Dielectric Strength & Capacitor Design

An insulator holds its electrons bound, but only up to a point. Raise the field and an electron freed by chance gains energy eEλ between collisions; once that exceeds the ionisation energy of the molecules around it, every collision liberates more carriers and the current avalanches. As an order of magnitude, 10 eV over a 1 nm mean free path needs about 10¹⁰ V m⁻¹, and measured strengths sit well below that because defects, trapped gas and local heating trigger the cascade first. Typical figures: dry air 3 × 10⁶ V m⁻¹, paper about 1.6 × 10⁷, Pyrex 1.4 × 10⁷, PTFE 6 × 10⁷, mica above 10⁸, and thin polypropylene film up to 6 × 10⁸. Treat these as design guides, not constants. Dielectric strength rises in thinner films, falls as temperature rises and as the stress is held for longer, and differs between DC, AC and impulse testing. Breakdown in a solid is also permanent: a puncture leaves a conducting channel, so the part does not recover the way an air gap does.

For parallel plates the two design equations pull in opposite directions: C = κε₀A/d rewards a thin dielectric and Vₘₐₓ = Eₘₐₓ d punishes it. Their product removes d entirely, Qₘₐₓ = C Vₘₐₓ = κε₀ A Eₘₐₓ, so the largest charge a design can hold is set by plate area and material, never by how thin the film is. Take metallised polypropylene, κ = 2.2 and Eₘₐₓ = 6.0 × 10⁸ V m⁻¹, rolled to A = 0.20 m² at d = 6.0 μm. Then C = 2.2 × 8.854 × 10⁻¹² × 0.20 / (6.0 × 10⁻⁶) = 0.65 μF, and the film punctures at V = 6.0 × 10⁸ × 6.0 × 10⁻⁶ = 3.6 kV. Halve the thickness to 3.0 μm at the same area and C doubles to 1.30 μF while the breakdown voltage halves to 1.8 kV. Neither is the better design; they are the same physics divided differently.

Put the two limits together: Uₘₐₓ = ½ C Vₘₐₓ² = ½ (κε₀A/d)(Eₘₐₓ d)² = ½ κε₀Eₘₐₓ² A d. Thickness cancels again, leaving energy proportional to the dielectric volume with uₘₐₓ = ½κε₀Eₘₐₓ² as the constant of proportionality. The polypropylene above gives uₘₐₓ = ½ × 2.2 × 8.854 × 10⁻¹² × (6.0 × 10⁸)² = 3.5 × 10⁶ J m⁻³, and its 1.2 cm³ of film holds 4.2 J at the instant it fails. That one expression settles which material to use. A Class-2 ceramic with κ ≈ 3000 has 1400 times the permittivity, but its usable field is nearer 1.5 × 10⁷ V m⁻¹, giving uₘₐₓ = 3.0 × 10⁶ J m⁻³ — slightly worse. Permittivity is not the figure of merit; κEₘₐₓ² is, and the square is what decides it.

No dielectric insulates perfectly. Model the loss as a resistance Rₗₑₐₖ = ρd/A in parallel with the capacitor, ρ being the bulk resistivity, and the stored charge decays as exp(−t/τ) with τ = Rₗₑₐₖ C = (ρd/A)(κε₀A/d) = ρκε₀. Area and thickness cancel: the self-discharge time constant is a property of the material alone, which is why datasheets specify insulation resistance as an RC product in ohm-farads rather than in ohms. Polypropylene at ρ ≈ 10¹⁶ Ω m and κ = 2.2 gives τ ≈ 1.9 × 10⁵ s, about 54 hours, so a charged film capacitor is still dangerous a day later. A lossier dielectric with ρ = 10¹² Ω m and κ = 3 gives τ ≈ 27 s, useless for holding a sampled voltage. Real parts fall short of the bulk figure, because surface films, humidity and terminations shunt the dielectric, and resistivity drops steeply as temperature rises.

Capacitance integrates over the whole plate area, so a few thin patches barely move it and the tolerance follows the mean. Write ΔC/C = Δκ/κ + ΔA/A − Δd/d; with independent spreads of ±2 % in κ, ±1 % in A and ±3 % in d, quadrature gives about ±3.7 %, which is why ±5 % film parts are routine while Class-2 ceramics are graded −20/+80 %. Breakdown obeys the opposite statistic. It occurs at the weakest point in the roll, so it follows the extreme of the thickness and defect distributions, not their mean — and doubling the area, which reliably doubles C, lowers the expected breakdown voltage, because there is twice as much film in which to find a flaw. Trapped gas is the usual flaw: across a flat void the normal component of D is continuous, so E_void = κE, and a pocket inside polypropylene sees 2.2 times the field the polymer feels while air withstands far less. Micron-scale voids survive better than that suggests, since Paschen's law raises the breakdown of very short gaps, but partial discharge still erodes film capacitors over years.

All of this is why ratings are derated in use: run at Eₘₐₓ/2 and the usable energy falls by four, so the 0.65 μF film above drops from 4.2 J to 1.05 J at 1.8 kV. Four further effects matter. Series resistance dissipates Iᵣₘₛ² × ESR, and that heat raises leakage and lowers dielectric strength, a loop that can run away. Class-2 ceramics have a strong voltage coefficient, and an X7R part can lose more than half its capacitance at rated voltage, so the printed value is a small-signal value. Dielectric absorption returns charge trapped in slow polarisation after a short, so a discharged high-voltage capacitor can recover hundreds of volts on its own. And metallised film self-heals: a puncture vaporises the thin electrode around it and isolates the fault, so the part loses a little capacitance instead of shorting, whereas foil electrodes and electrolytics fail outright, the latter also demanding correct polarity.