What is the frequency at which a series RLC circuit stores maximum energy?

Question

I was reading about the quality factor of a series RLC circuit. The quality factor is defined as

$$2\pi*\frac{\text{maximum energy stored}}{\text{total energy lost per period}}$$

Here maximum energy stored is calculated at the resonant frequency. My doubt is, is there any other frequency at which the maximum energy stored in the circuit is greater than that at resonance?

I have done the following calculations.

Energy in storage at any instant, $$ \mathrm{Energy}(t)=\frac{1}{2}*\textrm{L}i_{L}^2(t)+\frac{1}{2}*\textrm{C}v_{C}^2(t)$$

$$\text{Let source voltage}=v_{m}sin\omega t$$

After some calculations

Energy in storage at any instant = $$\frac{1}{2}*\frac{v_{m}^{2}}{[R^{2}+(\omega L-\frac{1}{\omega C})^2]}*[L*sin^2(\omega t-tan^{-1}\phi )+\frac{1}{\omega ^{2}C}*cos^2(\omega t-tan^{-1}\phi )]$$

At resonance, this reduces to a constant value, $$ \mathrm{Energy}(t)=\frac{1}{2}*\frac{v_{m}^2*L}{R^2}$$

This is the maximum energy stored at resonance.The question is, is there any other radian frequency at which the function ' Energy(t) ' peaks greater than that at resonance?

Image source: Boylestad, Robert L. Introductory Circuit Analysis. Pearson Education.

Answer 1

The question is, is there any other radian frequency at which the function ' Energy(t) ' peaks greater than that at resonance?

I was wrong in my "now deleted" answer and, I should have been less hasty in posting it. For a simple RLC series circuit you can look at the voltage across the capacitor just as you would when designing a simple 2nd order low-pass filter like this: -

If my brain had worked immediately, I'd have posted this picture from my basic website: -

• The maximum peak amplitude is: \$\dfrac{1}{2\zeta\sqrt{1-\zeta^2}}\$
• This occurs at a frequency of: \$\omega_n\sqrt{1-2\zeta^2}\$

There is a proof further down that linked page. For the peak inductor energy, this occurs at true resonance (\$\omega_n\$) because the reactances cancel and, current becomes a maximum.

Answer 2

For a overdamped series RLC, at 0 Hz (DC).

That's when energy is stored just in the capacitor. There is no current, so no power is wasted in the R and L.