# Solved Problems-9

Problems-9

Determine the current

*i*_{1}in the circuit using nodal analysis method and graph theory concepts.Solution

By source transformation technique, we convert the 19 V and 25 V voltage sources into current sources.

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

*R*_{1},*R*_{2}and*R*_{3}and finally let,*R*_{1}→ 0,*R*_{2 }→ ∝, and*R*_{3}→ ∝.The graph of the network is shown.

The complete incidence matrix is,

Reduced Incidence matrix is,

Branch admittance matrix is,

where,

*G*_{1}= ,*G*_{2}= ,*G*_{3}=∴

*AY*=_{b}*=*

∴

*AY*=_{b}A^{T}=

Now,

Thus, node equations are,

With

*R*_{1}→ 0,*G*_{1}→ ∝,*R*_{2}→ ∝,*G*_{2}→ 0,*R*_{3}→ ∝,*G*_{3}→ 0 the equations become:or

-(i)

-(ii)

-(iii)

Solving equations (i), (ii), and (iii), we get,

Hence, the current,