Power absorbed by placing 50 Hz appliance on 60 Hz source

Question

We know that we can calculate AC power using

$$ S=VI^{*}=|S| \angle \varphi . $$

Given that a grid works as constant voltage source, we can calculate the absorbed power in

$$ S = |I|^{2}Z = \frac {|V|^{2}}{Z^{*}} . $$

The problem is, I did not see any frequency on that equation. Thus, how we can approximate power consumption of electric appliances with a rating of 50 Hz connected to 60 Hz grid?

I want to know the relation on at least three kinds of appliance: synchronous motor, asynchronous motor, and resistive based appliances such as incandescent lamp. Also, how about an electronic such as laptop that has AC to DC converter?

I assume that there must be a general formula.

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EDIT:

As explained here, placing an appliance in under frequency environment cause reduction in mechanical power output. What I'm understand, the electrical power consumed will also decreased.

The reverse must be true, that is placing an appliance in over frequency environment.

So, how to calculate the power consumed of 50 Hz appliance that will be connected in 60 Hz grid? I know I can just go an measure, but what I want is any equation, explanation, or any material that I can read about this.

Answer 1

There is no frequency term because, if there is any frequency dependence, it's already accounted for by the relationship between V and I. Voltage and current give you power. For power to be dependent on frequency, Z would have to be dependent on frequency, but in many cases it isn't (or at least it doesn't vary enough for us to care about).

For a resistive load, frequency doesn't matter at all, unless it becomes so high that your resistive load starts to look more inductive than resistive, or so high that skin effect substantially alters the resistance — but at 50 or 60 Hz, that's not usually a concern.

For a switching power supply, the current draw depends on the input voltage, the output (DC) power, and the conversion efficiency. There might be a very small difference in efficiency between different frequencies, but you shouldn't expect frequency to matter significantly as long as you stay within the specs on the label / datasheet.

For a synchronous motor or an induction motor, changing the input frequency will change the speed of the motor, which will probably change the amount of torque it supplies to the mechanical load (the thing the motor is turning), which will change the current drawn by the motor. But we can't write a purely electrical equation for this, we need to know something about the mechanical system to figure out what the effect will be.

A universal motor will operate at the same speed and draw essentially the same power regardless of the input frequency (they also work fine on DC). There might be slight differences of efficiency at different frequencies, but all in all, the run speed is a function of the input RMS voltage, and the power draw is dependent on the mechanical power delivered and the efficiency.