The direct voltage that is a sawtooth

Before a regulator there is usually a transformer, a rectifier and a large capacitor, and the voltage they produce is called a direct voltage by everybody including the people who designed it. It is not one. It is a sawtooth with a mean, and this essay is about the difference and about which of the sawtooth’s properties the usual design expression can see.

What is being solved #

Two diodes from a centre-tapped source, a capacitor and a resistor. Every diode is an exponential — the same diodeElement the semiconductors field uses, with its own current law and its own derivative — and the whole thing is marched forward in time by the machinery the oscillator essays needed, with a Newton loop at every step.

The topology is centre-tapped rather than a bridge, and the reason is numerical rather than historical. A bridge’s two alternating-current nodes are held by nothing at all during the interval when all four diodes are off; each connects to the rest of the circuit only through devices whose companion conductance has fallen to the floor, so the matrix loses rank at every instant between conductions. A centre tap grounds the source and every node keeps a path to ground at every instant. The two circuits rectify identically and only one of them is solvable at every step.

That is a small thing and it is the sort of small thing this collection writes down, because the alternative — adding a large resistor across each node to keep the matrix invertible — is exactly the “stamp a wire as a milliohm” mistake the networks field refuses.

What it settles at #

A 17 V peak, 50 Hz, 1000 µF and 100 Ω:

• mean output 15.69 V
• peak-to-peak ripple 1.331 V
• load current 157 mA
• peak diode current 2.098 A
• conduction 28.8° of each half cycle

The first number already contains something. Seventeen volts of peak into a capacitor should charge it to seventeen volts, and the mean is 15.69 — about a volt of that is the ripple’s own shape and the rest is the diode. A silicon junction carrying two amps sits at about 0.8 V rather than the 0.7 V it is usually credited with, because the drop moves 59.5 millivolts per decade of current and two amps is two and a half decades above a milliamp.

That the drop is a solution rather than a constant is the semiconductors field’s opening result. A bias point is a solution, not a choice solves the same junction from a five-volt supply through a kilohm and gets 0.692544 V, and from forty-eight volts through the same kilohm 0.754459 V — the same part, the same equation, sixty millivolts apart because the current is. And the value everybody quotes is exact somewhere: the one current a constant is right at locates seven-tenths of a volt at 5.748 milliamperes and nowhere else. A rectifier’s diode is two and a half decades above that current for a twentieth of each cycle and far below it for the rest, so the constant-drop model is wrong in both directions within one period and there is no single current at which to pin it.

The ripple, and the expression for it #

Every design guide gives the ripple as

$$V_r = \frac{I_{dc}}{2 f C}$$

which comes from assuming the capacitor is discharged by a constant current for the whole of each half period. At these values it gives 1.569 V. The march gives 1.331 V — the expression is 15.1% high.

Across a capacitance sweep and a load sweep:

The last two columns are the essay’s first result. The amount the expression is high by is the fraction of the half cycle the diode is conducting for — the ratio of the two runs 0.942 to 0.963 across the capacitance sweep and a load sweep together, and the gate holds that it stays inside a fifteen per cent band over both.

That is the mechanism named rather than a fudge factor fitted. The expression assumes the capacitor discharges for the whole half period; it actually discharges for the part of the half period when the diode is off, which is one minus the conduction fraction. Multiplying the expression by that fraction recovers the measurement to within a few per cent, and the few per cent is the fact that the discharge current is not quite constant either, since the load is a resistor and the voltage across it is falling.

The ratio is not exactly one, and the reason is worth stating rather than hiding: the conduction fraction is measured at a stated threshold — the interval during which the diode current exceeds one per cent of its own peak — and a different threshold moves the number by a few per cent. What does not move is that the two quantities track each other across a decade of capacitance and a decade of load.

What the expression cannot see #

The ripple is not what breaks a supply. Look at the lower panel of the figure.

The diode conducts for 28.8° of each half cycle and carries 2.098 A at the peak, against a load drawing 157 mA. That is a crest factor of 13.4, and it gets worse as the capacitor gets larger:

The gate holds that every increase in the reservoir capacitance makes the crest factor worse, which is the design fact hiding behind the ripple expression: the charge the load takes out over a whole cycle has to be put back during the conduction interval, so shortening the interval raises the current in proportion. A designer who doubles the capacitor to halve the ripple has also raised the peak current by about 40%.

And the peak current is what sizes things. The transformer’s winding heats as the mean square of the current, not as its mean; the diode’s dissipation is the drop times the current at the instant it flows; the capacitor’s own series resistance sees the whole of that pulse. None of those is in $I/2fC$, which contains the load current and the capacitance and nothing about the shape.

Two larger reservoirs finish the sweep, and they are the ones that matter to a designer, because they are the values anybody actually fits. The ripple specification is met somewhere in here; the conduction angle is not on the specification at all.

The number a regulator cares about is the trough #

The mean is what a voltmeter reads and it is not the quantity the rest of the design is constrained by. A regulator in front of this supply needs its input to stay above its output by its dropout voltage at every instant, so what matters is the bottom of the sawtooth.

The peak barely moves — it is the transformer’s peak less a diode, and the diode’s drop changes only with the logarithm of the current. Everything else in the table is the reservoir deciding how far the voltage falls before the next pulse arrives.

So a five-volt regulator with two volts of dropout needs seven volts at the trough and has it in every row; a twelve-volt regulator with the same dropout needs fourteen and gets it only from 470 µF upwards. The capacitor is sized by the trough and the mean is a summary of it.

The same table read the other way is the reason a supply’s regulation looks bad when it is measured casually. Between 220 µF and 4700 µF the mean output moves by 1.98 V — fourteen per cent — with no change to the transformer, the diodes or the load. A measurement of “the supply voltage” that does not say which reservoir it was made with has not said much.

And the load sweep says the same thing in the other variable:

A lighter load gives less ripple and a worse crest factor, for the same reason a larger capacitor does: the charge per cycle falls, the conduction interval shortens faster, and the current in the interval goes up. A supply designed at full load and then run lightly is running its diodes and its transformer at a peak current they were never sized for — and the ripple measurement, which is the one everybody makes, looks better while it happens.

The capacitor is not a capacitor either #

The march models the reservoir as an ideal capacitance, and it is worth saying what that leaves out, because the frequency field has already measured it.

A real electrolytic has an equivalent series resistance — tens of milliohms for a part this size — and a two-amp pulse through 50 mΩ is 100 mV of additional ripple that appears at the instant of conduction rather than being spread through the cycle. It also has an equivalent series inductance, which at 100 Hz is nothing at all.

Adding it is not hard — a resistor in series with the capacitor, which the regulator essays do — and it is left out here so that the comparison with $I/2fC$ is a comparison of two models of the same circuit rather than of two different circuits.

What comes before and after #

The supply this essay describes sits between two things, and both are elsewhere in the collection.

In front of it is a transformer, whose band, winding resistance and saturation limit the magnetics field measures. The pulse current matters there too: a winding’s loss is the mean square of the current times its resistance, and a crest factor of 13 with a given mean current means a mean square several times what a sinusoid of the same average would give.

Behind it is a regulator, and the ripple measured here is exactly what that regulator is asked to remove. The next essays measure how much of it gets through, and the answer depends on the frequency: at a hundred hertz the loop still has most of its gain, and at the megahertz where a switching supply puts its ripple it has none.

Three measurements of that regulator exist and all three are about the same sawtooth arriving at a hundred hertz. A source below a frequency drives the output node and finds 0.43 milliohms at direct current, ten times worse by 27 hertz and 1.95 ohms at ten kilohertz — so the impedance the ripple current works against is already climbing at the frequency this supply delivers it. What gets through from the rail asks the complementary question and answers it in the direction that matters here: the rejection is good at ten hertz, bad at a kilohertz and negative at ten, where the regulator puts out 1.9 times what arrives. A hundred hertz is the fortunate case, and it is fortunate because of where mains sits rather than because of anything anybody designed.

The capacitor that follows is where the two halves of this essay meet. Two requirements pulling one capacitor finds the output capacitor’s series resistance to be a stability requirement and a transient requirement at once, pulling in opposite directions, with the best transient sitting ten per cent inside the region the loop must not be built in. The reservoir measured here has the same quantity in it and no loop to be unstable in — which is why its series resistance appears in this essay only as an underestimate of the ripple, and in that one as the parameter the whole design is decided by.

The cycle the table does not contain #

Every number above is read from the settled state. The march starts with an empty capacitor, runs twelve cycles and reports the last of them, which is the right thing to report and is also a decision to discard the largest current the circuit ever carries. The first cycle, which no steady state contains measures that discarded cycle on this same arrangement and finds 32.4 amperes into an empty reservoir against a repetitive peak of 1.23 — twenty-six times anything the circuit does afterwards, and dependent on where in the supply’s cycle somebody’s hand closed the switch. A crest factor of 13.4 is a statement about the steady state. The parts have to survive both, and only one of the two is in the tables above.

The two repairs for that first cycle pull opposite ways on everything measured here. The inductance that limits, and lifts puts a winding resistance in and finds the peak limited and the energy not reduced at all — two per cent apart over a factor of four in the resistance — while leakage inductance divides the peak by seven and the energy by five and dissipates nothing to do it. What the inductance charges for is the one thing the arithmetic above forbids: a rectifier whose output sits 29 per cent above the peak of its own supply, permanently. Every row of the peak-and-trough table here assumes the only reactance in the loop is the reservoir, and the moment that assumption is dropped the peak stops being the transformer’s peak less a diode.

The part usually fitted to hold the first cycle down stops being that part after the first cycle. The protection that is gone by the second time measures an inrush thermistor at ten ohms cold and 1.94 ohms once the load current has warmed it, with 198 seconds to recover half of what it started with — against a reservoir that empties in tens of milliseconds. A supply dip inside that window hands the rectifier an unlimited inrush into an empty capacitor, which is the event the part is on the bill of materials for.

What the pulse costs where nobody is measuring it #

The crest factor is a property of the current drawn from the supply, and the supply is shared. Three of this collection’s measurements are about what that shape does after it leaves the circuit, and none of them is visible from inside it.

An average-responding meter is the usual instrument on a mains current, and what a meter multiplies by measures it as exactly right for a sinusoid and 35.9 per cent low on a rectifier drawing its current in sixty degrees. The narrower the conduction interval — which is what every increase in the reservoir capacitance buys — the further wrong the reading goes, so the instrument most likely to be pointed at this circuit understates it by more as the design is improved in the direction the ripple table recommends.

The same shape has a consequence three phases up. The neutral that carries more than a line shows three balanced loads of exactly this kind putting √3 times a line current down a conductor sized on the assumption that balanced loads put nothing there, because the pulses are disjoint and the neutral is their union. The imbalance that would normally explain a current in a neutral is absent: the loads are identical, and the current is there because none of them draws a sinusoid.

And the winding in front pays for the harmonics twice. The resistance that grows with frequency computes a conductor’s resistance from the Kelvin functions and finds it already 2.05 per cent up at the frequency the rule of thumb names as the point where the effect begins — so a pulse, which is a sum of harmonics reaching well past the fundamental, meets a higher resistance than a direct-current measurement of the same winding suggests. The mean square of the current is the first factor and the resistance it meets is the second, and the crest factor raises both.

The gate #

The expression is high at every capacitance and every load tested — five capacitances and three loads, all in the same direction.

And the amount it is high by is the conduction fraction, with the ratio between them inside a fifteen per cent band across both sweeps.

The crest factor exceeds five everywhere and rises with capacitance, monotonically, over the capacitance sweep — which is the design fact the expression cannot express.

Current law holds at every step of every march, rebuilt from the diodes’ own exponentials rather than from the companions the steps used, at worst $2\times10^{-11}$ of the largest branch current.

The first two of those are the essay. An expression that is fifteen per cent high is not a bad expression — it is a good one used outside the assumption it was derived under, and the assumption is recoverable from the discrepancy, which is the most useful thing a discrepancy can be. What makes the third claim worth having beside them is that it is about a quantity the expression does not mention at all: no amount of care with the ripple arithmetic will tell a designer that the diode is carrying thirteen times the load current, and that is the number the parts list is decided by.

The collection’s usual sentence applies unchanged. The model is right inside a range, the range has an edge, and the figure carries it.