Transformers 1

A time-dependent (varying) magnetic fluxdensity in the magnetic core induces (time-dependent) electromotive forces (e.m.f.s) ineach winding

Single-phase transformer

2 windings (primary & secondary) around a magnetic core(magnetic coupling)

Electric energy conversion from one voltage level to another

Faraday’s law

The voltage in each winding (~) isproportional to the number of turns

Open or closed secondary ...

09/02/2020

2

General transformer relations

F¶+F¶+=F¶+= t11ft1111t1111 nniRniRu

11t1111 eiiRu +¶l+=

F¶+F¶--=F¶--= t22ft2222t2222 nniRniRu

22t2222 eiiRu +¶l--=

Useful flux F

1f1 F+F=F 2f2 F+F-=F

2f

222f

1f

111f R

inetRin

=F=F

Leakage reluctances

e.m.f in 1 due to F

e.m.f.in 2 due to F

Leakage flux

Primary winding equation

Secondary winding equation

Coupling equation

Rinin 2211 -

=F

Magnetic circuit reluctance

÷÷ø

öççè

æ¶-¶= 2t

1

21t

21

1 inni

RneÞ

nnn

ee

2

1

2

1 ==Transformation ratio magnetizing inductance lµ

Transformers

3

Transformer equivalent circuit

11t1111 eiiRu +¶l+=

22t2222 eiiRu +¶l--=

÷÷ø

öççè

æ¶-¶l= µ 2t

1

21t1 innie

2t22

222

21 in1ni

n1Rnune ¶l++=

2t22221 'i''i'R'ue ¶l++=

Secondary quantities as seen from the primary

( )2t1t1 'iie ¶-¶l= µ

(a)

(b)

(c)

(a) (b)(c)

nnn

ee

2

1

2

1 ==

Transformers

4

j

Complex formalism: phasors

)tcos(A)t(a j+w=

j= jeAA

( )tjj eeARe)t(a wj=

( ) ( ) ( )( )

)tcos(A)tsin(j)tcos(ARe

eAReeAReeeARe )t(jtjjtjj

j+w=j+w+j+w=

== j+ww+jwj

j+j=j sinjcose j

Complex representation of sinusoidal quantities

A maximum value (amplitude) of a(t) [V, A, ...]w pulsation of a(t) (2pf, f = frequency) [rad/s]j phase of a(t) at t = 0 [rad]Sinusoidal quantity

(voltage, current, ...)

with

Quantity as a phasor

A

Re[.]

Im[.]

a(t=0)

Complex plane

Transformers

5

Limit cases with phasorsNo-load case

Short-circuit case

111

122 UjXjXR

jXE'U0'I

µ

µ

++==Þ=

12 U'U » nUU2

1 =

µ<< XX,R 11

nominal1 III <<= µ

µ

µ++=Þ=

jXjX'jX'R

'II0'U 22

2

12

21 'II »n1

II2

1 =

µ<< X'X,'R 22

>>1I !

(ratios ~ 400 X, 4000 R)

(ratios ~ 400 X, 4000 R)

_ _ _ _

___

Transformers

6

Operating pointsSimplified equivalent circuit

Phasor representation Exterior characteristic

Transformers

7

Magnetic losses and saturationHysteresis & eddy currents

Saturation

FH

212

1magmag REEKp+

==

Eddy current lossesproportional to b2, hence to F2, hence to E1

2

Idem for hysteresis losses (approx.)

(frequency dependent!)

Transformers

8

Equivalent circuit parameters

Experimental determination of the parametersfrom the equivalent circuit

No-load test

Short-circuit test

FH

21

v RUP+

=µ

=XUQ21

v

( ) 2cc21cc I'RRP += ( ) 2cc21cc I'XXQ +=

<<Þ<< primaireJoulepI

<<Þ<< mag1 pU

Transformers

primaryprimary

9

Construction typesTransformer with separate columns

Transformer with concentric windings

High leakage flux

Series or parallel connection of windings1 : High Voltage, 2 : Low Voltage

Shell-type transformer (« cuirassé »)

1 : High Voltage, 2 : Low Voltage

Transformers

10

Three-phase transformer

3 primary windings and 3 secondarywindings with star or triangle connection

0cba =F+F+F

Three columns Five columns Shell-type

Geometric symmetry3 single-phase transformer

advantages!

Transformers