Harmonic driving of a non dissipatice circuit (LC for example): does the energy really sometimes goes back in the generator?

Question

Let's assume you have a non dissipative circuit represented by a box: it involves only capacitance and inductance for example.

You drive this box via a sinusoidal voltage generator.

I call $U(t)$ and $I(t)$ the voltage and total current in receptor convention.

At some instant of the evolution, you might have: $P(t)=U(t)I(t)>0$ (the box receives some power) and at some other: $P(t)=U(t)I(t)<0$: (the box releases some power).

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For example, we can consider a capacitance only in this box and I drive it via $U(t)=U_0 \cos(\omega_0 t)=Q/C$

$$P(t)=U(t)I(t)=C U(t) \dot{U}(t)=- C U_0^2 \omega_0 \sin(\omega_0 t) \cos(\omega_0 t) = -\frac{ C U_0^2 \omega_0}{2} \sin(2 \omega_0 t) $$

We see that sometimes the power is positive, sometimes it is negative, $0$ in average.

My question: When the box releases some power, does this power really goes back in the generator and then to the electric network ?

Or it could be the case conceptually but in practice it won't be true: i.e the power that goes back in the generator is not recycled with "standard" generator used in experiments. So it is dissipated somehow.

Answer 1

This oscillating reactive power is "converted" to some extent into real power due to resistors in transformer coils and power lines. This decreases the capacity to transfer real power. So the goal is to compensate this effect as close as possible near the source of this reactive power, called power factor correction.