How to find equivalent resistance in star connection?
2 Answers
To find the equivalent resistance in a star (Y) connection, follow these steps:
1. **Identify the resistors**: Label the resistors connected in the star configuration as \( R_1, R_2, R_3 \) corresponding to each leg of the star.
2. **Use the formula for equivalent resistance**: The equivalent resistance \( R_{eq} \) can be calculated using the formula:
\[
R_{eq} = \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_1 + R_2 + R_3}
\]
3. **Calculation**: Substitute the values of \( R_1, R_2, \) and \( R_3 \) into the formula to find \( R_{eq} \).
This formula derives from the relationship between star and delta configurations, and it allows you to effectively combine the resistances in the star connection into a single equivalent resistance.
1. **Identify the resistors**: Label the resistors connected in the star configuration as \( R_1, R_2, R_3 \) corresponding to each leg of the star.
2. **Use the formula for equivalent resistance**: The equivalent resistance \( R_{eq} \) can be calculated using the formula:
\[
R_{eq} = \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_1 + R_2 + R_3}
\]
3. **Calculation**: Substitute the values of \( R_1, R_2, \) and \( R_3 \) into the formula to find \( R_{eq} \).
This formula derives from the relationship between star and delta configurations, and it allows you to effectively combine the resistances in the star connection into a single equivalent resistance.
To find the equivalent resistance of a star (or wye) connection, you need to convert it into a delta (or triangle) connection or use the formulas directly. Here’s a detailed approach to both methods:
### 1. **Converting Star to Delta Connection**
In a star connection, resistors are connected as follows:
- \( R_A \) is connected between nodes \( A \) and \( B \)
- \( R_B \) is connected between nodes \( B \) and \( C \)
- \( R_C \) is connected between nodes \( C \) and \( A \)
The delta (or triangle) connection resistors are:
- \( R_{AB} \) between nodes \( A \) and \( B \)
- \( R_{BC} \) between nodes \( B \) and \( C \)
- \( R_{CA} \) between nodes \( C \) and \( A \)
The formulas for converting star resistors (\(R_A\), \(R_B\), and \(R_C\)) to delta resistors (\(R_{AB}\), \(R_{BC}\), and \(R_{CA}\)) are:
\[ R_{AB} = \frac{R_A \cdot R_B}{R_C} + R_A + R_B \]
\[ R_{BC} = \frac{R_B \cdot R_C}{R_A} + R_B + R_C \]
\[ R_{CA} = \frac{R_C \cdot R_A}{R_B} + R_C + R_A \]
### 2. **Direct Formula for Equivalent Resistance in Star Connection**
If you’re given the resistors in the star configuration and you want to find the equivalent resistance between any two points in the network, you can use these steps:
#### **Finding the Equivalent Resistance Between Two Points:**
1. **Between Nodes A and B:**
For resistors \( R_A \) and \( R_B \) in the star connection and \( R_C \) being the third resistor:
\[ R_{AB} = \frac{R_A \cdot R_B}{R_C} \]
2. **Between Nodes B and C:**
\[ R_{BC} = \frac{R_B \cdot R_C}{R_A} \]
3. **Between Nodes C and A:**
\[ R_{CA} = \frac{R_C \cdot R_A}{R_B} \]
### **Example Calculation**
Suppose you have a star connection with resistors \( R_A = 2 \ \Omega \), \( R_B = 3 \ \Omega \), and \( R_C = 6 \ \Omega \):
1. **Calculate \( R_{AB} \):**
\[ R_{AB} = \frac{2 \cdot 3}{6} = 1 \ \Omega \]
2. **Calculate \( R_{BC} \):**
\[ R_{BC} = \frac{3 \cdot 6}{2} = 9 \ \Omega \]
3. **Calculate \( R_{CA} \):**
\[ R_{CA} = \frac{6 \cdot 2}{3} = 4 \ \Omega \]
So, the equivalent delta resistances are \( R_{AB} = 1 \ \Omega \), \( R_{BC} = 9 \ \Omega \), and \( R_{CA} = 4 \ \Omega \).
You can now use these delta resistances to analyze the circuit further as needed.
### 1. **Converting Star to Delta Connection**
In a star connection, resistors are connected as follows:
- \( R_A \) is connected between nodes \( A \) and \( B \)
- \( R_B \) is connected between nodes \( B \) and \( C \)
- \( R_C \) is connected between nodes \( C \) and \( A \)
The delta (or triangle) connection resistors are:
- \( R_{AB} \) between nodes \( A \) and \( B \)
- \( R_{BC} \) between nodes \( B \) and \( C \)
- \( R_{CA} \) between nodes \( C \) and \( A \)
The formulas for converting star resistors (\(R_A\), \(R_B\), and \(R_C\)) to delta resistors (\(R_{AB}\), \(R_{BC}\), and \(R_{CA}\)) are:
\[ R_{AB} = \frac{R_A \cdot R_B}{R_C} + R_A + R_B \]
\[ R_{BC} = \frac{R_B \cdot R_C}{R_A} + R_B + R_C \]
\[ R_{CA} = \frac{R_C \cdot R_A}{R_B} + R_C + R_A \]
### 2. **Direct Formula for Equivalent Resistance in Star Connection**
If you’re given the resistors in the star configuration and you want to find the equivalent resistance between any two points in the network, you can use these steps:
#### **Finding the Equivalent Resistance Between Two Points:**
1. **Between Nodes A and B:**
For resistors \( R_A \) and \( R_B \) in the star connection and \( R_C \) being the third resistor:
\[ R_{AB} = \frac{R_A \cdot R_B}{R_C} \]
2. **Between Nodes B and C:**
\[ R_{BC} = \frac{R_B \cdot R_C}{R_A} \]
3. **Between Nodes C and A:**
\[ R_{CA} = \frac{R_C \cdot R_A}{R_B} \]
### **Example Calculation**
Suppose you have a star connection with resistors \( R_A = 2 \ \Omega \), \( R_B = 3 \ \Omega \), and \( R_C = 6 \ \Omega \):
1. **Calculate \( R_{AB} \):**
\[ R_{AB} = \frac{2 \cdot 3}{6} = 1 \ \Omega \]
2. **Calculate \( R_{BC} \):**
\[ R_{BC} = \frac{3 \cdot 6}{2} = 9 \ \Omega \]
3. **Calculate \( R_{CA} \):**
\[ R_{CA} = \frac{6 \cdot 2}{3} = 4 \ \Omega \]
So, the equivalent delta resistances are \( R_{AB} = 1 \ \Omega \), \( R_{BC} = 9 \ \Omega \), and \( R_{CA} = 4 \ \Omega \).
You can now use these delta resistances to analyze the circuit further as needed.