Three-phase connections and series RC phasor

Two ways of connecting a 3-phase system

1. Star (Y) connection: one end of each of the three phase windings is joined to a common point called the neutral (N); the other three ends go to the lines.
   • V_L = √3 V_ph (line voltage leads the phase voltage by 30°)
   • I_L = I_ph
2. Delta (Δ) connection: the three windings are joined end to end in a closed loop; the lines are taken from the three junctions.
   • V_L = V_ph
   • I_L = √3 I_ph (line current lags the phase current by 30°)

In both cases the total power is P = √3 V_L I_L cos φ.

Series RC circuit

A resistance R and capacitance C are in series across v = Vm sin ωt. The same current I flows through both, so I is taken as the reference phasor.

 o---[ R ]---| C |---o
     V_R      V_C
 <-------- V -------->

• Voltage across R: V_R = I R, in phase with I.
• Voltage across C: V_C = I X_C, lagging I by 90°, where X_C = 1/(ωC) = 1/(2πfC).

Phasor sum:

V   = V_R − j V_C = I (R − j X_C)
|V| = √(V_R² + V_C²) = I √(R² + X_C²)
Z   = V/I = R − j X_C
|Z| = √(R² + X_C²)
φ   = tan⁻¹(X_C/R)   (current leads voltage)

Phasor diagram:

        I, V_R
  o------------->       (reference)
  |\
  | \  φ
  |  \
V_C   \ V
  |    \
  v     v

The voltage V lags the current I by φ (i.e. current leads voltage).

If v = Vm sin ωt, then i = Im sin(ωt + φ) with Im = Vm/|Z|.

Power: power factor cos φ = R/|Z| (leading). Average power P = V I cos φ = I²R; the capacitor takes no average power, only reactive power Q = I² X_C.