Question

If all resistances in delta are equal to 30 $\Omega$, then each resistance in star is

• A. 30 $\Omega$
• B. 90 $\Omega$
• C. 10 $\Omega$
• D. 900 $\Omega$

Hint:

For any balanced network, the transformation simplifies drastically to a factor of 3: \[ R_{\text{Star}} = \frac{1}{3} R_{\text{Delta}} \quad \Longleftrightarrow \quad R_{\text{Delta}} = 3 \cdot R_{\text{Star}} \] Since Star connections distribute power across a neutral reference point, their equivalent branch resistances are always three times smaller than their corresponding Delta loop values!

Answer 1

The Correct Option is C

Concept: In electrical network analysis, Delta ($\Delta$) and Star ($Y$) configurations are two primary topological methods of interconnecting three terminal networks. Transforming resistances from a delta framework to a equivalent star framework is a powerful reduction technique. Let the three nodes of the network be labeled as $A$, $B$, and $C$.
• In a Delta ($\Delta$) network, three resistors are connected between the terminal pairs: $R_{AB}$, $R_{BC}$, and $R_{CA}$.
• In an equivalent Star ($Y$) network, three resistors branch out from a common neutral central node to the respective external terminals: $R_A$, $R_B$, and $R_C$. The comprehensive general mathematical formulas used to convert a delta network into its equivalent star counterparts are derived by matching terminal resistances: \[ R_A = \frac{R_{AB} \cdot R_{CA}}{R_{AB} + R_{BC} + R_{CA}} \] \[ R_B = \frac{R_{AB} \cdot R_{BC}}{R_{AB} + R_{BC} + R_{CA}} \] \[ R_C = \frac{R_{BC} \cdot R_{CA}}{R_{AB} + R_{BC} + R_{CA}} \]

Step 1: Applying the given conditions of balance. The problem explicitly states that all the resistances in the delta configuration are completely symmetric and equal to one another. Let this common resistance value be $R_{\Delta}$: \[ R_{AB} = R_{BC} = R_{CA} = R_{\Delta} = 30 \;\Omega \]

Step 2: Performing the systematic substitution. Since the circuit is entirely symmetrical, each star arm resistance ($R_A = R_B = R_C = R_Y$) will share the same value. Let us calculate $R_A$: \[ R_A = \frac{R_{\Delta} \cdot R_{\Delta}}{R_{\Delta} + R_{\Delta} + R_{\Delta}} = \frac{(R_{\Delta})^2}{3 \cdot R_{\Delta}} = \frac{R_{\Delta}}{3} \] This gives us the standard conversion rule for balanced networks: \[ R_Y = \frac{R_{\Delta}}{3} \]

Step 3: Evaluating the numerical result. Substituting the given value of $R_{\Delta} = 30 \;\Omega$ into our derived balanced equation: \[ R_Y = \frac{30 \;\Omega}{3} = 10 \;\Omega \] Therefore, each individual resistor in the equivalent star network has a value of precisely $10 \;\Omega$. This corresponds precisely to Option (3).