204
Practical MATLAB
® Applications for Engineers
Observe that by changing the value of R (R = 75, 36, and 3 Ω), the three solutions for
the second-order differential equation—over, critical, and underdamped cases—
are obtained. Also observe that the analytical solutions completely agree with the
MATLAB solutions.
Example 2.31
Steady-state conditions exist in the circuit of Figure 2.105 for t ≤ 0, while the voltage
source V 1 = 6 V is connected to the network. At t = 0
+
, the switch moves downward
disconnecting the source while connecting R 1 to the rest of the circuit.
1. Obtain analytically the loop differential equation set and the initial conditions
2. Using the MATLAB symbolic solver, create the script fi le transient_2loops that
returns the transient currents for each loop, and their respective plots, for t ≥ 0
3. Also obtain simplify and pretty expressions for each of the transient loop currents of
part 2
FIGURE 2.105
Network of Example 2.31.
V 1 = 6 V
R 3 = 6 Ω
R 2 = 3 Ω
R 1 = 4 Ω
Switch moves down at t = 0
+
−
+
−
L 1 = 1 H
i 2 (t )
i 1 (t )
L 2 = 2 H
+
−
+
−
ANALYTICAL Solution
The conditions at t = 0 are i 1 (0) = 3 A and i 2 (0) = 1 A. Applying KVL to the two-mesh
network of Figure 2.105, for t ≥ 0, results in the following set of simultaneous differential equations:
(
) ()
( )
( )
R R i t L
di t
di
R i t
1
2 1
1
1
2 2
0
ϩ
ϩ
ϭ
Ϫ
ϪR i t
R
R i t L
di t
di
2 1
3
2 2
2
2
0
( ) (
) ( )
( )
ϩ
ϩ
ϩ
ϭ
Replacing the elements by their values yields the following set of differential equations:
7
3
0
1
1
2
i t
di t
di
i t
( )
( )
( )
ϩ
ϭ
Ϫ
CRC_47760_CH002.indd 204
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7/23/2008 1:39:01 PM
7/23/2008 1:39:01 PM