Question

If a Delta circuit has branches \(R_{ab} = 10\ \Omega\), \(R_{bc} = 20\ \Omega\), and \(R_{ca} = 30\ \Omega\), what is the equivalent Wye resistor \(R_a\) connected to node?

• A. \(10\ \Omega \)
• B. \(20\ \Omega \)
• C. \(5\ \Omega \)
• D. \(6\ \Omega \)

Hint:

For \(\Delta \rightarrow Y\) transformation, remember: \(R_{\text{wye}} = \frac{\text{Product of neighbors}}{\text{Sum of all}}\). It saves conversion time during competitive examinations.

Answer 1

The Correct Option is C

Concept: Delta-to-Wye (\(\Delta\)-to-\(Y\)) transformation allows us to convert three resistors connected in a closed loop delta configuration into an equivalent star/wye network connected to a central common node. The general conversion formula for finding a resistor connected to a specific terminal node in the Wye network is given by: \[ R_{\text{node}} = \frac{\text{Product of the two adjacent Delta resistors sharing that node}}{\text{Sum of all three Delta resistors}} \] For node \(a\), the two adjacent delta branches that share this vertex are \(R_{ab}\) and \(R_{ca}\). Therefore, the expression for \(R_a\) is: \[ R_a = \frac{R_{ab} \times R_{ca}}{R_{ab} + R_{bc} + R_{ca}} \]

Step 1: Extracting given values and computing the total denominator sum.
We are given the individual values of the delta network branches:
• \(R_{ab} = 10\ \Omega\)
• \(R_{bc} = 20\ \Omega\)
• \(R_{ca} = 30\ \Omega\) Let us calculate the sum of all individual resistors in the delta loop: \[ \text{Sum} = R_{ab} + R_{bc} + R_{ca} = 10 + 20 + 30 = 60\ \Omega \]

Step 2: Applying the Delta-Wye transformation formula to calculate \(R_a\).
Substitute the relevant adjacent branch values and the total sum calculated above into the formula: \[ R_a = \frac{10 \times 30}{60} \] Multiply the values in the numerator: \[ 10 \times 30 = 300 \] Now substitute back and simplify the fraction: \[ R_a = \frac{300}{60} = 5\ \Omega \] Hence, the value of the equivalent Wye resistor connected to node \(a\) is exactly \(5\ \Omega\). This directly corresponds to Option (C).