The conversion from a delta (Δ) connection to a star (Y) connection can be accomplished using the following formulas. If the resistances in a delta connection are \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \), the equivalent resistances in the star connection (\( R_A \), \( R_B \), \( R_C \)) can be calculated as:
\[ R_A = \frac{R_{AB} \cdot R_{CA}}{R_{AB} + R_{BC} + R_{CA}} \]
\[ R_B = \frac{R_{AB} \cdot R_{BC}}{R_{AB} + R_{BC} + R_{CA}} \]
\[ R_C = \frac{R_{BC} \cdot R_{CA}}{R_{AB} + R_{BC} + R_{CA}} \]
Given the resistances in the delta connection:
- \( R_{AB} = 20 , \Omega \)
- \( R_{BC} = 80 , \Omega \)
- \( R_{CA} = 30 , \Omega \)
We first calculate the total resistance in the delta connection:
\[ R_{total} = R_{AB} + R_{BC} + R_{CA} = 20 , \Omega + 80 , \Omega + 30 , \Omega = 130 , \Omega \]
Now we can find \( R_A \), \( R_B \), and \( R_C \):
Calculate \( R_A \):
\[ R_A = \frac{R_{AB} \cdot R_{CA}}{R_{total}} = \frac{20 \cdot 30}{130} = \frac{600}{130} \approx 4.62 , \Omega \]
Calculate \( R_B \):
\[ R_B = \frac{R_{AB} \cdot R_{BC}}{R_{total}} = \frac{20 \cdot 80}{130} = \frac{1600}{130} \approx 12.31 , \Omega \]
Calculate \( R_C \):
\[ R_C = \frac{R_{BC} \cdot R_{CA}}{R_{total}} = \frac{80 \cdot 30}{130} = \frac{2400}{130} \approx 18.46 , \Omega \]
Now we can summarize the equivalent star-connected resistances:
- \( R_A \approx 4.62 , \Omega \)
- \( R_B \approx 12.31 , \Omega \)
- \( R_C \approx 18.46 , \Omega \)
Thus, the resistances of the equivalent star-connected resistors are approximately:
- \( R_A \approx 4.62 , \Omega \)
- \( R_B \approx 12.31 , \Omega \)
- \( R_C \approx 18.46 , \Omega \)
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