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Sang-Yong Jung

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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) 𝜇_r ↑ (required MMF for flux in Core : negligible) (∵ H =0)

5) Core loss : negligible

𝛷

N1 N2

+

e

𝜆

-

+

𝜐

𝜐

-

e₂₂ 𝜆₂

+ +

-

i₁

i₂

-

Ch2. Transformers

4

▣ Ideal Transformer

dt

e

 dλ

λ

 N

Φ

dt N d

e



 )

ignore (

υ

 e



 

dt

e

 dλ

λ

 N

Φ

dt N d

e



 )

ignore (

υ

 e



 

2 1 22

11 2

1

N N e

e υ

υ  

a

Ch2. Transformers

5

▣ Ideal Transformer

2 2 1

1

i υ i

υ 

L 2

2 1 2

2 2

2 1 1

1

Z

N N i

υ N

N i

υ  



 



 

 



 



 

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  



Z_L 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)

𝛷_m

+

- ^-

+

𝜐

𝜐

- + +

-

𝛷_𝓁1 𝛷_𝓁2

e₁₁=^𝑑𝜆_𝑑𝑡¹

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  R_{2 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

e

-

e

+ - +

L_𝓁2 i₁

i₂

cf ) ,

· ,

𝛷_m

+

- -

+

𝜐

𝜐

- +

e

L_𝓁1

R₂ R₁

Ch2. Transformers

8

· i

producing mutual flux(𝛷

) → Magnetizing Current

· N

i

= N

i

- N

i

→

·

2 1 2 m

1

i

N i N

i  

1 2

1  

m m 1

m i

Φ

L  N : Magnetizing Inductance

Ideal Transformer

+

- _-

+

𝜐

e

+

-

e

L_𝓁1 R₁

L_m

+

-

𝜐

+

-

e

-

+

e

L_𝓁2 R₂

𝑁₂ 𝑁₁𝑖₂

i₁

i_m

i₂

Ch2. Transformers

9

m2 m1

m 2 2 1

1

m Φ Φ

R i N i

Φ  N    Ideal transformer : Φ_m1  Φ_m2

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  N_{1 1}  _{1 l1}  _m  _{1 l1}  _m1  _m2 Φ ) Φ

Φ ( N Φ )

Φ ( Φ N

λ₂  N_{2 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

 L

, ,

, ,

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  R_{1 1}  ₁₁ ¹  ₁₂ ²

dt L di dt

L di i

υ₂  R_{2 2}  ₂₂ ²  ₁₂ ¹

Lm1

12 2 1 L N

 N ₁₂

1 2 L N

 N Ll2 L_m2

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

L_𝓁1 R₁

L_m1

+

-

𝜐`

L`<sub>𝓁2</sub> R`₂

i₁ I`₂

·

·