Capacitor discharge into a coil
Dump a charged capacitor into the coil: how big a current pulse, how fast, and does it ring?
- Ipk
- Peak current, A
- V0
- Charge voltage, V
- ωd
- Damped angular frequency √(ω₀² − α²), rad/s (the hyperbolic rate √(α² − ω₀²) when overdamped, and sin becomes sinh)
- L
- Coil inductance, H
- α
- Damping rate R ÷ 2L, per second
- tpk
- Time to peak, s
LaTeX
Ipk = \frac{V0}{ωd \cdot L} \, e^{-α \cdot tpk} \sin\left(ωd \cdot tpk\right)Method
- Find the undamped angular frequency ω₀ = 1/√(LC) and the damping rate α = R/2L. Their ratio ζ = α/ω₀ names the regime.
- Underdamped (ζ < 1): the current is (V₀/ω_d L)·e^(−αt)·sin(ω_d t) with ω_d = √(ω₀² − α²). It peaks at t = atan(ω_d/α)/ω_d and then reverses, ringing at ω_d/2π and decaying at α.
- Overdamped (ζ > 1): the current is (V₀/βL)·e^(−αt)·sinh(βt) with β = √(α² − ω₀²) — the same shape with sinh for sin — and peaks at ln(s₂/s₁)/(s₁ − s₂) where s₁,₂ = −α ± β. It never reverses.
- Critically damped (ζ = 1): the current is (V₀ t/L)·e^(−αt), peaking at t = 1/α with V₀/(αL)·e^(−1).
- The stored energy is ½CV₀², and since the capacitor ends up empty and the coil ends up with no current, every joule of it is dissipated in the circuit's resistance — mostly the coil's copper.
Assumptions
- Constant L, R and C throughout, and no initial current in the coil. A medium that saturates during the pulse lowers L as the current rises; an electrolytic bank's ESR rises with frequency. Both make the real peak a little different from this.
- The switch closes instantly and stays closed. A thyristor opens itself at the first current zero, which in the underdamped case cuts the ring off after its first half-cycle and leaves the capacitor charged the other way; an IGBT or a contactor does not.
- No freewheel or clamp diode. With a diode across the coil the reverse half-cycle is shorted out and the current decays through the diode at the coil's own L/R instead of reversing — which is what a pulse magnetiser does so the medium is not un-set by the swing back.
- Nothing here is Meyer's. The estate material does not describe how the EPG coils were driven beyond the pulse timings; a capacitor dump is what a workshop would use to get a low-resistance coil to a high current, and the page shows what that costs and what it does.