In electrical engineering, particularly in three-phase systems, the **Delta (Δ)** and **Star (Y)** connections refer to different ways of connecting resistances (or impedances) in a circuit. Understanding the relationship between these two configurations is crucial for analyzing three-phase circuits. Here's a detailed explanation of the relationship between resistances connected in delta and their equivalent star system:
### Basic Definitions
1. **Delta Connection (Δ)**:
- In a delta connection, the components (resistances) are connected in a loop (triangle shape). Each resistance connects to two other resistances.
- For example, if we have three resistances \( R_{A}, R_{B}, R_{C} \), they would connect at points A, B, and C forming a triangle.
2. **Star Connection (Y)**:
- In a star connection, all components connect to a common central point (the star point). Each resistance connects from this central point to a terminal.
- Using the same resistances \( R_{A}, R_{B}, R_{C} \), each would connect to a central node, forming a “Y” shape.
### Mathematical Relationship
To establish the relationship between the delta and star configurations, we can derive formulas to convert between the two systems.
#### Conversion from Delta to Star
Given resistances \( R_A \), \( R_B \), and \( R_C \) in a delta configuration, the equivalent resistances \( R_{A}' \), \( R_{B}' \), and \( R_{C}' \) in the star configuration can be calculated as follows:
\[
R_{A}' = \frac{R_A R_B}{R_A + R_B + R_C}
\]
\[
R_{B}' = \frac{R_B R_C}{R_A + R_B + R_C}
\]
\[
R_{C}' = \frac{R_C R_A}{R_A + R_B + R_C}
\]
- Here, \( R_{A}' \), \( R_{B}' \), and \( R_{C}' \) are the resistances in the star configuration.
#### Conversion from Star to Delta
Conversely, if you have resistances in a star configuration, the equivalent resistances in a delta configuration can be calculated using:
\[
R_A = \frac{R_{A}' R_{B}'}{R_{A}' + R_{B}' + R_{C}'}
\]
\[
R_B = \frac{R_{B}' R_{C}'}{R_{A}' + R_{B}' + R_{C}'}
\]
\[
R_C = \frac{R_{C}' R_{A}'}{R_{A}' + R_{B}' + R_{C}'}
\]
### Implications of the Relationship
1. **Voltage and Current Relationships**:
- In a delta connection, the phase voltage is equal to the line voltage, while the line current is divided among the phases.
- In a star connection, the phase voltage is the line voltage divided by \( \sqrt{3} \), and the line current is equal to the phase current.
2. **Power Calculations**:
- The total power in both configurations can be analyzed by converting from one to the other. The relationships between line and phase voltages and currents significantly affect power calculations.
3. **Impedance Matching**:
- Knowing how to convert between delta and star configurations helps in impedance matching in circuits, ensuring maximum power transfer.
### Example
Let’s consider an example where:
- Delta resistances: \( R_A = 30 \, \Omega \), \( R_B = 20 \, \Omega \), \( R_C = 10 \, \Omega \).
To find the equivalent star resistances:
1. Calculate \( R_{A}' \):
\[
R_{A}' = \frac{30 \times 20}{30 + 20 + 10} = \frac{600}{60} = 10 \, \Omega
\]
2. Calculate \( R_{B}' \):
\[
R_{B}' = \frac{20 \times 10}{30 + 20 + 10} = \frac{200}{60} \approx 3.33 \, \Omega
\]
3. Calculate \( R_{C}' \):
\[
R_{C}' = \frac{10 \times 30}{30 + 20 + 10} = \frac{300}{60} = 5 \, \Omega
\]
So, the equivalent star resistances are \( R_{A}' = 10 \, \Omega \), \( R_{B}' \approx 3.33 \, \Omega \), and \( R_{C}' = 5 \, \Omega \).
### Conclusion
The relationship between delta and star connected resistances is fundamental in circuit analysis, allowing engineers to easily switch between configurations for calculations. By using the derived formulas, one can determine the equivalent resistances in either configuration, facilitating the understanding of the electrical behavior of three-phase systems. This conversion is particularly useful in power distribution, load balancing, and circuit design.