Hi,
there are many good application notes available in the internet.
They discuss the problem in detail. (good AN: semiconductor manufacturer)
I guess none of your formuals is generally true.
So, average power = average gate current * average gate voltage.
Take this as example.
Let´s consider a power supply 10V DC, a switch 50% duty cycle, a 10 Ohms resistor. (almost ideal parts)
Do you agree that the power that comes out of the power supply has to be the same as the power that is dissipated in the resistor? (where else should be the difference?)
Hopefully Yes.
Do you also agree
* that the current during ON is 10V / 10 Ohms = 1A?
* that the average current of a 50% duty cycle of 1A = 0.5A?
* that the average current of the resistor is the same as the current of the power supply?
* that the average voltage of the power supply is 10V?
* that the average voltage on the resistor is 5V
so the power of the power supply then is: 10V x 0.5A = 5W (this is the correct one)
and the power of the resistor is: 5V x 0.5A = 2.5W
You see there is a mismatch in power.
Even RMS does not work (here the value for the resistor is correct but not for the power supply)
Klaus