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Tests on Single Phase Transformer
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Objective: To determent the efficiency and regulation of a single phase transformer by conducting (a) open circuit test and (b) short circuit test.

 

 

The physical basis of the transformer is mutual induction between two circuits  linked by a common magnetic field . Transformer is required to pass eletrical energy from one circuit to another, via the medium of the pulsating magnetic field, as efficiently and economically as possible. This could be achived using either iron or steel which serves as a good permeable path for the mutual magnetic flux.

 

Elementary Transformer: 

 

                                                              

                                                                                                        Figure 1

 

Let an alternating voltage V1 be applied to primary coil of N1 turns linking a suitable iron core. A current flows in the coil , establishing a

flux «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mo»§#8709;«/mo»«mi»p«/mi»«/msub»«/math»  in the core. This flux induces an emf e1 in the coil to counterbalance the applied voltage V1. This emf is

                                             

                                                                               «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»e«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«msub»«mi»N«/mi»«mn»1«/mn»«/msub»«mfrac»«mrow»«mi»d«/mi»«msub»«mo»§#8709;«/mo»«mi»p«/mi»«/msub»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo».«/mo»«/math»

                                         

 

Assuming sinusoidal time variation of the flux , let  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mo»§#8709;«/mo»«mi»p«/mi»«/msub»«mo»=«/mo»«msub»«mo»§#8709;«/mo»«mi»m«/mi»«/msub»«mi mathvariant=¨normal¨»sin«/mi»«mo»(«/mo»«mi»w«/mi»«mi»t«/mi»«mo»)«/mo»«mo».«/mo»«/math» Then,

                                             

                                                             «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»e«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«msub»«mi»N«/mi»«mn»1«/mn»«/msub»«mi»w«/mi»«msub»«mo»§#8709;«/mo»«mi»m«/mi»«/msub»«mi mathvariant=¨normal¨»cos«/mi»«mo»(«/mo»«mi»w«/mi»«mi»t«/mi»«mo»)«/mo»«mo»,«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»w«/mi»«mi»h«/mi»«mi»e«/mi»«mi»r«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»w«/mi»«mo»=«/mo»«mn»2«/mn»«mi»§#960;«/mi»«mi»F«/mi»«/math»

                                                          

 

 The r.m.s value of this voltage is given by : 

                                        «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»E«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mn»4«/mn»«mo».«/mo»«mn»44«/mn»«mi»F«/mi»«msub»«mi»N«/mi»«mn»1«/mn»«/msub»«msub»«mo»§#8709;«/mo»«mi»m«/mi»«/msub»«/math»

 

Now if there is a secondary coil of N2 turns, wound on the same core, then by mutual induction an emf e2 is developed therein. The

 

r.m.s value of this voltage is given by :

                                           

                                                                 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»E«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mn»4«/mn»«mo».«/mo»«mn»44«/mn»«mi»F«/mi»«msub»«mi»N«/mi»«mn»2«/mn»«/msub»«mo»§#8709;«/mo»«msub»«mo»§apos;«/mo»«mi»m«/mi»«/msub»«/math»

 

where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«msub»«mo»§apos;«/mo»«mi»m«/mi»«/msub»«/math» is the maximum value of the (sinusoidal) nflux linking the secondary coil («math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mo»§#8709;«/mo»«mi»s«/mi»«/msub»«/math»).

 

If it is assumed that «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mo»§#8709;«/mo»«mi»p«/mi»«/msub»«mo»=«/mo»«msub»«mo»§#8709;«/mo»«mi»s«/mi»«/msub»«mo»§nbsp;«/mo»«/math»then the  primary and secondary emf bear the following ratio:

                                                                            «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msub»«mi»e«/mi»«mn»1«/mn»«/msub»«msub»«mi»e«/mi»«mn»2«/mn»«/msub»«/mfrac»«mo»=«/mo»«mfrac»«msub»«mi»E«/mi»«mn»2«/mn»«/msub»«msub»«mi»E«/mi»«mn»1«/mn»«/msub»«/mfrac»«mo»=«/mo»«mfrac»«msub»«mi»N«/mi»«mn»2«/mn»«/msub»«msub»«mi»N«/mi»«mn»1«/mn»«/msub»«/mfrac»«/math» 

 

Note that in actual practice, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mo»§#8709;«/mo»«mi»p«/mi»«/msub»«mo»=«/mo»«msub»«mo»§#8709;«/mo»«mi»s«/mi»«/msub»«/math» since some of the flux paths linking the primary coil do not link the secondary coil and similarly

 

some of the flux paths linking the secondary coil do not link the primary coil. The fluxes which do not link both the coils are called

 

"Leakage Fluxes" of the primary and secondary coil.

 

Although the iron core is highly permeable, it is not possible to generate a magnetic field in it without the application of a small m.m.f

 

(magnato motive force), denoted by Mm. 

            «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»P«/mi»«mi»h«/mi»«/msub»«mo»=«/mo»«msub»«mi»K«/mi»«mi»h«/mi»«/msub»«msub»«msup»«mi»B«/mi»«mn»2«/mn»«/msup»«mrow»«mi»m«/mi»«mi»a«/mi»«mi»x«/mi»«/mrow»«/msub»«mo»§nbsp;«/mo»«msup»«mi»F«/mi»«mn»2«/mn»«/msup»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi»P«/mi»«mi»e«/mi»«/msub»«mo»=«/mo»«msub»«mi»K«/mi»«mi»e«/mi»«/msub»«msub»«msup»«mi»B«/mi»«mn»2«/mn»«/msup»«mrow»«mi»m«/mi»«mi»a«/mi»«mi»x«/mi»«/mrow»«/msub»«mo»§nbsp;«/mo»«msup»«mi»F«/mi»«mn»2«/mn»«/msup»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»a«/mi»«mi»n«/mi»«mi»d«/mi»«mo»§nbsp;«/mo»«msub»«mi»P«/mi»«mi»c«/mi»«/msub»«mo»=«/mo»«msub»«mi»P«/mi»«mi»h«/mi»«/msub»«mo»+«/mo»«msub»«mi»P«/mi»«mi»e«/mi»«/msub»«mo»§nbsp;«/mo»«/math»

 

 where Ph, Pe, Pc are hysteresis, eddy current and core losses respectively, Kh and Ke are constants which depend upon on the

 

magnetic material, and Bmax  is the maximum flux density in the core.

 

 

Equivalent Circuit of a Practical Transformer:

 

 

                                           

                                                       

                                                                              Figure 2(a).

 

 

Development of Transformer Equivalent Circuit:

                                          

                                               

                                          

                                                                                                           

                                                                                                                Figure 2(b).

 

The practical transforme has coils of finite resistance. Though this resistance is actually distributed uniformly, it can be conceived as

concentrated. Also, all the flux produced by the primary current cannot confined into a desired path completely as an eletric current.

On account of the leakage flux, both the windings have a voltage drop which is due to 'leakage reactance' . The transformer shown in

the figure 1 can be resolved into an equivalent circuit as shown in figure 2(a) in which the resistance and leakage reactance of primary

and secondary respectively are represented by lumped  R1, X1, R2, X2 . This equivalent circuit can be simplified by referring all quantities

in the secondary side of the transformer to primary side and is shown in figure 2(b).

 

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»R«/mi»«msub»«mo»§apos;«/mo»«mn»2«/mn»«/msub»«mo»=«/mo»«msub»«mi»R«/mi»«mn»2«/mn»«/msub»«mo»(«/mo»«mfrac»«msub»«mi»N«/mi»«mn»1«/mn»«/msub»«msub»«mi»N«/mi»«mn»2«/mn»«/msub»«/mfrac»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»X«/mi»«msub»«mo»§apos;«/mo»«mn»2«/mn»«/msub»«mo»=«/mo»«msub»«mi»X«/mi»«mn»2«/mn»«/msub»«mo»(«/mo»«mfrac»«msub»«mi»N«/mi»«mn»1«/mn»«/msub»«msub»«mi»N«/mi»«mn»2«/mn»«/msub»«/mfrac»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»I«/mi»«msub»«mo»§apos;«/mo»«mn»2«/mn»«/msub»«mo»=«/mo»«msub»«mi»I«/mi»«mn»2«/mn»«/msub»«mo»(«/mo»«mfrac»«msub»«mi»N«/mi»«mn»2«/mn»«/msub»«msub»«mi»N«/mi»«mn»1«/mn»«/msub»«/mfrac»«mo»)«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»V«/mi»«msub»«mo»§apos;«/mo»«mn»2«/mn»«/msub»«mo»=«/mo»«msub»«mi»V«/mi»«mn»2«/mn»«/msub»«mo»(«/mo»«mfrac»«msub»«mi»N«/mi»«mn»1«/mn»«/msub»«msub»«mi»N«/mi»«mn»2«/mn»«/msub»«/mfrac»«mo»)«/mo»«mo»§nbsp;«/mo»«/math»

 

Approximate Equivalent Circuit of Transformer:

                             

                                                             

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