PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

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 1 (t) i 2 (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 differen-

tial equations:

()()

()

RRitL ()

di t

di

121  11 Ri t 22 
^0

Ri t R R i t L

di t

21 3 2 2 (^2) di
() () ()^2 ()
0
Replacing the elements by their values yields the following set of differential equations:
730 it 1 1 2
di t
di
() it
()

 ()