In a three-phase system with a delta-connected load, the relationship between the line voltage (\( V_L \)) and the phase voltage (\( V_{ph} \)) is given by:
\[ V_L = \sqrt{3} \times V_{ph} \]
Here's how this relationship comes about:
1. **Delta Connection Basics**:
- In a delta connection, each load is connected between two phases of the three-phase supply.
- The phase voltage (\( V_{ph} \)) is the voltage across each load.
2. **Line Voltage**:
- The line voltage (\( V_L \)) is the voltage measured between any two of the three supply lines.
3. **Vector Relationship**:
- In a balanced three-phase system, the line voltages and phase voltages are related through vector geometry.
- Each phase voltage is separated from the adjacent phase voltages by 120 degrees.
4. **Derivation**:
- Consider a delta-connected system where the phase voltages are represented as vectors. Each vector forms an angle of 120 degrees with the other two.
- To find the line voltage, you can use the following vector equation derived from the geometry of the phase vectors:
\[
V_L = \sqrt{V_{ph}^2 + V_{ph}^2 - 2 \cdot V_{ph} \cdot V_{ph} \cdot \cos(120^\circ)}
\]
Since \(\cos(120^\circ) = -0.5\), this simplifies to:
\[
V_L = \sqrt{V_{ph}^2 + V_{ph}^2 + V_{ph}^2} = \sqrt{3 \cdot V_{ph}^2} = \sqrt{3} \cdot V_{ph}
\]
In summary, the line voltage in a delta-connected load is \(\sqrt{3}\) times the phase voltage. This factor of \(\sqrt{3}\) arises from the geometric relationship between the phase voltages and line voltages in a balanced three-phase system.