الخميس، 14 يوليو 2011

Effect of Winding Resistances

1. Effect OF Winding Resistances 
       A practical transformer windings process some resistances which not only cause the power losses but also the voltage drops. Let us see what is the effect of winding resistance on the performance of the transformer.
       Let              R1 = primary winding resistance in ohms
                           R2 = secondary winding resistance in ohms
       Now when current I1 flows through primary, there is voltage drop I1 R1 across the winding. The supply voltage V1 has to supply this drop. Hence primary induced e.m.f. E1 is the vector difference between V1 and I1 R1.
       Similarly the induced e.m.f. in secondary is E2. When load is connected, current I2 flows and there is voltage drop I2 R2. The e.m.f. E2 has to supply this drop. The vector difference between E2 and I2 R2 is available to the load as a terminal voltage.
The drops I1 R1 and I2 R2 are purely resistive drops hence are always in phase with the respective currents I1 and I2.
1.1 Equivalent Resistance
       The resistance of the two windings can be transferred to any one side either primary or secondary without affecting the performance of the transformer. The transfer of the resistances on any one side is advantageous as it makes the calculations very easy. Let us see how to transfer the resistances on any one side.
       The total copper loss due to both the resistances can be obtained as,
       total copper loss = I12 R1 + I22 R2
                                 = I12{ R1 +(I22/I12) R2}
                                  = I12{R1 + (1/K2) R2}                                             .......(3)
Where            I2/I1 = 1/K                                                 neglecting no load current.
       Now the expression (3) indicates that the total copper loss can be expressed as I12 R1 + I12 .R2/K2. This means R2/K2 is the resistance value of R2 shifted to primary side which causes same copper loss with I1 as R2 causes with. This value of resistance which R2 /K2 is the value of R2 referred to primary is called equivalent resistance of secondary referred to primary. It is denoted as R2'.
                                       R2' = R2 /K2                                                       ........(4)
       Hence the total resistance referred to primary is the addition of R1 and R2' called equivalent resistance of transformer referred to primary and denoted as R1e.
                                            = R1 + R2'= R1 + R2 /K2                                 .........(5)
       This resistance R1e causes same copper loss with I1 as the total copper loss due to the individual windings.
       total copper loss = I12 R1e = I12 R1 + I22 R2                                               ......(6)
       So equivalent resistance simplifies the calculations as we have to calculate parameters on one side only.
       Similarly it is possible to refer the equivalent resistance to secondary winding.
       total copper loss = I12 R1 + I22 R2
                                 = I22 {(I12/I22) R1 + R2}
                                  = I22 ( K2 R1 + R2)                                                            ........(7)
       Thus the resistance K2 R1 is primary resistance referred to secondary denoted as R1'.
                                R1' = KR1                                                                          .......(8)
       Hence the total resistance referred to secondary is the addition of R2 and R1' called equivalent resistance of transformer referred to secondary and denoted as R2e.
                                 R2e = R2 + R1' = R2 + KR1                                              .........(9)
       total copper loss = I22 R2e                                                                             ........(10)
       The concept of equivalent resistance is shown in the Fig. 1(a), (b) and (c).
Fig. 1 Equivalent resistance
Key Point : When resistance are transferred to primary, the secondary winding becomes zero resistance winding for calculation purpose. The entire copper loss occurs due to R1e. Similarly when resistances are referred to secondary, the primary becomes resistanceless for calculation purpose. The entire copper loss occurs due to R2e.
Important Note : When a resistance is to be transferred from the primary to secondary, it must be multiplied by K2. When a resistance is to be transferred from the secondary to primary, it must be divided by K2. Remember that K is N1 /N2
        The result can be cross-checked by another approach. The high voltage winding is always low current winding and hence the resistance of high voltage side is high. The low voltage side is high current side and hence resistance of low voltage side is low. So while transferring resistance from low voltage side to high voltage side, its value must increase while transferring resistance from high voltage side to low voltage side, its value must decrease.

Key point :
High voltage side → Low current side → High resistance side
Low voltage side → High current side → Low resistance side

Example 1 : A 6600/400 V single phase transformer has primary resistance of 2.5 Ω and secondary resistance of 0.01 Ω calculate total equivalent resistance referred to primary and secondary.
Solution : The given values are,
                     R1 = 2.5 Ω           R2 = 0.01 Ω
                     K = 400/6600 = 0.0606
      While finding equivalent resistance referred to primary, transfer to primary as,
                    R2'= R2 /K = 0.01/(0.0606)2 = 2.7225 Ω
                    R1e = R1 + R2' = 2.5 + 2.7225 = 5.2225 Ω
       It can be observed that primary is high voltage hence high resistance side hence while transferring from low voltage to on high voltage, its value increases.
       To find total equivalent resistance referred to secondary, first calculate ,
                    R1'= K R1 = (0.0606)2 x 25 = 0.00918 Ω
                    R2e = R2 + R1' = 0.01 + 0.00918 = 0.01918 Ω

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