The relation between power in a three-phase Delta (Δ) and Star (Y) connected system arises due to differences in how the phases are interconnected, but the total power delivered in both systems remains the same under balanced conditions. Here's a breakdown to explain the relationship between the power in Delta and Star configurations:
### 1. **Three-Phase Power Overview**
In a three-phase system, the total power is the sum of the power delivered by each phase. The total power can be calculated using the formula:
\[
P_{\text{total}} = \sqrt{3} \times V_L \times I_L \times \cos(\phi)
\]
Where:
- \( V_L \) is the line voltage (the voltage between two phases),
- \( I_L \) is the line current (the current through each line conductor),
- \( \cos(\phi) \) is the power factor.
### 2. **Delta (Δ) Connected System**
In a Delta connection, each phase is connected between two lines (not involving the neutral point). As a result:
- **Line Voltage** (\( V_L \)) is the same as the **Phase Voltage** (\( V_\text{ph} \)):
\[
V_L = V_\text{ph}
\]
- **Line Current** (\( I_L \)) is related to the **Phase Current** (\( I_\text{ph} \)) by:
\[
I_L = \sqrt{3} \times I_\text{ph}
\]
### 3. **Star (Y) Connected System**
In a Star connection, each phase is connected between a line and a common neutral point. Therefore:
- **Line Voltage** (\( V_L \)) is related to the **Phase Voltage** (\( V_\text{ph} \)) by:
\[
V_L = \sqrt{3} \times V_\text{ph}
\]
- **Line Current** (\( I_L \)) is the same as the **Phase Current** (\( I_\text{ph} \)):
\[
I_L = I_\text{ph}
\]
### 4. **Power Relation Between Star and Delta**
- The total power in both Star and Delta connections can be expressed in terms of **line voltage** and **line current** using the same power formula:
\[
P_{\text{total}} = \sqrt{3} \times V_L \times I_L \times \cos(\phi)
\]
This means that the total power delivered in a Star-connected system is **equal** to the total power delivered in a Delta-connected system for the same load, voltage, and current.
- However, the **phase voltages** and **phase currents** differ between Star and Delta configurations:
- In a **Delta** connection, the phase voltage is equal to the line voltage, and the line current is greater than the phase current by a factor of \( \sqrt{3} \).
- In a **Star** connection, the line voltage is greater than the phase voltage by a factor of \( \sqrt{3} \), while the line current is the same as the phase current.
Thus, while the voltages and currents differ in each configuration, the total power delivered remains the same, as long as the system is balanced.
### 5. **Example for Better Understanding**
- **Delta Connection:**
- Assume the line voltage \( V_L = 400 \, \text{V} \) and the phase current \( I_\text{ph} = 10 \, \text{A} \).
- Since \( I_L = \sqrt{3} \times I_\text{ph} \), the line current \( I_L = 17.32 \, \text{A} \).
- The total power is:
\[
P_{\text{total}} = \sqrt{3} \times 400 \times 17.32 \times \cos(\phi)
\]
- **Star Connection:**
- The same line voltage \( V_L = 400 \, \text{V} \) would result in a phase voltage \( V_\text{ph} = \frac{400}{\sqrt{3}} = 230 \, \text{V} \).
- The line current \( I_L = I_\text{ph} = 10 \, \text{A} \).
- The total power is:
\[
P_{\text{total}} = \sqrt{3} \times 400 \times 10 \times \cos(\phi)
\]
In both cases, the total power will be the same as long as the load and power factor remain the same.
### Conclusion
The relation between power in Delta and Star connected systems is that **the total power is the same** in both configurations, given balanced conditions. The main difference lies in how the voltages and currents are distributed between the phases and lines. In Delta, line voltage equals phase voltage, and the line current is \( \sqrt{3} \) times the phase current, while in Star, line voltage is \( \sqrt{3} \) times the phase voltage, and line current equals phase current.