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

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).

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.