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Sang-Yong Jung
( 23432B 031-299-4952 www.emlab.kr syjung@skku.edu ) - 3 week -
ELECTRIC MACHINERY
Ch2. Transformers
2
▣ Structure & Type
< Core Type > < Shell Type >
§.Transformers
- Michael Faraday (1791-1867)
- Electromagnetic coupling (current + induced electromotive force)
① ventilated dry-type transformer
② gas sealed dry-type transformer
③ Liquid-immersed transformer
Ch2. Transformers
3
▣ Ideal Transformer
1) Electric field produced by the windings : negligible 2) The winding resistances : negligible
3) All magnetic flux confined to the ferromagnetic core (no leakage flux) 4) 𝜇r ↑ (required MMF for flux in Core : negligible) (∵ H =0)
5) Core loss : negligible
𝛷
N1 N2
+
e
11𝜆
1-
+
𝜐
1𝜐
2-
e22 𝜆2
+ +
-
i1
i2
-
Ch2. Transformers
4
▣ Ideal Transformer
dt
e
11 dλ
1λ
1 N
1Φ
dt N d
e
11 1
)
ignore (
υ
1 e
11
1
dt
e
22 dλ
2λ
2 N
2Φ
dt N d
e
22
2 )
ignore (
υ
2 e
22
2
2 1 22
11 2
1
N N e
e υ
υ
=a
(Winding Ratio)Ch2. Transformers
5
▣ Ideal Transformer
2 2 1
1
i υ i
υ
(Input power = out power)L 2
2 1 2
2 2
2 1 1
1
Z
N N i
υ N
N i
υ
(sinusoidal input)L 2 L
2
2 1
L
Z a Z
N
Z N
· MMF in Core
·
·
0 i
N i
N11 2 2
a 1 N
N i
i
1 2 2
1
ZL referred to primary side
(converted primary load resistance )
Ch2. Transformers
6
▣ Real Transformer
1) The electric fields produced by the windings : negligible 2) Winding resistances : lumped parameter
3) 𝜇 : Constant (Core loss is later consideration)
m 2 2 1
1 m
i N i
Φ N
Linking
Flux
m l1
1 Φ Φ
Φ
m l2
2 Φ Φ
Φ
(primary) (secondary)
𝛷m
+
- -
+
𝜐
1𝜐
2- + +
-
𝛷𝓁1 𝛷𝓁2
e11=𝑑𝜆𝑑𝑡1
R1
R2
e22=𝑑𝜆2
𝑑𝑡
Ch2. Transformers
7
▣ Real Transformer
1 1
1 NΦ
λ λ2 N2Φ2 dt i dλ R e
i
υ1 R11 11 11 1
dt i dλ R e
i
υ2 R2 2 22 2 2 2
dt N dΦ dt
N dΦ dt
N dΦ dt
e11 dλ1 1 1 1 l1 1 m
dt N dΦ dt
N dΦ dt
N dΦ dt
e22 dλ2 2 2 2 l2 2 m
1 l1 1
l1 i
Φ L N
2 l2 2
l2 i
Φ L N
e1
e2
∴ 𝑒1
𝑒2 = 𝑁1 𝑁2
2 2 l2 2
2
2 e
dt L di i
υ R
1 1 l1 1
1
1 e
dt L di i
υ R
+
-
e
11e
22-
e
2+ - +
L𝓁2 i1
i2
cf ) ,
· ,
𝛷m
+
- -
+
𝜐
1𝜐
2- +
e
1L𝓁1
R2 R1
Ch2. Transformers
8
· i
mproducing mutual flux(𝛷
m) → Magnetizing Current
· N
1i
m= N
1i
1- N
2i
2→
·
2 1 2 m
1
i
N i N
i
N (NN i2) N2i2 01 2
1
m m 1
m i
Φ
L N : Magnetizing Inductance
Ideal Transformer
+
- -
+
𝜐
1e
1+
-
e
11L𝓁1 R1
Lm
+
-
𝜐
2+
-
e
22-
+
e
2L𝓁2 R2
𝑁2 𝑁1𝑖2
i1
im
i2
Ch2. Transformers
9
m2 m1
m 2 2 1
1
m Φ Φ
R i N i
Φ N Ideal transformer : Φm1 Φm2
m2 1 m1
l1 1
1
N ( Φ Φ ) N Φ
λ
m1 2 m2
l2 2
2
N ( Φ Φ ) N Φ
λ
Self Mutual
Φ ) Φ
Φ ( N Φ )
Φ ( Φ N
λ1 N1 1 1 l1 m 1 l1 m1 m2 Φ ) Φ
Φ ( N Φ )
Φ ( Φ N
λ2 N2 2 2 l2 m 2 l2 m1 m2 cf)
1
m1 l1
1
11 i
Φ ) Φ
(
L N
2
m2 l2
2
22 i
Φ ) Φ
( N
L -
2 m2 1
12 i
Φ L N
1 m1 2
21 i
Φ L N
cf)
m 1 1
m1 R
i Φ N
m 2 2
m2 R
i Φ N
m 2 1
12 R
N L N
m 1 2
21 R
N
L N
L
12 L
21, ,
, ,
Ch2. Transformers
10
2 12 1
11
1 L i L i
λ
1 12 2
22
2 L i L i
λ
dt L di dt
L di dt
e11 dλ1 11 1 12 2
dt L di dt
L di dt
e22 dλ2 22 2 12 1
dt L di dt
L di i
υ1 R1 1 11 1 12 2
dt L di dt
L di i
υ2 R2 2 22 2 12 1
Lm1
12 2 1 L N
N 12
1 2 L N
N Ll2 Lm2
cf) ,
1 m1 1 1
l1 1
11 i
Φ N i
Φ
L N
2 m2 2 2
l2 2
22 i
Φ N i
Φ
L N
Ll1
Ch2. Transformers
11
) i (i L i L
N i Lm N N i N L i L
i N Lm
i N L i λ L
' 2 1 m1 1
11
' 2 2 1 1 1 2 1 m1 1
11
2 1 1 2 1
m1 1
11 1
) i - (i L i L
i L - i L i L
N i L N N N N N
N i L N N i N λ L
N λ N
i N L
i N L i λ L
' 2 1 m1 '
2 '
l2
' 2 '
m1 '
2 '
l2 1 m1
' 2 2 1 m1 2
1 2 2 1
' 2 2 1 l2 2 1 1
m1 2
' 1 2
1 m1 1 2 2
m2 2
12 2
dt ) i L d(i
dt L di i dt R
i dλ υ R
' 2 1 m1 1
l1 1 1 1 1 1 1
) i L d(i
L di i dλ R
i υ R
' 2 1 m1 '
' 2 l2 ' 2 ' 2 '
' 2 2 ' 2 ' 2
∴ +
-
𝜐
1L𝓁1 R1
Lm1
+
-
𝜐`
2L`𝓁2 R`2
i1 I`2