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Phil,
If one monitors the average voltage and current into a directly modulated RF final amplifier both will remain constant with or without modulation. This is because both the current and voltage swing above and below the quiescent values (assuming linear operaton) but remember that power is not the product of average AC volts times average AC amps. Power is the product of the instantaneous voltage and the instantaneous current. This is, of course, instantaneous power and to get the average power this is integrated over a time interval and is divided by the time interval. When measured in this manner, the average input power will increase with modulation by the amount calculated in my post above. The sideband power comes from the power input to the final as does that of the carrier.
Hope this helps.
Neil