Query about the time average of an AC power supply

[Felipe good guy]

TL;DR

Considering purely resistive loads, the instantaneous electric power P(t) is investigated for the frequency dependence of its time average. This is relevant as different countries use different frequencies.

Hi there,
I have a query about electric power in AC systems. I am Chilean, and over here the power supply is 220 V (rms) at 50 Hz. Some countries use voltages in the range 200-230, but others use 110 V or so, at frequencies of 60 Hz. There is no uniqueness, but all work. My question is related to the frequency dependence of the time average of power supply:

Under the assumption of purely resistive load, V(t) and i(t) are in-phase, and the math it's much simpler than when capacitors/inductors exist. So for a pure resistor, let's say V(t)=V0.cos(w.t) and i(t)=i0.cos(w.t) (I will assume you know the variable-letter definitions).

Then the instantaneous power is P(t) = i0.V0.cos²(wt). This function is always positive, and looks like a time sequence of positive ''bells'' with amplitude i0.V0 and ''periodicity'' or ''frequency'' 2w.
So, my question is related to the dependence of P(t) on the frequency w. Beyond the instantaneous power here analytically given, I want to know about the average power per unit time, say per second.
If I consider 50 Hz, both i(t) and V(t) do 50 cycles(+,-) per second, but that implies that P(t) does 100 positive-bell-cycles per second. The average power will be the time integral (say over 1 second) of those 100 cycles (which gives me the energy delivered during that second), and then divided by 1 second. All good, so far so good.
Now, putting aside the amplitude (let's consider it fixed), let's consider a frequency of 60 Hz. In this case, P(t) does 120 positive-bell-cycles per second. The average power, following the aforementioned math, will be, surprisingly, the same !

This is the problem: The integral of P(t) = cos²(w.t) over one second of time is the same if w is 50 Hz, or 60 Hz, or 90 Hz, ..or 733 Hz ; cause when w increases, the individual positive-bell-cycle areas decrease proportionally, thus the integral of P(t) over time intervals that encompass integer numbers of cycles, is mathematically invariant under integer changes in w.

This means that the average power supply (on our appliances) is invariant to the delivery frequency w.

Am I right?

Felipe
Felipe

 

[Andrew Mason]

Correct. Root mean square of a sine wave is independent of frequency. It is $\sqrt{2}/2$ x amplitude.

AM

 

[Baluncore]

This means that the average power supply (on our appliances) is invariant to the delivery frequency w.
Am I right?

A resistive load will heat and partly cool on every half cycle of the AC supply. Light globe filaments have a short time constant so they flicker when the supply frequency is below 50 Hz where it is more than about 10 msec between energy peaks. So your observation is really only true for continuous sine waves that repeat faster than the cooling time of the resistive load.

With a single phase supply, the voltage falls to zero twice per cycle. If instead you have 3 phase AC then energy can flow continuously without ever falling to zero.