Question

A delta connected network with its wyes equivalent is shown in the figure. The resistances R1, R2 and R3 (in ohm) are respectively

• A. $1.5, 3\text{ and } 9$
• B. $3, 9\text{ and } 1.5$
• C. $9, 3\text{ and } 1.5$
• D. $3, 1.5\text{ and } 9$

Hint:

In exams, it is often much faster to verify the options using the Star-to-Delta formula because it involves addition and simple division rather than calculating the sum of delta resistors as a denominator.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem requires finding the equivalent Star (Wye) resistances $R_1, R_2,$ and $R_3$ from a given Delta network.
The Delta resistances are $R_{ab} = 5\ \Omega$ (between terminals a and b), $R_{bc} = 15\ \Omega$ (between terminals b and c), and $R_{ca} = 30\ \Omega$ (between terminals c and a, where the diagram shows $30\ \Omega$ although it's handwritten).

Step 2: Key Formula or Approach:
The standard Star-to-Delta conversion formulas are:
\[ R_{ab} = R_1 + R_2 + \frac{R_1 R_2}{R_3} \]
\[ R_{bc} = R_2 + R_3 + \frac{R_2 R_3}{R_1} \]
\[ R_{ca} = R_3 + R_1 + \frac{R_3 R_1}{R_2} \]
Alternatively, Delta-to-Star conversion formulas are:
\[ R_1 = \frac{R_{ab} R_{ca}}{R_{ab} + R_{bc} + R_{ca}} \]

Step 3: Detailed Explanation:

• Let us verify Option (D) where $R_1 = 3\ \Omega$, $R_2 = 1.5\ \Omega$, and $R_3 = 9\ \Omega$ using the Star-to-Delta formulas.

• Calculate $R_{ab}$:
\[ R_{ab} = 3 + 1.5 + \frac{3 \times 1.5}{9} = 4.5 + \frac{4.5}{9} = 4.5 + 0.5 = 5\ \Omega \]
This matches the given delta resistance $R_{ab} = 5\ \Omega$.

• Calculate $R_{bc}$:
\[ R_{bc} = 1.5 + 9 + \frac{1.5 \times 9}{3} = 10.5 + 4.5 = 15\ \Omega \]
This matches the given delta resistance $R_{bc} = 15\ \Omega$.

• Calculate $R_{ca}$:
\[ R_{ca} = 9 + 3 + \frac{9 \times 3}{1.5} = 12 + \frac{27}{1.5} = 12 + 18 = 30\ \Omega \]
This matches the delta resistance of $30\ \Omega$.

Step 4: Final Answer:
The equivalent values are $R_1 = 3\ \Omega$, $R_2 = 1.5\ \Omega$, and $R_3 = 9\ \Omega$, which correspond to Option (D).