Power formulae without phasors

Question

I wanted to derive the formulae for active, reactive and apparent power. While I understand the concept of phasors, I think I would find an analytical proof that doesn't involve phasors, more intuitive. I recently came across an article that explained how active power is basically the time average of the product of instantaneous voltage and current through the load. I could make sense of it.

I was wondering if there are similar derivations for reactive and apparent powers as well, that do not employ phasors.

What I thought of:

The derivation of active power led me to think that I could perhaps derive the other formulae too, using integrals.

I had the following equations with me:

$$\begin{align} I(t) &= I_m \sin(\omega t + \phi) \\ V(t) &= V_m \sin(\omega t) \end{align}$$

When I decomposed $I(t)$ in terms of $\sin(\omega t)$ and $\cos(\omega t)$, I immediately saw that while finding the average of $V(t)I(t)$ over a time period, the term with $\cos(\omega t)$ would not contribute.

So, the term $I_m \sin(\phi)\cos(\omega t)$ would have something to do with the reactive power. However, I still am unable to logically arrive at the pre-established formula for reactive power: $\dfrac{V_m I_m \sin(\phi)}{2}$.

I was unable to make a headway with the derivation of apparent power, either.

Any help or pointers would be greatly appreciated.

Answer 1

Instantaneous equations:

$$\begin{align} v(t) &=\ V_M \sin(\omega t) \\ i(t) &=\ I_M \sin(\omega t + \phi) \\ \end{align}$$

Using capitals for constants and lower case for things variable with time.

Instantaneous equation for power is just multiplication of instantaneous equations for voltage and current.

$$\begin{align} p(t) &= v(t)\ i(t) \\ p(t) &= V_M \sin(\omega t) \ I_M \sin(\omega t + \phi) \\ p(t) &= V_M I_M\ \sin(\omega t) \ \sin(\omega t + \phi) \\ p(t) &= P_M\ \sin(\omega t) \ \sin(\omega t + \phi) \\ \end{align}$$

Trigonometric identity:

$$ \sin \alpha \ \sin \beta = \frac {\cos(\alpha - \beta) - \cos(\alpha + \beta)}{2} $$

Applying trigonometric identity, we get:

$$\begin{align} p(t) &= \frac {P_M} {2}\ [{ \cos(\omega t - \omega t - \phi) - \cos(\omega t + \omega t + \phi)}] \\ p(t) &= P_{AVG} \ \cos(-\phi) - P_{AVG} \cos(2 \omega t + \phi) \\ p(t) &= P_{AVG} \ \cos(\phi) - P_{AVG} \cos(2 \omega t + \phi) \\ \end{align}$$

Both are Apparent Power with first component Real Power (constant) and second Reactive Power (sinewave at twice frequency of voltage or current).

Image shows instantaneous waveforms for $\phi$ = 30°.

As $\phi$ approaches 0 (a more resistive circuit) (pf increases), $P_{AVG} \ \cos(\phi)$ increases.