Question

What is the relationship between star and delta connection formula?

Answer 1

In three-phase electrical systems, star (Y) and delta (Δ) connections are two common methods of connecting loads or power sources. Understanding the relationship between these two configurations is crucial for electrical engineers, especially in the context of power systems, motor operations, and load balancing. Below, I will explain the basic principles of both connections, their relationships, and the relevant formulas involved.

### 1. **Star Connection (Y-Connection)**

In a star connection, one end of each of the three phase windings is connected together to form a common point (neutral), while the other ends are connected to the supply lines.

#### Characteristics of Star Connection:
- **Phase Voltage (Vₚ):** Each phase voltage is measured between a phase terminal and the neutral point. The line voltage (Vₗ) is the voltage measured between any two line terminals.
- **Voltage Relationship:**
  \[
  Vₗ = \sqrt{3} \cdot Vₚ
  \]
  where \( Vₗ \) is the line voltage and \( Vₚ \) is the phase voltage.

- **Current Relationship:**
  \[
  Iₗ = Iₚ
  \]
  where \( Iₗ \) is the line current and \( Iₚ \) is the phase current.

### 2. **Delta Connection (Δ-Connection)**

In a delta connection, the ends of the three windings are connected in a closed loop (triangle shape), forming the three phases connected in series with each other.

#### Characteristics of Delta Connection:
- **Phase Voltage (Vₚ):** In a delta connection, the phase voltage is equal to the line voltage.
- **Voltage Relationship:**
  \[
  Vₚ = Vₗ
  \]
  
- **Current Relationship:**
  \[
  Iₗ = \sqrt{3} \cdot Iₚ
  \]
  where \( Iₗ \) is the line current and \( Iₚ \) is the phase current.

### 3. **Relationship Between Star and Delta Connections**

To analyze the relationship between the star and delta configurations, we can derive the transformation formulas, often referred to as "star-delta transformation."

#### Transformation Formulas

1. **Star to Delta Transformation:**
   - For a three-phase system, the equivalent phase resistance or impedance in delta (ZΔ) can be calculated from the star impedance (ZY) as follows:
   \[
   Z_Δ = 3 \cdot Z_Y
   \]
   where:
   - \( Z_Y \) is the impedance of each phase in the star connection.
   - \( Z_Δ \) is the impedance of each phase in the delta connection.

2. **Delta to Star Transformation:**
   - Conversely, if you need to convert from delta to star, the equivalent impedance in star can be calculated as:
   \[
   Z_Y = \frac{Z_Δ}{3}
   \]

### 4. **Example Calculation**

Let’s consider an example to illustrate these transformations:

- Suppose we have a delta-connected load with a phase impedance of \( Z_Δ = 30 \, \Omega \).

To find the equivalent star impedance:
\[
Z_Y = \frac{Z_Δ}{3} = \frac{30 \, \Omega}{3} = 10 \, \Omega
\]

Conversely, if we have a star-connected load with a phase impedance of \( Z_Y = 15 \, \Omega \):
\[
Z_Δ = 3 \cdot Z_Y = 3 \cdot 15 \, \Omega = 45 \, \Omega
\]

### 5. **Practical Applications**

- **Motor Connections:** Star connections are often used during the starting of large motors (star-delta starters) to reduce the starting current.
- **Load Balancing:** Engineers may need to convert between these configurations to ensure that loads are balanced and operate efficiently within the system.

### 6. **Conclusion**

The relationship between star and delta connections is fundamental in three-phase systems. The transformation formulas allow engineers to switch between the two configurations depending on operational needs, voltage levels, and current requirements. Mastery of these concepts is essential for effective design and analysis in electrical engineering.

Answer 2

The star (Y) and delta (Δ) connections are two different methods for connecting three-phase electrical circuits. The relationship between the two is important for understanding how to convert between them. Here’s how they relate to each other:

### Star (Y) Connection:
- In a star connection, the three-phase loads or impedances are connected at a common point, which is often grounded.
- The line voltage (\( V_L \)) is higher than the phase voltage (\( V_{\phi} \)).
- The relationship between line voltage and phase voltage in a star connection is:
  \[
  V_L = \sqrt{3} \cdot V_{\phi}
  \]
- The line current (\( I_L \)) is the same as the phase current (\( I_{\phi} \)):
  \[
  I_L = I_{\phi}
  \]

### Delta (Δ) Connection:
- In a delta connection, the three-phase loads or impedances are connected in a triangular loop.
- The line voltage (\( V_L \)) is the same as the phase voltage (\( V_{\phi} \)).
- The relationship between line current and phase current in a delta connection is:
  \[
  I_L = \sqrt{3} \cdot I_{\phi}
  \]

### Conversion Formulas:
To convert between star and delta, you use the following formulas:

1. **Impedance Conversion**:
   - From Delta to Star:
     \[
     Z_Y = \frac{Z_{\Delta}}{3}
     \]
   - From Star to Delta:
     \[
     Z_{\Delta} = 3 \cdot Z_Y
     \]
   Where \( Z_Y \) is the impedance in the star connection and \( Z_{\Delta} \) is the impedance in the delta connection.

2. **Voltage and Current Relations**:
   - For voltages:
     \[
     V_{L, \Delta} = V_{L, Y}
     \]
   - For currents:
     \[
     I_{\phi, \Delta} = \frac{I_{L, Y}}{\sqrt{3}}
     \]
     \[
     I_{\phi, Y} = \sqrt{3} \cdot I_{L, \Delta}
     \]

These relationships are fundamental for analyzing and designing three-phase electrical systems, especially when converting between different connection types.