The definition. The voltage safety factor compares the thyristor's rated peak inverse voltage with the peak of the supply it must withstand:

\(V_f=\dfrac{V_{PIV}}{\sqrt2\times V_{rms}}\)

which is option 2.

Why the \(\sqrt2\) is there and why it multiplies rather than covers the whole denominator. Supply voltages are always quoted as RMS, but the device sees the instantaneous peak. For a sinusoid,

\(V_{peak}=\sqrt2\,V_{rms}\)

so the denominator \(\sqrt2\,V_{rms}\) is simply the peak voltage the thyristor must block. This is precisely what separates option 2 from option 1, where the radical is drawn over the entire denominator — that would give \(\sqrt{2V_{rms}}\), a quantity with the dimensions of the square root of a voltage, which cannot be compared with V_PIV at all. A dimensional glance settles the pair.

Why options 3 and 4 fail. Option 3 multiplies the maximum value by \(\sqrt2\) as well, converting the peak into \(\sqrt2\) times the peak — the factor is applied twice. Option 4 uses the average, which for a full sine is zero and for a rectified one is \(2V_m/\pi\); neither is the stress the device actually experiences, since blocking is an instantaneous matter, not an averaged one.

A worked example. On a 230 V RMS mains the peak is

\(\sqrt2\times230=325\ \text{V}\)

so a thyristor rated at 1200 V PIV gives

\(V_f=\dfrac{1200}{325}=3.7\)

comfortably inside the usual design range of 2 to 3 or more.

Why so generous a margin is needed. Line transients, switching surges from inductive loads and the reverse-recovery spike of the device itself can all momentarily exceed the nominal peak by a large factor. And the failure is unforgiving: exceeding V_PIV drives the thyristor into reverse avalanche and destroys it, unlike an overcurrent which a fuse can clear. Snubber networks and metal-oxide varistors are added for the same reason.

Hence, \(V_f=\dfrac{V_{PIV}}{\sqrt2\times\text{RMS value of the operating voltage}}\).