The math behind this is not so tricky as it may sound.
Has to do with that "amplitude" thing. As in Amplitude Modulation.
To get 100 percent modulation, the RF peaks will be twice the amplitude of the dead carrier. Amplitude here might be RF voltage, might be RF current. If you double one of them, you get double the other one, so it doesn't matter which of these two quantities you're measuring for this.
But here's where Ohm's law comes in. Put one Volt across one ohm, and you get one Amp of current.
Power is 1 Watt, right? Now boost it to two Volts. Same 1 ohm now has 2 Amps through it. But power is voltage times current. Two Volts times two Amps is now FOUR Watts. Double times double equals four.
Make it 3 Volts, times 3 Amps and the power in the 1-ohm resistor is now NINE Watts.
It's called a square law. Plenty of those found in physics. This is just one of them.
This is why a 100-percent modulated carrier has a PEP that's exactly four times the carrier power.
Double amplitude equals four times power.
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