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Breakdown Voltage (Insulator) = The minimum voltage that makes an insulator react as a conductor.
Breakdown Voltage (diode) = The minimum reverse voltage to make the diode conduct in reverse.
In Detail
Breakdown Voltage (Insulator)
Is a parameter of an insulator that defines the maximum voltage difference that can be applied across the material before the insulator collapses and conducts. This may create a weak point in the insulator from a molecular change created by the current flow.Two different breakdown voltage measurements of a material are the AC and impulse breakdown voltages. The AC voltage is the line frequency of the mains (either 50 or 60 Hz depending on where you live). The impulse breakdown voltage is simulating lightning strikes, and uses a 1.2 microsecond rise for the wave to reach 90% amplitude then drops back down to 50% amplitude after 50 microseconds.
Two technical standards governing performing these tests are ASTM D1816 and ASTM D3300 published by ASTM.
Breakdown Voltage (diode)
Is a parameter of a diode that defines the largest reverse voltage potential that can be applied without causing an exponential increase in the current in the diode. As long as the current is limited, exceeding the breakdown voltage of a diode does no harm to the diode. In fact, Zener diodes are essentially just heavily doped normal diodes that exploit the breakdown voltage of a diode to provide regulation of voltage levels.See also
"Breakdown voltage"