Question

In the circuit shown in Fig. determine the voltage at a frequency of 50 Hz to be applied across AB in order that the current in the circuit is 10 A. Draw the phasor diagram.

Answer 1

X_L1 = 2 π × 50 × 0.05 = 15.71 Ω ; X_L2 = − 2 π × 50 × 0.02 = 6.28 Ω , 

X_C = 1/2π × 50 × 400 × 10⁻⁶ = 7.95 Ω 

Z₁ = R₁ + jX_L1 = 10 + j15.71 = 18.6 ∠ 57°33′ 

Z₂ = R₂ + jX_L2 = 5 + j6.28 = 8 ∠ 51°30′ 

Z₃ = R₃ − jX_C = 10 − j7.95 = 12.77 ∠ – 38°30′ 

Z_BC = Z₂ | | Z₃ = (5 + j6.28) || (10 − j7.95) = 6.42 + j2.25 = 6.8 ∠ 19°18′ 

Z = Z₁ + Z_BC = (10 + j15.71) + (6.42 + j2.25) = 16.42 + j17.96 = 24.36 ∠ 47°36′ 

Let I = 10∠ 0º; 

∴ V = IZ = 10∠0º × 24.36∠47º36 = 243.6∠47º36′

V_BC = IZ_BC = 10 ∠ 0° × 6.8 ∠ 19°18′ = 68 ∠ 19°18′

I₂ = V_BC/Z₂ = (68 ∠ 19°18')/(8 ∠ 51°31') = 8.5 ∠ - 32°12

I₃ = V_BC/Z₃ = (68 ∠19°18')/(12.77 ∠- 38°30') = 5.32 57°48'; 

V_AC = IZ₁ = 10 0° x 18.6 57°33' = 186 57°33'

The phasor diagram is shown in Fig..