What gets through from the rail

The supply two essays back delivers 1.33 volts of sawtooth at a hundred hertz. The regulator in front of the load is supposed to remove it. This essay asks how much of it comes out, gets an answer that depends strongly on frequency, and then finds that the answer barely depends on the feedback loop at all.

The measurement #

The netlist is the previous essays’ regulator with its reference set to zero and its rail driven with one volt at each frequency. The output voltage is then the fraction that gets through, and twenty times its logarithm is the rejection in decibels.

Two features of that column need saying out loud.

It falls by twenty decibels a decade from ten hertz to a kilohertz, which is the loop gain running out at the same rate. That is expected and it is the same slope the output impedance rose at in the first essay of this rung — both quantities are an open-loop number divided by 1 + T, and both therefore track the loop.

At ten kilohertz it is positive. The regulator delivers 1.9 times as much ripple as arrives. That is the crossover peak again — the loop gain magnitude is 0.95 there with 46.5° of margin, so $1+T$ has magnitude less than one and dividing by it is multiplying. A hundred millivolts of switching spur at ten kilohertz on the input rail comes out as 195 millivolts.

And above thirty kilohertz the rejection settles at about −1.5 dB, which is not rejection at all: it is the rail arriving through the pass device with a divider’s worth of loss and nothing else.

Why one number is not a specification #

Datasheets quote rejection as a single figure, usually at 120 hertz, because that is twice the mains frequency in the market the convention came from and therefore where a rectifier’s ripple lands.

The table above says what a single figure omits. Between 120 Hz and 10 kHz this regulator’s rejection changes by 42 decibels and changes sign; between 10 kHz and 1 MHz it changes by another 7 and comes back. A part specified as “−35 dB at 120 Hz” is being described at one point of a curve that has a peak above unity two decades further up, and the peak is where the load it is powering does its switching.

This is the same complaint a picosecond, read as bits makes about an aperture figure quoted without an input frequency — ten picoseconds holds sixteen bits to 199 kHz and twelve bits to 3.18 MHz, falling at exactly twenty decibels a decade, so the number alone states no resolution at all — and the same one the probe is part of the circuit makes about an instrument’s bandwidth quoted without a source impedance, where a hundred and fifteen picofarads on a two-kilohm source is one per cent wrong at 6.8 kHz with nothing inaccurate anywhere. The pattern is worth naming because it recurs: a number that is a function of something is not a specification until the something is stated, and the convention of quoting it at one standard value survives because the standard value was once where the disturbance was.

The three cases differ in one way that decides how much trouble each causes. An aperture figure and a probe bandwidth are both quoted at the good end of their curve and get worse monotonically, so a reader who knows the slope can extrapolate. A rejection figure quoted at 120 hertz is on a curve that gets worse, changes sign, and then improves again — so extrapolation from the quoted point is not merely optimistic but wrong about the direction, and the peak lands two decades above the quoted frequency, which is where a switching load puts its current.

Where the hundredth-of-a-volt gets in #

The interesting question is not why the rejection falls but why it is only −67 dB at the bottom, where the loop gain is 11 563. Dividing anything by eleven thousand is 81 decibels, so something is getting through that the loop is not dividing.

The route is the pass device. It is a transconductance whose current is set by the voltage between the rail and its own control terminal. Hold the control terminal still against ground and move the rail, and the device’s control voltage moves by the whole of the rail movement — so it modulates its current by $g_m$ amperes per volt of rail, straight into the output node, with no reference to what the loop wants.

That is a feedforward path in parallel with the loop, and the loop can only correct what it can see at its own bandwidth. At direct current the feedforward is $g_m$ times the output impedance — one siemens into 4.97 Ω, which is a gain of about 5 — and the loop divides it by 11 563, giving $4.3\times10^{-4}$ or −67.3 dB. Exactly what the table says.

So the answer is not “the loop is not good enough”. The answer is that a rejection measurement is a race between a forward path and a loop, and the forward path here is enormous.

Move one node #

If the pass device’s control voltage is what matters, the repair is to give it a control voltage that does not move with the rail. Develop the error amplifier’s output against the rail rather than against ground — put its load resistor and its compensation capacitor from its output up to the rail instead of down to ground — and the gate follows the rail exactly, so the device’s own control voltage is unmoved and it modulates nothing.

The same netlist, one node moved:

Sixty decibels, at every frequency in the table, to a thousandth of a decibel. The difference is 60.009 dB at all eight frequencies measured and the gate holds that the spread across six decades is under half a decibel.

And the loop is untouched. The loop gain is the same function of frequency, the crossover is at the same 9.7 kHz, and the phase margin is 46.47° in both cases — the gate checks that the two margins agree to a millionth of a degree, because the whole claim is that this is not a stability trade.

What the sixty decibels is #

It is the pass device’s own intrinsic gain.

$$20\log_{10}(g_m r_o) = 20\log_{10}(1 \times 1000) = 60.0\ \mathrm{dB}$$

Measured at four different devices:

The reason is that with the gate referred to the rail the feedforward path is no longer the transconductance; it is the device’s own output resistance, which lets the rail through as a divider against the load. The ratio of the two paths is $g_m$ against $1/r_o$, and that is $g_m r_o$.

Which makes the improvement a property of the device, not of the design. A pass transistor with an intrinsic gain of 60 dB gives 60 dB of extra rejection and one with 40 dB gives 40, whatever the loop does, and the only way to buy more is a longer channel or a cascode — both of which cost dropout voltage, which is the parameter a low-dropout regulator exists to minimise.

The cascode the dropout forbids #

The paragraph above disposes of a cascode in half a sentence, and the semiconductors field has measured both halves of what it would buy and cost, so the dismissal can be priced rather than asserted.

The device that never sees the swing puts a second transistor between the first and the load and finds that the expression everybody quotes for what that buys has no ceiling in it and is wrong: the output resistance is not $g_m r_o^2$, which would be 248 megohms there, but $\beta r_o$, measured at 11.5 megohms — because the upper device’s base draws current and shunts the very node the improvement was supposed to appear at. Carried into this essay’s arithmetic that is the difference between a rejection improvement of about 108 decibels and one of about 81, and the smaller of the two is still twenty decibels better than the plain device gives.

What it costs is stated there in the currency this essay cares about: two volts of a five-volt supply. A regulator sold on a dropout of a few hundred millivolts cannot spend that, which is why the arrangement measured here is the one the market builds even though the mechanism is understood. The same trade appears one rung along in the source that holds to the supply, where cascoding a mirror takes the output resistance from 82 kΩ to 7.39 MΩ and raises the floor by 0.71 volts — and where forty-three per cent of the resistance that should be there turns out to be missing, in the reference branch rather than the output one. Both essays are the same lesson pointed at this circuit: the number a cascode buys is smaller than the expression says, the price is a headroom the specification is written in, and the two have to be compared at the values in front of the designer rather than in the limit.

That leaves the longer channel, which buys $r_o$ without buying headroom and costs area and gate capacitance instead — and gate capacitance is the pass device’s contribution to the loop’s own crossover, so a device chosen for rejection is a device that moves the frequency at which the rejection starts failing. The two mechanisms this essay separated so carefully are coupled again by the choice of transistor, which is the honest place to leave it.

The three things that make this essay’s point #

Rejection is not a loop property. Two circuits with identical loop gain, identical crossover and identical phase margin differ by sixty decibels. Anyone reasoning about rejection from the loop gain alone would conclude the two are the same.

The difference is a device parameter. Not a compensation choice, not a bandwidth, not a capacitor: $g_m r_o$, which is set when the transistor is drawn on the die.

The schematic does not show it. Both circuits have an error amplifier, a pass device, a divider and a reference. The distinction is which rail one resistor and one capacitor return to — two wires on a page, invisible in a block diagram, and worth a thousandfold in the specification the part is sold on.

That last is one of this collection’s standing themes and this is the sharpest instance of it so far.

The input nobody rejects #

There is a third path into the output and the loop does not attenuate it at all: the reference.

The output is held at $V_{ref}(1 + R_1/R_2)$, so whatever the reference does the output does, times four. That is not a leak the feedback can close — it is the feedback working, faithfully reproducing its input. A millivolt of noise on the reference is four millivolts on the output, and no amount of loop gain changes it.

Which reverses the usual reasoning about where to spend effort. Rejection of the rail improves with loop gain and with the pass device; rejection of the reference does not exist. So a regulator’s output noise at low frequencies is its reference’s noise multiplied by the divider ratio, and the standard repair is not a better loop but a capacitor across the divider — and across the upper resistor, from the output to the feedback node, which raises the fraction of the output fed back at high frequency and so brings the reference’s gain down towards one. Across the lower resistor the same capacitor does the opposite and raises it. The capacitor across the upper resistor measures both, and finds that the gain of four is not the gain above a kilohertz in any case: the loop peaks the reference’s gain to 5.68 near its crossover before any capacitor is added.

The collection has measured what the floor under all of this is.

What the improvement costs #

If referring the amplifier to the rail is worth sixty decibels and costs nothing in the loop, the question is why any part is built the other way.

The answer is dropout. An amplifier whose output is developed against the rail has to swing its output close to the rail to turn the pass device off and some way below it to turn it on, and the room it needs comes out of the difference between the rail and the output — which is the quantity a low-dropout regulator exists to make small. The grounded arrangement has the whole rail to swing in; the rail-referred one has the dropout voltage.

It also costs common-mode range at the amplifier’s own inputs, since those still sit near the output while its supply now sits near the rail, and it makes the amplifier’s bias currents functions of the rail voltage, which reintroduces a small part of what was removed.

None of that is measured here — the model’s amplifier is a transconductance with no supply rails at all, which is exactly the idealisation the semiconductors field spends its time bounding. What is measured is the mechanism and its size, which is what decides whether the engineering is worth doing.

What arrives, and whether it matters #

Putting the numbers together with the supply two essays back:

The unregulated supply delivers 1.331 V of ripple at 100 Hz. The grounded-reference regulator rejects 36.0 dB there, which would put 21 mV at the output, and the rail-referred one rejects 96.0 dB, which would put 21 µV there. Both figures treat the ripple as a sinusoid, and it is a sawtooth whose higher lines meet a rejection twice and three times as poor. The ripple that arrives as a comb sums every line with its phase and finds 60.6 mV and 60.6 µV — 2.87 times the one-line figures, in both arrangements.

Sixty millivolts on a five-volt rail is 1.2%, which for a logic supply is nothing and for the reference of a sixteen-bit converter spanning five volts is about 795 counts. The floor a converter sets makes the same point from the other side: measured against the Johnson noise of a kilohm source in a hundred kilohertz of bandwidth, the quantiser is the limit only up to 18.80 bits and the resistor is the limit above it, so past that crossing every further bit buys a more precise measurement of thermal noise. A rail contributing sixty millivolts sits far above both floors, and no amount of resolution in the converter reaches it.

That the 1.331 volts arrives at a hundred hertz rather than somewhere else is the one piece of luck in the chain. The direct voltage that is a sawtooth measures where it comes from and why its frequency is fixed by the mains rather than chosen: a full-wave rectifier delivers its ripple at twice the line frequency whatever the reservoir is, and every design choice in that essay moves the ripple’s amplitude and the diode’s crest factor while leaving its frequency exactly where it was. A switching supply in the same place puts its ripple at a hundred kilohertz, where the table at the top of this essay says the loop has nothing left, so the rejection curve is a strong argument for one kind of supply in front of a linear regulator and no argument at all for the other.

And the ripple is not the whole of what the regulator is asked to hold still. A source below a frequency measures the same output node driven from the load side and finds 0.43 milliohms at direct current climbing to 1.95 ohms at ten kilohertz, and two requirements pulling one capacitor finds the component that decides the shape of that climb to be under two conflicting requirements at once. The peak in the rejection curve at ten kilohertz and the peak in the output impedance at the same frequency are the same $1+T$ in the same denominator, so a design change that moves one moves the other, and the crossover frequency is the single number both are functions of.

The gate #

Referring the drive to the rail improves rejection by exactly the pass device’s intrinsic gain, to within 0.05 dB, over four devices spanning two decades of the product.

And by the same amount at every frequency, with less than half a decibel of spread across six decades — so it is a change in one path’s strength and not a change in a frequency response.

While the loop gain and the phase margin are identical, to a millionth of a degree, which is what makes the first two claims interesting rather than a restatement of a bandwidth.

And the grounded arrangement amplifies the rail somewhere in the sweep, +5.79 dB at 10 kHz, rather than merely failing to reject it — which is the one entry in the table that a reader would not have predicted from the phrase “rejection falls with frequency”.