Resonant Frequency

Using the delta wye transform , we can re-write the circuit with both capacitance and inductance in a parallel wye configuration, as shown below:

Then we can imagine hooking up a three phase AC voltage source to this circuit to do a frequency sweep. The resonance angular frequency, based on the per-phase inductance and capacitance is:

$\omega = \frac{1}{\sqrt{3 L C}} = \frac{1}{\sqrt{3}}$

Denoting 3 smaller loops' currents as $i_{1},\,i_{2}$ and $i_{3}$ , respectively, and making sure they have the same orientation, by Kirchhoff's voltage law we have for one loop: $\begin{aligned} Q_{1}/C + L\frac{d}{dt}\left(i_{1}-i_{2}\right ) + L\frac{d}{dt}\left(i_{1}-i_{3}\right ) &= 0 \\ i_{1} + 2LC\frac{d^{2}i_{1}}{dt^{2}} + LC\frac{d^{2}}{dt^{2}}\left(-i_{2} - i_{3} \right ) &= 0\end{aligned}$ By Kirchhoff's current law applied to the central node, we have $i_{1}+i_{2}+i_{3} = 0$ , so our equation becomes: $i_{1} + 3LC\frac{d^{2}i}{dt^{2}}= 0.$ Due to the symmetry of the circuit, we get the same equations for other two loops. We read off natural angular frequency of the circuit to be $\omega = 1/\sqrt{3LC}$ .