RLC circuit with constant inductance $L$ and time-dependent resistance $R(t)$ and capacitance $C(t)$.[]{data-label="fig:varRLC"}](FigSI6_McGuyer_MarxCoil)

Consider a lumped series RLC circuit as sketched in Fig. 6. Let us assume that the inductance $L$ is constant, but that the resistance $R = R(t)$ and capacitance $C = C(t)$ may vary in time. This follows from noting that the inductance is set mainly by the inductors $L_1$ to $L_{36}$ and $L_s$, which are not expected to vary during operation except when the solenoid $L_s$ is removed, in which case the increased current through the stage inductors may lead to saturation. Following Kirchoff’s voltage law, the circuit satisfies the differential equation $$\begin{aligned}
(LI)'+ R I + {Q}/{C} = 0. \end{aligned}$$ Here and subsequently a prime denotes differentiation with respect to time $t$. Using this with $I = Q'$, the charge $Q(t)$ then evolves according to the differential equation $$\begin{aligned}
   \label{RLC_Q}
Q'' + \left( \frac{R}{L} \right) Q' + \left( \frac{1}{LC} \right) Q = 0, \end{aligned}$$ which has the same form as that for an RLC circuit with constant parameters. The current and voltage, however, will not evolve according to this equation. Instead, multiplying (\[RLC\_Q\]) by $LC$ and differentiating gives the corresponding equation for the current $I(t)$, 