Relationship of Line and Phase Voltages and Currents in a Star Connection

Key learnings:

• Star Connected System Definition: A star connected system is a type of electrical circuit where each component connects to a common neutral point.
• Voltage Relationship: In a star-connected system, the line voltage is √3 times the phase voltage.
• Current Relationship: The line current is the same as the phase current in a star-connected system.
• Balanced System: In a balanced star system, the magnitude of voltage and current is the same in each phase.
• Power Calculation: The total power in a three-phase system is calculated using the phase voltage, phase current, and the angle φ.

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.

The presence of load impedance causes a phase shift, resulting in the current lagging behind the applied voltage by an angle of ϕ in each phase. In an ideal balanced system, the current and voltage magnitudes are identical across all phases. For instance, the voltage measured between the neutral point (N) and the red phase terminal (R) is denoted as VR.
Similarly, the magnitude of the voltage acrossyellow 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-connected system, the current flowing through each line is equal to the current flowing through each phase, denoted as I_L. This means that the magnitude of the line current and phase current is identical in all three phases.”
∴ 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, In a star-connected system, the line voltage is equal to the phase voltage multiplied by the square root of 3 (√3).
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