PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

436 Practical MATLAB® Applications for Engineers

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The system poles are:

ans =

5.3196

1.0051

1.0051

0.3721

Note that all the poles of I 1 (s) and I 2 (s) are in the left half of the s-plane, then the initial

and fi nal value theorems can be used. Note also that the initial currents are verifi ed,

and the fi nal currents are zero since the two capacitors for t = ∞ act as an open circuit.

Example 4.20

The switch in the circuit diagram of Figure 4.81 closes at t = 0, where it remains for

t > 0. All the storing elements are initially charged as indicated in Figure 4.81, at t = 0.

1. Redraw the equivalent circuit in the s-domain


2. Write the s-domain node equations


3. Write the matrix node equation


4. Use MATLAB and obtain expressions for V 1 (s) and V 2 (s)


5. Use MATLAB and obtain plots of V 1 (s) versus s and V 2 (s) versus s


6. Use MATLAB to evaluate and verify the initial and fi nal voltage for each node by
   applying the initial and fi nal value theorems
   The MATLAB solution is given as follows by the script fi le nodal _eqs:

ANALYTICAL Solution

Part 1

The time domain circuit diagram of Figure 4.80 is redrawn in the s-domain shown

in Figure 4.82, indicating the two nodal voltages V 1 (s) and V 2 (s) and the circuit ini-

tial conditions transformed into sources.

Part 2

The two nodal equations are

For node #V 1

ss

s

s

Vs

s

:()V

5

2

31

3

1

42

1

3

1

3

1

(^14)

  







22 ()s
For node #V 2 :
ss
s
ss
Is
ss
32 8
9
1
3
1
4
1
4
1
9
1
3
1
(^215)



 



() 



Vs 2 ()
Part 3
The node matrix equation in the s domain is given by
2
2
38
9
1
3
7
12 2
1
3
1
4
1
3
1
4
13
36
8
(^215)
s
s
s
s
s
s
s
s



















 


ss
Vs
Vs
























1
2
()
()
MATLAB Solution
% Script file: nodal _ eqs
syms s Ys Is Vs it y
Ys = [7/12+s/2+1/(3s) -(1/4+1/(3s));-(1/4+1/(3s)) 13/36+8/(15s)];