# Reactive Power Apparent Power Issue

1. Aug 8, 2011

### azaharak

This is not a HW question.

Assume you have a sinusoidally varying voltage, such that current lags by some phase angle phi

Then you can write the power as a function of time as

p(t)=i(t)v(t)=2*Vrms*Irms*cos(omega*t)*cos(omega*t-phi)

You can break this up via trig identity into the following

Eq1: p(t)=2*Vrms*Irms*( [1+cos(2*omega*t)]*cos(phi) + sin(2*omega*t)*sin(phi) )

If you average over 1 period, then you obtain that the average REAL power is

P=Vrms*Irms*cos(phi) This is average real power

The peak real power would be twice this value since (take t=0 in Eq1:)
real power is associated with cos(phi) term.

The average reactive power is zero, it bounces back and forth between inductive and capacitive loads (B and E fields).

But the peak reactive power is defined as Q=Irms*Vrm*sin(phi).

HERE IS THE QUESTion.

I see often that the Apparent power S = P + i*Q. Unfortunately it is never throuroughly defined. Why would would be adding the Average value of real power and Peak value of Reactice power and call it the apparent power.

Am I misunderstanding this? Shouldn't we be adding the PEAK real power and PEAK reactive power and call this the apparent power.

Also, other relations are P=(I^2)R Q=(I^2)X S=(I^2)Z

THANK YOU!!

Am i correct in saying that the reason why it is defined this way is so that "you" could say that the phase angle is given by

Phi = arctan(Q/P).

I'm just a little weired out by combining the Average Real power with Peak Reactive power.

Thanks

Last edited: Aug 8, 2011
2. Aug 8, 2011

### tiny-tim

hi azaharak!
no, that's the complex power

the apparent power is VrmsIrms … it's voltage times current
so far as i know, neither complex power nor apparent power have any precise physical significance

(but you need to know them because they come up in exam questions! )

from the pf library on impedance

Power:

Power = work per time = voltage times charge per time = voltage times current:

$$P = VI =\ V_{max}I_{max}\cos(\omega t + \phi/2)\cos(\omega t - \phi/2)$$
$$=\ V_{max}I_{max}(\cos\phi + \cos2\omega t)/2$$
$$=\ V_{rms}I_{rms}(\cos\phi + \cos2\omega t)$$
$$=\ (V_{rms}^2/|Z|)(\cos\phi + \cos2\omega t)$$​

So (instantaneous) power is the constant part, $P_{av} = V_{rms}I_{rms}\cos\phi$ (the average power), plus a component varying with double the circuit frequency, $V_{rms}I_{rms}\cos2\omega t$ (so a graph of the whole power is a sine wave shifted by a ratio $\cos\phi$ above the x-axis).

Apparent power, reactive power, and complex power, are convenient mathematical definitions with no precise physical significance: apparent power is the (constant) product of the r.m.s. voltage and current, $S=V_{r.m.s.}I_{r.m.s.}$: it is what we would expect the average power to be if we knew nothing about reactance!

Similarly reactive power is defined as $Q=S\sin\phi=P_{av}\tan\phi$, and complex power is defined as $Se^{j\phi}=P_{av}+jQ$.