Star Delta transformation

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STAR DELTA TRANSFORMATION 

 

 

ice screenshot 20160414 100948

                                   STAR                                                  DELTA

 Figure (a) shows a Y ( star or wye ) connected impedance circuit and figure (b) a delta (Δ) connected  impedance circuit , their Τ and π shape shape circuit also represented respectively in above figure 

The two system will be exactly equivalent if the impedance between any pair of terminal a . b and c in figure (a) for the star ,is same as that between the corresponding pairs for the delta connection in figure (b) .when the third terminal is isolated 

First take star connection : the impedance between the terminals a and is chart?cht=tx&chl=Z %7B%7B1%7D%7D%20%2BZ %7B%7B2%7D%7D .

Now take delta connection : between  the same terminal a and b there are two parallel path , one having a impedance chart?cht=tx&chl=Z %7B%7BA%7D%7D%20%2BZ %7B%7BC%7D%7D and other having an impedance of  chart?cht=tx&chl=Z %7B%7BB%7D%7D%20

so equivalent impedance between terminals a and b for delta connected network will be 

 chart?cht=tx&chl=Z %7B%7BB%7D%7D%20%5Csetminus%5Csetminus%20(Z %7BA%7D%20%2BZ %7BC%7D)  = chart?cht=tx&chl=%5Cfrac%7BZ %7BB%7D

For transformation, an impedance between terminals a and has to be same in both star and delta connected network ,so 

chart?cht=tx&chl=Z %7B%7B1%7D%7D%20%2BZ %7B%7B2%7D%7D  =  chart?cht=tx&chl=%5Cfrac%7BZ %7BB%7D               eq(1)

similarly we can write between terminals b and

chart?cht=tx&chl=Z %7B%7B2%7D%7D%2BZ %7B%7B3%7D%7D   = chart?cht=tx&chl=%5Cfrac%7BZ %7BC%7D               eq(2)

in similar fashion between terminal c and a

chart?cht=tx&chl=Z %7B%7B3%7D%7D%2BZ %7B%7B1%7D%7D      = chart?cht=tx&chl=%5Cfrac%7BZ %7BA%7D              eq(3)

DELTA TO STAR TRANSFORMATION:

ice screenshot 20160414 101822
DELTA

                                                                                        TO

ice screenshot 20160414 101756
STAR

For transformation from delta to star , impedance chart?cht=tx&chl=Z %7B%7BA%7D%7D ,chart?cht=tx&chl=Z %7B%7BB%7D%7D,chart?cht=tx&chl=Z %7B%7BC%7D%7D given in delta network , and we have to find equivalent impedance chart?cht=tx&chl=Z %7B%7B1%7D%7D,chart?cht=tx&chl=Z %7B%7B2%7D%7D,chart?cht=tx&chl=Z %7B%7B3%7D%7D between respective terminals in star network 

to find chart?cht=tx&chl=Z %7B%7B1%7D%7D , now subtracting eq(2) from eq(1) and adding the result to eq(3), we get 

     chart?cht=tx&chl=Z %7B%7B1%7D%7D = chart?cht=tx&chl=%5Cfrac%7BZ %7BA%7D                   eq(4)

in similar fashion 

     chart?cht=tx&chl=Z %7B%7B2%7D%7D  = chart?cht=tx&chl=%5Cfrac%7BZ %7BB%7D                 eq(5)

     chart?cht=tx&chl=Z %7B%7B3%7D%7D  = chart?cht=tx&chl=%5Cfrac%7BZ %7BC%7D                 eq(6)

NOTE:- from above eq(4), eq(5) ,eq(6) it is clear that equivalent impedance of each arm of the star is given by the product of the impedance of the two delta side that meet at its ends divide by the sum of three delta impedance

 

STAR TO DELTA TRANSFORMATION: 

 in this case impedance chart?cht=tx&chl=Z %7B%7B1%7D%7D,chart?cht=tx&chl=Z %7B%7B2%7D%7D and chart?cht=tx&chl=Z %7B%7B3%7D%7D of star connected network is given and we have to find impedance chart?cht=tx&chl=Z %7B%7BA%7D%7D,chart?cht=tx&chl=Z %7B%7BB%7D%7D and chart?cht=tx&chl=Z %7B%7BC%7D%7D in delta connected network as shown in figure above 

let us define

  chart?cht=tx&chl=Z %7B%7BK%7D%7D = chart?cht=tx&chl=Z %7B%7B1%7D%7D.Z %7B%7B2%7D%7D%2BZ %7B%7B2%7D%7D                 eq(7)

after substituting the respective values of chart?cht=tx&chl=Z %7B%7B1%7D%7D ,chart?cht=tx&chl=Z %7B%7B2%7D%7D ,chart?cht=tx&chl=Z %7B%7B3%7D%7D from eq(4), eq(5) , eq(6) in eq(7) we will get 

   chart?cht=tx&chl=Z %7B%7BK%7D%7D   =  chart?cht=tx&chl=%5Cfrac%7BZ %7BA%7D.Z %7BB%7D                               eq(8)          

from eq(4) amd eq(8)

   chart?cht=tx&chl=Z %7B%7BK%7D%7D = chart?cht=tx&chl=Z %7B%7B1%7D%7D

  so                  

   chart?cht=tx&chl=Z %7B%7BC%7D%7D  = chart?cht=tx&chl=%5Cfrac%7BZ %7BK%7D%7D%7BZ %7B1%7D%7D   =  chart?cht=tx&chl=Z %7B%7B2%7D%7D%2BZ %7B%7B3%7D%7D%20%2B%5Cfrac%7BZ %7B2%7D               eq(9)

  in similar fashion

     chart?cht=tx&chl=Z %7B%7BA%7D%7D  =  chart?cht=tx&chl=Z %7B%7B1%7D%7D%2BZ %7B%7B3%7D%7D%20%2B%5Cfrac%7BZ %7B1%7D                           eq(10)

      chart?cht=tx&chl=Z %7B%7BB%7D%7D  =  chart?cht=tx&chl=Z %7B%7B1%7D%7D%2BZ %7B%7B2%7D%7D%20%2B%5Cfrac%7BZ %7B1%7D                           eq(11) 

NOTE:-  Therefore , the equivalent impedance of each arm of the delta is given by the some of star impedance between those terminals plus the product of these two star impedances divided by the third impedance