What is the relationship between the line voltage and the phase voltage in a delta connected load?
2 Answers
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.
\[ V_L = \sqrt{3} \times V_{ph} \]
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.
In a **delta-connected load**, the relationship between the **line voltage** (\(V_L\)) and the **phase voltage** (\(V_\phi\)) is as follows:
- The **line voltage** is equal to the **phase voltage** in a delta connection.
Mathematically:
\[
V_L = V_\phi
\]
### Explanation:
- In a delta connection, each phase winding is connected between two line conductors. Since the phase winding is directly across two lines, the voltage across each phase winding is the same as the line voltage.
Unlike in a star connection (where the line voltage is \(\sqrt{3}\) times the phase voltage), the delta configuration has equal line and phase voltages.
This relationship holds true for both balanced and unbalanced delta systems.
- The **line voltage** is equal to the **phase voltage** in a delta connection.
Mathematically:
\[
V_L = V_\phi
\]
### Explanation:
- In a delta connection, each phase winding is connected between two line conductors. Since the phase winding is directly across two lines, the voltage across each phase winding is the same as the line voltage.
Unlike in a star connection (where the line voltage is \(\sqrt{3}\) times the phase voltage), the delta configuration has equal line and phase voltages.
This relationship holds true for both balanced and unbalanced delta systems.