Star-Delta Conversion Online

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5. Series Equivalent Resistance

6. Delta to star conversion

7. Star to delta conversion

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Star to Delta Conversion

Delta to star Conversion

Star–Delta (Y–Δ) Transformation

1. Star (Y) Structure

A star network consists of three resistors connected to a common center point \(N\).

R1 A ───/\/\/───┐ │ R2 │ B ───/\/\/───┤── N │ R3 │ C ───/\/\/───┘

Here, \(R_1\), \(R_2\), and \(R_3\) connect the three external terminals to the common point \(N\).

2. Delta (Δ) Structure

A delta network consists of three resistors connected in the form of a triangle.

RAB A ───/\/\/─── B \ / \ / \ / \ / \ / \ / \ / C

The three resistors directly connect the three external terminals.

3. Why Do We Use the Transformation?

Some circuits cannot be simplified directly using only series and parallel combinations.

In such cases, a star network can be converted into an equivalent delta network, or a delta network can be converted into an equivalent star network.

Important: The transformation preserves the electrical behavior seen from the three external terminals \(A\), \(B\), and \(C\).

4. Delta → Star (Δ → Y)

Suppose the delta resistors are:

• \(R_{AB}\) between A and B
• \(R_{BC}\) between B and C
• \(R_{CA}\) between C and A

The equivalent star resistors are:

$$ R_A= \frac{R_{AB}R_{CA}} {R_{AB}+R_{BC}+R_{CA}} $$

$$ R_B= \frac{R_{AB}R_{BC}} {R_{AB}+R_{BC}+R_{CA}} $$

$$ R_C= \frac{R_{BC}R_{CA}} {R_{AB}+R_{BC}+R_{CA}} $$

Easy way to remember:

For each Y resistor:

$$ R_A= \frac{\text{product of the two adjacent Δ resistors}} {\text{sum of all Δ resistors}} $$

5. Star → Delta (Y → Δ)

Suppose the star resistors are \(R_A\), \(R_B\), and \(R_C\).

The equivalent delta resistors are:

$$ R_{AB} = R_A+R_B+ \frac{R_AR_B}{R_C} $$

or equivalently:

$$ R_{AB} = \frac{ R_AR_B+R_BR_C+R_CR_A }{R_C} $$

$$ R_{BC} = \frac{ R_AR_B+R_BR_C+R_CR_A }{R_A} $$

$$ R_{CA} = \frac{ R_AR_B+R_BR_C+R_CR_A }{R_B} $$

Easy way to remember:

For each Δ resistor:

$$ R_{AB} = \frac{\text{sum of pairwise products}} {\text{opposite Y resistor}} $$

6. Example

Suppose we have a star network:

$$ R_A=2\Omega,\qquad R_B=3\Omega,\qquad R_C=6\Omega $$

First calculate:

$$ R_AR_B+R_BR_C+R_CR_A $$

$$ =(2)(3)+(3)(6)+(6)(2) $$

$$ =6+18+12=36 $$

Therefore:

$$ R_{AB}=\frac{36}{6}=6\Omega $$

$$ R_{BC}=\frac{36}{2}=18\Omega $$

$$ R_{CA}=\frac{36}{3}=12\Omega $$

Equivalent Delta:

$$ \boxed{ R_{AB}=6\Omega,\quad R_{BC}=18\Omega,\quad R_{CA}=12\Omega } $$

7. How Does the Transformation Work?

The transformation works because the voltage-current relationships at the three external terminals remain the same.

The internal currents inside the original and transformed networks may be different, but an external circuit connected to terminals \(A\), \(B\), and \(C\) cannot distinguish between the two equivalent networks.

Key idea: A Star–Delta transformation changes the internal arrangement of resistors while maintaining the same electrical behavior at the external terminals.