Question

The series and parallel circuits shown in Fig. have the same impedance and the same power factor. If R = 3 Ω and X = 4 Ω, find the values of R₁ and X₁. Also, find the impedance and power factor.

Answer 1

Series Circuit [Fig.]

Y_S = 1/(R + jX) = (R - jX)/(R² + X²) = R/(R² + X²) - j(X/(R² + X²))

Parallel Circuit [Fig.]

Y_P = 1/(R₁ + j0) = 1/(0 + jX₁) = 1/R₁ + 1/jX₁ = 1/R₁ - j/X₁

∴ R/(R² + X²) - j(X/(R² + X²)) = 1/R₁ - j/X₁

∴ R₁ = R + X² /R and X₁ = X + R²/X 

∴ R₁ = 3 + (16/3) = 8.33 Ω; X₁ = 4 + (9/4) = 6.25 Ω 

Impedance = 3 + j 4 = 5 ∠53.1º ; 

Power factor = cos 53.1º = 0.6 (lag)