Generation of High Alternating Voltages

The grid runs on alternating current, so the apparatus that serves it — transformers, cables, switchgear, insulators — is built for AC and must be proven at AC. Standards demand it: before a piece of equipment enters service it faces a withstand test, held for a minute or more at a voltage well above its rating, to confirm the insulation survives. Generating that test voltage is the task of this chapter.

There is a twist that shapes everything. The test object is rarely a resistor that draws real power; it is overwhelmingly a capacitor. A long cable, a bushing, a length of gas-insulated bus — each is two conductors separated by a dielectric, which is the very definition of a capacitor. When we raise such a load to high AC voltage, almost no power is consumed, but a large charging current flows in and out every cycle. The source's burden is therefore not watts but reactive volt-amperes, and as we will see, that distinction is what makes high-AC generation interesting — and what the resonant circuit later exploits so cleverly.

The workhorse for moderate test voltages is the high-voltage testing transformer: a step-up transformer with a low-voltage primary and a heavily insulated high-voltage secondary, usually with one end of the secondary solidly earthed so that only the live terminal needs full insulation. It looks like a power transformer but is bred for a different life. A power transformer carries large continuous current at modest voltage; a test transformer carries tiny current at enormous voltage, for short periods, with its winding insulation carefully graded from the earthed end to the live end.

Because the load is a capacitance \(C\) raised to voltage \(V\) at angular frequency \(\omega = 2\pi f\), the current it draws is the capacitive charging current, and the apparent power the transformer must furnish follows at once:

That \(V^{2}\) is the sting. Doubling the test voltage quadruples the volt-ampere demand, so a single transformer's rating climbs steeply with voltage. This is manageable up to a point — but to reach the highest voltages, a single unit becomes impossibly large and hard to insulate, and we must turn to cascading.

Above roughly a megavolt, building one transformer with secondary insulation rated for the full voltage becomes prohibitive. The cascade solves this by stacking several identical units of modest voltage so that their high-voltage windings add in series. Each transformer sits on an insulating support raised to the potential of the stage below it, so unit number two floats at \(V\), unit three at \(2V\), and the top terminal reaches \(nV\) for \(n\) stages — yet no single transformer is insulated for more than \(V\) between its own windings.

The cascade hides a subtlety that governs its design. The excitation for the upper units must be passed up through the lower ones, so the lower transformers carry not only their own load but also the throughput power of every stage above them. If each secondary delivers power \(P = VI\), the transformer at the bottom must transmit \(nP\), the next \((n-1)P\), and so on up to the top unit's \(P\). The total installed capacity is therefore far larger than the delivered power:

Cascading conquers the voltage, but it does nothing about the current. Recall \(S = \omega C V^{2}\): for a long high-voltage cable, \(C\) can be hundreds of nanofarads, and at a few hundred kilovolts the charging current runs to tens of amperes and the apparent power to several megavolt-amperes. Supplying that with transformers means a vast, heavy installation whose entire output is reactive — it is shuttled in and out of the cable every cycle and never consumed.

This is wasteful in the extreme, and it invites a question: if the load is a capacitor that only ever stores and returns energy, why supply that energy from the source at all? Why not give the energy somewhere local to slosh into, an inductor, so the source need only top up the small real losses? That single idea — pairing the capacitive load with an inductor at resonance — is the most important development in modern high-AC testing.

Place a variable reactor (a high-voltage inductor) in series between the supply and the capacitive test object, and tune its inductance \(L\) until its reactance exactly equals that of the load capacitance \(C\) at the supply frequency. At that point the two reactances cancel — the inductor's appetite for current is met precisely by the capacitor's surplus — and the circuit is in series resonance:

With the reactances cancelled, the supply no longer sees a megavolt-ampere capacitor; it sees only the small resistance \(R\) that represents the reactor and dielectric losses. A modest low-voltage supply, providing just enough to make up those losses, drives a large resonant current around the \(L\)–\(C\) loop — and that current develops a very high voltage across the test capacitor. The reactive energy now sloshes back and forth between \(L\) and \(C\), borrowed from the source only once and thereafter recycled each cycle.

How high does the test voltage rise above the supply? The answer is the circuit's quality factor \(Q\), the ratio of reactance to resistance, which here doubles as the voltage magnification:

A reactor with \(Q = 50\) turns a \(5~\mathrm{kV}\) supply into a \(250~\mathrm{kV}\) test voltage, and the supply provides only \(1/Q\) of the apparent power the load would otherwise demand. Four virtues follow, and together they explain why resonant sets have displaced transformer banks for cable and GIS testing:

Practical resonant test systems come in two flavours. Variable-inductance sets keep the frequency at mains value and tune \(L\) to match each load. Variable-frequency sets keep a fixed reactor and instead sweep the supply frequency (typically \(20\)–\(300~\mathrm{Hz}\)) until resonance is found — lighter and now standard for on-site testing of long cables, where a tunable HV inductor would be impractical to carry.

One more AC source deserves mention, for high frequency rather than power frequency. The Tesla coil is an air-cored, doubly-tuned resonant transformer: a primary circuit with a capacitor and spark gap rings at high frequency and couples loosely into a secondary tuned to the same frequency, which builds up a very high oscillating voltage. Because the coupling is loose and the gap interrupts the drive, the output is a train of damped high-frequency oscillations rather than a steady sine.

Tesla coils are little used for routine withstand testing today, but they remain valuable for high-frequency insulation studies and as a vivid demonstration of resonant voltage magnification — the same principle as the series-resonant set, pushed to radio frequencies. They close the loop on a theme that runs through this whole chapter: at high voltage, resonance is not a curiosity but the central tool, because it lets a small source command a large voltage.

For each item, first identify what it tests — the charging-power formula, the cascade loading, the resonance condition, the \(Q\) magnification, or the power saving — then apply it. Difficulty rises down the list.