In a balanced three-phase, star-connected system, understanding the relationship between phase voltages and line voltages is crucial. This relationship includes both their magnitudes and their phase difference.

Three-Phase System Overview

A three-phase system is a common and efficient method for generating, transmitting, and distributing electrical power. It involves three alternating currents or voltages that are of the same frequency but are displaced from each other by a phase angle of \(120^\circ\).

Star-Connected System Configuration

In a star-connected system (often referred to as a Y-connection), one end of each of the three phase windings is joined at a common point, which is known as the neutral point. The other ends of these windings are connected to the three line terminals, typically denoted as R, Y, and B (or A, B, and C).

• Phase Voltages (\(V_P\)): These are the voltages measured between any line terminal and the neutral point (e.g., \(V_{RN}\), \(V_{YN}\), \(V_{BN}\)). In a balanced three-phase system, their magnitudes are equal, and they are separated by \(120^\circ\) in phase.
• Line Voltages (\(V_L\)): These are the voltages measured between any two line terminals (e.g., \(V_{RY}\), \(V_{YB}\), \(V_{BR}\)).

Voltage Relationships in Star Connection

For a balanced star-connected system, there's a specific relationship between the magnitude of the line voltage (\(V_L\)) and the phase voltage (\(V_P\)):

\[V_L = \sqrt{3} V_P\]

This means that the line voltage magnitude is \(\sqrt{3}\) times the phase voltage magnitude.

Phase Difference Calculation

To determine the phase difference between phase voltages and their respective line voltages, we use phasor diagrams. Consider the phase voltages \(V_{RN}\), \(V_{YN}\), and \(V_{BN}\). The line voltage \(V_{RY}\) is the phasor difference between \(V_{RN}\) and \(V_{YN}\).

\[V_{RY} = V_{RN} - V_{YN}\]

This can also be written as a phasor sum:

\[V_{RY} = V_{RN} + (-V_{YN})\]

Let's assume \(V_{RN}\) is our reference phasor at \(0^\circ\). In a balanced system:

• \(V_{RN} = V_P \angle 0^\circ\)
• \(V_{YN} = V_P \angle -120^\circ\)

The phasor \(-V_{YN}\) will have the same magnitude as \(V_{YN}\) but will be \(180^\circ\) out of phase with \(V_{YN}\). Therefore, the angle of \(-V_{YN}\) will be \(-120^\circ + 180^\circ = 60^\circ\).

Now, we need to find the resultant of \(V_{RN}\) (\(V_P \angle 0^\circ\)) and \(-V_{YN}\) (\(V_P \angle 60^\circ\)). These two phasors have equal magnitudes (\(V_P\)) and an angle of \(60^\circ\) between them.

This shows that the line voltage \(V_{RY}\) leads the phase voltage \(V_{RN}\) by an angle of \(30^\circ\). This same principle applies to other line-to-phase voltage pairs in a balanced star-connected system (e.g., \(V_{YB}\) leads \(V_{YN}\) by \(30^\circ\), and \(V_{BR}\) leads \(V_{BN}\) by \(30^\circ\)).

Conclusion on Phase Shift

In a balanced three-phase, star-connected system, the phase difference between any phase voltage and its respective line voltage is consistently \(30^\circ\). The line voltage always leads its corresponding phase voltage by this angle.