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Solved Problems-18

Problems-18
For the circuit shown in the figure construct a tree in which 10Ω and 20Ω are in tree branches. Using node analysis, solve for V1 and V2.
 
Description: 3258.png
Solution
Here, we have on e current source without parallel resistance and one voltage source without series resistance. Therefore, we connect a parallel resistance R1 in parallel with the 2A current source and a series resistance R2 in series with the 20V voltage source. Finally, we will let R1 → ∞ and R2 → 0.
 
Description: 3264.png
 
Now, we construct the graph of the network as shown below. A tree, in which 10Ω and 20Ω are in tree branches, is selected.
Description: 3271.png Description: 3278.png
 
The complete incidence matrix is,
Description: 5616.png
 
Reduced incidence matrix is,
Description: 5623.png
 
Branch admittance matrix is,
Description: 5629.png
Description: 5636.png
∴ Description: 5642.png
Description: 5652.png 
Now,
Description: 5658.png Description: 5664.png 
Description: 5670.png
 
Thus, node equations are,
Description: 5679.png
 
With R1 → ∞, the equations become
Description: 5685.png
 
Solving equations, we get,
Description: 5691.png 
 
With R2 = 0,
Description: 5700.png
 
Similarly, with R2 = 0, we get,
Description: 5706.png 
 
and Description: 5712.png
 




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