Proff of that in RLC circuit in series where AC source works in resonant frequency the sum energy of the inductor and capacitor is zero at all time

can anyone help me please prove the statement that: In RLC circuit in series where AC voltage source works in resonant frequency the sum energy of the inductor and capacitor is zero at all time. I know that the impedance is pure ohm, and that there is on difference in phase shift between the current and the voltage source, but i can't figure what else.. Thanks.

• Sounds like you're being asked to derive $\omega_r=\dfrac{1}{\sqrt{LC}}$ – Matt Young Jul 3 '14 at 15:48

When you apply a voltage to a series resonant circuit of R, L and C, the reactances of L and C are equal but have opposite signs. This means their impedances totally cancel out and the only impedance that is left (at resonance) is R.

Regards the energy - this oscillates between L and C so at any one point the energies will not necessarily be equal but, if you averaged those energies out over time, you will find that they do become equal.

Because C and L reactances are equal, it means that the peak voltage attained on both is the same magnitude AND because they share the same current (series resonant) it duly follows that each will store the same instantaneous peak energy (but of course at different times in the AC cycle).

EDIT - reasons why the energies are the same: -

• Being a series tuned circuit at resonance, XL and XC MUST have the same magnitude because this is a definition of series resonance of an R, L and C.
• Because they both share the same current, the magnitude of the voltage across each must also be identical.

Here are what the waveforms look like for voltage and current in a capacitor: - Here are what the waveforms look like for voltage and current in an inductor: - Pictures taken from here

If you look at the voltage across the capacitor you will see that it lags the current by 90 degrees i.e. when the current is at a peak the voltage is zero. For the inductor, when the current is at a peak the voltage is also zero but, compared to the capacitor's voltage, it is 180 degrees different.

Because reactances are equal and current is equal the two voltages cancel each other out leaving only the resistor in circuit to take the full AC voltage of the driving source. It's like having two 9V batteries in series - one way they produce 18 volts and the other way they produce zero volts and you would not be able to distinguish two batteries wired series antiphase from a very low value ohmic resistor.

Also if you multiplied the capacitor's and inductor's respective current and voltage waveforms to produce a power waveform you'd find this: -

• The average power equals zero for both L and C (intuitively this is well-known)
• The peak power is the same

This means the peak energy stored in both L and C is the same.

• Is there a way to "see" it from the power equations? how do you know for sure that the reactances have opposite signs? – user3921 Jul 3 '14 at 18:05
• I'll try - catch my answer in a moment – Andy aka Jul 3 '14 at 18:24

Hope this helps. RLC circuits can easily be proven with the Impedance (Z) calculation. Z = resistance of the resistor squared plus the product squared of the indictor equivalent resistance (impedance) minus the capacitor equivalent resistance (impedance).  