The resistance that lowers the ripple

The direct voltage that is a sawtooth marched a bridge rectifier and a reservoir capacitor with the diodes in the netlist and found the textbook ripple expression high by about a third, because it assumes the capacitor discharges for the whole cycle when it is being recharged for a tenth of it. The current that sizes the transformer found what that recharging costs: a peak diode current 13.374 times the load’s direct current, a root-mean-square 3.0069 times it, and nine times the copper loss a direct current would produce.

Both marched a perfect capacitor. A thousand-microfarad aluminium electrolytic has something between twenty and a hundred milliohms of series resistance, and it is the least ideal part in the circuit — the diodes are modelled with their exponentials, the transformer with its winding resistance, and the capacitor with nothing at all.

The obvious expectation is that adding it makes the ripple worse. The peak diode current is thirteen times the load current; put fifty milliohms in its path and the resistive step is a hundred millivolts on top of a 1.33-volt ripple, which is eight per cent.

The march says the ripple goes down.

Two effects, in opposite directions #

The resistance does two things and only one of them is usually counted.

It adds a step. The capacitor’s terminal voltage is its internal voltage plus $I R_{esr}$, and the current through it is the diode current during conduction and the load current otherwise. So the output rises by $R_{esr}$ times the peak diode current at the top of the charging pulse, which is where the peak of the ripple already is, and the peak-to-peak grows.

It limits the charging current. The loop during conduction is the transformer’s winding, the diode’s dynamic resistance and now the capacitor’s series resistance. Adding to that loop reduces the peak current, which lengthens the conduction angle, which means the capacitor is recharged over a longer interval and discharges for a shorter one — so the capacitive part of the ripple falls.

The two are about the same size over a decade of resistance and they nearly cancel:

Read the ripple column and it falls, reaches a minimum, and rises. Golden-sectioned on the marched ripple rather than read off the sweep, the minimum is at 17.0 milliohms, where the ripple is 1.3303 V against 1.3312 with a perfect capacitor.

The minimum is small and it is not the point #

Seven hundredths of a per cent is not a design margin and nobody should choose a capacitor for it. What the minimum establishes is that the resistance is not a straightforward degradation, and it disposes of the intuition that a better capacitor is always a better capacitor here.

It is worth being clear about what is and is not being claimed. A capacitor with lower series resistance is better in every other respect — it dissipates less, runs cooler and lasts longer, which is the actual reason to buy one — and its effect on the ripple is nil to within a tenth of a per cent over the range where ordinary parts live. The design decision the series resistance participates in is not the ripple at all.

The useful column is the third one.

What the resistance actually buys #

The crest factor falls monotonically and by a lot: 13.374 with a perfect capacitor and 6.550 at an ohm, which is more than a halving of the peak current that sizes the transformer, the diodes and the fuse.

That is worth money. The essay before it measured what the crest factor costs — a root-mean-square winding current 3.0069 times the direct current, and therefore 9.04 times the copper loss a direct current would produce — so anything that lowers the peak lowers the transformer’s dissipation. Marched, the root-mean-square diode current falls from 0.4717 A to 0.3484 at an ohm, which is a factor of 1.354 in current and 1.83 in copper loss.

The price is 380 millivolts of mean output out of 15.686, which is 2.4 per cent, and 380 millivolts of extra ripple. For a supply feeding a regulator that is an excellent trade: the regulator throws away the mean anyway and rejects the ripple, and the transformer is the expensive part.

This is what a surge resistor is, and the usual justification for one is the switch-on transient — the first cycle, which no steady state contains prices that — with the steady-state benefit thrown in as an afterthought. The arithmetic here says the steady-state benefit is the larger of the two for a supply that runs continuously, because it is paid every cycle for the life of the equipment rather than once at switch-on.

And the resistance does not have to be a separate component. A transformer with a higher winding resistance does the same thing, which is why a small transformer running near its rating has a lower crest factor than a large one lightly loaded, and why substituting a bigger transformer for a marginal one sometimes makes the diodes run hotter.

Why the conduction angle is doing all of it #

Every column in the table moves and one quantity explains all of them: the conduction angle, which is the fraction of each half cycle the diodes are forward-biased for.

With no series resistance the diodes conduct for 28.80° of each 180° half cycle — a sixth of it — and the capacitor supplies the load for the other five sixths. At an ohm they conduct for 43.02°, at three ohms for 58.14°. Adding resistance to the charging loop lowers the peak current, so the charge per cycle has to be delivered over a longer interval, and the only way to lengthen the interval is for the diodes to turn on earlier.

Everything else follows from that one number.

The crest factor is very nearly the ratio of a half cycle to the conduction angle, because the same charge is delivered either way: 180/28.8 = 6.25 against a measured 13.37, the factor of two between them being the shape of the pulse rather than its width. Open the angle by 1.5 and the peak falls by 1.5.

The ripple’s capacitive part is the load current times the non-conducting time over the capacitance. At 28.8° the capacitor discharges for 84 per cent of the half cycle; at 43.02° for 76 per cent. That is a nine per cent reduction in the capacitive ripple, and it is what pays for the resistive step.

The mean falls because the diodes conduct further down the sinusoid’s flank — the capacitor is recharged to a peak that is lower than $V_{pk}$ by the drop across the loop at the moment conduction ends, and a longer conduction means that moment is further from the crest.

So the resistance’s whole effect is to open the conduction angle, and the three columns are three consequences of one geometric change. The essay before it made the same observation about the capacitance: the textbook expression is high by the fraction of the cycle the diode conducts for, measured at 0.94 to 0.96 across two sweeps. The conduction angle is the quantity these essays keep finding underneath whatever is being measured, and it is the quantity neither closed form contains.

Neither expression has a term for any of this #

The two closed forms those essays measured against are

$$V_{\text{ripple}} \approx \frac{I_{dc}}{2fC}, \qquad V_{\text{mean}} \approx V_{pk} - \frac{I_{dc}}{4fC}$$

and neither of them contains a resistance. Not a small term in one, not a correction factor — the quantity is absent.

So across a range over which the marched ripple moves by thirty per cent, the expression moves by 2.4 per cent, and every bit of that movement comes in through the measured mean current rather than through any awareness of the resistance. The figure states that as a refusal: the expression cannot see the minimum, cannot see the rise past it, and has nothing at all to say about the crest factor, which is the quantity the resistance is actually for.

That is the third time this field has found the same shape. The ripple expression is high by a third because it has no term for the conduction angle. The mean expression is wrong in three places whose signs differ. And now both are blind to the one component parameter that a designer can actually choose, because the derivation assumed the capacitor was a capacitor.

The pattern is worth naming: a closed form derived by assuming a part is ideal cannot be corrected for that part being real, because the correction is not a term in it. The only available route is to solve the circuit, which is what every number in this essay is.

Where the dissipation goes #

The series resistance carries the whole of the diode current during conduction and the whole of the load current the rest of the time, and its dissipation is the root-mean-square of that times its resistance.

At the marched values with a perfect-capacitor waveform, the capacitor’s own root-mean-square current is about 2.85 times the load’s direct current — a little below the diode’s 3.0069, because the capacitor supplies the load during the discharge while the diodes carry nothing. At 157 mA of load that is 448 mA in the capacitor, and at fifty milliohms it is 10 mW.

Ten milliwatts is nothing thermally and it is everything for the part’s life. An aluminium electrolytic’s life halves for every ten kelvin of core temperature, and the core temperature above ambient is the ripple current squared times the resistance times a thermal resistance — so the ripple current rating on a capacitor’s data sheet is a life specification rather than an electrical one.

That closes a loop with the essay before it, and it is the same accounting the average a square root pulls low does for a meter: the peak, the mean and the root-mean-square of one waveform are three different numbers that size three different things, and the mistake is using one of them for a part it does not size. Its crest factor of 13.374 and its root-mean-square of 3.0069 were presented as facts about the transformer, and they are equally facts about the capacitor: the same waveform runs through both, and the part that is sized by the peak is the transformer while the part that is sized by the root-mean-square is the capacitor.

So a series resistance that lowers the crest factor extends the capacitor’s life as well as the transformer’s, and the two benefits arrive from the same change — at the price of the resistance’s own dissipation, which at an ohm is 121 mW rather than 10 and is beginning to matter.

What one marched value does not cover #

That the minimum is worth designing to. It is 0.07 per cent deep and it is at a resistance most capacitors of this value already have. Its value is as evidence about the mechanism, not as a target.

That the capacitor’s own root-mean-square current is measured here. It is estimated from the marched diode current and the load current, not read off a probe in the capacitor’s branch, and the 2.85 figure is a consequence of charge conservation rather than a measurement — which is a weaker statement than everything else in this essay and is flagged as such.

That the series resistance is a constant. It is not — an aluminium electrolytic’s rises steeply below about zero degrees and falls with frequency up to a point, which is the same part written two ways’s subject seen from the loss side. The march here uses one value, and the interesting consequence is that a supply’s crest factor is worse when cold, at exactly the moment the switch-on surge is largest.

That the diode’s own dynamic resistance is separable from the capacitor’s. They are in the same loop during conduction and the march does not distinguish them; what is swept is the capacitor’s, because that is the one a designer chooses. Two exponentials, and where they meet is where the diode’s own contribution to the same loop is measured.

That a perfect capacitor is a useful limit. The marched numbers say it is not even the best case for ripple, and it is unbuildable — a capacitor with no series resistance would also have no inductance and no dielectric loss, which is three idealisations rather than one.

That the crest factor’s benefit is free of the mean. It is not: 380 mV out of 15.686 is 2.4 per cent of the output, and for a supply feeding a regulator with a fixed dropout it is 380 mV of headroom that has to come from somewhere. The trade is real and it is priced above.

The minimum searched on the march, and the expression that cannot see it #

Every point is marched with the diodes’ exponentials in the netlist and the current law checked at every step, to better than $10^{-8}$.

The ripple minimum is golden-sectioned on the marched ripple rather than read off the sweep, and required to be below the perfect capacitor’s — which is the essay’s headline and would be easy to lose to a coarse scan.

The crest factor is checked to fall monotonically at all nine resistances marched, which is worth checking rather than assuming precisely because the ripple does not.

The mean is checked to fall by between 0.2 and 0.5 volts at an ohm, so that the price of the crest factor is measured rather than described.

And the expression is refused: across a range where the marched ripple moves by thirty per cent the closed form moves by 2.4, and only through the mean it is computed from.

A parasitic that is a design variable #

The usual finding here about a parasitic is that it sets an edge — a capacitor stops being a capacitor, an amplifier stops being ideal, a wire stops being a node. This one is different and the difference is worth stating.

The reservoir’s series resistance is not a limit on anything. It is a knob, and it happens to be supplied as a side effect of choosing a capacitor rather than as a component in its own right. Turned up, it lowers the peak current, the root-mean-square current, the transformer’s copper loss and the capacitor’s own heating; it raises the ripple slightly and lowers the mean; and the ripple has a minimum inside the useful range, so the direction of the small effect is not even monotone.

A designer who wants that knob can have it explicitly — a resistor in series with the transformer secondary, which is a part costing pennies and is fitted for a different reason — two requirements pulling one capacitor is the same situation where the two requirements are on the same component instead of on two. The arithmetic here says what it is worth in the steady state, and the answer is a factor of 1.83 in transformer copper loss for 2.4 per cent of output voltage.

Which suggests the general form. A quantity that appears in no expression is not thereby unimportant; it is unexamined, and the first thing to do with it is to sweep it and see whether the curve has a shape. This one has a minimum, two monotone columns and a conduction angle that explains all of them, and none of that is visible from the two formulas the field is taught with.

Still open: the resistance in the other place, the capacitor bank, and the cold start #

A resistor in series with the secondary rather than with the capacitor. They are in the same loop during conduction and are not the same element: the capacitor’s resistance also carries the discharge current and the winding’s does not. Marching both separately would say whether the ripple minimum survives the move, and whether the crest-factor benefit is the same per milliohm in the two places — which decides where a designer should put the deliberate one.

Two capacitors in parallel, which is not one capacitor of twice the value. Paralleling halves the series resistance and doubles the capacitance, so it moves both variables at once and in directions that oppose in the ripple. Whether the pair is better or worse than one part of the same total capacitance is a question the march can answer and the expressions cannot, and the floor a second capacitor removes is the same question asked about a decoupling bank.

And the cold start, where every number here is different. An electrolytic’s series resistance at −20 °C is several times its room-temperature value, so a supply switched on cold has a much lower crest factor and a much higher ripple than the same supply warm — and the inrush, which is the one place the resistance is universally acknowledged, is smallest at exactly the moment the capacitance is also lowest. Marching the pair of temperature dependences together would say whether the cold start is really the worst case it is designed as.