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Star Delta Transformation

Star Delta Transformation:

There are some networks in which resistors are neither in series nor in parallel. Such networks are star-connected or delta-connected resistors.A familiar case is a three terminal network e.g delta network or star network.It is not possible to simplify such networks by employing series or parallel technique.In order to solve them we transfer them into one another and then apply series parallel techniques.We can transfer star-connected into delta-connected resistances by keeping the two networks electrically equivalent and vice versa. By electrically equivalent  networks means 

 “ Resistance between any two terminals of one network is equal to the resistance between the corresponding the second network.”

Delta to star transformation:
Resistance between terminals A and B (Δ-connected)
RAB=RAB||(RBC+RCA)
RAB=RAB(RBC+RCA)/RAB+RBC+RCA
Resistance between terminals A and B (Star-connected)
RAB=RA+RB
As two networks are electrically equivalent so
Resistance between terminals A and B (Δ-connected)=  Resistance between terminals A and B (Star-connected)
RAB (RBC+RCA)/RAB+RBC+RCA =RA+RB  ………………..(1)
Similarly
Resistance between terminals B and C (Δ-connected)=  Resistance between terminals B and C (Star-connected)
RBC(RAB+RCA)/RAB+RBC+RCA = RB+RC   …………………(2)
Resistance between terminals C and A(Δ-connected)=  Resistance between terminals C and A (Star-connected)
RCA(RAB+RBC)/RAB+RBC+RCA=RC+RA   ………………………(3)
Subtract (2) from (1)
RAB (RBC+RCA)/RAB+RBC+RCA - RBC(RAB+RCA)/RAB+RBC+RCA =RA+RB - RB+RC
On solving we get
RA-RC = RAB (RBC+RCA)- RBC(RAB+RCA)/ RAB+RBC+RCA   ………..(4)
Adding (3) & (4)
RC+RA +  RA-RC = RCA(RAB+RBC)/RAB+RBC+RCA+ RAB (RBC+RCA)- RBC(RAB+RCA)/ RAB+RBC+RCA
On solving we get
RA=RCA RAB/RAB+RBC+RCA
Similarly,
RB=RBCRAB/ RAB+RBC+RCA
RC=RBCRCA/ RAB+RBC+RCA
Here we provide an easy formula to calculate equivalent resistance for star-connection by simply looking at a diagram in which star-connected resistors are connected to a delta-connection.
Resistance of any arm of star-connection =                                                                                                                       Product of two adjacent arms of Δ connected to satr-connected arm/Sum of all arms of Δ
According to above formula
RB=RBCRAB/ RAB+RBC+RCA
RC=RBCRCA/ RAB+RBC+RCA  



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