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.
Suppose due to load impedance the current lags the applied voltage in each phase of the system by an angle ϕ. As we have considered that the system is perfectly balanced, the magnitude of current and voltage of each phase is the same. Let us say, the magnitude of the voltage across the red phase i.e. magnitude of the voltage between neutral point (N) and red phase terminal (R) is VR.
Similarly, the magnitude of the voltage across yellow phase is VY and the magnitude of the voltage across blue phase is VB.
In the balanced star system, magnitude of phase voltage in each phase is Vph.
∴ VR = VY = VB = Vph
We know in the star connection, line current is same as phase current. The magnitude of this current is same in all three phases and say it is IL.
∴ IR = IY = IB = IL, Where, IR is line current of R phase, IY is line current of Y phase and IB is line current of B phase. Again, phase current, Iph of each phase is same as line current IL in star connected system.
∴ IR = IY = IB = IL = Iph.
Now, let us say, the voltage across R and Y terminal of the star connected circuit is VRY.
The voltage across Y and B terminal of the star connected circuit is VYB The voltage across B and R terminal of the star connected circuit is VBR.
From the diagram, it is found that
VRY = VR + (− VY)
Similarly, VYB = VY + (− VB)
And, VBR = VB + (− VR)
Now, as angle between VR and VY is 120o(electrical), the angle between VR and – VY is 180o – 120o = 60o(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