Parallel RL and RC branches

Circuit: Branch 1 is R₁ = 30 Ω in series with L = 75 mH. Branch 2 is R₂ = 20 Ω in series with C = 100 μF. Supply V = 240∠0° V, 50 Hz (reference). ω = 314.16 rad/s.

Reactances and branch impedances

• X_L = ωL = 314.16 × 0.075 = 23.56 Ω
• X_C = 1/(ωC) = 1/(314.16 × 100 × 10⁻⁶) = 31.83 Ω
• Z₁ = 30 + j23.56 = 38.15∠38.15° Ω
• Z₂ = 20 − j31.83 = 37.59∠−57.86° Ω

Branch currents

• I₁ = 240/38.15 = 6.29 A, ∠−38.15° (lagging) = 4.95 − j3.89 A
• I₂ = 240/37.59 = 6.38 A, ∠+57.86° (leading) = 3.40 + j5.41 A

Total current

I = I₁ + I₂ = 8.34 + j1.52 A

|I| = 8.48 A, at 10.32° (leading the voltage)

Power factor

cos φ = cos(10.32°) = 0.984 leading

Power

• Active power: P = VI cos φ = 240 × 8.48 × 0.984 = 2003 W (check: I₁²R₁ + I₂²R₂ = 6.29² × 30 + 6.38² × 20 = 1187 + 815 = 2002 W)
• Reactive power: Q = VI sin φ = −365 VAR (net capacitive; inductive branch 933 VAR lagging, capacitive branch 1297 VAR leading)
• Apparent power: S = VI = 240 × 8.48 = 2036 VA

Phasor diagram

                 I2 (+57.9°, 6.38 A)
                ↗
              I (+10.3°, 8.48 A)
   ─────────────────────────► V (0°)
                ↘
                 I1 (−38.2°, 6.29 A)

I₁ lags V, I₂ leads V, and their sum I is slightly ahead of V.