Hmm, there would be a few regimes there, where different things might happen:
1. Low current. The voltage remains low until the last contact point lifts off. For the first for nanometers, current tunnels across the gap. Resistance then rises, capacitance charges and inductance begins to discharge. If the peak voltage is small (under 20V, say),
By the way, it is essentially impossible to ignore inductance here. Contact closures have rise/fall times in the subnanosecond domain. (Yes, the lowly relay, is actually one of the fastest, cheapest components you can buy! It's just incredibly sloppy...) So, a single inch of wire has enough inductance to matter, even in a high impedance circuit.
Or more accurately, wires are transmission lines. When a step change occurs, there is not an unlimited rise in voltage, or change in current -- the ratio is fixed, and this ratio is the characteristic impedance of the line. The impedance of a typical contact circuit will vary all over the place, but the important part is it is perfectly well defined, physically speaking.
The impedance might vary from, say, below 50 ohms within the contacts, to ~100 ohms over the springs/wires leading from the contacts, to 200+ ohms for loose wires traveling through space, and so on up and down, going along the rest of the wiring. But it is a perfectly knowable, bounded value -- just not a convenient one! (RF relays are basically made so that this impedance is nearly constant; they're also made with better shielding between poles.)
2. Medium current. As contact points lift off (and some smoosh back together, because it's a messy sliding and rotating and flopping motion, not a rigid one way movement!), resistance rises, and voltage rises. A lot of heat is dissipated in some areas, but not enough to cause melting and sparking. As the contacts further separate, at some point the resistance shoots way up in a fraction of a nanosecond, and the step current change induces a step voltage change on the contacts. This is enough (~20V+) to break down what little air has entered the gap (reminder: the mean free path in air is 68nm, so until the contacts are about that far apart, they are effectively filled with vacuum!), and a spark is maintained until the voltage runs out (the transmission line discharges its current) and the gap opens wide enough to clear what voltage remains.
This is the regime where EFT (electrical fast transient) occurs. The transmission line can act like a resonant circuit, and the arc has an overall negative incremental resistance. As the TL discharges, it ringing voltage "slaps" across the opening contacts, emitting a series of sparks. The sparks are rapid, on the order of 1ns, while the repeat rate is determined by the TL length. Each spark launches a wavefront of energy up the TL, where it couples into nearby wires, or radiates into space.
A typical EFT model is described in IEC 61000-4-4: The pulse has an amplitude of 2kV (adjustable depending on what you're testing), the repeat rate is 100kHz (which corresponds to a pretty long equivalent transmission line -- which might be typical of a contact opening on facility wiring plus motor/transformer inductance), rise time 5ns, and an exponential decay (FWHM 50ns). The source impedance is 50 ohms and the signal is coupled onto EUT (equipment under test) cables with a "capacitive clamp", which is really a transmission line structure itself, which is important because it makes a single-peak waveform in one direction and a double-peak waveform in the other (because of reflection).
These pulses would usually be generated from, hmm, I suspect a hydrogen thyratron really, but there's a couple of possibilities; but that doesn't matter. In the real world, any damn spark will do! (They wouldn't use actual spark gaps in test equipment -- too inconsistent.)
3. High current. As contact points lift off, resistance rises, and voltage rises. A lot of heat is dissipated in some areas, causing melting. The hot points produce thermionic emission, and as air enters, it is ionized by the electron stream. (If air doesn't enter quickly enough, or the current is even higher, the material itself can vaporize, due to further heating (vaporization) and ion bombardment (sputtering). An arc will then be sustained on the ejected material.)
Supported by emission, the arc can grow to quite long lengths before clearing. In the extreme -- when a lot of voltage is available -- a medium must be provided to extinguish it. Power line equipment has solved this problem with, for example: sand-filled fuses (the high purity quartz sand, melts and vaporizes without becoming conductive, absorbing arc energy and supplying more gas to snuff out the arc), gas filled switches (sometimes explosively triggered to be sure there's enough gas), or oil filled reclosers (the oil is vaporized and decomposed by the arc, generating gas; the oil also has a very high breakdown voltage, so less distance is required).
(There are also vacuum reclosers, which suffer from the melting and self-arcing problem, but once enough distance is cleared, they go back to an open state of absolutely nothing at all. Downside: all that blasted-out metal condenses on the walls, eventually turning it conductive..)
Incidentally, the same process in reverse, give or take a hearty dose of inrush current to boost things further, gets you welded contacts. Contacts normally weld anyway; it only becomes a problem when the coil/spring force isn't enough to break them apart!
So in summary,
0. Contact closures and openings are stupidly fast. There's always some final point of liftoff that makes a really fast edge (~1ns give or take).
1. If there's not enough current/voltage to do anything interesting, that's that. You get the edge, but it just goes out wherever.
2. If there is enough, then the contacts spark. The sparking makes multiple repeat edges, and they're bigger because they're supplied by what the line current was at, multiplied by the line impedance.
3. If there's a lot of current, then the contacts melt and arc. (You probably don't get so much EFT in this case, but still one big spike when the arc finally goes out.)
And so we come to snubbers.
What are we doing?
We are terminating the source end of the transmission line.
That's all there is to it!
So, R ~= characteristic impedance, and C ~= equivalent capacitance of the line. (If you don't have a direct measurement, use:
\$ C = \epsilon_0 \times (\textrm{Length}) \times (\textrm{Impedance of free space} / \textrm{Characteristic impedance})^2 / (\textrm{Velocity factor})
\$
Ballpark values, \$ \epsilon_0 \$ is 8.84pF/m, impedance of free space is 377Ω, characteristic impedance is 50-150Ω (varies with geometry -- twisted pair is around 100, coax is down around 50, twin lead is up at 300, etc.), and velocity factor is 0.67 (coax) to 0.8ish (twisted pair) to ~1 (bare wires in air). This gets you the rough capacitance of any wire in space or insulator -- very handy, it's just a bunch of ratios.
And you can always use more, up until the leakage current through the capacitance is excessive in the off state (since we're talking mains frequency, often). Or for DC, you have the problem of, when will it actually be "off"? -- you get the exponential tail of the cap charging.
If you prefer to work in lumped inductances rather than transmission lines, that's perfectly fine, too! Then you work with the figure sqrt(L/C), which has units of ohms -- the characteristic resonant impedance. You can set this so that, at peak load current, peak voltage (Vpk ~= sqrt(L/C) * Ipk) stays reasonable. Then put R = sqrt(L/C) in series, and you're done!
It's all about impedances. Impedance is just the ratio between voltage and current. Important stuff!
Tim
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