Electromechanical, thermal and treeing breakdown

Electromechanical breakdown

When voltage is applied to a solid between electrodes, the electrodes attract each other. In soft materials this electrostatic pressure compresses the solid from its original thickness $d_0$ to $d$. Equilibrium between electrical and mechanical stress:

$$\frac{\varepsilon_0\varepsilon_r V^2}{2d^2} = Y\ln\frac{d_0}{d}$$

where $Y$ is Young's modulus. The equilibrium becomes unstable at $d/d_0 = 0.6$; any further increase in voltage crushes the material and breakdown follows. The highest apparent stress is

$$E_{max} = \frac{V}{d_0} = 0.6\sqrt{\frac{Y}{\varepsilon_0\varepsilon_r}}$$

It occurs in about $10^{-8}$ s and matters for soft polymers (e.g. polythene at raised temperature).

Thermal breakdown

Dielectric losses heat the insulation:

$$W_{dc} = \sigma E^2, \qquad W_{ac} = \frac{E^2 f\varepsilon_r\tan\delta}{1.8\times10^{12}}\ \text{W/cm}^3$$

Since conductivity and $\tan\delta$ increase with temperature, losses grow as the material heats. Heat balance:

$$C_v\frac{dT}{dt} + \text{div}(K\,\text{grad}\,T) = \sigma E^2$$

If heat generated is less than heat conducted away, a stable temperature is reached. Above a critical field, heat generated always exceeds dissipation; temperature rises without limit (thermal runaway) and the material melts or burns. Thermal breakdown strength is lower under AC than DC, and falls with higher ambient temperature, poor cooling and greater thickness.

Breakdown due to treeing and tracking

Treeing: At sharp points, protrusions, voids or impurities inside the solid the local field is very high. Partial discharges there erode fine channels, which branch out like a tree. Electrical trees grow by repeated discharges; water trees grow in polymer cables with moisture. When a branch reaches the other electrode, the insulation fails.

Tracking: On the surface of an insulator, a film of dirt and moisture carries leakage current. Uneven drying forms dry bands, small arcs strike across them and carbonize the organic surface. The carbon track grows with each cycle until it bridges the electrodes, causing surface flashover.