Switching Surges and Travelling Waves

Chapter 22 traced overvoltages to an external cause — the lightning flash. But the system is also its own troublemaker. Every time a circuit breaker opens or closes, it suddenly rearranges where energy is stored in the network's inductances and capacitances, and the network responds with a transient: a switching surge. Energizing a long line, reclosing onto a trapped charge, clearing a fault, or switching a capacitor bank all launch such transients, and on the longest extra-high-voltage lines these internally generated surges, not lightning, decide how much insulation the line needs.

Whatever its origin, a surge obeys the same travel law. A disturbance applied at one point on a line does not raise the whole line at once; it sets off a wave of voltage and current that moves along the conductors at close to the speed of light, and behaves at every change of line — a junction, an open end, a transformer — like a wave meeting a boundary. Mastering that wave behaviour is the key that unlocks both switching and lightning surge analysis, so we build it first.

Model the line as a chain of series inductance \(L\) and shunt capacitance \(C\) per unit length. A voltage step applied at one end charges the first capacitor through the first inductor, which then charges the next, and so on — the disturbance is handed down the line at a finite speed. Solving the resulting wave equation gives two fundamental quantities. The velocity of propagation is

For an overhead line in air this works out to almost exactly the speed of light, \(3\times10^{8}~\mathrm{m/s}\); in a cable, where the dielectric slows it, it falls to roughly half that. The second quantity is the surge (characteristic) impedance, the fixed ratio of voltage to current within a single travelling wave:

Typical values are \(300\)–\(500~\Omega\) for an overhead line and only \(20\)–\(60~\Omega\) for a cable, whose large capacitance lowers \(Z\). The surge impedance is not a resistance that dissipates energy; it is the ratio the line demands between the voltage and current of a wave passing along it. Every wave carries this fixed partnership — and that is exactly why a wave is disturbed when it reaches a place where the line wants a different ratio.

Suppose a wave travelling on a line of surge impedance \(Z_1\) reaches a junction with a second line of impedance \(Z_2\) — an overhead line meeting a cable, say, or a line meeting a transformer. The incoming wave cannot simply continue, because the two sides demand different voltage-to-current ratios. Nature reconciles them by splitting the wave: part refracts (transmits) onward into \(Z_2\), and part reflects back along \(Z_1\), the two adjusting so that voltage and current are continuous across the junction. Enforcing that continuity gives the two governing coefficients:

The reflection coefficient \(\Gamma\) lies between \(-1\) and \(+1\). When the wave passes into a higher impedance (\(Z_2>Z_1\)), \(\Gamma\) is positive and the transmitted voltage rises above the incident — it can approach double. When it passes into a lower impedance, \(\Gamma\) is negative and the transmitted voltage falls. The current coefficients are the mirror image (the current reflection coefficient is \(-\Gamma\)), because a voltage that doubles at a high-impedance end must be met by a current that nearly vanishes there. This single pair of formulas governs every junction a surge can meet.

Three terminations are worth committing to memory, because the whole network is built from them. Setting \(Z_2\) to its extreme values in the coefficient formulas gives each one immediately.

At an open circuit (\(Z_2\to\infty\)) the reflection coefficient is \(+1\): the reflected wave adds to the incident, and the voltage at the open end doubles to \(2V\). This is the most important single fact in surge protection — a surge running into an open breaker, an unloaded transformer terminal or a line end is amplified twofold, which is why those points are so vulnerable. At a short circuit (\(Z_2=0\)) the coefficient is \(-1\): the reflected wave inverts and cancels the incident, holding the voltage at zero (while the current doubles). At a matched end (\(Z_2=Z_1\)) the coefficient is \(0\): nothing reflects, and the wave is absorbed as if the line continued forever — the ideal that surge-absorbing terminations strive for.

A real surge does not reflect once and stop; it bounces between the two ends of a line, reflecting at each with that end's coefficient, the voltage at any point building up from the overlap of all the waves present at that instant. Keeping track of this by hand is the job of the Bewley lattice diagram — a simple but powerful bookkeeping chart. Distance along the line runs across the top; time runs downward. Each travelling wave is a sloped line crossing the chart in one transit time \(T=\ell/v\); at each end it turns back, multiplied by that end's reflection coefficient. Reading across at any chosen time and place, and summing the waves that have arrived, gives the voltage there and then.

The lattice makes visible why surges are dangerous and why timing matters. Successive reflections can add, pushing the voltage at a vulnerable point well above the original wave before the transients finally damp out through line losses. It also shows that the worst stress often appears not at the instant the surge arrives but a transit time or two later, once a reflection has returned — a subtlety that pure peak-value thinking misses.

With the wave tools in hand, return to the surges the system makes itself. Switching overvoltages arise whenever a breaker abruptly changes the network, and a few mechanisms dominate:

Two features set switching surges apart from lightning. First, their magnitude depends on the instant in the power-frequency cycle at which the contacts close — the point on the wave — so the surge is essentially random, varying from operation to operation, and must be described by a statistical distribution rather than a single value. Second, the most violent cases involve energy already trapped in the system: reclosing onto a line that still holds charge from a previous opening can superimpose the new surge on the old, and capacitor or reactor switching can pump energy up through repeated restrikes, each one ratcheting the voltage higher. Magnitudes commonly reach \(2\)–\(3~\mathrm{p.u.}\) — lower in crest than a direct lightning stroke, but applied to the whole line at once and lasting far longer.

Because switching surges are slow-front and statistical, they are tested with their own standard waveform: the \(250/2500~\mu\mathrm{s}\) switching impulse, with a time-to-crest of \(250~\mu\mathrm{s}\) and a time-to-half-value of \(2500~\mu\mathrm{s}\) — hundreds of times slower on the front than the \(1.2/50\) lightning impulse.

Why a separate waveform? Because long air gaps are weakest against slow-front stress. The breakdown strength of a large air gap, plotted against the front time of the applied impulse, sags to a minimum near the switching-impulse range — a gap can withstand a fast lightning spike better than a slow switching hump of the same crest. As a result, on systems at \(300~\mathrm{kV}\) and above, and decisively at UHV, it is the switching impulse that sets the insulation level, while lightning is held in check by arresters. The withstand itself is statistical: the air gap has a \(50\%\) flashover voltage \(U_{50}\) with a spread, so insulation coordination works in probabilities — a theme taken up fully in Chapter 24.

Unlike lightning, switching surges are made by equipment the utility controls, so they can be attacked at the source. The classic remedy is the pre-insertion closing resistor: a resistor briefly inserted in series as the breaker closes, damping the energizing transient before it is shorted out, which can cut the surge from around \(2.5\) toward \(1.5~\mathrm{p.u.}\). The modern alternative is controlled (point-on-wave) switching, in which the breaker is commanded to close each pole at the instant its voltage is near zero, so almost no transient is launched in the first place. Shunt reactors help bleed away trapped charge so that reclosing does not find a charged line, and over everything sits the surge arrester, which clips whatever overvoltage survives to a safe ceiling.

These measures, together with the travelling-wave understanding of how surges build and where they double, are precisely the inputs the next chapter needs. Knowing the size and shape of both lightning and switching surges, and how arresters and resistors limit them, is what allows the insulation of a whole system to be coordinated — strong enough to survive, economical enough to build.

For each item, first identify what it tests — the wave parameters, the reflection/refraction coefficients, a special termination, the lattice idea, switching-surge causes, or the switching impulse — then apply it. Difficulty rises down the list.