The resistance that belongs in the winding

The resistance that lowers the ripple put a series resistance inside the reservoir capacitor of a marched rectifier and found two effects pulling in opposite directions. The resistance adds a step to the output, which raises the ripple, and it limits the charging current, which opens the conduction angle and lowers it. Over the first seventeen milliohms the second wins by a hair, and the ripple bottoms out at 1.3303 volts against the 1.3312 a perfect capacitor gives. Past that the step wins, and at an ohm the ripple is 1.711 volts.

What the resistance bought was never the ripple. It was the peak current: the crest factor fell from 13.374 to 6.550 at an ohm, which more than halves the current that sizes the transformer, the diodes and the fuse. That essay closed on the observation that the resistance does not have to be a separate part, because a transformer with more winding resistance does the same thing.

It does the same thing to the peak current. Whether it does the same thing to anything else is a question the schematic cannot answer, because on a schematic the two resistances are in the same loop and a reader tends to treat a loop as one place. This essay marches both.

Same loop, different currents #

The circuit is the one every earlier essay here marched: a centre-tapped secondary at seventeen volts peak, two diodes with their exponentials in the netlist, a thousand-microfarad reservoir and a hundred-ohm load drawing about 157 milliamperes. The resistance goes in one of two places.

Inside the capacitor. The capacitor becomes an ideal capacitance behind a resistance, and the output node — where the diodes and the load meet — sits on the resistive side of it.

In the winding. A resistance in series with each half of the secondary, between the winding and its diode, so it is on the source side of the node where the load and the capacitor meet.

During conduction both are in the loop that recharges the capacitor, which is why they look like one place. But they are not in the same branch, and a branch is what a current is counted in. Follow the two currents that exist in this circuit through the two positions, one interval at a time.

While a diode conducts, the transformer supplies the load and recharges the capacitor at once. The winding’s resistance carries both: the whole of the diode current. The capacitor’s resistance carries only the part going into the capacitor, which is the diode current less the load’s.

While both diodes are off, the capacitor supplies the load alone. The winding’s resistance carries nothing — there is no current in the secondary at all. The capacitor’s resistance carries the whole load current, flowing out.

So each resistance carries a current the other does not. The winding’s carries the load’s current during conduction; the capacitor’s carries the load’s current during discharge. Everything that differs between the two placements comes from that asymmetry, and everything that does not differ comes from the part the two share, which is the charging current.

The part they share: the peak #

The peak diode current is set by the loop’s resistance at the moment the transformer’s voltage is highest above the capacitor’s, and at that moment both resistances are carrying charging current. The load’s share of the diode current at the peak is 157 milliamperes out of about two amperes, which is under eight per cent, so to the first order both placements put the same resistance in the path of the same pulse.

The march agrees to better than first order. The crest factor — the peak diode current over the load’s direct current — at seven resistances from a milliohm to an ohm, in each place:

The two crest columns agree to two parts in a thousand at every row, and the figure checks that they do. The winding’s is always the lower by a few parts in ten thousand, which is the load’s eight per cent share of the pulse flowing through one resistance and not the other, weighed by a pulse that is mostly charging current.

That settles the half of the question a designer usually asks first. If the resistance is there to lower the peak current, it does not matter which side of the capacitor it is on. Per milliohm, the crest-factor benefit is identical.

The part they do not share: the ripple #

The ripple columns part company from the second row, and not by a little. At a hundred milliohms the capacitor’s ripple is 1.3349 volts, already above the perfect capacitor’s, while the winding’s is 1.3175 — a per cent below it. At an ohm the capacitor gives 1.711 and the winding 1.172. The difference between them at an ohm is 539 millivolts, on a ripple of about 1.3.

The winding’s ripple has no minimum. It falls at every resistance marched, and the figure checks that too.

The asymmetry explains why. The ripple the output shows is the capacitor’s own voltage swing plus whatever step a resistance between the capacitor and the output adds to it.

With the resistance in the winding there is no such resistance. The output node is the capacitor, so the output’s ripple is exactly the capacitance’s ripple — and the resistance only ever acts on that through the conduction angle. More resistance in the charging loop means a lower peak, a longer conduction, a shorter discharge and a smaller swing. That is one effect, it has one sign, and the curve is monotone because nothing opposes it.

With the resistance in the capacitor the same effect is there, and so is the step. At the top of each charging pulse the output sits above the capacitance by the charging current times the resistance; in the discharge interval it sits below the capacitance by the load current times the resistance. So the output’s swing is the capacitance’s swing widened at both ends. The widening at the bottom is the load current flowing through a resistance it only meets in this placement, and it is paid for the whole of the discharge interval, which is five sixths of each half cycle.

So the minimum that essay found is not a property of a resistance in the charging loop. It is a property of a resistance that is in the charging loop and in the discharge path, where the first role lowers the ripple and the second raises it. Take the second role away and the minimum goes with it.

Moving a fixed resistance #

The cleanest version of the comparison holds the total constant and moves it. Three hundred milliohms is about what a small mains transformer’s secondary and an ordinary electrolytic contribute between them, so it is a realistic total, and the question a designer faces is not whether to have it but where it ends up.

Marched with a quarter, a half and three quarters of it in the winding and the rest in the capacitor, the ripple goes 1.386, 1.346, 1.315, 1.293 and 1.282 volts as the resistance moves from the capacitor to the winding. Every ohm moved lowers it. The crest factor over the same five marches is 9.0722, 9.0701, 9.0689, 9.0687 and 9.0692: flat to three parts in ten thousand, which is the marching tolerance rather than a trend.

That is the whole finding in one figure. The peak current depends on how much resistance is in the charging loop and not on where it sits. The ripple depends on how much of it the load’s current also has to cross.

The consequence for a component choice is backwards from the usual instinct. A designer who is specifying a capacitor with low series resistance “for ripple”, on a supply whose transformer already has a few hundred milliohms of winding, is buying a smaller share of a total that the ripple does not care about except through its distribution — and the distribution improves as the capacitor’s share goes down. Nothing in this argues for a worse capacitor: a low-resistance part dissipates less and lasts longer, which is the real reason to buy one. It argues that the ripple is not the reason.

What the winding charges for it #

The winding placement is not free, and it is worth being exact about what it costs, because the asymmetry that gives it the lower ripple is the same one that makes it more expensive.

Mean output. At an ohm the capacitor’s resistance costs 380 millivolts of mean output and the winding’s costs 534. At three hundred milliohms the two are 84 and 131. The winding’s is always the larger, and the asymmetry says why: during conduction the winding’s resistance carries the load’s current as well as the charging current, so the drop across it at the moment the capacitor stops charging is larger, and the capacitor is left charged to a lower peak. The capacitor’s resistance carries the load’s current too, but only during discharge, and the mean of the output across a whole cycle is the mean of the capacitance’s own voltage — the step during discharge and the step during charging average out against each other, because the capacitor’s current has no mean.

Heat. The dissipation in each resistance is the mean square of the current in its own branch times its value, and here the branches matter again.

In the winding, exactly one half of the secondary carries the rectified current at any instant, so the two halves together dissipate the root-mean-square of the rectified current squared times the resistance. Marched at an ohm that current is 0.3449 amperes and the dissipation is 119 milliwatts.

In the capacitor, the resistance carries the capacitor’s current, which is the rectified current less the load’s. Because the capacitor’s current has no mean, its mean square is the rectified current’s mean square less the load’s, to within a correction the size of the resistance over the load. At an ohm the rectified current is 0.3484 amperes and the load’s is 153 milliamperes, so the capacitor’s current is about 0.313 and the dissipation about 98 milliwatts.

So the winding costs a fifth more heat and two fifths more mean output, for a ripple a third lower. At the realistic hundred milliohms the differences are a tenth as large and so is the ripple advantage: 1.3175 volts against 1.3349.

Which place a deliberate resistance goes #

A supply that needs its peak current limited — to protect the diodes, to cut the transformer’s copper loss, or to hold the power factor of the line current inside a limit — has three places to put a resistor and this march prices two of them. The third, a resistor between the reservoir and the load, is a different circuit: it filters rather than limits, and it does nothing to the peak.

Of the two, the winding side wins on ripple at every value and loses on mean output and heat at every value. For the supplies these essays have been describing — a reservoir feeding a linear regulator — the ripple matters only as far as the regulator’s dropout, and the dropout is set by the bottom of the ripple. The bottom of the output is the mean less half the ripple, so the quantity to compare is that trough:

• capacitor, one ohm: 15.306 − 1.711/2 = 14.45 volts;
• winding, one ohm: 15.152 − 1.172/2 = 14.57 volts.

The winding’s larger mean loss is more than paid back by its smaller ripple. At three hundred milliohms the troughs are 14.91 and 14.91 to two decimals, and at a hundred 15.00 and 14.99. Below about three hundred milliohms the placement is a wash on dropout; above it the winding side leaves the regulator more room, and the gap widens with the resistance.

So a deliberate surge resistor belongs on the source side of the capacitor — where a series resistor before the rectifier or a transformer specified with a higher winding resistance puts it — and not in series with the capacitor itself. The second arrangement buys the same crest factor and pays for it twice in ripple: once through the step the load’s current puts across it for five sixths of every half cycle, and again through the step the charging pulse puts across it at the top.

There is a practical reason the capacitor side gets chosen anyway, and it is worth stating so that the recommendation is not naive. A resistor in the secondary has to be rated for the winding’s current, and in a centre-tapped design there are two of them; a resistor in series with the reservoir is one part. But the one part carries the capacitor’s ripple current, which at an ohm is about 0.313 amperes, while each winding resistor carries its own half of the rectified current — 0.3449 amperes divided by $\sqrt2$, or 0.244 — so the single part has to be rated for more current than either of the pair, and it costs a ripple half as large again. The count of parts is the only thing the capacitor side saves.

The inrush argument is a separate one and this march does not settle it. The first cycle, which no steady state contains prices switch-on, where the capacitor starts empty and the whole transformer voltage appears across the loop resistance. There the two placements again carry the same charging current, and there is no discharge interval for the difference to show up in, so the peak inrush should be the same in both places as well. What differs is afterwards.

What a schematic calls one place #

The general point is a small one, and it recurs across these essays. A schematic draws a loop, and a reader totals the resistance around the loop, because for the one question a loop answers — how much current flows when a voltage drives it — the total is all that matters. The peak current here is that question, and the two placements give it the same answer.

But a circuit with a load hung off one node has more than one loop, and a resistance belongs to the loops that pass through its branch. The capacitor’s resistance is in the charging loop and the discharge loop; the winding’s is in the charging loop and the load’s supply loop. Two resistances of equal value in “the same loop” are in different pairs of loops, and every quantity that is not purely about the charging loop tells them apart.

The same part written two ways made the matching point about a capacitor’s loss drawn in series or in parallel: a pair of conversion expressions that is exact at one frequency and at no other. Here it is placement rather than topology, and the conversion is exact for one quantity — the peak — and wrong for every other.

It is also why the textbook expressions for this circuit can have no term for either resistance. The direct voltage that is a sawtooth found the ripple expression high by about a third because it has no conduction angle in it. A resistance acts on the ripple only through the conduction angle and through the step, so an expression without the angle cannot see the first effect and an expression without a branch structure cannot see the second.

Still open #

Load regulation, which the two placements should answer differently. An unregulated supply’s output moves with its load, and the slope of mean output against load current is its output resistance. The winding’s resistance carries the load’s current during conduction and the capacitor’s carries it during discharge, so each should appear in the output resistance weighted by the fraction of the cycle it carries the load in — roughly a sixth for the winding and five sixths for the capacitor, which is the opposite of what the mean-output cost above suggests. Marching the mean at two loads in each placement would say whether that weighting is right, and whether the winding’s larger cost at one load is really a smaller slope across loads.

The capacitor bank, where the placement question is asked of several capacitors at once. Two parts in parallel halve the series resistance and double the capacitance, so they move the capacitor’s share of the loop’s resistance down without moving the winding’s. Whether a pair beats one part of the same total capacitance is exactly a question of how much resistance the discharge current has to cross, and this essay’s asymmetry is what should decide it. Two requirements pulling one capacitor is the regulator-side version, where an output capacitor’s series resistance is wanted for stability and unwanted for the droop, and the part is chosen for it rather than despite it.

The leakage inductance, which sits in the winding and nowhere else. A real secondary is a resistance and an inductance in series, and the inductance limits the peak by slowing the current’s rise rather than by dropping a voltage. It has no counterpart inside a capacitor at fifty hertz. It should lower the crest factor without dissipating anything, and the question is how much of the winding resistance’s ripple advantage it keeps. The inductance that limits and lifts measured it dividing the switch-on pulse by seven and lifting the output rather than lowering it; how it shares the settled ripple with a winding resistance beside it, when the two are split in the proportions a real transformer has, is unmeasured. The capacitor’s own inductance is no rival, because the capacitor that is an inductor finds it mattering only at frequencies this circuit never reaches.