Question

How to find equivalent resistance in star delta connection?

Answer 1

Finding the equivalent resistance in a star-delta (Y-Δ) connection is a common task in electrical engineering, especially when analyzing complex circuits. Here’s a detailed explanation of how to perform the conversion and calculate equivalent resistance.

### Star (Y) Connection

In a star connection, three resistors \( R_a \), \( R_b \), and \( R_c \) are connected at a single point (the star point). The resistances are arranged as follows:

```
     R_a
    /    \
   /      \
  Rb       Rc
   \      /
    \    /
     ------
```

### Delta (Δ) Connection

In a delta connection, three resistors \( R_{ab} \), \( R_{bc} \), and \( R_{ca} \) are connected in a triangular shape. The arrangement looks like this:

```
     R_ab
    /    \
   /      \
  R_ca    R_bc
   \      /
    \    /
     ------
```

### Conversion Formulas

To convert between star and delta connections, you can use the following formulas:

1. **From Star to Delta:**
   - \( R_{ab} = \frac{R_a R_b + R_b R_c + R_c R_a}{R_c} \)
   - \( R_{bc} = \frac{R_a R_b + R_b R_c + R_c R_a}{R_a} \)
   - \( R_{ca} = \frac{R_a R_b + R_b R_c + R_c R_a}{R_b} \)

2. **From Delta to Star:**
   - \( R_a = \frac{R_{ab} R_{ca}}{R_{ab} + R_{bc} + R_{ca}} \)
   - \( R_b = \frac{R_{ab} R_{bc}}{R_{ab} + R_{bc} + R_{ca}} \)
   - \( R_c = \frac{R_{bc} R_{ca}}{R_{ab} + R_{bc} + R_{ca}} \)

### Steps to Find Equivalent Resistance

1. **Identify the Configuration:**
   - Determine if your resistors are in a star or delta configuration.

2. **Use the Appropriate Conversion Formula:**
   - If you need to find the equivalent resistance in a different configuration, apply the appropriate conversion formula.

3. **Calculate Total Resistance:**
   - If the resistors are in series or parallel after conversion, use:
     - **Series:** \( R_{total} = R_1 + R_2 + R_3 \)
     - **Parallel:** \( \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \)

### Example

Let’s say you have a star connection with resistances:

- \( R_a = 10 \, \Omega \)
- \( R_b = 20 \, \Omega \)
- \( R_c = 30 \, \Omega \)

Using the formulas for conversion to delta:

1. Calculate \( R_{ab} \):

\[
R_{ab} = \frac{10 \times 20 + 20 \times 30 + 30 \times 10}{30} = \frac{200 + 600 + 300}{30} = \frac{1100}{30} = \frac{110}{3} \approx 36.67 \, \Omega
\]

2. Calculate \( R_{bc} \):

\[
R_{bc} = \frac{10 \times 20 + 20 \times 30 + 30 \times 10}{10} = \frac{1100}{10} = 110 \, \Omega
\]

3. Calculate \( R_{ca} \):

\[
R_{ca} = \frac{10 \times 20 + 20 \times 30 + 30 \times 10}{20} = \frac{1100}{20} = 55 \, \Omega
\]

Now, you can use these \( R_{ab} \), \( R_{bc} \), and \( R_{ca} \) values in your circuit as needed.

### Conclusion

Understanding how to convert between star and delta configurations and calculating equivalent resistances is crucial for analyzing circuits effectively. Always ensure that you identify the correct configuration before applying the formulas. If you have specific values or a particular circuit in mind, feel free to share, and I can assist you further!

Answer 2

Finding the equivalent resistance between two points in a star-delta (also known as wye-delta) network involves converting the network from one configuration to another. The star (Y) and delta (Δ) connections are two common ways to connect three resistors in a circuit, and sometimes it's necessary to convert between these configurations to simplify the analysis.

### Understanding Star and Delta Configurations

1. **Star (Y) Configuration:**
   - In a star configuration, three resistors are connected to a central node, with each resistor connecting to one of the three outer points.
   - The resistors are labeled \( R_A \), \( R_B \), and \( R_C \) for the resistors connecting to nodes \( A \), \( B \), and \( C \) respectively.

2. **Delta (Δ) Configuration:**
   - In a delta configuration, three resistors form a closed loop, with each resistor connecting to two of the three nodes.
   - The resistors are labeled \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \) for the resistors between nodes \( A \) and \( B \), \( B \) and \( C \), and \( C \) and \( A \) respectively.

### Converting Star to Delta and Vice Versa

**1. Star to Delta Conversion:**

To convert a star network to an equivalent delta network, use the following formulas:

- Let \( R_A \), \( R_B \), and \( R_C \) be the resistors in the star configuration.
- The resistors in the delta configuration \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \) can be calculated using the formulas:

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

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

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

**2. Delta to Star Conversion:**

To convert a delta network to an equivalent star network, use the following formulas:

- Let \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \) be the resistors in the delta configuration.
- The resistors in the star configuration \( R_A \), \( R_B \), and \( R_C \) can be calculated using the formulas:

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

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

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

### Example Problem

Suppose you have a star network with resistors \( R_A = 4 \ \Omega \), \( R_B = 6 \ \Omega \), and \( R_C = 12 \ \Omega \), and you want to find the equivalent delta network.

1. Compute \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \):

\[
R_{AB} = \frac{4 \times 6 + 6 \times 12 + 12 \times 4}{12} = \frac{24 + 72 + 48}{12} = \frac{144}{12} = 12 \ \Omega
\]

\[
R_{BC} = \frac{4 \times 6 + 6 \times 12 + 12 \times 4}{4} = \frac{144}{4} = 36 \ \Omega
\]

\[
R_{CA} = \frac{4 \times 6 + 6 \times 12 + 12 \times 4}{6} = \frac{144}{6} = 24 \ \Omega
\]

So the equivalent delta resistors are \( R_{AB} = 12 \ \Omega \), \( R_{BC} = 36 \ \Omega \), and \( R_{CA} = 24 \ \Omega \).

### Summary

To find the equivalent resistance in a star-delta network, you typically need to convert the network from one configuration to the other. Use the conversion formulas to switch between star and delta configurations, and then find the equivalent resistance based on the simplified network. This approach is useful in complex circuits where direct calculation of equivalent resistance is challenging.