Relationship between Line and Phase Values

Definition

Line values are the electrical quantities measured between any two line conductors in a three-phase system, while phase values are the quantities associated with a single phase of the source or load. In a three-phase system:

Line voltage

• is the voltage between any two line terminals.

Phase voltage

• is the voltage across one phase winding or one phase load element.

Line current

• is the current flowing in each line conductor.

Phase current

• is the current flowing through each phase element.

The exact relationship between line and phase values depends on whether the three-phase system is connected in star (Y) or delta (Δ).

Main Content

1. Three-Phase Star (Y) Connection

In a star-connected system, one end of each phase is joined to a common point called the neutral point, and the other ends are connected to the three line conductors.

Voltage relationship

In a star connection, the line voltage is greater than the phase voltage by a factor of √3.
$$V_L = \sqrt{3}\,V_{ph}$$ Therefore, $$V_{ph} = \frac{V_L}{\sqrt{3}}$$ This happens because line voltage is the vector difference of two phase voltages, which are 120° apart.

Current relationship

In a star connection, the line current is equal to the phase current.
$$I_L = I_{ph}$$ This is because each line conductor is directly in series with one phase winding or phase load.

Example:
If a star-connected system has a line voltage of 415 V, then the phase voltage is: $$V_{ph} = \frac{415}{\sqrt{3}} \approx 240 \text{ V}$$ So each phase gets about 240 V, while the line-to-line supply is 415 V.

2. Three-Phase Delta (Δ) Connection

In a delta-connected system, the three phase elements are connected end-to-end to form a closed loop, and the three line conductors are connected at the three junctions.

Voltage relationship

In a delta connection, the line voltage is equal to the phase voltage.
$$V_L = V_{ph}$$ This is because each phase is directly connected across two line conductors.

Current relationship

In a delta connection, the line current is greater than the phase current by a factor of √3.
$$I_L = \sqrt{3}\,I_{ph}$$ Therefore, $$I_{ph} = \frac{I_L}{\sqrt{3}}$$ This occurs because the line current is the vector sum of two phase currents meeting at each line terminal.

Example:
If a delta-connected load has a phase current of 10 A, then the line current is: $$I_L = \sqrt{3}\times 10 \approx 17.32 \text{ A}$$ So the line conductors carry more current than each phase element.

3. Power Calculation Using Line and Phase Values

Three-phase power can be calculated using either line values or phase values, depending on the connection type. The total power is the sum of power in all three phases.

Using phase values

$$P = 3V_{ph}I_{ph}\cos\phi$$ where $\phi$ is the phase angle between voltage and current.

Using line values

Since the line and phase relationships differ in star and delta, the total power can also be written as: $$P = \sqrt{3}V_LI_L\cos\phi$$ This formula is valid for balanced three-phase systems.

Importance of balance

These relationships are most accurate when the load is balanced, meaning each phase has the same impedance and each current is equal in magnitude.

Example:
If a balanced three-phase system has $V_L = 400\text{ V}$, $I_L = 20\text{ A}$, and power factor = 0.8: $$P = \sqrt{3}\times 400\times 20\times 0.8 \approx 11.09 \text{ kW}$$

Working / Process

1. Identify the type of connection

• Determine whether the three-phase system is connected in star or delta.
• This is the first and most important step because the relationships change with the connection type.

2. Apply the correct line-phase formulas

• For star: $V_L=\sqrt{3}V_{ph}$, $I_L=I_{ph}$
• For delta: $V_L=V_{ph}$, $I_L=\sqrt{3}I_{ph}$

3. Use the values in power or load calculations

• Once line and phase values are converted correctly, calculate current, voltage, power, or impedance as required.
• Always ensure the system is balanced before applying standard formulas directly.

Advantages / Applications

• Helps in solving three-phase circuit problems quickly and accurately.
• Essential for designing and analyzing motors, alternators, transformers, and industrial loads.
• Used in power distribution systems to choose proper line voltage and current ratings.
• Useful in calculating total three-phase power and understanding system efficiency.
• Important in selecting protective devices, cables, and switchgear based on correct current values.

Summary

• In a three-phase system, line values are measured between lines, and phase values are measured across one phase.
• In star connection, line voltage is $\sqrt{3}$ times phase voltage, and line current equals phase current.
• In delta connection, line voltage equals phase voltage, and line current is $\sqrt{3}$ times phase current.
• This relationship is fundamental for three-phase power calculations and practical electrical system analysis.