Current phase between inductor and capacitor in LCR-circuit

[Order]

At resonance ($\omega_0^2= 1/LC$) in an LCR-Circuit the phase angle given by
$$\theta=\tan^{-1}\left( \frac{\omega L - 1/\omega C}{R} \right)$$
obviously is zero. And still there are other phases to deal with. This I don't understand. Let me elaborate.

For example when calculating the amount of stored energy at resonance, then you can visualize that the energy goes back and forth between capacitor and inductor. So they are not in phase, but are in fact out of phase by $\pi /2$, or rather the current is.

Now my question is: In what equation (or diagram) is this clearly marked?

 

[wotanub]

This shows the problem with perfect resonance. You know the phase angle is π/2, so plug that into the equation you used, and you'll find that must mean the parameter of the arctan function is some really huge or really small number (let's say... infinity).

Notice the parameter goes to infinity as R goes to 0. But R can't be zero in a real circuit.

 

[Baluncore]

At resonance the CURRENT flowing is in quadrature with the VOLTAGE.

For the inductor; v = L * di/dt
If di/dt is a sine wave then v must be a cosine. Hence the quadrature.

 

[Order]

Ok, let's see if I got things right:

1. The current is (for an ideal inductor with no capacitance) equal throughout the LCR-Circuit. (This is so because of Kirchoffs first rule.)

2. The voltage is not in phase between the different parts of the Circuit but at resonance it all adds up to zero.

3. The energy in the inductor is dependent of the current, whereas the energy of the capacitor is dependent of the voltage. And when the voltage is $\pi /2$ out of phase to the current (especially in the capacitor), the effect is that energy goes back and forth between inductor and capacitor. (In the book I read they put the current out of phase, when calculating the energy, which confused me.)

 

[sardar]

The phase between the inductor and capacitor in an LCR-circuit is an important concept to understand in order to fully grasp the behavior of the circuit. At resonance, the phase angle between the inductor and capacitor is indeed zero, as shown by the equation given. This means that the voltage and current in the circuit are in phase with each other.

However, as you mentioned, there are other phases to consider. When calculating the amount of stored energy at resonance, it is important to take into account the phase difference between the current and voltage. This is where the $\pi /2$ phase difference comes into play. This phase difference is due to the fact that the current and voltage are out of phase with each other, as you correctly pointed out.

To answer your question, the phase difference between the current and voltage can be clearly seen in a phasor diagram. In this diagram, the voltage and current are represented by vectors at an angle to each other, with the angle representing the phase difference between them. At resonance, the voltage and current vectors will be aligned, indicating a phase difference of zero. However, when the circuit is not at resonance, the voltage and current vectors will be at an angle to each other, indicating a phase difference of $\pi /2$.

Additionally, the equation you provided also takes into account the phase difference between the current and voltage, as it includes the term $\tan^{-1}\left( \frac{\omega L - 1/\omega C}{R} \right)$ which represents the phase angle between the current and voltage. So, while the phase angle between the inductor and capacitor may be zero at resonance, the overall phase difference between the current and voltage is still present and must be considered in calculations.

In conclusion, the phase between the inductor and capacitor is an important aspect of an LCR-circuit, and can be clearly seen in a phasor diagram or represented by the phase angle in the equation you provided. It is crucial to understand and consider this phase difference in order to fully understand the behavior of the circuit.