The inductance that flattens the supply

The resistance a supply’s regulation counts measured an unregulated supply’s output resistance — the slope of its mean output against its load — and found the right way to count each resistance in it. A winding resistance carries all of the load’s mean current and adds to the output resistance at about its own value; a capacitor’s series resistance carries none of the mean current and adds almost nothing. The rule is the mean current each element carries, not the share of the cycle it conducts in.

It ended by naming the element that splits the rule. A transformer’s leakage inductance sits in the winding, so it carries the load’s whole mean current, like the winding resistance. But an inductance has no mean voltage across it in a steady state, like the capacitor’s resistance, which drops no mean voltage for the mean current it does not carry. By the rule’s own logic the inductance should belong with the capacitor and count for almost nothing. The inductance that limits, and lifts had already found leakage inductance lifting a rectifier’s output 29 per cent above the peak of its own supply on switch-on, so “nothing” was never likely.

This essay marches the supply with the inductance in the winding and finds it counting for less than nothing: it flattens the regulation, sharply, and then less sharply as there is more of it.

The supply, marched with the inductance in it #

The supply is the one the earlier measurements used: a centre-tapped rectifier from a 17-volt-peak winding into a thousand-microfarad reservoir, the diodes in the netlist as exponential junctions, the whole circuit marched forward through a dozen mains cycles until it settles and read over the last cycle. Every run here carries fifty milliohms of winding resistance, because the solver needs a resistance beside an inductance to hold the diode’s anode, and an inductance of zero to three millihenries in series with it.

The output resistance is measured as before: the mean output at two loads ten per cent either side of the load of interest, the difference divided by the difference in load current. A slope rather than a ratio, because the supply’s output is a curve against its load and a following circuit sees its tangent.

Less than nothing #

Without inductance the output falls from 15.98 volts at 200 ohms to 14.29 at 25, 1.69 volts over the range. With 0.3 millihenries it starts higher, at 16.14, and falls only 0.62 volts; with a millihenry it falls 0.42. The inductance does not leave the regulation alone, as the mean-voltage argument predicted; it improves it by a factor of three to four.

The local slope says the same at every load. Without inductance the output resistance runs from 4.21 ohms at light load to 2.86 at heavy, below the textbook’s five ohms and falling, as the earlier essay found. With 0.3 millihenries it is 0.87 ohms at light load and 0.67 at 157 milliamps — a sixth of the bare supply’s — rising to 1.97 at the heaviest load.

The mean-current rule was right about resistances and is silent about this, and the reason is that the rule describes a dissipative drop. A resistance in the winding lowers the output by its mean voltage, the mean current times the resistance, and that is what it adds to the slope. An inductance has no mean voltage, so it adds no drop of that kind. What it does instead is change the shape of the charging current, and through the shape it changes how high the reservoir is charged on each cycle. That is a different mechanism, and at light load it raises the output more than at heavy load, which is a negative contribution to the slope.

The size of the effect is the surprise, more than its sign. The winding resistance beside the inductance is fifty milliohms, which by the mean-current rule adds about fifty milliohms to the slope; the inductance takes more than three ohms away. A parasitic whose dissipative contribution is nothing has a reactive contribution sixty times larger than the resistance it sits beside, and a model of the supply that kept the winding resistance and dropped the leakage would be wrong about the regulation by a factor of five.

How the inductance raises the light-load output #

Without inductance the reservoir is charged by a narrow pulse each half-cycle, starting when the winding’s voltage rises above the reservoir’s and ending near the winding’s peak. With inductance the current cannot stop the instant the winding falls below the reservoir: the inductor’s stored energy keeps it flowing for a while after the peak, carrying the reservoir above the point where a resistive charge would have stopped. That is the swing the swing a diode keeps measured in a resonant charge, arriving here uninvited: the leakage inductance and the reservoir form a resonant circuit, the diode stops the current at its first zero, and the reservoir keeps part of the overshoot.

The overshoot is largest when the load draws least, because a light load drains little of the lifted charge before the next pulse, so the lift raises the light-load output more than the heavy-load output. That is a slope in the right direction, and it is larger than the inductance’s small contribution to the heavy-load drop, so the net output resistance falls.

An optimum, not a monotone #

If more inductance meant more lift, the regulation would keep improving. It does not.

The output resistance at a hundred ohms is not monotonic in the inductance. A very small inductance, 0.03 millihenries, makes it slightly worse, 4.245 ohms against 3.883: too little to ring, it only delays the pulse and lowers its peak. It is least at about 0.3 millihenries, 0.666 ohms, where the resonance between the inductance and the reservoir rings the charge furthest above the winding’s peak. Beyond that it rises again: at three millihenries it is back to 2.856 ohms.

The rising branch is the rectifier changing kind. As the inductance grows the diodes conduct for longer — 30, 34 and then 64 degrees of each half-cycle in the pulse figure — and in the limit of a large inductance the current flows continuously and the rectifier becomes a choke-input supply, whose output is set by the average of the rectified wave, $2V_{pk}/\pi$, rather than by its peak. Between the two regimes the output has to fall from near the peak to near the average, and that transition is a slope of its own. A designer who wanted a choke-input supply’s excellent regulation needs an inductance well past this transition; a leakage inductance of a fraction of a millihenry sits in the capacitor-input regime and helps.

What it does to the headroom #

The quantity a following linear regulator depends on is the ripple’s trough, not the mean: the regulator needs its input above its output plus its dropout at the bottom of every ripple.

At full load the trough rises from 12.10 volts without inductance to 13.25 with 0.3 millihenries and 13.52 with one: over a volt of headroom recovered at the load where headroom is scarcest. The ripple itself is not reduced — it is slightly larger at 0.3 millihenries, 4.82 volts against 4.16 — because the lifted charge also lifts the peak; but the whole waveform sits higher, and the trough is what the regulator sees. The resistance that belongs in the winding found a winding resistance lowering the ripple while lowering the output; the leakage inductance does the opposite trade, raising the output at the cost of a little ripple, and for a regulator’s headroom that is the better trade.

The winding’s heating, which the inductance also lowers #

The pulse figure carries a second consequence that the regulation figures do not. The current that sizes the transformer found that a reservoir’s charging current, narrow and tall, heats the winding by the square of its root-mean-square value, which for a narrow pulse is several times the load’s direct current. A wider, lower pulse carrying the same charge has a smaller root-mean-square value, for the same reason a constant current carries a charge with the least heat.

At a hundred ohms the bare supply’s diode pulse peaks at 1.89 amperes over 30 degrees of each half-cycle; with 0.3 millihenries it peaks at 1.46 amperes over 34 degrees, and with three millihenries at 0.74 amperes over 64. The charge per cycle is nearly the same in all three, since it is the load’s current times the period, so the heating in the winding falls as the pulse widens. The leakage inductance therefore buys three things at once in this regime — flatter regulation, more headroom and a cooler winding — and costs the switch-on overshoot and, past the optimum, regulation again. Few parasitics are this generous, and it is worth knowing which side of the optimum a given transformer sits on before treating its leakage as something to be minimised.

Reading it against the reservoir’s own rules #

The sawtooth that the direct voltage that is a sawtooth measured on the bare supply — 1.33 volts of ripple on 15.69 at a hundred ohms — is the baseline every one of these figures departs from. The leakage inductance does not change what sets the ripple’s fall between charges: the reservoir still discharges into the load at the load’s current for most of each half-cycle, so the downward slope of the sawtooth is the same. What changes is where the sawtooth starts each cycle — higher, by the lift the ringing charge provides — and how long the charge takes to deliver, which moves the start of the fall later in the cycle.

That is why the ripple is nearly unchanged while the whole waveform rises. A regulator following the supply rejects the ripple by its own power-supply rejection, which what gets through from the rail measured for a regulator’s loop; the leakage inductance leaves the ripple the regulator has to reject almost where it was, and gives it more volts to reject it with. The two numbers that decide whether a regulator holds its output — the trough’s height and the ripple’s amplitude — move independently here, and only the first moves.

The figures are also a caution about how a supply’s regulation is quoted. A data sheet’s “output voltage at full load” for an unregulated supply is a mean, and a supply’s mean can be propped up by leakage inductance while its trough is not raised by the same amount; at 0.3 millihenries the mean rose 1.23 volts at full load while the trough rose 1.15. The trough is the number to design to, and the mean is the number that is usually printed.

A design rule that uses the parasitic #

The practical reading is that leakage inductance in a mains transformer, usually treated as a nuisance, is worth having in a capacitor-input supply up to a point. A transformer with a fraction of a millihenry of leakage referred to the secondary — the range a small laminated transformer typically has — gives better regulation and more headroom than an ideal one would, and does it without dissipating anything. A toroidal transformer, with very little leakage, behaves like the zero-inductance curve: the stiffest-looking transformer gives the softest supply. The comparison is worth making before a toroid is chosen for a capacitor-input supply on the grounds of its lower losses and smaller field: it will also give a supply whose output moves more with its load.

The rule has two limits. Too little inductance does nothing useful and a very small amount makes things marginally worse. Too much moves the supply out of the capacitor-input regime and the regulation degrades again, toward the transition to choke input. And the inductance carries a cost the regulation figures do not show: the first cycle, which no steady state contains and the inductance that limits, and lifts found the same inductance lifting the output above the peak at switch-on, which is a stress on the reservoir capacitor’s voltage rating that a design must allow for.

Where the rule from the last measurement stands #

The mean-current rule is not wrong; it is incomplete. It counts dissipative drops correctly, and every resistance in the circuit is counted by it as before. An inductance contributes through a different channel — the energy it stores and returns changes the reservoir’s charge per cycle — and that channel depends on the load through the conduction angle, so it appears in the slope. A complete account of the output resistance is therefore the mean-current sum of the resistive drops plus a reactive term that the rule does not see, and for a winding with a fraction of a millihenry of leakage the reactive term is larger than all the resistive ones together.

That is a pattern this field has met before. Two capacitors that are one found a reservoir’s series resistance lowering the ripple to an interior minimum before raising it, for the same reason: a parasitic that reshapes the charging current has an optimum, and the first-order rule that counts its dissipation alone misses it.

What the march leaves out #

The leakage inductance is a single lumped inductance in series with each half of the winding, and the transformer has no magnetising branch, no core loss and no interwinding capacitance. A real transformer’s leakage is distributed, and at the frequencies of the charging pulse’s harmonics the winding capacitance resonates with it, which can add ringing to the diode current at turn-off that this march does not contain.

The mains is a perfect source. A real supply’s own impedance is mostly resistive at the mains frequency and adds to the winding resistance; a long supply cable adds inductance of its own, which by this measurement would also lift the regulation, by an amount set by how it compares with the leakage already present.

And the diodes are ideal junctions without stored charge. A real rectifier diode’s reverse recovery, which the transients field measures, matters more with an inductance in series, because the current at the end of conduction falls at a finite rate set by the inductance and the recovery current then flows through the inductor, adding a snap at turn-off.

The number to keep #

A leakage inductance in a capacitor-input rectifier’s winding carries the load’s mean current and drops no mean voltage, and instead of counting for nothing it flattens the regulation: 0.3 millihenries takes the output resistance at 100 ohms from 3.88 to 0.67 ohms and the fall from 200 to 25 ohms of load from 1.69 volts to 0.62. It does so by ringing the reservoir above the winding’s peak, more at light load than heavy. The effect has an optimum — slightly worse at 0.03 millihenries, best near 0.3, back to 2.86 ohms at 3 — and it raises the full-load trough by over a volt, which is the headroom a regulator after the supply needs.

Still open: the headroom budget, the cable’s inductance, and the inductance that resonates with the winding #

The smallest reservoir for a regulator. A regulator’s input trough must clear its output plus dropout at the heaviest load and the lowest mains. With leakage inductance raising the trough, the smallest reservoir that meets all three conditions is smaller than without, and finding it for a stated regulator puts this essay’s figures into the decision they serve.

The mains cable. The supply’s own source inductance adds to the leakage and is outside the designer’s control. Whether a supply tuned near the optimum is pushed past it by a long cable — and its regulation degraded by a longer lead — is a march with a second inductance in the source.

Leakage against winding capacitance. At the harmonics of the charging pulse the leakage inductance resonates with the winding’s own capacitance. Adding that capacitance to the march would show whether the ringing it causes at diode turn-off changes the charge the reservoir keeps.