Effect of Winding Resistances

Effect of Winding Resistances

AU: May-14

• A practical transformer windings possess some resistances which not only cause the power losses but also the voltage drops. Let us see what is the effect of winding resistances 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 V₁ 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. E has to supply this drop. The vector difference between E₂ and I₂ R₂ is available to the load as a terminal voltage V₂.

• The drops I₁ R₁ and I₂ R₂ are purely resistive drops hence are always in phase with the respective currents I₁ and I₂.

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 calcuations 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

. Now the expression (6.9.3) 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 12. This value of resistance R₂/K² which is the value of R2 referred to primary is called equivalent resistance of secondary referred to primary. It is denoted as R '₂.

R '₂ = R₂ / K_2...........(6.9.4)

• Hence the total resistance referred to primary is the addition of R₁ and R '₂ called equivalent resistance of transformer referred to primary and denoted as R_1e

R_1e = R₁ + R '₂ = R₁ + R₂ / K₂ (6.9.5)

• This resistance R_1e causes same copper loss with I₁ as the total copper loss due to the individual windings.

Total copper loss = I₁² R_1e = I₁² R₁ +I₁² R₂...(6.9.6)

• So equivalent resistance R_1e 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 lossv =

• Thus the resistance K² R₁ is primary resistance referred to secondary denoted as R'₁.

R'₁ = K² R₁ .....(6.9.8)

• Hence the total resistance referred to secondary is the addition of R₂ and R'₁ called equivalent resistance of transformer referred to secondary and denoted as R_2e

R_2e = R₂+ R'₁ =  R₂+ K²  .................... (6.9.9)

.. Total copper loss = I₂² R_2e ......(6.9.10)

• The concept of equivalent resistance is shown in the Fig. 6.9.1 (a), (b) and (c).

Key Point: When resistances are transferred to primary, the secondary winding becomes resistance winding for calculation purpose. The entire copper loss occurs due to R_1e. Similarly when resistances are referred to secondary, the primary becomes resistanceless for calculation purpose. The entire copper loss occurs due to R_2e.

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 N₂/N₁.

• Thus the resistance K² R₁ is primary resistance referred to secondary denoted as R '₁.

R '₁ = K² R₁ ............. (6.9.8)

• Hence the total resistance referred to secondary is the addition of R₂ and R'₁ called equivalent resistance of transformer referred to secondary and denoted as R_2e

R_2e = R₂+ R'₁ = R₂+ K² R₁ .............(6.9.9)

 Total copper loss = I²₂ R_2e  ...............(6.9.10)

• The concept of equivalent resistance is shown in the Fig. 6.9.1 (a), (b) and (c).

Key Point: When resistances are transferred to primary, the secondary winding becomes resistance winding for calculation purpose. The entire copper loss occurs due to R_1e. Similarly when resistances are referred to secondary, the primary becomes resistanceless for calculation purpose. The entire copper loss occurs due to R_2e.

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 N₂/N₁.

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

Ex. 6.9.1 A 1-phase transformer has 360 turns and 180 turns respectively in its secondary and primary windings. The respective resistance are 0.233 and 0.067. Calculate the equivalent resistance of

1) The primary in terms of the secondary winding.hong

2) The secondary in terms of the primary winding, and

3) The total resistance of the transformer in terms of the  primary. AU May-14, Marks 8

Sol.: N₂ = 360, N₁ = 180, R₂ = 0.233 Ω, R₁ = 0.067 Ω.

K = N₂ / N₁  = 360 / 180 = 2

 As secondary turns are more, secondary winding is high voltage side. Hence on secondary the resistance values are high.

Review Question

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. 

[Ans.: 5.2225Ω, 0.01918 Ω]