Do we get 2x the frequency when doing P=VI for AC Circuits?

[yosimba2000]

I know that when you get a current from a voltage (like calculating current through resistor), the current and voltage equations have the same frequency. Does this still hold for power P = VI ?

So assume V in Euler form, V(t) = Vcos(wt+theta) + jVsin(wt+theta) = V*e^j(wt+theta) and in Phasor, V<theta = Vcos(wt+theta)

And assume I in Euler form, I(t) = Icos(wt+phi) + jIsin(wt+phi) = I*e^j(wt+phi) and in Phasor, I<phi = Icos(wt+phi)

Then P = V(t)*I(t) = V*e^j(wt+theta) * I*e^j(wt+phi)

To get P = VI*e^{jwt+jwt+jphi+jtheta}, call jphi + jtheta = jpsi
Then P = VI *e^2jwt+jpsi
Then P = VI *e^j(2wt+psi)
Then P = VIcos(2wt+psi)+jVIsin(wt+psi)

Real component is what we want, so P = VIcos(2wt+psi)

2wt (power frequency) isn't the same as wt (the frequency of current and voltage). Did I get the right result?

 

[meBigGuy]

Think of the power as peaking on the positive voltage peak and the negative voltage peak, which is twice the frequency.

 

[jim hardy]

meBigGuy answered it

here's a picture to help

any time volts and amps have same sign their product is positive
and that happens twice per line cycle

see - you learned that in first year algebra. Much of learning is really just discovering what we already know.

 

[meBigGuy]

That's the picture I was looking for!

 

[Averagesupernova]

Not to hijack the thread but notice that there is now also a DC component. This is like AM modulation in that 2 signals are multiplied together to get sum and difference frequencies. In this case it so happens that the two input signals are the SAME frequencies.

 

[meBigGuy]

The trig identity for squaring a sinewave (since power is E^2/R) shows the DC and the 2x frequency
sin2(x) = ½[1 – cos(2x)]

 

[jim hardy]

½[1 – cos(2x)]

I envy you who are fluent in math.