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1
First Order Circuits, First Order Differential Equations
EECS 215
Jamie Phillips
A first order circuit will contain one energy storage element (capacitor or inductor),
where the circuit response will be represented by a first order differential equation. A
general configuration for such a circuit is given by the following schematic. Note that the
voltage source v
s
may be time dependent. Also note that any linear circuit (combination
of independent sources, dependent sources, and resistors) attached to a capacitor may be
“converted” into the circuit below through determination of the Thevenin equivalent (a
powerful technique!). Similar analysis may be done for a first order circuit with an
inductor, current source, etc.
C
R
v
s
(t)
v
+

1. Set up differential equation
From KCL, the differential equation for this circuit is
RC
t
v
RC
v
dt
dv
s
)
(
=
+
The solution is
f
n
v
v
v
+
=
, where
v
n
and
v
f
are the natural (unforced, homogeneous) and
forced (particular) solutions, respectively.
2. Natural solution
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This note was uploaded on 04/04/2008 for the course EECS 215 taught by Professor Phillips during the Winter '08 term at University of Michigan.
 Winter '08
 Phillips

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