1. The document describes different electrical components including resistive, inductive, and capacitive elements as well as three-phase winding with mutual inductances. Equations for voltage, current, and flux linkage are provided for each element. 2. It then discusses induction motors in the rotor reference frame including equations for torque, magnetic flux, and rotor angle. A block diagram is also presented showing the motor model. 3. Parameters for simulating an induction motor are listed at the end.

1. Elementos R, C, L e M

2. Elementos Resistivos
0
1
0 0
1
0
0 0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0

qd s
abcs abcs abcs
qd s s s s qd s
r
s
s s s qd s s
s
qs s qs
qd s qd s qd s ds s ds
s s s
r
r
r
v r i
v r i
v r i
v r i
v K r K i
K r K r
v r i

3. Elementos Indutivos
1
0
0
1
01
0 0
abcs
abcs
qd s s
qd s s
s qd s
s qq s d s sd
d
dt
d
dt
d d
dt dt
λ
v
λ K
v K
K λ
Kv λ K

4. 01
0 0
1
1
2 2
cos cos cos
sin cos 03 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
qd s qd s
s
s ss
s
s
dd
dt
d
dt
dt
λ
K KK
K
K
λ
K
v
1
2
os 0
3
0 1 0
1 0 0
0 0 0
s
s
d
dt
K
K
PRt

5. 0
0
1
0
0
0
01
0
0
0
0 1 0
1 0 0
0 0 0
1 0 0
0 1 0
0 0 1
qd s
qd s
qs
qs ds
ds
qd s ds qs
s
s
s
s qd s
qs
ds
qd s
s s
qs
s
s
ds
d
dt
d
v
dt
d
v
d
d
dt
d
dt
d
dt
t
d
t
t
d
v
d
d
λ
K Kλ
v
K
Kv
v
2 60

6. Elementos Capacitivos
abcs
abcs
d
dt
q
i
1
0
0
1
1 0
00
s qd s
qd s s
s qd s
s qd sqd s s
d
dt
d d
dt dt
i
K q
i K
K q
K q K

7. 
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 dt
d
dt
dq
i q
dt
dq
i q
dt
dq
i
dt
q
K q
i K q K K
i
q
q
1 0 1 0
1 0 0
0 0 0
s
s
d
dt
K
K

8. Enrolamento trifásico com
indutâncias mútuas
0 0
0 0
0 0 0
s
s s
s
s s
s
as asR asL
asbc bs bsR bsL
cs csR csL
r
r
L M M
L M L M
M M L
v v v
v v v v
v v v
r sr
sL
sr
sL
sr
sL
csi
bsi
asi
M M
M
asv
bsv
csv

9. 

 0
0
0
0
1
0 0
2
0 0 0 0 0 0
s qs
s ds
s s
qs
qs ds
ds
qd s ds qs
s
s
qd s s s s qd s
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
d
v
dt
d
v
dt
d
v
dt
d
v v v r i
dt
d
v v v r i
dt
v v v r i
v
λ K L K i
qsv
qsi
ds
sL M
sr
dsv
sr
qs
sL M
dsi
0sv
sr
2sL M
dsi
1
0 0
0 0
0 0 2
s
s s s s
s
L M
L M
L M
K L K

10. Exercício
sr
sL
sr
sL
sr
sL
csi
bsi
asi
M M
M
asv
bsv
csv
qsv
qsi
ds
sL M
sr
dsv
sr
qs
sL M
dsi
0sv
sr
2sL M
dsi

11. Próxmia Aula

12. Motores de Indução no Sistema
qd0

13. 0 0
0 0
0 0
1 1
2 2
1 1
2 2
1 1
2 2
s
abcs s
s
ls ms ms ms
abcs ms ls ms ms
ms ms ls ms
r
r
r
L L L L
L L L L
L L L L
r
L
sr
sL
sr
sL
sr
sL
csi
bsi
asi
M M
M
asv
bsv
csv
A
A’
C’
C
B B’

14. 0
1
1
0
1 1
2 2
1 1
2 2
1 1
2 2
3
0 0
2
3
0 0
2
0 0
abcs s abcs
dq s dq s
ls ms ms ms
s ms ls ms ms
ms ms ls ms
s s s
s
ls ms
ls ms
ls
s s
L L L L
L L L L
L L L L
L L
L L
L
K L K
K L
λ
K
λ L i
i
L
indutância de dispersão
indutância de magnetização
ls
ms
L
L

15. -1.5
-1
-0.5
0
0.5
1
1.5
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
B C A
A
A’
C’
C
B B’
Indutância Mútua em MITs

16. 0
0 0
0
0 0
qs
qs s qs ds
ds
ds s ds ds
s
s s s
qr
qr r qr r dr
dr
dr r dr r qr
r
r r r
v r i
v r i
v r i
v r i
v r
d
dt
d
i
v
dt
d
dt
d
dt
d
dt
d
r
d
i
t
0 0
0 0
qs ls qs M qs qr
ds ls ds M ds dr
s ls s
qr lr qr M qs qr
dr lr dr M ds dr
r lr r
L i
L i
L i i
L i i
L i i
L i
L i
L i
L
L i i
i

17. 1 0 00 0 0
0 00 0 0
0 00 0 0
0 0 0 0
2
0 0 0 0





r L L LL L L s ls M Mi ls M Mqs
L L r LL L L ls M s Mi ls M Mds
L r L LL L Li r M r r lr r MM lr Mqr
L L L L LM lr Mi r M r lrdr
J
r P
...
0
3 3
0 0 0
4 4
10 0 0
0 0 0
0 0 0
... 0
0 0 0
0
2
0 0 0 0
i
qs
i
ds
i
qr
L r ir M dr
P P
L i L i rM dr M qr
L L L
ls M M v
qs
L L L
ls M M v
dsL L L
M lr M
L L L
M lr M
TJ L
P
0 0
0 0 0
0 0
0 0 0
0 00 0 0
0 0 0
0 0
0 0 0 0
3 32 0 0 0
4 4





r L L L
s ls M Mi S Mqs
L L r L
ls M s MS Mi
ds
L r L LM Ri r M r r lr r M
qr M R
L L L ri r M r lr r MPdr
P PJ L i L ir M dr M qr
0 0 0
0 0 0
0 0 0
0
0 0 0
0
0 0 0 0
2





i vS Mqs qs
S Mi vds dsM Ri
qr M R
i
Pdr
T
J Lr
i
qs Sr M
si
ds
i
qr
i
d
S L L M L
ls
r
M r M
r
0 0 0
0
0 0 0
0 0 0
0
0 0 0
0
0 0 0 0
2

S L M L L
M r lr r M i
i vS Mqs qsr
r S M
v
dsM Ri
qr M R
P
T
J Lr
i Sr i S L L M L i M
qs s qs ls M r M
ds
r
ds
i
d
r i S L M L L i S v
r qr M r lr r M dr qs
M lr
M lr M ls lr ls
M
M lr M ls lr
M ls
M lr M ls lr ls
L L
S
L L L L L L
L
M
L L L L L L
L L
R
L L L L L L
S L L M L i Mr i S L M L L i S v
ls M r M ds r qr M r lr r M dr qs

18. Ls
0
M
0
0
0
S
0
M
0
M
0
R
0
0
0
M
0
R
0
0
0
0
0
J2P
R
M
2
R S
0
M
M
2
R S
0
0
0
R
M
2
R S
0
M
M
2
R S
0
M
M
2
R S
0
S
M
2
R S
0
0
0
M
M
2
R S
0
S
M
2
R S
0
0
0
0
0
1
J2P

19. Motor de indução no referencial
rotórico
2
1 m
s r
L
L L

20. Torque
• Torque eletromagnético
• Torque mecânico
• Equação de estado
'
2
m dr qs
e PP
r
m
pp qs ds
r
L i L
T N N i i
LL
r
m n r d
d
T J B
dt
1 10
0 01 0
er
dr
n
r
r
B T
J JJ
1 1 1 1 1x = A x + B u



21. • Fluxo Magnético
• Ângulo do Rotor
r
m ds
r
r
m
r
ds
r
r
L d
L i
R dt
R
L
L
I
s
R
L
0
1 q
s me
r d
i
dt
i

23. 6
Iabc
5
wr
4
Phidr
3
Te
2
Ids
1
Iqs
sin
seno
dq0
sin_cos
abc
dq02abc
cos
cos
1/J
s+B/J
Transfer Fcn5
Lm*Rr/Lr
s+Rr/Lr
Transfer Fcn4
1
0.013311s+1.25
Transfer Fcn
(with initial outputs)
1
d*Le.s+Re
Transfer Fcn
Product
1
s
Integrator
[Ids]
Goto1
[Iqs]
Goto
ws*Le*d
Gain2
ws*Le*d
Gain1
Np*Lm*Lm/Lr
Gain
[Ids]
From6
[Iqs]
From5
[Ids]
From4
[Ids]
From3
[Iqs]
From2
[Ids]
From1 [Iqs]
From
Divide1
Divide
0.0
Constant2
Lr/Rr
Constant1
2
Vds
1
Vqs
Corrente de eixo em quadratura (A)
Corrente de eixo direto (A)
Tensão q
Tensão d
Densidade de f luxo magnético mútuo
Torque Eletromagnético
Rotação do Rotor (rd/s)
Rotação do Rotor (rd/s)
Correntes abc
Correntes dq0
Escorregamento
(rd/s)
Rotação do
Campo Girante
(rd/s)
Angulo (rd)
eT
Modelo do MIT

24. SaturationRamp
Product
PIDPID
PIDPID
MIT
dq0
Vds
Vqs
Iqs
Ids
Te
Phi_dr
wr
MIT - dq
[Vqs]
[Vds]
[Iabc]
[Te]
[wm]
[PHIdr]
[Rampa]
[Ids]
[Iqs]
[PHIdr]
[Iqs]
[Ids]
[wm]2*pi*60/3
Constant3
0.7
Constant1
Controle externo do MIT

25. Parâmetros do motor a simular
Parâmetro Valor nominal

26. 2 2
cos cos cos
cos sin 13 3
2 2 2 2 2
sin sin sin cos sin 1
3 3 3 3 3
1 1 1 2 2
cos sin 1
2 2 2 3 3
L M Ms
M L Ms s
M M Ls
L
2 2 2 2 2 2
cos cos cos cos cos cos cos cos cos
3 3 3 3 3 3
2 2 2 2 2
sin sin sin in sin sin s
3 3 3 3 3
L M M M L M M M L
L M M M L M Ms
S S S
s SL
cos sin 1
2 2 2 2
in sin sin cos sin 1
3 3 3 3
1 11 1 1 1 2 2
cos sin
1 1 1
2 2 2 2
1
2 2 2 3 3
cos cos co
2
3
2 2
S
S S S
S
M L
L M M M L M M M L
L
s
SM L
L
2 2 2 2
s cos cos cos
cos sin 13 3 3 3
2 2 2
sin sin sin sin sin sin cos sin
3 3 3 3 3
1 1 1
2 2 2
M M L
L M L M M L
L M L M L
S
s S S
S S S M
2
1
3
2 2
cos sin 1
3 3
2 2 2 2 2 2
cos cos cos cos s2 2 2 s in cos sin cos cos
3 3 3 3
in
3 3
2
3
L M L M L M L M L M L MS S S S S S L MS
sL
2 2
cos cos
3 3
2 2 2 2 2 2 2
cos sin cos sin cos sin sin sin sin sin sin
3 3 3 3 3 3 3
2 2 2sin
S S
S S S S
L M L M
S S SL M L M L M L M L M L M L SM L LS M M
2
3
1 1 2 1 2 1 1 2 1 2 3
cos cos cos sin sin sin 3
2 2 3 2 3 2 2
3
2
3
3 2 3
0
2
2
0 0
2
0
3
3
0 0
L M L M L M L M L M L M L M
L M
L
S S S S S S S
S
S MsL
3
2
0 0
0
0 2
0
0
L M
L M
L Ms
L M
S
S
S
S
L

28. 2
1
1dr ds dss
qr qs q
ds dss
qs qsm m
dr ds
r m
qr
ds
qs
dr ds
r m
qr qs
dr ds
sm
ds
qs
s rr
qr m m
m
m
qs
i
L L
i
i
L
i iL
i i
L
iL
i i
iL L
L
LL
L L
L
L
i
i
L
2
dr m
qr
qs
ds dsm s r
qs qsr r
i
iL L L L
iL L

29. 

1
0 1
1 0
0 1
1 0
0
0
ds dss
qs qsm m
r
ds ds dsds
s
qs qs qsqs
dr dr drdr
i
r sl
qr qr qrqr
dr
r
qr
L
iL L
v i
R
v i
v i
R
v i
i
R
i




2
2
0 1
1 0
0
0
dr ds dsm s rr
qr qs qsm m
dr ds mr
qr qsm
dr
qr
i
d
L L LL
iL L
L LL
L
rds
r
qrqs
ds dsr s r
qs qsm m
iR L R
iL L
 
 





 2 2
0 1
1 0
0
0
dr dsm m s r dsdrm m s r
qr qsr r
d
qsq
ss r
q
r
s
r
m
r
dr
r
qr
dsr s
iL
iL
dsr
qsm m
ds
qs
iL L L L iL L L L
iL L i
r
q
L
sm
L
iR L R
iL
L
LL

 

 


22 2
2
0 1
1 0
0
0
dr ds dsdrr r m s r r m r m s r
qr
dsm s r
qsm
dsm s r
qsm
dr
r
qr
dss r
qsq ms qsqrr m r m r m r
i iR R L L iL R
iL
iL L L
iL
iL L L
iL
L L L L L L L
i iL L L L L L L




 
 
2 22
2
0
1
0 1
1 0
r
ds drdsdr r m s r m s r
sl
qs qrqsqr m r
r m s r
m r
r m s r
m
dsdr
i iL L L L L LR L L L
L
L
i iL L L
i
L
R L L L
 
 
 0 drsl

30. 2 2
0 1
1 0
dr dr ds dsds r m r m s r m s r
sl
qr qr qs qsqs r r r r
i iR L R L L L L L R
i iL L L L



31. Modelo do Motor Referenciado no
Fluxo Rotórico
 
Estator
Total Rotors ls mL L
ds s
dr ds
m m
ds m ls ds m
L
drL L i L i
L
i i
L L

 
Estator
Total Rotors ls mL L
qs s
qr qs
qs m
m m
l qs
L
s m qr
i
L L i L i
L
i
L L

 Total
2
dr r m ds
dr r m ds
m s
dr
dr
ds s
m
ds ds
m m
m
rr
L L i
L
i
L
L L
L i
L L LL
i
L L
2
qs s
q
qr r qr m qs
qr r qs m qs
m s
qr qs qs
s
m m
m
r
m
r
L i L i
L L ii
L L LL
i
L
L
i
L
L
L

32. r
r
r
L
R
2
1 m
s r
L
L L
2
rr m s
dr ds ds
m m
L L LL
i
L L
2
2
1
m s m
s
ds dsds r s r s r s r
dr
qs qsqs PP r m sm s
s
s rr s r s
v
vN
L R L
ii L L L L L
ii L LL R
L LL L L


Estator Rotor
qs
qs s qs ds
ds
ds s ds ds
d
v R i
dt
d
v R i
dt
0
0
qr
r qr r dr
dr
r dr r qr
d
dt
d
t
i
d
R
R i
qs ls qs M qs qr
ds ls ds M ds dr
L i i
L i i
L i
L i
qr lr qr M qs qr
dr lr dr M ds dr
L i
L
L i i
L ii i
ds s
dr ds
m m
L
i i
L L

33. 2 2
cos cos cos
cos sin 13 3
2 2 2 2 2
sin sin sin cos sin 1
3 3 3 3 3
1 1 1 2 2
cos sin 1
2 2 2 3 3
L M Ms
M L Ms s
M M Ls
L
2 2 2 2 2 2
cos cos cos cos cos cos cos cos cos
3 3 3 3 3 3
2 2 2 2 2
sin sin sin sin sin sin
3 3 3 3 3
L M M M L M M M L
L M M
S S S
M L Ms S MsL
cos sin 1
2 2 2 2
sin sin sin cos sin 1
3 3 3 3
1 1 1 2 2
cos
1 1 1 1 1 1
2 2 2
sin 1
2 2 2 3 3
cos cos c
2
3
2 2 2
M L
L M M M L M M M L
L M L
s
S
S S S
S S
L
2 2 2 2
os cos cos cos
cos sin 13 3 3 3
2 2 2
sin sin sin sin sin sin cos sin
3 3 3 3 3
1 1 1
2 2 2
M M LS
s S S
S S S
L M L M M L
L M L M L M
2
1
3
2 2
cos sin 1
3 3
2 2 2 2 2 2
cos cos c2 2 2 sinos cos sin cos sin cos cos
3 3 3 3 3 3
2
3
L M L M L M L M LS S S S S S SM L M L M
sL
2 2
cos cos
3 3
2 2 2 2 2 2 2
cos sin cos sin cos sin sin sin sin sin sin
3 3 3 3 3 3
2s
3
2 2in
L M L M
L M L M L M L M L M L M L M L M L M
S S
S S S S S S S S S
2
3
1 1 2 1 2 1 1 2 1 2 3
cos cos cos sin sin sin 3
2 2 3 2 3 2 2 3 2 3 2
0
3
2
3
0
2
0
2
0
3
0 0
S S S S S S SL M L M L M L M L M L M L M
L M
Ls
S
S ML
3
3
2
0 0
0
0 2
0
0
S
S
L M
L M
L MS
MS
s
L
L

34. 0 0 0
0 0 0
0 0 0
0





L L LM lr M
L L L L L L L L L L L LM lr M ls lr ls M lr M ls lr ls
L L Li M lr Mqs
L L L L L L L L L L L LM lr M ls lr ls M lr M ls lr lsids
L LL M lsMiqr
L L L L L L L L L L L LM lr M ls lr ls M lr M ls lr lsidr LM
r
0 0
0 0
0 0
0 0
3 30 0 0 0 0
4 4
0 0 0 0
2
r L L Ls ls M M
L L r Lls M s M
L r L Lr M r r lr r M
L L L rr M r lr r M
L L P PM ls
L i L iM dr M qrL L L L L L L L L L L LM lr M ls lr ls M lr M ls lr ls
P
J
0 0 0
0 0 0
0
L L LM lr M
L L L L L L L L L L L LM lr M ls lr ls M lr M ls lr ls
L L Li M lr Mqs
L L L L L L L L L L L LM lr M ls lr ls M lr M ls lr lsids
L LL M lsi Mqr
L L L L L L L LM lr M ls lr ls M lridr
r
0 0 0
0
0 0 0
0 0 0 0
2
vqs
vds
L L L LM ls lr ls
L LL TM lsM L
L L L L L L L L L L L LM lr M ls lr ls M lr M ls lr ls
P
J