Second Order Circuits

Second order circuits are circuits that have two energy storage

elements, resultingin second-order differential equations. The circuits have to be cast in terms of irreducible elements (i.e. combine L's and C's into one when possible).

An unforced RLC circuit

Consider an un-forced RLC circuit. We want to find $V_{C}$.

First, we can use KVL and KCL

$V_{R}+V_{L}+V_{C}=0$

$iR+L{\frac {di}{dt}}+V_{C}=0$

Next, we can use $i=C{\frac {dV_{C}}{dt}}$ and substitution to get

$RC{\frac {dV_{C}}{dt}}+L{\frac {d}{dt}}{\frac {CV_{C}}{dt}}V_{C}=0$

Changing the order and moving the constants,

$LC{\frac {d^{2}V}{dt^{2}}}+RC{\frac {dV_{C}}{dt}}+V_{C}=0$

Moving constants away from the first term to get a second-order differential equation,

${\frac {d^{2}V_{C}}{dt^{2}}}+{\frac {R}{L}}{\frac {dV_{C}}{dt}}+{\frac {1}{LC}}V_{C}=0$