Question

STAR – DELTA TRANSFORMATION

Answer 1

Star connection and delta connection are the two different methods of connecting three basic elements which cannot be further simplified into series or parallel.

The two ways of representation can have equivalent circuits in either form. Assume some voltage source across the terminals AB.

\(R_{eq}=R_{a}+R_{b}\)

\(R_{eq}=R_{1}(R_{2}+R_{3})/(R_{1}+R_{2}+R_{3})\)

Therefore     \(R_{a}+R_{b}=R_{1}(R_{2}+R_{3})/(R_{1}+R_{2}+R_{3})\).........(1)

Similarly       \(R_{b}+R_{c}=R_{3}(R_{1}+R_{2})/(R_{1}+R_{2}+R_{3})\).........(2)

                   \(R_{c}+R_{a}=R_{2}(R_{3}+R_{1})/(R_{1}+R_{2}+R_{3})\)

Subtracting (2) from (1) and adding to (3) ,

\(R_{a}=R_{1}R_{2}/(R_{1}+R_{2}+R_{3})\) ........(4)

\(R_{b}=R_{1}R_{3}/(R_{1}+R_{2}+R_{3})\) .........(5)

\(R_{c}=R_{2}R_{3}/(R_{1}+R_{2}+R_{3})\) .........(6)

A delta connection of can be replaced by an equvivalent star connection with the values from equations (4),(5),(6).

Multiply (4)(5) ; (5)(6) ; (4)(6) and then adding the three we get,

\(R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}=R_{1}R_{2}R_{3}/(R_{1}+R_{2}+R_{3})\)

Dividing LHS by \(R_{a}\) gives \(R_{3}\), by R_b gives R_2, by R_c gives R₁

_{\(R_{1}=(R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a})/R_{c}\)}

_{\(R_{2}=(R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a})/R_{b}\)}

_{\(R_{3}=(R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a})/R_{a}\)}