Dielectric Test Cell (2D)

Paschen's law

In a uniform field a gas gap breaks down when electrons gain enough energy between collisions to sustain an avalanche. Townsend's criterion gives V_b = B p d / [ln(A p d) - ln(ln(1 + 1/γ))], with gas constants A, B (air: 15 per cm Torr and 365 V per cm Torr) and secondary-emission coefficient γ. The curve has a minimum at p d = e ln(1 + 1/γ) / A (about 0.57 Torr cm in air, 327 V); to the left of it the gas is too thin to ionise, to the right the electrons collide too often.

Series dielectric

With a solid slab of thickness t and relative permittivity εr in series with an air gap d, the normal component of D is continuous, so εr E_s = E_air and V = E_air (d + t / εr). The air gap sees E_air = V / (d + t/εr), more than the average field, and so can break down first and stress the slab (the origin of partial discharge in voids).

Non-uniform field

Sharp electrodes concentrate the field. With utilisation factor η = E_mean / E_max the critical local field is reached at V_b,eff ≈ η V_b. This first-order rule is only a rough guide: real point-plane gaps also show corona stabilisation and polarity effects that are not modelled.

Charging and the arc

The source charges the gap capacitance C = ε0 A / d_eff through a resistor: R C dV/dt = V_s - V. At breakdown the stored energy ½ C V^2 is released in the spark and the voltage collapses.

Safety note

Educational model of a laboratory test cell; real high-voltage work needs interlocks, grounding sticks and trained staff.

• Sweep the gap from 0.05 mm to 30 mm at 101 kPa and watch the operating point move along the Paschen curve.
• Lower the pressure to a few kPa and see the breakdown voltage fall, then rise again for very low pressure.
• Insert a 3 mm PE slab: the air gap field rises above the average and the arc moves to the gap.
• Switch the gas to helium or argon and compare the minimum voltage.