Question

Write the expressions for delta connected resistors in terms of star connected resistors.

Answer 1

To express delta-connected resistors in terms of star-connected resistors, we can use the following relationships. Let's denote the resistors in the star configuration as \( R_A \), \( R_B \), and \( R_C \), and the resistors in the delta configuration as \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \).

The relationships are given by:

1. \( R_{AB} = \frac{R_A R_B}{R_A + R_B + R_C} \)
2. \( R_{BC} = \frac{R_B R_C}{R_A + R_B + R_C} \)
3. \( R_{CA} = \frac{R_C R_A}{R_A + R_B + R_C} \)

These equations allow you to convert between star (Y) and delta (Δ) configurations of resistors.

Answer 2

To express delta-connected resistors in terms of star-connected resistors, or vice versa, you can use the following formulas. These formulas are useful in circuit analysis, especially when dealing with the conversion between delta (Δ) and star (Y) configurations of resistors.

### Delta to Star Conversion

If you have resistors \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \) in a delta configuration, and you want to find the equivalent star resistors \( R_A \), \( R_B \), and \( R_C \), you use the following formulas:

1. **For \( R_A \) (the resistor connecting node A to the center of the star):**

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

2. **For \( R_B \) (the resistor connecting node B to the center of the star):**

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

3. **For \( R_C \) (the resistor connecting node C to the center of the star):**

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

### Star to Delta Conversion

If you have resistors \( R_A \), \( R_B \), and \( R_C \) in a star configuration, and you want to find the equivalent delta resistors \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \), you use the following formulas:

1. **For \( R_{AB} \) (the resistor between nodes A and B in the delta configuration):**

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

2. **For \( R_{BC} \) (the resistor between nodes B and C in the delta configuration):**

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

3. **For \( R_{CA} \) (the resistor between nodes C and A in the delta configuration):**

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

### Summary

- **Delta to Star:**
  \[ R_A = \frac{R_{AB} \cdot R_{CA}}{R_{AB} + R_{BC} + R_{CA}} \]
  \[ R_B = \frac{R_{AB} \cdot R_{BC}}{R_{AB} + R_{BC} + R_{CA}} \]
  \[ R_C = \frac{R_{BC} \cdot R_{CA}}{R_{AB} + R_{BC} + R_{CA}} \]

- **Star to Delta:**
  \[ R_{AB} = \frac{R_A \cdot R_B + R_B \cdot R_C + R_C \cdot R_A}{R_C} \]
  \[ R_{BC} = \frac{R_A \cdot R_B + R_B \cdot R_C + R_C \cdot R_A}{R_A} \]
  \[ R_{CA} = \frac{R_A \cdot R_B + R_B \cdot R_C + R_C \cdot R_A}{R_B} \]

These conversions are crucial for simplifying and analyzing electrical circuits in both configurations.