Question

You analyze the voltages across different branches of a simple circuit (using Kirchoff's Laws), and you find the following equations relating the voltages across different resistors 4V 1+4V 2 90 6.1 V 2+7 v 3 72 where vi represents current through wire 1 2 through wire 2, etc. All voltages are in V (Volts). Using matrix methods, determine the magnitude of vi1 in Volts Answer Check

Answer 1

The following equations relating the voltages across different resistors which are given below as :

# (4) V₁ + (4) V₂ = (90 V)

V₁ = [(90 V) - (4) V₂] / (4)                                               

V₁ = [(45 V) - (2) V₂] / (2)                                                                { eq.1 }

# (2) V₁ + (3) V₂ + (6) V₃ = (100 V)

(2) {[(45) - (2) V₂] / (2) } + (3) V₂ + (6) V₃ = (100 V)

(45) - (2) V₂ + (3) V₂ + (6) V₃ = (100 V)

(1) V₂ + (6) V₃ = [(100 V) - (45 V)]

(1) V₂ + (6) V₃ = (55 V)                                                                     { eq.2 }

# (6.1) V₂ + (7) V₃ = (7.2 V)                                                              { eq.3 }

Equating eq.2 & 3, we get

(1) V₂ + (6) V₃ = (55 V)                                                            x 7

(6.1) V₂ + (7) V₃ = (7.2 V)                                                        x 6

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(7) V₂ + (42) V₃ = (385 V)

(36.6) V₂ + (42) V₃ = (43.2 V)

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- (29.6) V₂ = (341.8 V)

V₂ = - [(341.8 V) / (29.6)]

V₂ = - 11.5 V

Inserting the value of 'V₂' in eq.1, then we get

V₁ = [(45 V) - (2) V₂] / (2)  

V₁ = [(45 V) - (2) (-11.5 V)] / (2)  

V₁ = [(45 V) + (23 V)] / (2)  

V₁ = [(68 V) / (2)]

V₁ = 34 V