How is the average power calculated in a series circuit?

[blender3d]

I have a derivation from a book that says

Q = 2π Es / Ed

Where Es is the energy stored in the resonant components. Dividing both by the period at resonance gives...

Q = ωo Es / Pd

This is where I'm stuck. The book says Es = 1/2 LI^2 at the instant that all of the energy is being stored in the inductor. Then it goes on to say the power dissipated in the series resistance, Pd is equal to 1/2 I^2 R. Why is it the average power? Of course these both simplify down to ωoL/R.

And, how do you go from 1/2 LI^2 as the energy of the inductor to ωLI^2 as the power? The power is 4πf times the energy?

I think I posted this in the wrong section, sorry.

 

[AlephZero]

The power dissipated in the resistor at any instant in time is $I^2R$.

The circuit is oscillating so $I^2 = I_{\text{max}}^2 \sin^2 \omega t$.

When you take the average power by integrating over one cycle of the oscillation, you get $I_{\text{max}}^2 R/2$.

For the second question, average power = energy / time. The energy in the inductor changes between $0$ and $LI_{\text{max}}^2/2$ every half cycle, or in time $\pi / \omega$ seconds.

 

[blender3d]

Thank you very much, that helps a lot. Too bad my teacher could not explain this to me.