Power dissipated in resistor

Question

Suppose one has a circuit consisting of an inductor $L$ and resistor $R$ in series where $L$ and $R$ are known, passes an alternating voltage of frequency $\omega$ through it and that one wishes to calculate the mean power dissipated in the resistor.

Let the RMS voltage across the series combination be $V_0$. Then the RMS current through the components will be $I=\frac{V_0}{i\omega L + R}$ and the mean power dissipated in $R_2$ will be $\overline{P}=I^2R$. However, at this point, $I$ involves a complex quantity. How do you calculate the mean power? Do you calculate the magnitude of the complex current?

With very many thanks,

Froskoy.

Answer 1

okay, This was really cool and I got some help from my physics professors on this one (apparently I won't learn this until next semester) and to find the magnitude of the square of a complex number you take it times it's complex conjugate. So in this case $$\frac{V_0}{i \omega L +R}$$ is multiplied with $$\frac{V_0}{-i \omega L + R}$$ leaving you with simply $$\frac{V_0^2}{w^2L^2+R^2}$$, then just take it times your $$R_2$$ giving you $$\frac{V_0^2}{\omega^2 L^2+R^2} \times R_2$$ for the average power. Sorry that my equations aren't very pretty, LaTex isn't working on my ubuntu install yet..... Hope this helps!!