Biomedical Engineering Reference
In-Depth Information
In general, the order of the differential equation relating a single output variable and
the inputs is equal to the number of energy storing elements in the circuit (capacitors
and inductors). In some circuits, the order of the differential equation is less than the num-
ber of capacitors and inductors in the circuit. This occurs when capacitor voltages and
inductor currents are not independent; that is, there is an algebraic relationship between
the capacitor—specifically, voltages and the inputs, or the inductor currents and the inputs.
This occurs when capacitors are connected directly to a voltage source or when inductors
are connected directly to a current source.
Example Problem 9.15 involved a circuit with zero initial conditions. When circuits
involve nonzero initial conditions, our approach remains the same as before except that
the initial inductor currents are included when writing the node-voltage equations.
EXAMPLE PROBLEM 9.16
Write the node equations for the following circuit for
t
0 assuming the initial conditions are
i
L
1
ð
0
Þ¼
8 A and
i
L
2
ð
0
Þ¼
4A
:
i
L
1
i
L
2
1 H
2 H
10
Ω
+
−
v
s
1 F
2 F
3 F
Solution
With the reference node at the bottom of the circuit, there are three essential nodes, as shown in
the redrawn circuit that follows.
i
L
1
i
L
2
1
2
3
1 H
2 H
+
+
+
10
Ω
+
−
v
s
v
1
v
2
v
3
1 F
2 F
3 F
−
−
−
