## Frequencies of Laddered LC Circuits

The voltages and currents in circuits containing inductors and capacitors can oscillate. Suppose we have repeating capacitance - inductance circuits as in the diagram.

The voltage across
$V_L$
across an inductance
$L$
is given by
$V_L = L \frac{dI}{dt}$
and the voltage
$V_C$
across a ca[capacitance
$C$
holding a charge
$Q$
is given by
$V_C =\frac{Q}{C}$
.
Since there is no battery or power supply in the circuit,
$L \frac{dI_1}{dt}+\frac{Q_1}{C}=0$
by Kirchoff's Voltage Law.
Differentiation gives
$L \frac{d^2I_1}{dt^2}+\frac{1}{C}\frac{dQ_1}{dt}=0$

But
$\frac{dQ_1}{dt}=I_1-I_2$
so
$L \frac{d^2I_1}{dt^2}+\frac{I_1}{C}-\frac{I_2}{C}=0 \rightarrow L \frac{d^2I_1}{dt^2}=-\frac{I_1}{C}+\frac{I_2}{C}$
.
Similarly for the second and third loops.
$L \frac{d^2I_2}{dt^2}=\frac{I_1}{C}-2\frac{I_2}{C}+\frac{I_3}{C}$
.
$L \frac{d^2I_3}{dt^2}=\frac{I_2}{C}-\frac{I_3}{C}$
.
In simple LC circuits with a battery or power supply, current is periodic and simple harmonic, like a mass spring system. Hence
$\frac{d^2I_1}{dt^2}=-\omega_1^2 I_1, \: \frac{d^2I_2}{dt^2}=-\omega_2^2 I_2, \: \frac{d^2I_3}{dt^2}=-\omega_3^2 I_3$
.
The differential equations become
$L \omega^2 I_1 =-\frac{I_1}{C}+\frac{I_2}{C}$

$L \omega^2 I_2 =\frac{I_1}{C}-2\frac{I_2}{C}+\frac{I_3}{C}{C}$

$L \omega^3 I_3 =\frac{I_2}{C}-\frac{I_3}{C}$

Rewrite these equations as
$I_1-I_2=LC \omega^2 I_1$

$-I_1+2I_2-I_3=LC \omega^2 I_2 =$

$-I_2+I_3=LC \omega^2 I_3 =$

In matrix form this is
$\left( \begin{array}{ccc} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 &-1 & 1 \end{array} \right) \begin{pmatrix}I_1\\I_2\\I_3\end{pmatrix}= \omega^2 LC \begin{pmatrix}I_1\\I_2\\I_3\end{pmatrix}$
.
The natural frequencies of the circuit are the square roots of the eigenvalues.
Solve
$det \left( \begin{array}{ccc} 1- \omega^2 LC & -1 & 0 \\ -1 & 2- \omega^2 LC & -1 \\ 0 &-1 & 1- \omega^2 LC \end{array} \right) = \omega^2 LC (\omega^2 LC -1)( \omega^2 LC -3)=0$

Hence the frequencies of the system are
$\omega^2 LC=0 \rightarrow \omega =0, \: \omega^2 LC=1 \rightarrow \omega =\sqrt{\frac{1}{LC}}, \: \omega^2 LC=3 \rightarrow \omega =\sqrt{\frac{3}{LC}}$
.