The rule that reads the sign of the gain

The regulator that pushes past the nose found the failure at the centre of voltage regulation on a line. A regulator that raises a turns ratio whenever its load’s voltage is low is a stabiliser only while raising the ratio raises the voltage, and on a line with resistance that stops being true at exactly the nose: the load’s voltage, $nV_0R_L/(n^2R + R_L)$, peaks at a ratio of $\sqrt{R_L/R}$ and falls beyond it. Ask for more than the line’s maximum power and the regulator raises the ratio through the peak and keeps going, and the voltage collapses — in 256 seconds at 101 per cent of the nose for the integrating regulator of that essay.

The essay closed with the protection that real regulators carry, named and not modelled: block the regulator when the voltage fails to respond to a tap change in the expected direction. That rule is a measurement made on the running system — a one-step estimate of the sign of the regulator’s own loop gain — and the question it leaves is how well an estimate does a job that the earlier essay showed to need the gain’s sign exactly at the place where the gain is zero.

A tap changer, and the rule #

The line is the regulator essays’ own: a 10 V source behind 1 Ω, feeding a load through an ideal ratio $n$, the load’s resistance chosen so that at the 10 V setpoint it takes a stated fraction of the most the line can deliver, $V_0^2/4R = 25$ W. Where the earlier essay’s regulator integrated the error continuously, this one is a tap changer, which is how such regulators are actually built: every 5 seconds, if the measured voltage is more than 20 mV below the setpoint, it raises the ratio by one tap of 1.25 per cent.

The blocking rule is added on top. Before each tap the regulator remembers the voltage it measured; after the tap it measures again; and if the voltage went down, it stops raising for good. Raising the ratio lowers the voltage only past the nose, so the rule fires exactly when the regulator has crossed it — in the noiseless case, at the first tap beyond $\sqrt{R_L/R}$.

At 105 per cent of the nose that is what happens. The nose ratio is 1.9518; the tap to 1.9625 is the first past it, the voltage falls by a few millivolts, and the regulator stops at 385 seconds with the load at 9.759 V. That is within a tap of the most this line can give this load, $V_0\sqrt{R_L/R}/2$, also 9.759 V to the third figure. Without the rule the tap changer goes on: the voltage falls a little with every tap, and at 900 seconds the ratio is 3.25 and the load is at 8.61 V and still sinking. The setpoint is out of reach either way — no ratio gives a load that wants 105 per cent of the nose its 10 V — and the rule’s whole achievement is to convert a collapse into a stall at the best voltage available.

Where each regulator leaves the load #

Swept over the demand, the two regulators agree everywhere below the nose: both raise the ratio until the load is within the dead band of its setpoint and stop, and the rule never fires because every tap raised the voltage. Above the nose they part. With the rule, the regulator stops at the top of the voltage curve, so the load gets the most this line can deliver it — 9.90 V at 102 per cent, 9.53 at 110, 8.77 at 130 — falling as the demand rises, as it must. Without it, the regulator drives the ratio past the peak and the load sits below its best by more than a volt after fifteen minutes, and lower the longer it runs.

This is the noiseless result, and it is a strong one. A rule that compares two readings, with no model of the line and no knowledge of where the nose is, finds the nose to within a tap. It does so because it measures exactly the quantity that changes sign there.

A loop gain measured by the loop itself #

The rule is worth seeing as what it is: a measurement of the regulator’s loop gain, made by the regulator on the running system. The regulator’s loop runs from its ratio, through the line, to the voltage it measures and back to its decision about the ratio, and the line’s contribution to the loop gain is $dV/dn$. A tap is a small injected perturbation and the voltage change it produces is the loop’s response to it — exactly the cut-and-inject measurement what is left at crossover made on an amplifier, here made once per tap with a step instead of a sinusoid.

That framing explains both halves of the result. The rule works because a loop that needs a positive gain to be stable can test for one directly, and the nose is precisely where that gain passes through zero and changes sign — the one-parameter version of the conditional stability stable, and unstable with less gain found in an amplifier whose phase crossed −180° at three frequencies. And the rule fails near the nose for the reason any loop-gain measurement fails near a zero of the gain: the response to a finite perturbation shrinks to the size of the measurement’s own error. Two wrong readings and one loop gain found a bench measurement wrong by the impedance ratio at its cut; this one is wrong by its noise, and there is no second reading to combine it with, only more of the same one.

Why a stall at the peak is the best available #

The rule’s success above the nose deserves a sentence of its own, because it is easy to undervalue. A load asking for more than the nose power cannot be served at its setpoint by any ratio, and the load that takes the most showed what the line’s maximum is: the power it delivers into a load matched to its source resistance, here the ratio that makes $n^2R$ equal $R_L$. Stopping at the first tap past the nose leaves the load within one tap of that match — 9.759 V against a best of 9.759 at 105 per cent, the difference below the third figure. At 110 per cent the stop comes one tap later, at n = 1.925 against a nose of 1.907, because the tap that straddles the nose can still raise the voltage a little and the rule only reads a fall; the loss is a second-order one, since the curve is flat at its top. No regulator, however clever, could deliver more to that load from that line, and the rule finds the optimum with no model of the line at all.

The cost of not stopping is not only a lower voltage. Past the nose every further tap moves the operating point down the lower branch of the curve, where the load draws more current for less voltage and the line dissipates more of the source’s power. The unguarded regulator at 900 seconds is running the line harder than the guarded one to deliver less.

The difference that vanishes at the nose #

The same fact is the rule’s weakness. What it measures is the change a tap makes, $\Delta V = (dV/dn)\,\Delta n$, and $dV/dn$ is zero at the nose. On the way up from a low ratio the first taps move the voltage by tens of millivolts; the last taps before a setpoint close to the nose move it by almost nothing. A real measurement carries noise — the load fluctuating, a transducer’s resolution, a converter’s quantisation — and the difference of two noisy readings has a spread of $\sqrt 2\,\sigma$. When a tap’s true $\Delta V$ is comparable with that, the comparison returns the wrong sign, and the rule stops a raise that was still on the right side of the nose.

Each tap is a coin toss weighted by $\Phi(-\Delta V_k/(\sqrt 2\,\sigma))$, and a raise survives only if every toss comes up right, so the probability of a false block is one minus the product over the taps on the way. With 3 mV of noise the product is indistinguishable from one at 95 per cent of the nose, because every tap there moves the voltage by much more than 4 mV. At 98 per cent the seeded raises are wrongly stopped 0.3 per cent of the time; at 99, 4.8; at 99.5, 15. The seeded share runs a little below the closed form because a noisy reading can also end a raise a tap early — the regulator sees the setpoint reached when it is not — which gives the rule fewer chances to fire, but the two agree on the shape: the rule fails more often the closer the operating point is to the edge it guards.

That is the cruel geometry of this protection. It is exact far from the nose, where it is not needed, and unreliable near it, where it is. A false block leaves the load below its setpoint — not collapsed, but under-regulated — and on a real network, where demand drifts up through the day, it is the operating points just below the nose that see the most taps.

The margin noise costs #

The useful number is how close to the nose the regulator can be run while keeping false blocks rare.

With 1.25 per cent taps and a millivolt of noise, the rule is reliable to within a fraction of a per cent of the nose. At 3 mV it is reliable only up to 97.9 per cent; at 10 mV, 79.9; at 30 mV, 10.7 per cent — at which point the rule fires on noise during almost any raise, and the regulator might as well have no rule. The dependence is steep because both the tap’s signal and the number of taps change with the demand: higher demand means a longer raise and smaller steps at its end.

Larger taps help in proportion. With taps of 2.5 per cent the same 10 mV of noise allows 94.8 per cent of the nose rather than 79.9; with taps of 0.625 per cent, only 36.7. A tap is both the regulator’s resolution and the rule’s test signal, and making it finer for better regulation makes the test weaker in the same proportion.

A tap coarse enough to read #

At a fixed operating point — 95 per cent of the nose, 3 mV of noise — the false-block probability against the tap size falls off a cliff. Taps of 0.3 per cent are blocked falsely on nearly every raise; 0.5 per cent, half the time; 0.75 per cent, 3.3 per cent of the time; 1 per cent, 0.14; and above that, never in practice. A 1 per cent tap moves the voltage by 49 mV at the start of the raise, and the rule needs the last taps’ few millivolts to stand clear of the noise. The cliff is a consequence of the normal distribution’s tail: once the signal is a few standard deviations of the difference, each toss is nearly certain, and the product over twenty of them barely moves.

So the blocking rule constrains the regulator’s design in a direction its main job does not. Regulation wants fine taps, to hold the setpoint closely; protection wants taps coarse enough that a single step’s effect can be read through the measurement’s noise. The choice of tap size is a choice of how close to the nose the protection can be trusted, and it is usually made for regulation alone.

What the rule could measure instead #

The comparison of two single readings is the crudest estimate of the gain’s sign, and its failure is a statistical one with statistical cures. Averaging each reading over $m$ samples divides $\sigma$ by $\sqrt m$ and moves every curve on the margin figure to the right by that factor, at the price of a slower regulator. Requiring two consecutive wrong-signed taps before blocking squares the false-block probability per tap and delays a true block by one tap. And a regulator that knows the line — that estimates $R$ from its own taps and computes the nose ratio — can block on a model rather than on a difference, which removes the noise problem and adds a model’s errors in its place.

The load that has two voltages or none found that near the nose every power below the limit is delivered at two voltages; a regulator that falsely blocks below the nose sits on the upper branch, short of its setpoint, which is the safe one of the two. That is the asymmetry that makes a false block tolerable where a missed block is not: the first costs regulation, the second costs the supply. A rule that errs on the side of blocking, as this one does near the nose, errs in the right direction, and the question the figures answer is only how much regulation it gives up to do so.

What the march leaves out #

The load here is a fixed resistance at each demand. A load that restores its power — a motor drive, a regulated supply downstream — draws more current as its voltage falls, which makes the voltage curve steeper past the nose and the rule’s signal larger there, and makes the collapse without the rule faster. The headroom that is the line’s own charge found that a real line’s shunt capacitance moves the nose and changes how the far end’s voltage reads relative to it; a regulator watching a voltage that rises with the line’s charging current sees a different slope, and the rule’s test signal is read off that slope.

And the line is resistive. On a reactive line feeding a leading load, the far end that rises found the far end able to read above its source at the nose itself, so a regulator watching that voltage may never see a sag to correct until it is past the edge — the case in which the rule’s test is needed most and gets its first signal latest.

The numbers #

On a 10 V, 1 Ω line asked for 105 per cent of its 25 W nose, a 1.25 per cent tap changer with the blocking rule stops at n = 1.9625, the first tap past the nose ratio 1.9518, and holds 9.759 V, the most the line can give; without the rule it reaches n = 3.25 and 8.61 V by 900 s. Below the nose a noiseless rule never fires. With 3 mV of measurement noise it falsely stops a raise 0.3 per cent of the time at 98 per cent of the nose, 4.8 at 99 and 15 at 99.5. Kept under 1 per cent, it allows 97.9 per cent of the nose at 3 mV and 79.9 at 10 mV with 1.25 per cent taps, 94.8 at 10 mV with 2.5 per cent taps; at 95 per cent and 3 mV it needs taps of at least 1 per cent.

Still open: a rule that averages, a load that restores, and a reactive line #

A rule that averages. Averaging each reading over several samples, or requiring two wrong-signed taps in a row, trades the regulator’s speed for the rule’s reliability. The response time against the margin, for a stated noise, is the curve a protection engineer would want, and it has the normal tail in it.

A load that restores its power. A load with its own time constant, restoring its power after a voltage change, makes the voltage after a tap depend on when it is read. Whether the rule then fires on the load’s recovery rather than on the line’s gain — reading the sign right at the wrong moment — is a march with two slow states.

The reactive line with a leading load. The far end of a reactive line feeding a leading load reads high at the nose, so the regulator’s raise may begin only after the nose has been passed. Whether the rule then blocks immediately, at the first tap, and leaves the load on the lower branch — the dangerous one — is the case that would test the rule where its design assumption fails.