Relationship of Line and Phase Voltages and Currents in a Star Connected System

In a balanced star (wye) system, line current equals phase current and line voltage is √3 times phase voltage. The relations between line and phase currents and voltages of a star connected system follow from the phasor diagram below.

With a lagging load, current in each phase lags that phase voltage by ϕ. In a balanced set the voltage and current magnitudes are the same on every phase. The red-phase voltage, from neutral N to terminal R, is V_R.
Yellow-phase voltage is V_Y and blue-phase voltage is V_B.
Each of those phase voltages equals V_ph.
∴ V_R = V_Y = V_B = V_ph

In a star connection, the line current is the same as the phase current. The magnitude of this current is the same in all three phases, denoted as I_L.
∴ I_R = I_Y = I_B = I_L, Where, I_R is line current of R phase, I_Y is line current of Y phase and I_B is line current of B phase. Again, phase current, I_ph of each phase is same as line current I_L in star connected system.
∴ I_R = I_Y = I_B = I_L = I_ph.

Let’s denote the voltage across the R and Y terminals of the star connected circuit as V_RY.
The voltage across Y and B terminal of the star connected circuit is V_{YB<!–
The voltage across B and R terminal of the star connected circuit is VBR}.
From the diagram, it is found that
V_RY = V_R + (− V_Y)
Similarly, V_YB = V_Y + (− V_B)
And, V_BR = V_B + (− V_R)
Now, as angle between V_R and V_Y is 120^o(electrical), the angle between V_R and – V_Y is 180^o – 120^o = 60^o(electrical).

Thus, for the star-connected system line voltage = √3 × phase voltage.
Line current = Phase current
As, the angle between voltage and current per phase is φ, the electric power per phase is

So the total power of three phase system is