the resistance between them. At high frequency the feedback node follows the output directly, the fraction fed back rises towards one, and the reference’s gain falls towards one — the behaviour the rejection essay wanted. With a nanofarad there the gain never exceeds 4.000 and is down to 1.913 at ten kilohertz.

The capacitor has to be large enough to reach the band the peak is in. Its zero, where its reactance equals the upper resistor, is the frequency above which it takes over from that resistor, and a hundred picofarads puts the zero at 53 kilohertz — above the loop’s 9.73-kilohertz crossover, where the peak lives. At that value the capacitor lowers the gain at ten kilohertz a little, to 4.214, and leaves the peak standing at 5.15. A nanofarad’s zero is at 5.31 kilohertz, below the crossover, and the peak is gone. So the rule has a size in it as well as a position: across the upper resistor, with its zero below the crossover.

Across the lower resistor, from the feedback node to ground, the capacitor shorts the feedback node down. At high frequency the fraction fed back falls, so the gain the loop is trying to establish rises, and the reference is amplified more. With a nanofarad there the gain peaks at 11.226 and is 10.239 at ten kilohertz, more than twice what it was with no capacitor at all.

A divider with two ratios is the passive version of the same fact: a divider with capacitance in it has one ratio at low frequency, set by its resistors, and another at high frequency, set by its capacitors, and which way the ratio moves depends on which arm the capacitance is in. The regulator’s feedback divider is that divider, inside a loop that then amplifies by its reciprocal.

Ten times the capacitance makes the upper position better still — 1.163 at ten kilohertz, closer to one — and turns the lower position’s peak into 27.417 at 5.19 kilohertz. A regulator whose reference gain is designed to be four and which amplifies reference noise twenty-seven times at five kilohertz is a closed loop close to its own edge. The capacitor across the lower resistor removes feedback exactly in the band where the loop most needs it, and the phase margin of that arrangement is not drawn here; the peak is the measurement, and a peak of that height does not come from a comfortable margin.

The capacitor is in the loop, not only on the reference #

The capacitor across the upper resistor was introduced for the reference, and it is also in the only path by which the output reaches the amplifier. So it changes the loop gain, and a change to the loop gain is a change to everything the regulator does.

In the loop the capacitor makes a pole-zero pair. The zero is where the capacitor’s reactance equals the upper resistor, 1/2πR₁C, and the pole is where it equals the two resistors in parallel, 1/2π(R₁‖R₂)C; their ratio is (R₁ + R₂)/R₂, which is the divider’s four. Between them the fraction fed back rises by four times and its phase leads, and a lead placed near the crossover is what a phase margin wants.

The reactances sit where the arithmetic puts them. At the zero, 6.34 kilohertz, 836.5 picofarads is thirty kilohms, the upper resistor; at the pole, 25.4 kilohertz, it is the 7.5 kilohms of the two resistors in parallel; and at the crossover the search settled on, 16.9 kilohertz — above the pair’s geometric mean of 12.7 — it is 11.3 kilohms.

At 836.5 picofarads the zero is at 6.34 kilohertz and the pole at 25.4, and the margin is at its greatest, 90.929 degrees, at a crossover that has moved from 9.73 kilohertz to 16.9. A pole and a zero four apart can lead by at most arcsin(3/5), 36.87 degrees, at their geometric mean, which is 12.7 kilohertz here; the margin rose by 44.46 degrees. The difference is not the lead pair’s alone, and the essay does not separate where the remaining seven and a half degrees come from. What is measured is the total, found by searching the capacitance for the largest margin rather than by placing the pair where the textbook would.

Past the optimum the margin falls back, and slowly. At ten nanofarads it is 66.34 degrees; the same loop solved at other values gives 79.46 degrees at two nanofarads, 74.30 at three, 63.94 at thirty and 63.09 at a hundred. From the optimum to a hundred nanofarads the margin never falls below 63 degrees, which is more than the regulator had with no capacitor at all.

The impedance the load sees #

The output impedance is the open-loop impedance divided by 1 + T, and a source below a frequency found its peak at the crossover, where 1 + T is smallest. With no capacitor that peak is 1.95 ohms at ten kilohertz.

With the capacitor at its best value the peak at the crossover is gone. The impedance is 828 milliohms at ten kilohertz rather than 1.95 ohms, and its highest value, 841 milliohms, is now at 562 kilohertz, where the loop has nothing left to do with it and the output capacitor’s own series resistance decides. What does not change is the bottom of the curve: the floor is 0.430 milliohms and the impedance has doubled by 4.81 hertz with or without the capacitor, because the loop gain at low frequency is untouched by a capacitor whose reactance there is enormous.

Two requirements pulling one capacitor found the output capacitor’s series resistance caught between a stability requirement and a load-step requirement, and the window between them was set by the margin. A capacitor across the divider that adds forty degrees of margin moves the stability requirement, and so moves that window; how far is not measured here.

The rail #

The rejection essay’s table had one entry nobody would have predicted: +5.79 decibels at ten kilohertz, the regulator amplifying the ripple it was placed to remove. With a nanofarad across the upper resistor that entry is gone. The worst rejection anywhere is −1.50 decibels, at 518 kilohertz, where the rail is arriving through the pass device and the loop has stopped mattering, and the peak near the crossover has been flattened with the output impedance’s.

The rejection essay’s other result survives the change exactly: the rail-referred arrangement is 60.00 decibels better at every frequency, because the improvement is the pass device’s intrinsic gain and the capacitor is in neither of the paths that ratio compares.

The comb #

The ripple that arrives as a comb found the reservoir’s ripple reaching the output as 60.62 millivolts, its lines flattened by a rejection that worsens with frequency. A capacitor that removes the rejection’s peak near the crossover and improves it above a kilohertz acts on exactly the lines the one-line estimate left out.

At the value that maximises the margin the ripple falls from 60.62 millivolts to 52.33. The hundred-hertz line barely moves — 7.731 millivolts without the capacitor and 7.730 with it, because at a hundred hertz 836.5 picofarads is 1.90 megohms, sixty-three times the upper resistor — and what falls is the share above a kilohertz, from 10.4 per cent to 8.6.

Ten nanofarads takes the ripple to 35.40 millivolts, forty-two per cent less than with no capacitor, and the comb is drooping: its second line is 89 per cent of its first and its third 74, where without the capacitor they were 93 and 83. The lines above a kilohertz carry two per cent of what arrives. The one-line estimate is now short by 1.705 rather than 2.872, because the capacitor has made the regulator’s rejection less steep in exactly the band the comb occupies.

How much that is worth depends on the supply in front. The capacitor acts on the lines above a kilohertz, and the comb essay found those lines carrying 5.9 per cent of the regulated ripple behind 220 microfarads and 17.0 per cent behind 4700. So the same capacitor is worth more to a regulator behind a large reservoir than behind a small one — a comparison across reservoirs that this essay does not draw, and one the two figures together would make.

The amplifier’s own noise sees the same gain #

The reference is not the only thing inside the loop that the divider’s reciprocal multiplies. The error amplifier’s own input-referred noise enters at the same place the reference does, between the amplifier’s two inputs, and it reaches the output by the same path — so every gain drawn above for the reference is also the gain the amplifier’s noise sees. The feedback field calls that quantity the noise gain and measures it in an operational amplifier, where the gain the loop closes against puts a capacitor across a feedback resistor to cancel a zero that the input’s own capacitance created.

The regulator’s case differs from that one in a way that makes it simpler to read. Nothing here has to be cancelled: the divider has no input capacitance to fight, and the capacitor across the upper resistor creates its lead pair from nothing. So what it buys is visible on its own terms — a lower gain for the reference and for the amplifier’s noise above a kilohertz, and a flatter closed loop — and what the lower position does is the same mechanism run backwards, raising the gain that every noise source at the amplifier’s input sees. A regulator built with the capacitor on the wrong side has a noisier output from its own amplifier as well as from its reference.

Two optima, and which one binds #

The margin and the ripple want different capacitors. The margin is greatest at 836.5 picofarads; the ripple, and the reference’s gain at ten kilohertz with it, keep improving past that value — 1.913 at a nanofarad, 1.163 at ten.

On this regulator the trade is not a hard one, because the margin falls back slowly. At ten nanofarads it is 66.34 degrees, and it stays above 63 degrees out to a hundred. So the reading for this circuit is that the capacitor should be chosen for the reference’s noise and the output ripple, and the margin is comfortably met anywhere past the optimum. That is a statement about this regulator’s loop and not a rule: a loop with less margin to start from, or with another pole close to the crossover, would not fall back so gently.

The comparison with the lower position is the part worth carrying. The same capacitor, in a divider that looks the same on a schematic, gives a loop with ninety degrees of margin and a reference gain that never exceeds four, or a reference gain of twenty-seven at five kilohertz. The rejection essay called the choice of which rail a resistor returns to “two wires on a page, invisible in a block diagram”, and this is the same thing one component over.

What a single noise figure hides #

A regulator’s output noise is usually quoted as one number: a voltage integrated over a band such as ten hertz to a hundred kilohertz. The gains above say what that number is an integral of. With no capacitor the reference’s noise and the amplifier’s are multiplied by four at low frequency and by up to 5.681 near eight kilohertz, so a band that includes the crossover is weighted towards it. With a nanofarad across the lower resistor the weight near nine kilohertz is 11.226, and an integrated figure would be dominated by a few kilohertz either side of the loop’s crossover. With the same capacitor across the upper resistor the weight at ten kilohertz is 1.913.

A single integrated figure can be the same number for a regulator whose noise is spread across the band and for one whose noise is concentrated at its crossover, and only the second is changed by which resistor one capacitor is soldered across. That is the rejection essay’s complaint about a rejection figure quoted at one frequency, arriving at the noise specification from the other direction: a number that is an integral hides the spectrum it was integrated from.

What the start-up was said to cost #

The rejection essay attached a price to its recommendation: a slower start-up. Nothing measured here supports that for the capacitor across the upper resistor. A start-up is a large-signal event — a reference ramping, an amplifier slewing, a pass device leaving its limit — and the regulator here is its small-signal model, in which the capacitor raises the loop’s crossover from 9.73 kilohertz to 16.9 and so makes the loop faster rather than slower. Whether the capacitor slows a real start-up, by charging through the divider while the output rises, is a question this model cannot ask, and the essay leaves it open rather than repeat the claim.

What is not in the model #

The reference here has no noise of its own: the figures give the gain its noise would see, not the noise. The bandwidth noise sees and the floor a circuit has are the tools for turning such a gain into a noise voltage, and the reference’s own spectrum is what they would need. The capacitor is ideal, with no series resistance and no tolerance, and the amplifier’s input has no capacitance of its own at the feedback node — which in a real part sits in parallel with the lower resistor and so does, in a small way, exactly what the rejection essay recommended.

Still open: the noise, the window, and the step #

The reference’s noise, as a voltage. The gains measured here are what a reference’s noise is multiplied by, frequency by frequency. A reference with a flat density and a flicker corner, pushed through the three arrangements and integrated over a load’s bandwidth, would give an output noise voltage for each — and would say whether the peak near the crossover or the flat region below it dominates, which decides whether the capacitor is buying a factor or a decibel.

The series-resistance window with the capacitor in. The output capacitor’s series resistance had to be above 939 milliohms for 45 degrees of margin with no capacitor across the divider. With ninety degrees available, that floor should fall a long way, and re-solving the window with the capacitor present would say how much of the output capacitor’s specification the divider capacitor removes.

The load step. A higher crossover and a flattened impedance peak should shorten the recovery from a load step and reduce its overshoot, and two requirements pulling one capacitor already marches that step. Marching it again with the capacitor present would put a time on the improvement the impedance curve implies.