How to Calculate Breakdown Voltage

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An electric field is a region of space around a charged particle that exerts a force on other charged particles. The direction of this field is the direction of the force the field would exert on a positive test electric charge. The strength of the electric field is volt per meter (V/m). Technically, insulators do not conduct electricity but if the electric field is large enough, the insulator breaks down and conducts electricity.

This can sometimes be seen as an electric discharge or arc in air between the two electrodes. The breakdown voltage of a gas can be calculated from ​Paschen's Law.​ The physics is different for semiconducting diodes where the breakdown voltage is the point where the device starts conducting in reverse-bias mode.

The Breakdown Voltage

Diodes and Semiconductors

Diodes are typically made of semiconducting crystals, usually silicon or germanium. Impurities are added to create a region of negative charge carriers (electrons) on one side creating an n-type semiconductor, and positive charge carriers (holes) to make a p-type semiconductor on the other.

When the p-type and n-type materials are brought together, a momentary charge flow creates a third region or depletion region where no charge carriers are present. A current flows when a sufficiently higher potential difference is applied to the p-side than the n-side.

A diode typically has a high resistance in the reverse direction and does not allow electrons to flow in this reverse-biased mode. When the reverse voltage reaches a certain value, this resistance drops and the diode conducts in reverse-biased mode. The potential at which this occurs is called the ​breakdown voltage​.

Insulators

Unlike conductors, insulators have electrons that are tightly bound to their atoms which resists free electron flow. The force holding these electrons in place is not infinite and with enough voltage those electrons can gain enough energy to overcome those bonds and the insulator becomes a conductor. The threshold voltage at which this occurs is known as the breakdown voltage or ​dielectric strength​. In a gas, the breakdown voltage is determined by ​Paschen's Law​.

Paschen's Law is an equation that gives the breakdown voltage as a function of atmospheric pressure and gap length and is written as

V_b=\frac{Bpd}{\ln{(Apd)}-\ln{(\ln{(1+1/\gamma_{se})})}}

where ​Vb is the DC breakdown voltage, ​p​ is the pressure of the gas, ​d​ is the gap distance in meters, ​A​ and ​B​ are constants that depend on the surrounding gas, and ​γse is the secondary electron emission coefficient. The secondary electron emission coefficient is the point where incident particles have enough kinetic energy that when they strike other particles, they induce the emission of secondary particles.

Calculating the Breakdown Voltage of Air Per Inch

An air gap breakdown voltage table can be used to look up the breakdown voltage for any gas. Where a reference manual is not available, the dielectric strength calculation for two electrodes separated by one inch (2.54 cm) can be calculated using Paschen's Law where

A​ = 112.50 (kPacm)−1

B​ = 2737.50 V/(kPa.cm)-1

γse = 0.01

P​ = 101,325 Pa

Plugging those values into the above equation yields

V_b=\frac{2737.50 \times 101,325 \times 2.54 \times 10^{-2}}{\ln{(112.50 \times 101,325 \times 2.54 \times 10^{-2})}-\ln{(\ln{(1+1/0.01)})}}=20.3\text{ kV}

From engineering and physical tables, the typical range for the breakdown voltage in air is expected to be 20 kV to 75 kV. There are other factors that influence the breakdown voltage in air, e.g., humidity, thickness, and temperature, hence the wide range.