The far end that rises

A supply feeds a load through something with impedance in it — a cable, a transformer winding, a length of overhead line — and the voltage at the load is lower than the voltage at the source. How much lower is called the voltage regulation, and it is quoted as a percentage of the no-load value.

A percentage is a single number, and the quantity it describes is a function of two things. One of them is the load’s size, which the percentage does contain, at one stated load. The other is the load’s angle, which it does not contain at all — and past a certain angle the sign changes and the far end is higher than the source.

What the four curves are #

Each curve is one load angle, swept over three and a half decades of load resistance. The angle is the load’s power factor angle: positive is lagging — a motor, a transformer, anything inductive — and negative is leading, which in practice means capacitive.

At unity power factor the load voltage rises monotonically towards the source’s own and gets to 0.9675 of it at the largest load drawn. It cannot exceed it: a resistive load and a source with any impedance at all form a divider, and a divider of two impedances cannot have a magnitude ratio above one when one of them is purely resistive and in series.

At thirty degrees lagging it is lower everywhere, which is the case everybody has in mind when “voltage regulation” is quoted. An inductive load’s current lags, and it lags in the same direction the line’s own reactance drops voltage in, so the two add.

At thirty and sixty degrees leading the curve crosses one. The load’s capacitive current leads, its drop across the line’s inductive reactance is in the opposite sense, and past a certain size of load it more than makes up the resistive drop.

That is the same phenomenon that raises the voltage at the end of a long lightly-loaded transmission line, and it is the reason a line’s terminal voltage is a thing to be managed rather than merely tolerated.

The condition, which contains the load size #

Writing the source impedance as $R_s + jX_s$ and the load as $R_L(1 + j\tan\varphi)$, the load voltage exceeds the source voltage when $|Z_L| > |Z_s + Z_L|$, which reduces to

$$2R_L\left(R_s + X_s\tan\varphi\right) + R_s^2 + X_s^2 < 0$$

Two readings of that are worth having.

As a condition on the angle, it depends on the load’s size. Solving for $\varphi$ gives a crossing at $-\arctan\!\big((R_s + (R_s^2+X_s^2)/2R_L)/X_s\big)$, so a larger load needs less lead. Only in the limit of a very large load does it become the tidy $-\arctan(R_s/X_s)$ that a textbook prints.

That distinction is not academic and it caught the first version of this figure. Bisecting the crossing on a sweep that stopped at thirty-one times the source resistance gave 17.60° where the asymptote says 14.04° — a twenty-five per cent disagreement between a measurement and a closed form, which looks exactly like an error and is the finite load being finite.

As a condition on the line, it needs reactance. With $X_s = 0$ the left-hand side is positive for every angle and every load: a resistive source cannot be raised. The rise is a partial resonance between the line’s reactance and the load’s, and a resonance needs two reactances of opposite sign.

The measured column comes from bisecting the solved family; the asymptote column is the closed form above. They agree to a per cent at the top of the table and diverge towards the bottom exactly as the finite-load term predicts.

Why the reactance-to-resistance ratio decides it #

The asymptote $-\arctan(R_s/X_s)$ says the crossing angle is set by the line’s own angle and nothing else, which is worth turning round: the far end rises whenever the load’s angle is more leading than the line’s is lagging.

That is a statement about matching two angles, and it is the same shape as the maximum-power-transfer result the field’s neighbouring essay measures, where the best resistive load is the magnitude of the source impedance and a conjugate-matched load extracts more.

The ratio $X_s/R_s$ is a strong function of what the line is:

• a short length of building wiring is nearly all resistance, ratio well below one, and no practical load angle will raise its far end;
• a transformer winding is a few times more reactive than resistive, so the crossing sits in the range a power-factor-corrected load can actually reach;
• a long overhead line is dominated by its reactance, ratio of five or more, and the crossing is at ten or fifteen degrees of lead — which is a light load on a line whose own shunt capacitance is supplying it.

The two things a load’s angle changes #

It is worth separating them, because they are usually merged into “a bad power factor causes voltage drop” and they behave differently.

The current is larger for the same real power. At a power factor of 0.8 the current is 1.25 times what it would be at unity for the same watts, so the resistive drop along the line is 25 per cent larger. That part is symmetric: leading and lagging by the same angle cost the same, and it is the part a correction capacitor is bought to fix.

The current is at an angle to the drop it produces. The line’s own impedance is $R_s + jX_s$, and the voltage lost along it is that impedance times the current — a phasor multiplication, so the direction of the loss depends on the current’s angle. A lagging current turns the drop towards being in phase with the source voltage, which subtracts from its magnitude efficiently; a leading one turns it away, and past far enough the subtraction becomes an addition.

The second effect is the one this page is about, and it is the one that has a sign. It is also the reason the two curves at plus and minus thirty degrees in the figure are not mirror images of each other about the unity-power-factor one: at plus thirty the load is 0.87 of the source voltage at the largest load and at minus thirty it is 1.05, which is not a symmetric pair.

What a resistive line does instead #

The zero-reactance frame is worth a paragraph of its own, because it is the case most low-voltage wiring is in and its behaviour is different in kind rather than in degree.

With $X_s = 0$ the drop is $I R_s$ and its direction is the current’s direction. A leading current and a lagging current of the same magnitude then produce drops of the same magnitude at mirror-image angles, and both reduce the terminal voltage by the same amount. There is no asymmetry to exploit and no crossing to find.

Which is why power-factor correction on a short resistive run buys almost nothing in voltage — it reduces the current, and the drop with it, but it cannot turn the drop around. The rise this page is about is a property of reactive lines, and correcting for it on a resistive one is solving a problem that is not there.

Correction, and the amount that is too much #

Power-factor correction is the deliberate use of this effect, and the interesting number is where the deliberate part stops.

A lagging load is corrected by putting capacitance across it, which cancels part of its reactive current. Cancel exactly and the load looks resistive, the current is at its minimum for the power being delivered, and the voltage drop along the line is as small as a resistive drop can be.

Cancel too much and the load is leading. The current starts to rise again, the drop reappears with the opposite sign, and past the crossing above the terminal voltage climbs above the source. On a distribution network that shows up as an over-voltage at light load, which is why correction capacitors are switched out at night, and why the sizing question is not “how much capacitance cancels the inductance” but “how much cancels it at the load that will be there”.

This field’s correction essay measures the same trade from the other side: a capacitor exact at one load is 0.23 leading a few times the resistance away, which is the same statement in the same units.

The quantity a percentage cannot carry #

Putting the pieces together, a regulation figure quoted as a percentage is a measurement of one point of a two-dimensional surface, and neither coordinate is stated.

It assumes a load size, usually the rated one, and the curves above are not flat: at a tenth of rated load the drop is a tenth of the size, and the crossing angle is different too.

It assumes a load angle, usually unity or the rated power factor, and the difference between unity and thirty degrees lagging on this line is several per cent of the terminal voltage.

And it hides the sign change entirely. A specification that says “regulation better than 5%” is satisfied by a line whose far end sits six per cent above the source at light leading load, which is a different problem with a different fix.

The honest form of the specification is the surface, or at least two points of it, and the reason it is rarely given that way is that it takes a figure rather than a number.

Where the same effect is a hazard rather than a curiosity #

Three cases, all of which are the figure above with different numbers.

A lightly loaded cable. A long cable’s own shunt capacitance is a leading load, and with nothing at the far end it is the only load. The terminal voltage then rises, which is a well-known problem on transmission systems and a smaller one on any long run of screened cable.

A capacitor bank left switched in. The classic overnight over-voltage, and the reason correction is switched with load.

A supply with a capacitive input filter at the end of a long lead. The rectifier’s reservoir capacitor is not a linear load, but the fundamental component of its current leads, and the same arithmetic applies to that component — which is one reason the field’s rectifier essays measure the true power factor rather than the displacement factor.

What to do with the number #

The crossing angle is a design quantity rather than a curiosity, and it is used in three ways.

As a limit on correction. Size the capacitance so that the load angle at the lightest expected load stays on the lagging side of the crossing. That is a stricter condition than “do not over-correct at full load”, and it is the one that keeps the terminal voltage below the source’s at every hour of the day.

As a warning about long lightly-loaded runs. A cable’s own capacitance is a leading load that cannot be switched out, so a long run with nothing on the end of it is permanently past its own crossing. The remedy is a shunt reactor, which is a deliberate lagging load added to bring the angle back — the opposite operation to correction, done for the same reason.

As the first check on a regulation specification. If the line’s reactance-to-resistance ratio is above about two, the crossing sits inside the range of load angles the installation will actually see, and a regulation figure quoted at one power factor is not enough information.

None of those needs the figure to be redrawn: they need the two numbers in the table, which are a property of the line and are available before any load is connected.

What is checked #

The figure holds three claims, and the second is the two-routes one.

That at unity power factor the load voltage never exceeds the source’s — asserted to a part in $10^{12}$, so that a sign error anywhere in the phasor arithmetic would show up as a rise where none can exist.

That the crossing angle from bisecting the solved family matches the closed form $-\arctan\!\big((R_s + |Z_s|^2/2R_L)/X_s\big)$ to two parts in a thousand, at every source reactance the slider offers. Those two routes share the netlist and nothing else: one is a search over solves, the other is algebra.

And that with no source reactance no load angle down to 89.5° of lead produces a rise at all — which is the refusal that makes the rest of it a statement about the line rather than about the load.

Where the leading load comes from #

A load leading by more than 28 degrees is not a natural load, and the reason this essay matters is that the commonest way to make one is a repair for a different problem.

The capacitor that was right once is that repair: cancelling a load’s reactive power needs one division and no iteration, and the answer is exact for the load it was computed from at the frequency it was computed at. A feeder correctly corrected at full load is over-corrected at light load, because the capacitor’s current does not fall with the load and the load’s lagging current does — so the installation walks into leading territory every night, which is where this essay’s rise happens.

The reactance cancelled, and the resonance it buys is the series version of the same repair and produces a second mechanism on the same feeder: a resonance at the line frequency times the root of the fraction cancelled, so a quarter compensation resonates at exactly half the line frequency and a ninth at exactly a third. Those are harmonic orders that exist — a rectifier’s supply current is full of them, as the neutral that carries more than a line shows — so a feeder can be simultaneously over-corrected into a voltage rise and resonant at a harmonic that is being driven.

Which is why the angle measured here is worth having as a number rather than as a caution. It is computable from the line’s own impedance, it has no load in it, and it is the threshold that says whether a correction scheme’s light-load behaviour is a small inefficiency or a voltage the equipment at the far end was not specified for.

A percentage that carries neither of the things that decide it #

The opening complaint of this essay is worth restating at the end, because the measurement has made it concrete. Regulation is quoted as a percentage — how far the far-end voltage falls when the load is applied — and the percentage contains neither the load’s angle nor the line’s reactance, which are the two quantities that decide whether the fall is a fall at all.

That is the same defect several other specifications in this collection have, and it is worth recognising by its shape. A picosecond, read as bits is a figure quoted without the input frequency that turns it into a resolution; the probe is part of the circuit is a bandwidth quoted without the source impedance that turns it into an error; and the constant that is a window is a derivative quoted without the current it was taken at. In every case the number is exactly true about one condition and the condition is the thing omitted.

What makes this one worse than those three is that the omission changes the sign. An aperture figure and a probe bandwidth are optimistic in a known direction; a regulation percentage quoted for a lagging load and applied to a leading one is describing a fall where there is a rise. A specification that is wrong about direction cannot be used with a safety margin, which is the usual remedy for a specification that is merely optimistic. The remedy here is the one this collection asks for everywhere: quote the percentage with the load angle it was measured at, and the angle at which it changes sign.