Q16Basic Electrical Engineering

Question

Q.6. Derive an expression for conversion of a resistive network from star to delta.

Answer 1

To simplify complex circuit networks, it is often necessary to convert a three-terminal Star (Y) connected resistor network into an electrically equivalent Delta ( $Δ$ ) network. 'Equivalent' means the resistance measured between any two corresponding terminals must be exactly identical in both configurations.

Let the resistors in the Star network, connected to a central neutral node, be $R_{1}$ , $R_{2}$ , and $R_{3}$ . We want to find the equivalent Delta resistors $R_{12}$ (between terminals 1 & 2), $R_{23}$ (between 2 & 3), and $R_{31}$ (between 3 & 1).

The standard transformation formula derived from circuit equivalency states that any resistor in the Delta network is equal to the sum of the two adjacent Star resistors, plus the product of those two adjacent resistors divided by the third (opposite) Star resistor.

R_{12} = R_{1} + R_{2} + \frac{R _{1} R _{2}}{R _{3}}

R_{23} = R_{2} + R_{3} + \frac{R _{2} R _{3}}{R _{1}}

R_{31} = R_{3} + R_{1} + \frac{R _{3} R _{1}}{R _{2}}

If all three Star resistors are equal in value (e.g., $R_{Y}$ ), the calculation drastically simplifies. The equivalent Delta resistors will all be equal, and exactly three times the value of the Star resistors: $R_{Δ} = 3 \cdot R_{Y}$ .

Q16Basic Electrical Engineering

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