Question

How do you find the resistance of a star and Delta Connection?

Answer 1

To understand the resistance in star (Y) and delta (Δ) connections, we first need to grasp how these configurations work in electrical circuits, particularly in three-phase systems. Each connection has a different way of linking three resistors (or impedances), affecting how we calculate their overall resistance.

### Star (Y) Connection

1. **Configuration**:
   - In a star connection, three resistors (R1, R2, R3) are connected at a single central point, forming a "Y" shape. Each resistor connects one terminal to this central point, while the other terminal connects to the supply line.

2. **Total Resistance Calculation**:
   - In a star configuration, the total or equivalent resistance (R_Y) seen from the supply lines can be calculated using the formula:
     \[
     R_Y = \frac{R1 \times R2 + R2 \times R3 + R3 \times R1}{R1 + R2 + R3}
     \]
   - This formula effectively allows us to find the equivalent resistance from one phase to the neutral point.

3. **Example**:
   - If R1 = 10Ω, R2 = 20Ω, and R3 = 30Ω, the equivalent resistance can be calculated as:
     \[
     R_Y = \frac{10 \times 20 + 20 \times 30 + 30 \times 10}{10 + 20 + 30} = \frac{200 + 600 + 300}{60} = \frac{1100}{60} \approx 18.33Ω
     \]

### Delta (Δ) Connection

1. **Configuration**:
   - In a delta connection, each resistor connects end-to-end, forming a closed loop or triangle shape. Each resistor (R1, R2, R3) connects two terminals of the supply, meaning there is no neutral point in the same way as in a star configuration.

2. **Total Resistance Calculation**:
   - To find the equivalent resistance (R_Δ) of the delta connection, the formula is:
     \[
     R_Δ = \frac{R1 \times R2 + R2 \times R3 + R3 \times R1}{R1 + R2 + R3}
     \]
   - This is actually the same formula as for the star connection, but remember, this calculates the equivalent resistance across the terminals of the delta.

3. **Example**:
   - Using the same resistor values as before (R1 = 10Ω, R2 = 20Ω, R3 = 30Ω):
     \[
     R_Δ = \frac{10 \times 20 + 20 \times 30 + 30 \times 10}{10 + 20 + 30} = \frac{200 + 600 + 300}{60} = \frac{1100}{60} \approx 18.33Ω
     \]
   - Interestingly, while the equivalent resistance from each configuration can be calculated, the actual values may differ when seen from different terminals.

### Conversion Between Star and Delta

There are situations when you might need to convert from a star to a delta connection (or vice versa). The formulas for conversion are as follows:

- **From Star to Delta**:
  \[
  R1(Δ) = \frac{R1(Y) \times R2(Y)}{R3(Y)}
  \]
  \[
  R2(Δ) = \frac{R2(Y) \times R3(Y)}{R1(Y)}
  \]
  \[
  R3(Δ) = \frac{R3(Y) \times R1(Y)}{R2(Y)}
  \]

- **From Delta to Star**:
  \[
  R1(Y) = \frac{R1(Δ) \times R2(Δ)}{R3(Δ) + R1(Δ) + R2(Δ)}
  \]
  \[
  R2(Y) = \frac{R2(Δ) \times R3(Δ)}{R1(Δ) + R2(Δ) + R3(Δ)}
  \]
  \[
  R3(Y) = \frac{R3(Δ) \times R1(Δ)}{R1(Δ) + R2(Δ) + R3(Δ)}
  \]

### Summary

In conclusion, the resistance calculations for star and delta connections are crucial in understanding how circuits behave, especially in three-phase systems. The formulas allow for effective analysis and conversion, enabling engineers to optimize circuits for various applications. When working with actual circuits, always ensure proper identification of each configuration and apply the corresponding formulas to achieve accurate results.

Answer 2

To find the resistance in a star (wye) or delta connection, you'll need to understand how the resistances are configured in each type of connection. Here’s a step-by-step guide on how to find the equivalent resistance for both star and delta connections:

### Star (Wye) Connection

In a star (or wye) connection, three resistors are connected to a common central point, forming a Y shape. Let’s denote these resistors as \( R_A \), \( R_B \), and \( R_C \).

**To find the equivalent resistance between any two points of the star configuration:**

1. **Identify the Resistances**: Suppose you want to find the equivalent resistance between two points (say \( A \) and \( B \)).

2. **Calculate the Equivalent Resistance**:
    - The resistances in the star connection between any two points can be found using the formula:

      \[
      R_{AB} = \frac{R_A \cdot R_B}{R_A + R_B + R_C}
      \]

      where \( R_A \), \( R_B \), and \( R_C \) are the resistances in the star configuration.

### Delta Connection

In a delta connection, three resistors are connected in a loop, forming a triangle. Let’s denote these resistors as \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \), where each resistor is between the points A, B, and C respectively.

**To find the equivalent resistance between any two points of the delta configuration:**

1. **Identify the Resistances**: Suppose you want to find the equivalent resistance between two points (say \( A \) and \( B \)).

2. **Calculate the Equivalent Resistance**:
    - The resistances in the delta connection between any two points can be found using the formula:

      \[
      R_{AB} = \frac{R_{AB} \cdot R_{BC}}{R_{AB} + R_{BC} + R_{CA}}
      \]

      where \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \) are the resistances in the delta configuration.

### Converting Between Star and Delta

Sometimes, you may need to convert a star (wye) network into a delta network or vice versa. This is done using the following formulas:

**Star to Delta Conversion:**

If you have resistances \( R_A \), \( R_B \), and \( R_C \) in the star configuration, the equivalent delta resistances \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \) can be found using:

\[
R_{AB} = \frac{R_A \cdot R_B}{R_A + R_B + R_C}
\]
\[
R_{BC} = \frac{R_B \cdot R_C}{R_A + R_B + R_C}
\]
\[
R_{CA} = \frac{R_C \cdot R_A}{R_A + R_B + R_C}
\]

**Delta to Star Conversion:**

If you have resistances \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \) in the delta configuration, the equivalent star resistances \( R_A \), \( R_B \), and \( R_C \) can be found using:

\[
R_A = \frac{R_{AB} \cdot R_{CA}}{R_{AB} + R_{BC} + R_{CA}}
\]
\[
R_B = \frac{R_{AB} \cdot R_{BC}}{R_{AB} + R_{BC} + R_{CA}}
\]
\[
R_C = \frac{R_{BC} \cdot R_{CA}}{R_{AB} + R_{BC} + R_{CA}}
\]

These formulas allow you to analyze and simplify circuits involving resistors in star or delta configurations.