Single-phase transformer
2 windings (primary & secondary) around a magnetic core
(magnetic coupling)
Faraday’s law
A time-dependent (varying) magnetic flux
density in the magnetic core induces (time-
dependent) electromotive forces (e.m.f.s) in
each winding
The voltage in each winding (~) is
proportional to the number of turns
Open or closed secondary ...
Electric energy conversion from one voltage level to another
21/02/2017 Transformers 1
General transformer relations
Useful flux F
Primary winding equation
F1 = F + F f 1 F 2 = -F + F f 2
u1 = R1 i1 + n1 ¶ t F1 = R1 i1 + n1 ¶ t Ff 1 + n1 ¶ t F
e.m.f in 1
u1 = R1 i1 + l1 ¶ t i1 + e1 due to F
Secondary winding equation
u 2 = -R 2 i 2 - n 2 ¶ t F 2 = -R 2 i 2 - n 2 ¶ t F f 2 + n 2 ¶ t F
Leakage flux
ni n i u 2 = -R 2 i 2 - l 2 ¶ t i 2 + e 2 e.m.f.
F f 1 = 1 1 et F f 2 = 2 2 in 2
Rf1 Rf 2 due to F
Coupling equation
Leakage reluctances
n i - n 2i 2 n12 æ n ö
F = 11 Þ e1 = çç ¶ t i1 - 2 ¶ t i 2 ÷÷
e1 n1 R Rè n1 ø
= =n
e2 n 2 Transformation ratio Magnetic circuit reluctance magnetizing inductance lµ
Transformers 2
Transformer equivalent circuit
u1 = R1 i1 + l1 ¶ t i1 + e1 (a)
u 2 = -R 2 i 2 - l 2 ¶ t i 2 + e 2 1 1
e1 = n u 2 + n 2 R 2 i 2 + n 2 l 2 ¶ t i 2
n n
e1 n1
= =n Secondary quantities as seen from the primary
e2 n 2
e1 = u '2 +R '2 i'2 +l'2 ¶ t i'2 (b)
æ n ö
e1 = lµ çç ¶ t i1 - 2 ¶ t i 2 ÷÷ e1 = lµ (¶ t i1 - ¶ t i'2 ) (c)
è n1 ø
(a) (c) (b)
Transformers 3
Complex formalism: phasors
Complex representation of sinusoidal quantities
a(t ) = A cos(wt + j) A maximum value (amplitude) of a(t) [V, A, ...]
w pulsation of a(t) (2pf, f = frequency) [rad/s]
Sinusoidal quantity
(voltage, current, ...) j phase of a(t) at t = 0 [rad]
(
a ( t ) = Re A e j j e j wt )
jj Im[.] Complex plane
with e = cos j + j sin j
A = A e jj A
j
Quantity as a phasor Re[.]
a(t=0)
( ) ( ) ( )
Re A e j j e j wt = Re A e j j+ j wt = Re A e j (wt + j)
= Re(A cos(wt + j) + j sin(wt + j) )
= A cos(wt + j)
Transformers 4
Limit cases with phasors
No-load case _ _ _ jXµ _
I'2 = 0 Þ U'2 = E1 = U1
R1 + jX1 + jXµ
R1, X1 << Xµ (ratios ~ 400 X, 4000 R)
U1
U'2 » U1 =n
U2
I1 = Iµ << Inominal
Short-circuit case _ _
I R '2 + jX'2 + jXµ
U'2 = 0 Þ _1 =
I'2 jXµ
R '2 , X'2 << Xµ (ratios ~ 400 X, 4000 R)
I1 1
I1 » I'2 =
I2 n
I1 >> !
Transformers 5
Operating points
Simplified equivalent circuit
Phasor representation Exterior characteristic
Transformers 6
Magnetic losses and saturation
Hysteresis & eddy currents
Eddy current losses
proportional to b2, hence to F2, hence to E12
Idem for hysteresis losses (approx.)
E12
p mag = K mag E12 =
R H+F
(frequency dependent!)
Saturation
Transformers 7
Equivalent circuit parameters
Experimental determination of the parameters
from the equivalent circuit
No-load test I << Þ p Joule primary
primaire <<
U12 U12
Pv = Qv =
R H+F Xµ
Short-circuit test U1 << Þ p mag <<
Pcc = (R1 + R '2 ) Icc
2
Qcc = (X1 + X'2 ) Icc
2
Transformers 8
Construction types
Transformer with separate columns
High leakage flux
Transformer with concentric windings
Series or parallel connection of windings
1 : High Voltage, 2 : Low Voltage
Shell-type transformer (« cuirassé »)
1 : High Voltage, 2 : Low Voltage
Transformers 9
Three-phase transformer
3 single-phase transformer Geometric symmetry
advantages!
3 primary windings and 3 secondary
windings with star or triangle connection
Fa + F b + Fc = 0
Three columns Five columns Shell-type
Transformers 10
