Circuits that do a job, and the range they do it over

The ripple that arrives as a comb

The rejection essay multiplied two numbers: 1.331 volts of reservoir ripple and the regulator's 36.0 decibels of rejection at a hundred hertz, for 21 millivolts at the output. The ripple is a sawtooth, a comb of lines at every multiple of a hundred hertz, and the rejection worsens at twenty decibels a decade — so the second line arrives at nearly the size of the first, and the third too. Summed with their phases, the output is 60.62 millivolts, 2.87 times the estimate, and the hundred-hertz line carries only 23.4 per cent of it. A reservoir twenty-one times larger cuts the rail's ripple 14.8 times and the regulated ripple 5.95.

Assumes: A source below a frequency · The direct voltage that is a sawtooth

What gets through from the rail measured how much of a wiggle on a regulator’s input reaches its output, found the answer falling by twenty decibels a decade of frequency from ten hertz to a kilohertz, and then put the result to work. The unregulated supply in front of it delivers 1.331 volts of ripple at a hundred hertz; the regulator rejects 36.0 decibels there; so 21 millivolts comes out.

That arithmetic treats the ripple as a sinusoid at a hundred hertz with a peak-to-peak of 1.331 volts. The ripple is not a sinusoid. The direct voltage that is a sawtooth drew it: a capacitor discharging almost linearly into its load for most of each half cycle and then recharged in a steep pulse at the top, which is a sawtooth, and a sawtooth is a comb of lines at every multiple of its own frequency. The question this essay asks is what the regulator does to all of them.

The ripple 1000 µF leaves, through the regulator: 60.62 mV rather than 21.11 mV. One settled cycle of the reservoir's output — 1.331 V peak to peak across 1000 µF — split into 1000 Fourier lines, each passed through the solved regulator's rail-to-output response, and summed. The output is 60.62 mV peak to peak. The ripple times the rejection at 100 Hz, -36.00 dB, gives 21.11 mV, which is 2.872 times too little: the rejection worsens at twenty decibels a decade, so each harmonic of the sawtooth arrives at nearly the size of the first. The 100 Hz line carries 23.4% of the output's mean square and the lines above 1 kHz 10.4%; the first three arrive at 7.731 mV at 100 Hz, 7.200 mV at 200 Hz, 6.388 mV at 300 Hz. The dashed curve is the 100 Hz line alone. — Fig. 1 One settled cycle of the reservoir’s output — 1.331 V peak to peak across 1000 µF — split into 1000 Fourier lines, each passed through the solved regulator’s rail-to-output response, and summed. The output is 60.62 mV peak to peak, where the ripple times the rejection at 100 Hz gives 21.11 mV: 2.872 times too little. The 100 Hz line carries 23.4% of the output’s mean square; the dashed curve is that line alone.

A sawtooth is a comb

1000 µF across a 100 Ω load, rectified from 17 V peak. The output sits at 15.69 V with 1.331 V of ripple, against the 1.569 V the expression I/2fC gives — 15.1% high, because the capacitor is being recharged for part of the cycle rather than discharging throughout it. The lower panel is why: the diode conducts for 28.8° of each half cycle and carries 2.10 A at the peak, which is 13.4 times the 157 mA the load draws. — Fig. 2 The rail in front of the regulator: a rectifier, 1000 µF and a 100 Ω load, marched with the diodes in the netlist. The capacitor discharges for most of each half cycle and is recharged in a pulse lasting a small fraction of it, so the output is a sawtooth with a steep edge rather than a sinusoid.

The rail’s shape is set by how long the diodes conduct. At a thousand microfarads each recharging pulse lasts 28.8 degrees of a half cycle, and the waveform the regulator sees is a slow ramp down followed by a fast climb back. A shape with a fast edge has harmonics that fall slowly, and measured by taking the settled cycle apart into its lines, the rail carries 487.6 millivolts of peak amplitude at a hundred hertz, 227.0 at two hundred and 134.2 at three hundred. The second line is 47 per cent of the first and the third 28 per cent.

Taking a cycle apart that way is exact rather than approximate. The settled output of the marched rectifier is two thousand samples over one twenty-millisecond mains cycle, and a discrete Fourier transform of two thousand samples has a thousand lines that rebuild every one of those samples to rounding, and they do so before anything is built from them. What the lines cannot contain is anything faster than the samples — the record is ten microseconds a sample, so nothing above fifty kilohertz — and whether the rail has content up there that matters to the output is not measured here.

A rejection that rises to meet the lines

Rejection is not a property of the loop. The same regulator with the same loop gain and the same 46.5° of phase margin, drawn twice. With the amplifier's output referred to ground the rail reaches the output at -67.3 dB at the bottom and 5.79 dB at 10.0 kHz — where it is amplified. Referred to the rail instead, every point is 60.00 dB lower, which is 20 log(gm·ro) for the pass device and not a design choice. — Fig. 3 How much of a wiggle on the rail reaches the output, against frequency, for the regulator drawn twice — its error amplifier referred to ground and referred to the rail. The grounded arrangement lets through −67.3 dB at 1 Hz, rises twenty decibels a decade through the band the ripple occupies, and amplifies the rail near 10 kHz.

Each line meets a different rejection, and the rejection is getting worse at exactly the rate that would cancel a line amplitude falling as one over its harmonic number. The regulator passes 1.59 per cent of the rail at a hundred hertz, 3.17 per cent at two hundred and 4.76 per cent at three hundred — a line twice the frequency is let through twice as freely — because the loop gain that does the rejecting is falling at twenty decibels a decade through the whole of that range.

So the lines that arrive are nearly equal. The regulated output carries 7.731 millivolts of peak amplitude at a hundred hertz, 7.200 at two hundred and 6.388 at three hundred: the second line is 93 per cent of the first, where on the rail it was 47. The regulator has flattened the comb. A sinusoid would have come through as a sinusoid; a sawtooth comes through as something whose lines are close to equal for the first several harmonics, and whose shape is closer to the recharging pulse than to the ramp.

The flattening has a short account, and it says how flat to expect. A loop whose gain falls at twenty decibels a decade rejects in proportion to frequency, so it passes the nth line n times as freely as the first. A sawtooth made of straight ramps has lines that fall as one over n, so the two cancel and every line would arrive at the same size. The rail here is not quite that sawtooth — its recharging pulse has a width, 28.8 degrees of each half cycle, and a pulse with a width has lines that fall faster than one over n once the harmonic is high enough to resolve it — so the comb arrives flattened but drooping: 93 per cent at the second line, 83 at the third. The narrower the pulse, the longer the comb stays flat, which is the whole of the next two figures.

That is why the one-line estimate is short. It keeps the rail’s peak-to-peak — a number set by all the lines together — and multiplies it by the rejection of the first line only, which is the best rejection any line gets. Every other line was let through more freely, and the sum of lines that arrive nearly equal is much larger than the first one scaled.

The sum, with its phases

The output is not the sum of the line amplitudes. Each line arrives with its own phase, shifted by the regulator’s response at that frequency as well as by where it sat in the sawtooth, and the peak-to-peak of the sum is decided by how those phases line up. Synthesised from all thousand lines, the output is 60.62 millivolts peak to peak. The hundred-hertz line alone — the dashed curve — would be 15.46.

Measured as power rather than as a peak, the hundred-hertz line carries 23.4 per cent of the output’s mean square, and the lines above a kilohertz 10.4 per cent. The line the estimate was built on is a quarter of the ripple the load receives, and a tenth of it lives above a kilohertz, which is a frequency no rectifier specification mentions.

Which of the two measures matters depends on what the ripple is disturbing. The peak-to-peak is what a converter sampling the rail at an arbitrary instant can see, and what a comparator threshold or a logic level has to stay clear of; there the estimate is short by 2.872. The root-mean-square is what a noise budget adds, and what a filter downstream integrates: the output carries 11.30 millivolts of it, against the 5.467 millivolts the hundred-hertz line alone would carry, so the estimate is short by 2.07 in that currency. Both corrections are large. They are not the same size, because the lines line up at the ramp’s steep edge and a peak is where alignment counts.

The synthesis is superposition used in the one place it is exact. Two solves that add found every voltage in a linear network the sum of the per-source solves to the last bit, and the rail’s lines are separate sources for exactly that purpose; the regulator here is the linear small-signal network the rejection essay solved, so its output is the sum of what each line produces. What is not additive, as that essay found, is power — which is why the shares above are computed from the lines that arrive and not from the lines that left.

A second route, and what the two disagreed about

A synthesis from lines is one route. The other is to march the regulator’s netlist forward in time, driven by the rail’s own settled waveform, until it settles too, and read its output. The two share the netlist and nothing else.

At two thousand steps a cycle the march gives 61.03 millivolts, 0.67 per cent above the synthesis. That is too large to be rounding and too small to be a mistake, and it has a precise cause. The trapezoidal rule the march uses does not respond to a sinusoid at its true frequency: it responds at a warped one, tan(πfh)/πh for a step h, which is slightly higher and is most visible near a crossover, where the response changes fastest. Re-running the synthesis with every line’s frequency warped that way reproduces the march to seven thousandths of a nanovolt — so the march and the synthesis agree exactly about the network, and differ only in which frequencies they evaluated it at.

The march converges on the synthesis as the step shortens: 61.00 millivolts at four thousand steps a cycle, 60.91 at eight thousand and 60.89 at sixteen thousand. It converges slowly, because the lines near fifty kilohertz are the ones the warping moves most and they are close to the frequency at which the regulator stops rejecting at all. So the synthesis, which evaluates the network at each line’s true frequency, is the number to quote, and the march is what confirms it is the right network.

A smaller reservoir

The ripple 220 µF leaves, through the regulator: 130.7 mV rather than 73.05 mV. One settled cycle of the reservoir's output — 4.607 V peak to peak across 220 µF — split into 1000 Fourier lines, each passed through the solved regulator's rail-to-output response, and summed. The output is 130.7 mV peak to peak. The ripple times the rejection at 100 Hz, -36.00 dB, gives 73.05 mV, which is 1.789 times too little: the rejection worsens at twenty decibels a decade, so each harmonic of the sawtooth arrives at nearly the size of the first. The 100 Hz line carries 47.2% of the output's mean square and the lines above 1 kHz 5.9%; the first three arrive at 29.49 mV at 100 Hz, 22.00 mV at 200 Hz, 13.25 mV at 300 Hz. The dashed curve is the 100 Hz line alone. — Fig. 4 The same regulator behind a 220 µF reservoir. The rail ripple is 4.607 V peak to peak and the output 130.7 mV, against 73.05 mV from the ripple times the rejection at 100 Hz — 1.789 times. The 100 Hz line carries 47.2% of the output and the lines above 1 kHz 5.9%.

A fifth of the capacitance lets the rail sag further between pulses and makes each pulse longer, 54.2 degrees of the half cycle. The rail is closer to a sinusoid: its second line is a smaller fraction of its first, and the estimate built on the first line is correspondingly less wrong. It is still short by a factor of 1.789, and the hundred-hertz line is still less than half of what arrives.

A larger one

The ripple 4700 µF leaves, through the regulator: 21.98 mV rather than 4.945 mV. One settled cycle of the reservoir's output — 311.8 mV peak to peak across 4700 µF — split into 1000 Fourier lines, each passed through the solved regulator's rail-to-output response, and summed. The output is 21.98 mV peak to peak. The ripple times the rejection at 100 Hz, -36.00 dB, gives 4.945 mV, which is 4.444 times too little: the rejection worsens at twenty decibels a decade, so each harmonic of the sawtooth arrives at nearly the size of the first. The 100 Hz line carries 12.4% of the output's mean square and the lines above 1 kHz 17.0%; the first three arrive at 1.723 mV at 100 Hz, 1.689 mV at 200 Hz, 1.635 mV at 300 Hz. The dashed curve is the 100 Hz line alone. — Fig. 5 Behind 4700 µF. The rail ripple is 311.8 mV peak to peak and the output 21.98 mV, against 4.945 mV from the one-line estimate — 4.444 times. The 100 Hz line carries 12.4% of the output and the lines above 1 kHz 17.0%.

A larger reservoir does what it is bought for to the rail and less than that to the output. The rail’s ripple falls to 311.8 millivolts, but the diodes now conduct for 17.3 degrees and the recharge is a spike, so the rail’s lines fall more slowly still. The regulated output is 21.98 millivolts, 4.444 times the estimate, and more of it now lives above a kilohertz than at a hundred hertz: 17.0 per cent against 12.4.

Across the rectifier essays’ capacitors

Fig. 6 For five reservoirs, the regulated ripple synthesised from every line divided by the one-line estimate (upper), and the 100 Hz line’s share of the output (lower). The ratio rises from 1.789 at 220 µF through 2.256, 2.872 and 3.655 to 4.444 at 4700 µF; the share falls from 47.2% through 33.3, 23.4 and 16.5 to 12.4%. Across the sweep the rail ripple falls 14.77 times and the regulated ripple 5.95 times.

Read as a design curve, this is the figure that matters. Twenty-one times the reservoir capacitance buys 14.77 times less ripple on the rail, which is what the ripple expression suggests and what a designer sizing the reservoir expects. It buys 5.95 times less ripple at the regulator’s output, which is where the ripple was going to matter. The rest was spent sharpening the sawtooth, and a sharper sawtooth is exactly the waveform the regulator passes worst.

The share at a hundred hertz also says which lever a designer should reach for. Behind 220 microfarads nearly half of the output is the first line, and the first line is set by the loop gain at a hundred hertz — so more loop gain at low frequency buys most of the improvement there. Behind 4700 microfarads an eighth is the first line and a sixth lives above a kilohertz, set by how far the loop’s bandwidth extends and by what happens near its crossover. The same regulator is limited by two different properties of its loop according to the reservoir in front of it, and a single rejection figure at one frequency cannot say which.

And the capacitance is not free in another currency. The current that sizes the transformer found that a larger reservoir shortens the conduction and raises the root-mean-square current in the winding, so the copper dissipates more for the same load. The capacitor that does less than expected for the regulated output does more than expected to the transformer.

Sixty decibels better, by the same factor

The ripple 1000 µF leaves, through the regulator: 60.56 µV rather than 21.09 µV. One settled cycle of the reservoir's output — 1.331 V peak to peak across 1000 µF — split into 1000 Fourier lines, each passed through the solved regulator's rail-to-output response with the amplifier referred to the rail, and summed. The output is 60.56 µV peak to peak. The ripple times the rejection at 100 Hz, -96.00 dB, gives 21.09 µV, which is 2.872 times too little: the rejection worsens at twenty decibels a decade, so each harmonic of the sawtooth arrives at nearly the size of the first. The 100 Hz line carries 23.4% of the output's mean square and the lines above 1 kHz 10.4%; the first three arrive at 7.724 µV at 100 Hz, 7.193 µV at 200 Hz, 6.382 µV at 300 Hz. The dashed curve is the 100 Hz line alone. — Fig. 7 The same 1000 µF rail through the regulator with its error amplifier referred to the rail. The output is 60.56 µV peak to peak against 21.09 µV from the one-line estimate at −96.00 dB — 2.872 times, the grounded arrangement’s factor exactly — and the 100 Hz line carries 23.4% of it.

The rejection essay’s second arrangement, with the amplifier’s output developed against the rail, rejects sixty decibels better at every frequency. Because the improvement is the same at every frequency, it multiplies every line by the same thousand, and the comb arrives with the same shape: 60.56 microvolts, 2.872 times the one-line estimate, with the same 23.4 per cent at a hundred hertz. So the corrected figure for that arrangement is about sixty microvolts rather than twenty-one, and the correction is the same factor of 2.87 whichever arrangement is built.

At the current the reservoir was sized for

The ripple 1000 µF leaves, through the regulator: 63.53 mV rather than 21.11 mV. One settled cycle of the reservoir's output — 1.331 V peak to peak across 1000 µF — split into 1000 Fourier lines, each passed through the solved regulator's rail-to-output response with a 31.9 Ω load, and summed. The output is 63.53 mV peak to peak. The ripple times the rejection at 100 Hz, -35.99 dB, gives 21.11 mV, which is 3.010 times too little: the rejection worsens at twenty decibels a decade, so each harmonic of the sawtooth arrives at nearly the size of the first. The 100 Hz line carries 23.3% of the output's mean square and the lines above 1 kHz 10.8%; the first three arrive at 7.732 mV at 100 Hz, 7.201 mV at 200 Hz, 6.389 mV at 300 Hz. The dashed curve is the 100 Hz line alone. — Fig. 8 The regulator loaded with 31.9 Ω, which draws the 157 mA the reservoir’s own 100 Ω load drew. The output is 63.53 mV peak to peak against 21.11 mV from the one-line estimate — 3.010 times — with 23.3% at 100 Hz.

The regulator in the rejection essay drives five ohms, and the reservoir in front of it was marched into a hundred; the two were solved separately and joined only by the ripple. Loading the regulator with 31.9 ohms makes it draw the 157 milliamps the reservoir was designed for, and the output rises to 63.53 millivolts. The load changes the regulator’s output impedance and so the rejection a little at every line, and it moves the answer by five per cent. It does not change the finding: the one-line estimate is short by a factor of three.

A real supply joins the two more tightly than that. The reservoir’s ripple depends on the current drawn from it, and the current a linear regulator draws is its load current, so a regulator in front of a varying load presents the reservoir with a varying load and the ripple changes with it. The synthesis here holds both loads fixed, and that coupling is not in it.

What the number is for

Sixty millivolts on a five-volt rail is 1.2 per cent, which is nothing to a logic supply and everything to an analogue reference. A sixteen-bit converter spanning five volts has a step of 76.3 microvolts, so sixty millivolts of ripple on its reference is about 795 counts. The floor a converter sets measured the floors underneath a converter’s resolution — the quantiser’s and the source’s thermal noise — and a rail contributing hundreds of counts sits far above both.

The error in the one-line estimate is also the kind of error a picosecond, read as bits found in a jitter specification: a quantity that is a function of frequency, quoted at one frequency, and then used as though it described all of them. A rejection figure at a hundred and twenty hertz is honest about a hundred and twenty hertz. Applied to a waveform with lines at every multiple of it, it describes a quarter of the answer.

The comb also explains what a source below a frequency and two requirements pulling one capacitor would each make of this. The rejection and the output impedance share the same 1 + T in their denominators, so the lines that the rejection lets through most freely are the lines at which the output impedance is highest. A regulator’s ripple and its response to a load step fail in the same band, and the band is where the reservoir’s comb still has amplitude.

What the synthesis leaves out

The regulator is its small-signal model, solved at each line’s frequency. The rail’s direct level is not in it, and neither is the pass device’s headroom: at the bottom of each ramp the rail is lowest, and if the regulator ran short of dropout voltage there the model would be describing a circuit that was not regulating. The two figures were built at different direct voltages and joined by their ripple alone, which is legitimate for the small-signal question and says nothing about the large-signal one.

The lines stop at fifty kilohertz, set by the sample spacing of the marched rail, and the rail’s recharging edge has content above that which is not included. The regulator’s rejection has stopped improving by then and is close to unity, so that content is passed nearly whole; how large it is, and so how much it adds to the sixty millivolts, is not measured here.

Still open: the reference, a switcher, and the edge

The capacitor across the upper divider resistor. The rejection essay ended on the one input the loop does not reject — the reference, reproduced at the output times the divider’s four — and recommended a capacitor across the lower divider resistor to bring that gain down at high frequency. The capacitor across the upper resistor measures that recommendation, finds it the wrong resistor, and finds the same capacitor changing the loop’s margin, its output impedance and the comb this essay synthesised.

A switching pre-regulator’s comb. A switching supply in front of a linear regulator puts its ripple at a hundred kilohertz, where the rejection essay’s table shows the regulator letting the rail through almost whole, and its ripple is a triangle or a spike train with lines of its own. The same synthesis pointed at that rail would say how much of a switcher’s ripple a post-regulator actually removes, and the table suggests the answer is very little.

The recharging edge itself. The lines above fifty kilohertz are where the rejection has nothing left and where the rail’s recharging pulse has its fastest content. Marching the rectifier finely enough to resolve that edge, with the diodes’ own recovery in it, would say whether the ripple a load sees is dominated by the comb measured here or by the spike at the top of each ramp.