Which expression to use for electrical power?

Question

We know that electric power can be written as $P=VI$, or $P=\frac{V^2}{R}$, or $P=I^2R$.

But when to use which one? Sometimes two different formulas give different results! Please explain with some examples.I 'm feeling very confused!

Answer 1

The product of the instantaneous voltage across and current through a circuit element gives the instantaneous power delivered to the circuit element (assuming passive sign convention).

$$p(t) = v(t) \cdot i(t)$$

This holds regardless. For particular circuit elements, one can eliminate one of the variables, e.g.

Resistor: $v = Ri$

$$p(t) = Ri^2(t) = \frac{v^2(t)}{R}$$

Inductor: $v = L\frac{di}{dt}$

$$p(t) = Li\frac{di}{dt}$$

Capacitor: $i = C \frac{dv}{dt}$

$$p(t) = Cv\frac{dv}{dt} $$

Answer 2

All 3 formulas are true for a $V$ and $I$ that are constant in time. Imagine there is a constant voltage $V$ across a resistor $R$. By Ohms Law this causes a current $I={V\over R}$ to flow through the resistor. The power being dissipated by the resistor is $$ P=IV $$ Ohms Law says $I={V\over R}$ so substitute this into the first equation for $I$ to get $$ P={{V^2}\over R} $$ Ohms Law also says $V=IR$ so substitute this into the first equation for $V$ to get $$ P=I^2 R $$ Now, for a time varying voltage, look at a single frequency $V=V_0 cos(\omega t)=RealPart(V_0e^{j\omega t})$ and $I=I_0 cos(\omega t-\phi)=RealPart(I_0e^{j(\omega t-\phi)})$ . The angle $\phi$ is the phase shift between $V$ and $I$ caused by the complex $Z=|Z|e^{j\phi}$ in Ohms Law. The formulas for Ohms Law and time-averaged (dissipated power) are still true just in slightly modified form (the hats indicate complex numbers) and the brackets <> mean time-averaged. $$ \hat {V}= \hat{I} \hat{Z} $$ $$ <P>={1\over2}I_0 V_0 cos(\phi) $$ $$ <P>={1\over 2}{{V_0^2}\over {|Z|}} cos(\phi) $$ $$ <P>={1 \over 2}I_0^2 |Z| cos(\phi) $$ where Z is the complex impedance of the circuit you want the power dissipation for. You make Z by adding up complex impedances $R$, $j\omega L$, and ${1 \over {j\omega C}}$ of components as if they were resistors using your series and parallel formulas. The ${1\over2}cos(\phi)$ come from integrating the product of two cosines when doing the time- average.

Complex impedances are described in many text books and web links ... here is one link that also derives the $<P>$ formulas.