# RLC DIFFERENTIAL EQUATION

Cuthbert Nyack

Setting the sum of the voltages around the circuit equal to zero and
after some slight rearranging, we get:-
Introducing the following substitutions
and taking the Laplace Transform gives the Laplace Transform Q(s) of q(t) as:-
The 3 parameters R, L and C are thus condensed into 2,
w_{n} and z. To determine R, L or C from these
parameters at least one of R, L or C must be known.

If R is known then
L = R/(2zw_{n})

and C = 2z/(Rw_{n}).

If L is known then R = 2Lzw_{n} and

C = 1/(Lw_{n}^{2}).

The Laplace Transform of the Capacitor, Resistor and Inductor Voltages
are shown below:-

With V(t) equal to a step function of magnitude V, V(s) = V/s, we get the
following expressions for the voltages across the components in the circuit.
Where

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COPYRIGHT © 1996 Cuthbert A. Nyack.