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But how is that the Watt/hour meter (I mean the meter which supplies data for consumption charging purposes to the energy supplier), is driven, at the supply voltage and frequency, by only real power?
Is it because the current which is the basis of the apparent power is out-of-phase with the current basis of the real power?
If so, why does the power developed by the out-of-phase current not do work in the meter's current coils?
The watt-hour meter (not watt/hour. that's watts per hour) performs real-time vector multiplication of current and voltage to resolve only the in-phase, real power component. In a traditional spinning-disc meter this is done magnetically by the interaction of the two fields set up by the voltage and current coils. Only the in-phase component of the field created by the current coil generates nett torque on the disc. In a smart meter the same multiplication is done electronically.
It is equally possible to arrange the meter coils to register only the reactive component. Both instantaneous kVAr meters and kVAr-hour meters are used on industrial supplies to monitor reactive power flow, because this is chargeable on large consumers' tarriffs. Not as energy supplied, of course, but to defray the cost of installing oversize transformers and cables to carry the reactive component of the current that does not pay for itself with energy supplied. Chargeable reactive power incentivises customers to maximise their power factor, i.e. minimise the current needed to supply their real power demand.