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

To derive the relations between line and phase currents and voltages of a star connected system, we have first to draw a balanced star connected system.

Due to load impedance, the current lags the applied voltage in each phase by an angle ϕ. In a perfectly balanced system, the magnitude of current and voltage in each phase is the same. For example, the voltage across the red phase, or between the neutral point (N) and red phase terminal (R), is V_R.
Similarly, the magnitude of the voltage across yellow phase is V_Y and the magnitude of the voltage across blue phase is V_B.
In the balanced star system, magnitude of phase voltage in each phase is 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