Krause2

(1)

(2)

0 30 60 90 120 150 180 210 240 270 300 330 360 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 v a v b v c

Análise Fasorial

1 2 3 4 5 6 7 8 9 10 11 12 wt v(t) 0,866 �0 ,5 00 1∠ 0�

(3)

Transformada abc-dq0

• É importante não confundir , e com fasores.

Elas são valores instantâneos que dependem

do tempo.

Obs: pode ser , ou outra variável.

•

(4)

Aplicação Mecânica da Transformada

Plano de Referência

 









0 0

cos cos cos

1cos 2 sin si 2 2 3 3 60 30 0,866 2 2 60 30 120 0,500 3 3 60 30 120 n sin 1cos 2 1cos 2 1 1 1 0,000 2 2 2 qd s abc qd t t t                 _ _              _{  } _               _ _ _ _  _{ } _  _ _ _ _{ } _    _ _ _ _{ }_ _  _{ }_ __       _ _  v K v v













60 30 1 cos 2 1 cos 2 1 cos 60 30 120 60 3 2 0 120 a b c t t v v v t                             PR

t

 w



(5)

�

_��

=

60 Hz

0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5

1 Tensões no Sistema abc

T en sã o ab c (p u) va v_b v_c 0 10 20 30 40 50 60 70 80 90 100 -0.5 0 0.5 1 X: 23.6 Y: 0.866 Tensões no Sistema qd0 Tempo (ms) T en sã o qd 0 (p u) X: 42.7 Y: 0.5 X: 64.4 Y: -1.665e-016 v_q v_d v₀

(6)

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 1 0 11 12

(7)

� = 0 Hz

0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5

T en sã o ab c (p u) va v_b v_c 0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5 1 Tensões no Sistema qd0 Tempo (ms) T en sã o qd 0 (p u) vq v_d v₀

(8)

�=60 Hz

0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5

T en sã o ab c (p u) va v_b v_c 0 10 20 30 40 50 60 70 80 90 100 -0.5 0 0.5 1 X: 23.6 Y: 0.866 Tensões no Sistema qd0 Tempo (ms) T en sã o qd 0 (p u) X: 42.7 Y: 0.5 X: 64.4 Y: -1.665e-016 v_q v_d v₀

(9)

�= 90 Hz

0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5

T en sã o ab c (p u) va v_b v_c 0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5 1 Tensões no Sistema qd0 Tempo (ms) T en sã o qd 0 (p u) vq v_d v₀

(10)

0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5 1

Tensões no Sistema abc

T en sã o ab c (p u) va v_b v_c 0 10 20 30 40 50 60 70 80 90 100 -0.5 0 0.5 1 X: 23.6 Y: 0.866 Tensões no Sistema qd0 Tempo (ms) T en sã o qd 0 (p u) X: 42.7 Y: 0.5 X: 64.4 Y: -1.665e-016 v_q v_d v₀ 0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5

T en sã o ab c (p u) va v_b v_c 0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5 1 Tensões no Sistema qd0 Tempo (ms) T en sã o qd 0 (p u) vq v_d v₀ 0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5

T en sã o ab c (p u) va v_b v_c 0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5 1 Tensões no Sistema qd0 Tempo (ms) T en sã o qd 0 (p u) vq v_d v₀ Hz 0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5 1

Tensões no Sistema abc

T en sã o ab c (p u) va v_b v_c 0 10 20 30 40 50 60 70 80 90 100 -1 -0.5 0 0.5 1 Tensões no Sistema qd0 Tempo (ms) T en sã o qd 0 (p u) vq v_d v₀ Hz Hz Hz

(11)

Elementos Resistivos

abcs



s abcs

i

v

r

 

1 0 0 qd s s s s

i

qd s 



v

K r K

 

1

0 0

0

0 0 0

s s s s s s s

r











  











K r K

r

 

1 0 0 0 0

0 0

0

0 0 0

qs s qs qd s s s s qd s ds s ds s s s

v

r

i

v

r

i

v

r

i



  

  

  

  





_{  }



_{  }

  

  

  

  

v

K r K

i

(12)

Elementos Indutivos

abcs

d

dt



λ

v

 

1 0 0 0 1 1 0 0 qd s s qd s s s _{qd s} s qd s s s qd s

d

dt

d

_d

dt

  















_

_













λ

K

v

K

_λ

Kλ

K K

v

(13)

 

1 1 0 0 1 0 2 2

cos cos cos

sin cos 0

3 3

2 2 2 2 2

sin sin sin sin cos 0

3 3 3 3 3 1 1 1 ₂ sin c 2 2 2 ₃ qd s q s s s d s qd s s s s d dt d dt d dt               w              _   _       _ _ _ _ _    _ _ _ _ _ _ _ _  _ _  _ _  __  _  _ _  _             _ _           _ _     K K λ K λ v K K K

 

1 2 os 0 3 0 1 0 1 0 0 0 0 0 s s d dt   w               _ _    _ _           _ _     K K

(14)

 

1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 s _{qd s} _{qd s} qd s s qd s s s dqs qs qs qs ds qs ds ds qd s ds ds qs s s s s d _d _d dt dt dt d d v dt dt d d v dt dt d d v dt w   w    w  w  _ _     _ _                                _ _{ } _{ } _ _ _                  _{ } _      K _λ _λ v Kλ K K λ v dt         

(15)

Matriz indutância válida para motores trifásicos

 

1 0 0 1

1

2

1

2

1

2

3

0

2

3

0

2

0

abcs s abcs dq s s s s dq s ls ms ms ms s ms ls ms ms ms ms ls ms ls ms s s s ls ms ls

L

 







 





_

_















 

_





_







_

_













_

















_



_

















λ

L i

λ

K L K

i

L

K L K

indutância de dispersão indutância de magnetização ls ms L L  

(16)

Elementos Capacitivos

abcs

d

dt



q

i

 

1 0 0 1 1 0 0 0 s qd s qd s s s _{qd s} s qd s s s qd s

d

dt

d

_d

dt

  















_

_













K

q

i

K

_q

K

q

K K

i

(17)

 



0 1 1 0 0 0 0 0 0 0 qs dqs ds s d s _{qd s} qd s s qd s s s qd s qds q q q qs qs ds ds ds qs s s q s

d

_d

dt

d

dt

dq

i

q

dt

dq

i

q

dt

dq

i

dt

w

       _ _    





_

_

























  







 





q

K

_q

i

K

q

K K

i

q

 

1 0 1 0 1 0 0 0 0 0 s s d dt w       _ _     K K

(18)

Exemplo 3B

Transformada dq0 em um ernolamento trifásico

1 1 2 2 1 1 2 2 1 1 2 2

0 0

0

0 0 0 0

 _ _ _       _ _ _       _ _ _     









 



















 













ls ms ms ms ms ls ms ms ms ms ls ms s s s s s s L L L L L L L L L L L L

r

L

M

L

M

L

s

r

L

(19)

(20)

 

1 3 0 0 2 3 0 0 2 0 0

0

2

ls ms ls ms ls s s s s s s L L L L L

L

M

L

M

L

M

  _       _           













_



_











K L K



 



 



 



2 0 0 0 0 0 0 0 s qs s ds s s as asR asL bs bsR bsL cs csR csL L M i L M i qs qs qsR qsL s qs ds ds ds dsR dsL s ds qs L M i s sR sL s s

v

d

v

r i

dt

d

v

r i

dt

v

r i



w 



w

w 

  







































(21)

 



 



 



2 0 0 0 0 0 s qs s ds s s L M i L M i qs qs s qs ds ds ds s ds qs L M i s s s

d

v

r i

dt

d

v

r i

dt

v

r i



w 



w

w 

  















(22)

Tensões Equilibradas

Transformação abc-dq0

Até aqui apenas transformamos um sistema trifásico em um sistema bifásico

 

0 0 2 2

cos cos cos _{2 cos}

3 3

2 2 2 2

sin sin sin 2 cos

3 3 3 3

1 1 1 2

2 cos

2 2 2 3

2

2 cos cos 2 cos

3 2 3 s ef qd s ef s ef s s ef q ef d f f f f f      _                _   _         _ _ _ _        _ _ _ _  _ _  _ _  _ _  __{ } _  __              _ _     _ _          _     f f

 

2 2 2

cos 2 cos cos

3 3 3

2 2 2 2

2 cos sin 2 cos sin 2 cos sin

3 3 3 3

2 2 2 2 2

cos cos cos

2 2 3 2 3 s ef s ef s ef s ef s ef s ef s ef f f f f f f f                       _ _ _ _ _ _       _ _ _ _ _ _    _ _ _ _ _ _ _ _         _ _ _ _ _ _ _ _             _ _ _ _    _  _ _  _       









0 2 cos 2 sin 0 s ef s qd ef f f        _               f

(23)

Mas se w do plano de referência for igual a velocidade dos fasores!!!

Transformamos um circuito trifásico em duas componentes contínas

equivalentes!!!









constante constante 0

2 cos

2 sin

0

s ef qd s ef

f









_















 

_



_





















f













(24)

Relações Fasoriais equilibradas em Estado

Estacionário

 

  0 2 0 3 2 0 3 2 cos 0 Re 2 2 2 cos 0 3 Re 2 2 2 cos 0 3 Re 2 ef e ef e ef e as s e ef j j t s bs s e ef j j t s cs s e ef j as bs c j t s s F F t F e e F F F F F t F e e F F t F e e  _w   w   w w   w   w   _       _         _  _    _ _    _   _               _   _           





 

     





 

      0 0 0 0 2 cos 0 0 Re 2 2 sin 0 Re 2 ef _e ef e qs s e ef j j t s bs s e ef j qs j s bs t F F F F t F e e F F t j F e e   w w   _{w w} w w   w w     _            _    _    _ _       _   _    _ _         0 0 0 ef ef j as s j qs s ds qs F F e F F e F jF               

(25)

Caso Especial 1

 

0 0 ef ef j as s j qs s qs as

F

F e

F

F e

F

 





(26)

Caso Especial 2

e

 





 





0 0 0 0

Re

2 Re

2 2 cos

0 2 sin

0

ef e ef e j e qs s j e qs s e qs ef e ds ef

F

F e

F

j

F e

F

   



         







_

_











_

_







 

 





 

sabemos que:

cos

sin

logo:

2 2 cos

2 sin

2

ef j as s s ef ef s ef s ef e e as qs d as as s

F

F e

F

j

F

j

F

j

F

















(27)

Equações de Tensão



as as as s as as e as LI as e as CV as s as

V

r I

V

j

I

j

Q

V

Z I

w







 







qs s qs qs qs e qs ds e qs qs e qs qs s qs

V

r I

V

j

V

j

I

j Q

V

Z I

w

w w

w



  













(28)

Exemplo 3D









as as bs cs as s as s I bs cs as as s as s as as s as s as s as e s as

dI

V

r I

L

M

dt

d I

I

dI

V

r I

L

M

dt

dI

V

r I

L

M

dt

V

r I

j

w

L

M I









_

_





_

_

_



















_



_















_

_

























(29)

 



 



s qs s ds L M i L M i qs qs s qs ds ds ds s ds qs

d

v

r i

dt

d

v

r i

dt



w 



w

 











(30)

        0 0 0 0 2 2 L r s M lr M r M r M r M lr L L L L L L L L L L L L M ls M lr ls lr M ls M lr ls lr M ls M lr ls lr M ls M lr ls lr r L L L_{M r} _{s M} _lr _{M r} _M _lr _{M r}r L L L L L L L L L L M ls M l r L L L L L L L L L L L L L L L L r ls lr M ls M L L L L L L L L L L L _lr L_{ls lr} L_{M ls} L_{M lr} L_{ls lr} L_{M l}_s w w w w w w                           A            0 0 0 0 0 0 0 0 0 2 0 0 L L M lr ls lr s ls L L L L L M r M ls r M ls r M lr M ls M s M L L L L L L L L L L L L L L L M ls M lr ls lr M ls M lr ls lr M ls M lr ls lr M ls M lr ls l L L r L L L L r L L L r M ls r L L L L L L L L L L L L L _{LM lr lL} L L L L L L L s lr L r M r M ls M s L L L L M ls M lr ls lr M l w w w w w                          0 0 0 0 0 0 0 0 0 3 3 0 0 0 0 0 8 8 2 2 2 L_{M r} _r _M _ls r L L L L L L L L s M lr ls lr M ls M lr ls lr M ls M lr ls lr r lr M qr M r L L L L L L L L L L r L L P i L P i J J dr w w w                                        _ _ _ _ _ _                      0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 M M M M M M M M lr lr ls ls ls L L L L L L L L L L L L L L L L L L L L L L L L L M ls M lr ls lr M ls M lr ls lr L L L L L L M ls M lr ls lr M ls M lr ls lr L L L L L L M ls M lr ls l L L L L L L L L L L r M ls M lr ls lr L L L L M l L L L _s L_{M lr} L_{ls lr} L_{M l}_{s L}_M  _      _                 B 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 lr L_lr L_lL L P J s lr                              _             _     

(31)

(32)

Sistema Elétrico













3 2 qs qs s qs qs ds ds s ds qs qr qr r qr r dr qr dr r dr r qr e ds qs qs ds d V R i dt d V R i dt d V R i dt d V R i dt T p i i



w



_w



w w 



w w 



                         

(33)

Sistema Mecânico





1 2 m e m m m m d T F T dt H d dt

w

_w



_w

   

(34)

Plano de Referência das Tensões

cos

3 sin

cos

2 ₂

3 _sin

_{3 cos}

sin

2 cos

3 sin

cos

2 ₂

3 _sin

_{3 cos}

sin

2

qs abs ds bcs qr abr dr bcr

V







_







 

_

_







 

_

_







 















_













_

_









_



_

_



_





















r

  

 

(35)

Plano de Referência das Correntes

Referência Rotor Estator Síncrono Referência Rotor Estator Síncrono

cos

sin

cos

3 sin

3 cos

sin

2

2 cos

sin

cos

3 sin

3 cos

sin

2

as qs bs ds ar qr br dr cs as bs cr ar br

i









 

_

_

 



 

_





_

 

 

_

_

 











 

_

_

 



 

_

_







_

 

_

 

_

_

 





  



  



(36)

Dynamic Simulation of electric

machinery

(37)

Exemplo 3B

Transformada dq0 em um ernolamento trifásico

          a b c asgs a a aa ab ac ag argr grgs a b c a b c a b c asgs a a aa ab ac ag argr g a b c gg ag bg cg asgs a g a b c g aa di dt d di di di v i r L L L L v v dt dt dt d i i i d i d i d i di d i di di dt dt i di v i r L L L L v r i i i L L L L dt dt dt dt dt dt dt v r r t L d                    _ _                2 gg ag a g gg ab bg ag b g gg ac cg ag c d d d d d d d d d d d L L i r L L L L i r L L L L i dt dt dt dt dt dt dt dt dt dt dt  _ _  _ _ _ _ _  _ _ _ _ _              2 2 2 a g g g g a g g g g a g aa gg ag ab gg bg ag ab gg cg ag bb gg ag bg bb gg bg bb gg cg bg cc gg ag cg cc gg bg cg cc gg cg p r r r r r r r r r r r r L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L          _  _  _                 _         _  _ _ _ _ _ _ _ _    s r v v Ri Li R L

(38)

2 2 a g g g s m m g a g g m s m g g a g m m s s gg ab gg bg ag ab gg cg ag bb gg ag bg bb gg bg bb gg cg bg cc gg ag cg cc gg bg cg cc gg p r r r r r r r r r r r r r r r r r r r r r L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L          _{ } _  _  _{ } _  _ _{ }_ __                           s r v v Ri Li R L 2 s m m m s m cg m m s L L L L L L L L L L  _{ } _  _{ }_ _  _{ } _  _ _{ }_ __  

(39)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 -1.5 -1 -0.5 0 0.5 1 1.5 B C A

(40)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(41)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(42)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(43)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(44)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(45)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(46)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(47)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(48)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(49)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(50)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(51)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(52)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(53)

0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 -1.5 -1 -0.5 0 0.5 1 1.5

(54)

Exemplo 3A (Krause)

Transforme abc para dq0:

Assumindo

(55)

Transformando

É importante não confundir , e com fasores. Elas são valores instantâneos

que dependem do tempo.

Obs: pode ser , ou outra variável.

(56)

(57)

(58)

(59)