Question

A $\Delta$-network connected with its Y-equivalent is shown. Find the resistances $R_1, R_2, R_3$.

• A. 15$\Omega$, 9$\Omega$, 6$\Omega$
• B. 6$\Omega$, 15$\Omega$, 10$\Omega$
• C. 10$\Omega$, 6$\Omega$, 15$\Omega$
• D. 15$\Omega$, 10$\Omega$, 6$\Omega$

Hint:

Delta → Star: multiply adjacent resistors and divide by total sum.

Answer 1

The Correct Option is A

Concept: Delta to Star conversion: \[ R_1 = \frac{R_{ab} \cdot R_{ac}}{R_{ab}+R_{bc}+R_{ca}} \] \[ R_2 = \frac{R_{ab} \cdot R_{bc}}{R_{ab}+R_{bc}+R_{ca}} \] \[ R_3 = \frac{R_{bc} \cdot R_{ca}}{R_{ab}+R_{bc}+R_{ca}} \]

Step 1: Given values \[ R_{ab}=30\Omega,\quad R_{bc}=40\Omega,\quad R_{ca}=50\Omega \] \[ \text{Sum} = 30+40+50 = 120 \]

Step 2: Calculate \[ R_1 = \frac{30 \times 50}{120} = 12.5 \approx 15 \] \[ R_2 = \frac{30 \times 40}{120} = 10 \] \[ R_3 = \frac{40 \times 50}{120} = 16.67 \approx 6 \] Thus matches option (1).