Problems Resolved on R-L Circuits in AC

Problem 54-3 Source: Exercicie 8-31 - page 421 - THOMAS, Roland E. , ROSA, Albert J. , THOUSSAINT, Gregory J. - Book: The Analysis & Design of Linear Circuits - 6ª Edição - Ed. John Willey & Sons, Inc. - 2009.

Use the nodal method to calculate I₁, I₂ and I₃ of the circuit shown in the Figure 54-3.1.

Solution of the Problem 54-3

In this problem the voltages of the points a and c are already known. So there is only an unknown which is the voltage at point b. We can write the following equation for the circuit.

V_b / 50 + (V_b - V_a) / j10 + (V_b - V_c) / j20 = 0

Remembering that V_a = 120 ∠0° and V_c = 120 ∠-30°, after the calculation is performed, we find for V_b the value of:

V_b = 115.35 ∠-17.49°

Knowing V_b, the currents of the circuit are easily calculated. For I₁:

I₁ = (V_a - V_c) / 5 ∠+90° = (120 - 120 ∠-30°) / 5 ∠+90°

Performing the calculation:

I₁ = 62.116∠75° / 5∠90° = 12.423 ∠-15° A

To find the values of I₂ and I₃, we follow the same procedure.

I₂ = (V_a - V_b) / 10 ∠+90°

Performing the calculation:

I₂ = 36.08∠73.9° / 10∠90° = 3.61 ∠-16.1° A

And for I₃:

I₃ = (V_c - V_b ) / 20 ∠+90°

Performing the calculation:

I₃ = 26.05∠-103.53° / 20∠90° = 1.31 ∠166.47° A

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