The resistance a supply's regulation counts

The resistance that belongs in the winding moved a series resistance between two places in a rectifier — inside the reservoir capacitor and inside each half of the transformer’s secondary — and found that the peak current did not care where it was, while the ripple and the mean output did. It measured all of that at one load. It closed by asking what the two placements do to load regulation, and it offered a guess: that each resistance should count in proportion to the share of the cycle it carries the load’s current in, roughly a sixth for the winding, which conducts only near the crests, and five sixths for the capacitor, which supplies the load for the rest.

Load regulation is the number a designer actually needs from an unregulated supply. The mean output at one load decides whether the circuit works at that load; the slope of the mean output against load decides whether it works across the range the load spans, and the slope of the trough decides whether a linear regulator after the supply keeps its headroom when the load is heaviest. This essay marches the same rectifier at seven loads, in three configurations, and measures those slopes directly.

The guess turns out to be wrong in both places, and wrong for a reason that leaves a better rule behind it.

A supply whose resistance is not a constant #

The circuit is the one the direct voltage that is a sawtooth began with: a centre-tapped secondary of 17 V peak at 50 Hz, two diodes and a thousand microfarads, marched with the diodes’ own exponential in the netlist until the waveform is settled. The load is now swept, from 200 Ω down to 25 Ω, which takes the load current from 80 mA to 573 mA.

With perfect parts the mean output falls from 15.99 V to 14.33 V across that range, an average slope of 3.38 Ω. The expression most references give for the mean output of a capacitor-input rectifier is the peak less half the ripple, and with the textbook ripple of $I/(2fC)$ that is

$$V_{\text{mean}} \approx V_{\text{pk}} - \frac{I}{4fC},$$

a straight line of slope $1/(4fC)$ = 5 Ω here. The measured slope is two-thirds of that, and it is not constant. Its tangent — the output resistance a following circuit sees at its operating point — was measured by marching ten per cent either side of each load.

It is 4.17 Ω at 80 mA, 3.83 at 157, 3.39 at 304 and 2.80 at 575 mA. The textbook constant is above all of them and the gap widens with load, for the reason the first essay on this circuit found the ripple expression high: it assumes the capacitor discharges for the whole of each half cycle, when in fact it is being recharged for part of it. That part is the conduction angle, and it grows with the load — 22.4° of each half cycle at 80 mA, 29.0 at 157, 38.3 at 304 and 51.3° at 575. Each extra milliampere of load is drawn from a capacitor that spends less of its cycle discharging, so each costs less ripple than the one before, and the supply regulates better the harder it is worked.

Two other slopes are in the same figure. The peak’s is small — 0.15 Ω at light load and 0.05 at heavy — which is the diodes’ forward voltage rising with their peak current, and nothing else. And the trough’s is about twice the mean’s at every load: 8.57 Ω against 4.17, 6.05 against 2.80, a ratio between 2.06 and 2.16. The trough sits a whole ripple below the peak and the mean about half a ripple, so nearly all of the regulation is the ripple growing. An unregulated supply’s output resistance is, to within a tenth of an ohm, the slope of its ripple.

The guess, and what the march says instead #

Now put a resistance somewhere. Rather than read the added slope off the whole curve, which mixes in the curvature above, the useful measurement is the weight: how many ohms of output resistance each ohm of series resistance adds, at a given load, with a small resistance so that the answer describes the placement rather than the size.

The winding’s weight is 0.94 at 80 mA, 0.98 at 157, 1.06 at 304 and 1.18 at 575 mA. The capacitor’s is −0.14, −0.17, −0.20 and −0.25. The share-of-the-cycle guess predicted 0.12 rising to 0.29 for the winding and 0.88 falling to 0.72 for the capacitor. Both measured curves are on the far side of their predictions: the winding counts at about its whole value, not a sixth of it, and the capacitor counts at slightly less than nothing.

The guess was reasonable, and it is worth seeing exactly where it goes wrong. It treats each resistance as dropping a voltage only while it carries the load, and weights that drop by the time it is carried. But the output’s mean is a mean over voltage, and the mean voltage across a linear resistance is its resistance times the mean current through it, whatever the current’s waveform and whenever it flows.

The winding, between its two halves, carries every coulomb the load takes. Over a settled cycle the capacitor’s charge returns to where it started, so the diodes deliver exactly the load’s mean current, and all of it passes through the winding resistance. It flows in narrow pulses near the crests rather than steadily, but the mean is the load current, so the winding’s mean drop is $r\,I$ to first order — a weight of one. What the pulses do change is how the drop reaches the output, which is why the measured weight drifts from 0.94 to 1.18 as the conduction angle widens.

The capacitor’s resistance carries the capacitor’s current, and the capacitor’s mean current in a settled state is exactly zero. So the mean voltage across it is exactly zero too, and the mean output is the mean capacitor voltage and nothing else:

$$\overline{v_{\text{out}}} = \overline{v_C} + R_{\text{ESR}}\,\overline{i_C} = \overline{v_C}.$$

The capacitor’s resistance cannot touch the mean output through its own drop at all. It can only change it by changing how the capacitor charges, and that route is the one the resistance that lowers the ripple found: a small resistance slows the charging pulse and lengthens the conduction. That does cost some mean output — the hero figure’s capacitor curve sits below the perfect parts’ at light load, as the essay on the winding found at a single load — but the cost shrinks as the load grows and the conduction widens, and the curve closes on the perfect one at 573 mA. A cost that shrinks with load is a slope removed from the supply. So the weight is small and negative: a slight improvement in regulation, of about a fifth of the resistance’s own value, bought from the same widening of the conduction that gave the ripple its interior minimum.

So the rule is not time-weighting. A resistance counts in the mean’s regulation by the mean current it carries per ampere of load: one for anything in the charging path, zero for anything that carries only alternating current, and a residue from the waveform in both.

What the regulator after it sees #

The mean is what a meter reads. A linear regulator after the supply sees something else: its dropout is violated at the lowest instant of the input, which is the ripple’s trough, at the heaviest load. The same measurement made on the trough gives a different pair of weights.

For the trough the winding weighs 0.08 at 80 mA, 0.17 at 157, 0.34 at 304 and 0.55 at 575 mA, and the capacitor weighs −0.36, −0.35, −0.39 and −0.57. Both are lower than for the mean, and the winding’s has dropped most.

The reason is that the trough is the peak minus the ripple, and a winding resistance moves the two in the same direction. It lowers the peak, by 0.76 to 1.20 of itself across the loads — close to its mean weight, since the peak is where it drops its voltage — and it also lowers the ripple, which is the finding the resistance that belongs in the winding made at one load, and a smaller ripple lifts the trough back towards the peak. At light load the two nearly cancel and the winding costs the trough a twelfth of itself. At heavy load the peak’s loss wins, and it costs half.

The capacitor’s resistance lowers the trough’s slope at every load, by a third to more than half of itself. That is the ripple minimum again: while the resistance is small, it spreads the charging pulse and lowers the ripple by more than its own step adds, and the trough is where the ripple’s reduction is worth the most.

This is the half of the original guess that survives in spirit. The winding’s resistance does matter less to the trough than to the mean, for a reason connected to its conduction — though the measured weight is not the conduction share, being below it at light load and above it at heavy. And where the guess expected the capacitor to dominate, it helps.

The size of the resistance #

Every weight above was measured with a tenth of an ohm, small enough that it describes the placement and not the resistance. A real winding is larger: a transformer for a few hundred milliamps at this voltage has a secondary resistance of the order of an ohm. The weights are not constant with size, and the last figure measures them from 30 mΩ to an ohm at the 100 Ω load.

The winding’s weight on the mean is 0.99 at 30 mΩ, 0.98 at 100 mΩ, 1.07 at 300 and 1.77 at an ohm; on the trough it is 0.17, 0.17, 0.29 and 1.16. So a small winding resistance is simply in series with the output, which is what the mean-current argument says. A large one counts for more than itself: at an ohm the drop it takes from the charging pulse is about a volt at the crest, the pulse it carries changes shape as the load changes, and the first-order argument, which assumed the waveform fixed, stops being the whole story. It remains the right starting point: across a thirty-fold range of resistance the measured weight stays between 0.98 and 1.77, and nowhere near the sixth the time-share guess gave it.

The capacitor’s weight on the mean is −0.16 at 30 mΩ, −0.17 at 100 and −0.07 at 300, and it has turned positive, +0.64, by an ohm. On the trough the change is steeper: from −0.66 to +0.80. That crossing is the same crossing the resistance that lowers the ripple found in the ripple itself, where the resistance lowered it to a minimum at 17 mΩ and raised it beyond. A capacitor resistance helps a supply regulate while it is small enough to help the ripple, and past that it becomes an ordinary series element in the part of the waveform that decides the trough.

The hero figure shows the two placements at an ohm across the whole load range. The average slope with perfect parts is 3.38 Ω; with an ohm in each half-winding it is 4.49, and with the same ohm in the capacitor it is 3.50. At three tenths of an ohm the two are 3.64 and 3.32, and the capacitor’s curve has a shallower slope than perfect parts, even though it sits a little below them.

Where a designer’s milliohms go #

Read as purchasing advice, the weights reorder two familiar choices.

A low-resistance capacitor is usually bought for regulation, on the reasoning that its resistance is in series with the output. For the mean it is not, by the exact identity above, and for small values the march finds it slightly helpful rather than harmful: replacing a 100 mΩ part with a 30 mΩ one makes the trough’s slope worse, by about 0.015 Ω. What the lower resistance genuinely buys is elsewhere — the current that sizes the transformer found the capacitor’s heating set by its ripple current, and two capacitors that are one found a pair dividing that heat. A capacitor’s resistance is a thermal specification in this circuit, not a regulation one, until it approaches an ohm.

Heavier copper in the secondary is usually bought to keep the transformer cool, and it is also the regulation purchase. Every milliohm taken out of the winding comes off the mean’s output resistance at about a milliohm, and off the trough’s at between a twelfth and a half of one depending on load. At the heaviest load in this sweep, where a following regulator’s headroom is decided, half an ohm less winding resistance takes at least a quarter of an ohm off the trough’s slope — at least, because the size figure shows the weight rising as the resistance grows.

And the conduction angle does part of the work for free. Because the supply’s own slope falls from 4.17 Ω to 2.80 as the load grows, a design checked for its dropout at light load and then extrapolated with a constant slope will overestimate the trough’s fall at heavy load. That is the safe direction to be wrong in, but it is not the textbook’s error: it is a second, independent conservatism stacked on top of the first.

Why the textbook constant survives #

The expression $V_{\text{pk}} - I/(4fC)$ overstates the slope by a fifth at light load and three-quarters at heavy, says nothing about where a resistance goes, and it remains the expression people use, for a defensible reason. It is conservative. Every correction measured here — the lengthening conduction, the small capacitor resistance — makes the real supply regulate better than it predicts, and a design that meets the regulation it states will meet the real one.

What it gets dangerously wrong is only the thing it leaves out. It has no term for the winding, and the winding counts at its whole value in the mean and at up to its whole value in the trough. At the 100 Ω load an ohm in each half-winding adds 1.77 Ω to an output resistance of 3.83 — nearly half again — and no estimate built from the capacitance alone will show it.

It also treats the supply as having one output resistance, a straight line with a slope, which is how a linear source model describes any source — the model the source that is not a source found has an edge where it stops describing the object at all. The march says there are three — one for the mean, one for the trough and one for the peak — and that they differ by a factor of two and more, because the ripple grows with load and they sit at different places on it. A Thévenin description is exact only for a linear source, and a rectifier and reservoir is linear in none of the quantities that matter. The regulator that follows is designed against the trough’s resistance, and if it is designed against the mean’s instead it is designed against half the true slope.

How the numbers were obtained #

Each point is a settled march: twelve cycles at two thousand steps a cycle, with the diodes’ exponential in the netlist, a current-law residual below 10⁻⁸ at every step, and the mean, trough and peak read from the last cycle. An output resistance is the fall in each between marches at 1.1 and 1/1.1 times the load resistance, divided by the rise in the load current between them. A weight is the output resistance with the series resistance less the output resistance without it, divided by the resistance. The conduction share is the fraction of each half cycle for which the diode current exceeds a hundredth of its own peak — the same definition the earlier essays on this circuit used — averaged over the two marches.

What it leaves out #

The source’s own regulation. The transformer’s primary and the mains behind it have impedance too, and on a real bench the peak is not a fixed 17 V but falls with the load before any of this begins. That enters like the winding resistance, at about its own value, referred through the turns ratio.

The leakage inductance. A real secondary’s reactance limits the charging pulse without dissipating, and it lengthens the conduction in a way no resistance does. The inductance that limits and lifts found it dividing the switch-on pulse and lifting the output rather than lowering it; its effect on the slope has not been measured.

The temperature. An electrolytic’s resistance rises several-fold when cold, which by the size figure moves it from helping the trough to hurting it.

Still open: the source behind the winding, the leakage and the regulator’s margin #

The mains impedance, through the turns ratio. The primary’s resistance and the supply’s own impedance appear at the secondary divided by the square of the turns ratio. Whether they enter the regulation at weight one, as the winding does, or differently because they are in series with the magnetising current as well, is a march with the primary in the netlist.

Leakage inductance and the slope. An inductance carries the load’s mean current like the winding resistance, but its mean voltage is zero, like the capacitor’s resistance — it belongs, by this essay’s rule, with the capacitor. Whether a leakage inductance therefore improves regulation, and by how much against the winding resistance that always comes with it, is the natural test of the rule on an element that splits it.

The headroom budget. A linear regulator needs its input trough above its output plus its dropout at the heaviest load and the lowest mains. The trough slope measured here, what gets through from the rail for the ripple the regulator then rejects, and the mains tolerance together decide how much reservoir a design needs, and the smallest capacitor that meets all three is a question none of these essays has yet put in one place.