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Solved Problems-9

Determine the current i1 in the circuit using nodal analysis method and graph theory concepts.
Description: 2738.png
By source transformation technique, we convert the 19 V and 25 V voltage sources into current sources.
Description: 2745.png
Since the 30 V voltage source, the 4 A current source, and controlled current source are not accompanied by the passive elements, we consider three resistors R1R2 and R3 and finally let, R1 → 0, R2 → ∝, and R3 → ∝.
Description: 2751.png
The graph of the network is shown.
Description: 2757.png
The complete incidence matrix is,
Description: 5177.png
Reduced Incidence matrix is,
Description: 5187.png
Branch admittance matrix is,
Description: 5193.png
where, G1 = Description: 5199.png, G2 = Description: 5205.pngG3 = Description: 5211.png
 AYb = Description: 5220.png
Description: 5245.png
 AYbAT = Description: 5234.png
Description: 5251.png
Description: 5263.png
Description: 5275.png
Thus, node equations are,
Description: 5281.png
With R1 → 0, G1 → ∝, R2 → ∝, G2 → 0, R3 → ∝, G3 → 0 the equations become:
Description: 5287.png
Description: 5293.png
Description: 5305.png
Description: 5311.png -(i)
Description: 5317.png -(ii)
Description: 5323.png -(iii)
Solving equations (i), (ii), and (iii), we get,
Description: 5329.png
Hence, the current, Description: 5343.png 

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