Elementos R, C, L e M

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

Elementos Indutivos

10

0

101

0 0

abcsabcs

qd s s

qd s s

s qd s

s qq s d s sd

d

dt

d

dt

d d

dt dt

λv

λ Kv K

K λKv λ K

010 0

1

1

2 2cos cos cos

sin cos 03 3

2 2 2 2 2sin sin sin sin cos 0

3 3 3 3 3

1 1 1 2sin c

2 2 2 3

qd s

qd s qd s

s

s ss

ss

dd

dt

d

dt

dt

λK KK

KK

λK

v

1

2os 0

3

0 1 0

1 0 0

0 0 0

ss

d

dt

KK

PRt

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

qsqs ds

dsqd s ds qs

ss

s

s qd s

qs

ds

qd s

s s

qs

ss

ds

d

dt

dv

dtd

vd

d

dtd

dtd

dt

t

d

t

td

v

d

d

λK Kλ

v

KKv

v

2 60

Elementos Capacitivos

abcsabcs

d

dtq

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 qi K

K qK q K

0

1

1 0

0 0

00

00

qs

dqs ds

s

d

s qd s

qd s s qd s s s

qd sqds

q

q

q

qsqs ds

dsds qs

s

s

q

s

d d

dt dtd

dt

dqi q

dtdq

i qdt

dqi

dt

q

K qi K q K K

iq

q

1 0 1 0

1 0 0

0 0 0

ss

d

dt

KK

Enrolamento trifásico comindutâncias 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

0

0

00

10 0

2

0 0 0 0 0 0

s qs

s ds

s s

qsqs ds

dsqd s ds qs

ss

qd s s s s qd s

L M iL M i

qsqs qsR qsL s qs ds

dsds dsR dsL s ds qs

L M i

s sR sL s s

dv

dtd

vdt

dv

dt

dv v v r i

dtd

v v v r idt

v v v r i

v

λ K L K i

qsv

qsi

ds

sL M

sr

dsv

srqs

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

Exercício

Encontre , , e . Considere , , e .

sr

sL

sr

sL

sr

sL

csi

bsi

asi

M M

M

asv

bsv

csv

qsv

qsi

ds

sL M

sr

dsv

srqs

sL M

dsi

0sv

sr

2sL M

dsi

Próxmia Aula

Motores de Indução no Sistema qd0

0 0

0 0

0 0

1 1

2 21 1

2 21 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

Lsr

sL

sr

sL

sr

sL

csi

bsi

asi

M M

M

asv

bsv

csv

A

A’

C’

C

B B’

0

1

1

0

1 1

2 21 1

2 21 1

2 2

30 0

23

0 02

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 magnetizaçãols

ms

L

L

Matriz indutância válida para motores trifásicos no sistema

Matriz indutância válida para motores trifásicos no sistema

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

-1.5

-1

-0.5

0

0.5

1

1.5

B C A

A

A’

C’

C

B B’

Indutância Mútua em MITs

• Porque é ?

00 0

00 0

qsqs s qs ds

dsds s ds ds

ss s s

qrqr r qr r dr

drdr r dr r qr

rr r r

v r i

v r i

v r i

v r i

v r

d

dtd

i

v

dtd

dtd

dtd

dtd

rd

it

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

1 0 00 0 0

0 00 0 0

0 00 0 0

0 0 0 0

20 0 0 0

r L L LL L L s ls M Mi ls M MqsL 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 MqrL L L L LM lr Mi r M r lrdr

Jr P

...

0

3 30 0 0

4 410 0 0

0 0 0

0 0 0... 0

0 0 0 02

0 0 0 0

iqs

idsiqr

L r ir M drP PL i L i rM dr M qr

L L Lls M M v

qsL L Lls M M v

dsL L LM lr M

L L LM lr M

TJ LP

0 00 0 0

0 00 0 0

0 00 0 0

0 0 00 0

0 0 0 0 3 32 0 0 04 4

r L L Ls ls M Mi S Mqs

L L r Lls M s MS Mi

dsL r L LM Ri r M r r lr r Mqr M R

L L L ri r M r lr r MPdrP PJ L i L ir M dr M qr

0 0 0

0 0 0

0 0 00

0 0 00

0 0 0 02

i vS Mqs qsS Mi vds dsM Ri

qr M Ri Pdr

TJ Lr

iqs Sr M

sidsiqr

id

S L L M Lls

r

M r M

r

0 0 00

0 0 0

0 0 00

0 0 00

0 0 0 02

S L M L LM r lr r M i

i vS Mqs qsrr S M v

dsM Riqr M R

PTJ Lr

i Sr i S L L M L i Mqs s qs ls M r M

ds

r

ds

id

r i S L M L L i S vr qr M r lr r M dr qs

M lr

M lr M ls lr ls

M

M lr M ls lr ls

M ls

M lr M ls lr ls

L LS

L L L L L L

LM

L L L L L L

L LR

L L L L L L

i Sr i S L L M L i Mr i S L M L L i S vqs s qs ls M r M ds r qr M r lr r M dr qs

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

M2

R S

0

M

M2

R S

0

0

0

R

M2

R S

0

M

M2

R S

0

M

M2

R S

0

S

M2

R S

0

0

0

M

M2

R S

0

S

M2

R S

0

0

0

0

0

1

J2P

Motor de indução no referencial rotórico

2

1 m

s r

L

L L

Torque

• Torque eletromagnético

• Torque mecânico

• Equação de estado

'

2m dr qs

e PPr

mpp qs ds

r

L i LT N N i i

LL

rm n r d

dT J B

dt

1 10

0 01 0

er

dr

nr

r

B TJ JJ

1 1 1 1 1x = A x +B u

• Fluxo Magnético

• Ângulo do Rotor

rm ds

r

rm

rds

r

r

L dL i

R dt

RLL

IsR

L

01 q

s mer d

idt

i

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

1s

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 doCampo Girante

(rd/s)

Angulo (rd)

eT

Modelo do MIT

wm

Te

Scope4

Scope2

Scope1

SaturationRampProduct

Phidr

PIDPID

PIDPID

MITdq0

Vds

Vqs

Iqs

Ids

Te

Phi_dr

wr

MIT - dq

Iqs

Ids

[Vqs]

[Vds]

[Iabc]

[Te]

[wm]

[PHIdr]

[Rampa]

[Ids]

[Iqs]

[Ids]

[Iqs]

[wm]

[wm]

[PHIdr]

[PHIdr]

[Iabc]

[Iqs]

[wm]

[Rampa]

[Te]

[Te]

[Ids]

[wm]2*pi*60/3

Constant3

0.7

Constant1

Controle externo do MIT

Parâmetros do motor a simularParâmetro Valor nominal

2 2cos cos cos

cos sin 13 32 2 2 2 2

sin sin sin cos sin 13 3 3 3 3

1 1 1 2 2cos sin 1

2 2 2 3 3

L M MsM L Ms sM M Ls

L

2 2 2 2 2 2cos cos cos cos cos cos cos cos cos

3 3 3 3 3 3

2 2 2 2 2sin 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 S

L

cos sin 1

2 2 2 2in sin sin cos sin 1

3 3 3 3

1 11 1 1 1 2 2cos sin

1 1 1

2 2 2 21

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 2s cos cos cos

cos sin 13 3 3 3

2 2 2sin 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

21

3

2 2cos sin 1

3 3

2 2 2 2 2 2cos cos cos cos s2 2 2 s in cos sin cos cos

3 3 3 3in

3 3

2

3

L M L M L M L M L M L MS S S S S S L MS

s

L

2 2cos cos

3 3

2 2 2 2 2 2 2cos sin cos sin cos sin sin sin sin sin sin

3 3 3 3 3 3 32 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 3cos cos cos sin sin sin 3

2 2 3 2 3 2 2

3

23

3 2 3

02

2

0 0

20

33

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 Ms

L

32

0 0

0

0 2

0

0

L M

L M

L MsL M

S

S

S

S

L

Dedução da matriz indutâcia de MITs para o sistema

2

1

1dr ds dss

qr qs q

ds dss

qs qsm m

dr ds

r mqr

ds

qs

dr ds

r mqr qs

dr ds

sm

ds

qs

s rr

qr m m

m

m

qs

iL L

i

iL

i iLi i

LiLi i

iL L

L

LL

L L

L

L

i

i

L

2dr m

qr

qs

ds dsm s r

qs qsr r

i

iL L L LiL L

1

0 1

1 0

0 1

1 0

0

0

ds dss

qs qsm m

r

ds ds dsdss

qs qs qsqs

dr dr drdr

i

r slqr qr qrqr

dr

rqr

LiL L

v iR

v i

v iR

v i

iR

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 LLiL L

L LL

L

rdsr

qrqs

ds dsr s r

qs qsm m

iR L RiL 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

rqr

dsr s

iL

iL

dsr

qsm m

ds

qs

iL L L L iL L L LiL L i

r

q

L

sm

L

iR L RiL

L

LL

22 2

2 0 1

1 0

0

0dr ds dsdrr r m s r r m r m s r

qr

dsm s r

qsm

dsm s r

qsm

dr

rqr

dss r

qsq ms qsqrr m r m r m r

i iR R L L iL RiL

iL L LiL

iL L LiL

L L L L L L Li iL L L L L L L

2 22

2

0 1

1 0

0 1

1 0

dr

rqr

ds drdsdr r m s r m s rsl

qs qrqsqr m r

r m s r

m r

r m s r

m

d

m r

sdr

qsqr

i iL L L L L LR L L L

L

Li iL L L

i

i

L

R L L L

L L

0

0drsl

qrsl

2 20 1

1 0dr dr ds dsds r m r m s r m s r

slqr qr qs qsqs r r r r

i iR L R L L L L L Ri iL L L L

Modelo do Motor Referenciado no Fluxo Rotórico

Estator

Total Rotors ls mL L

ds sdr ds

m m

ds m ls ds m

L

drL L i L i

Li i

L L

Estator

Total Rotors ls mL L

qs sqr qs

qs m

m m

l qs

L

s m qr

i

L L i L i

Li

L L

Total

2

dr r m ds

dr r m ds

m sdr

dr

ds s

m

ds dsm m

m

rr

L L i

L

i

L

L LL i

L L LLi

L L

2

qs sq

qr r qr m qs

qr r qs m qs

m sqr qs qs

sm m

m

r

m

r

L i L i

L L ii

L L LLi

L

Li

L

L

L

rr

r

L

R

2

1 m

s r

L

L L

2rr m s

dr ds dsm m

L L LLi

L L

2

2

1m s m

sds dsds r s r s r s r

drqs qsqs PP r m sm s

ss rr s r s

v

vN

L R Lii L L L L L

ii L LL RL LL L L

Estator Rotor

qsqs s qs ds

dsds s ds ds

dv R i

dtd

v R idt

0

0

qrr qr r dr

drr dr r qr

d

dtd

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 sdr ds

m m

Li i

L L

2 2cos cos cos

cos sin 13 32 2 2 2 2

sin sin sin cos sin 13 3 3 3 3

1 1 1 2 2cos sin 1

2 2 2 3 3

L M MsM L Ms sM M Ls

L

2 2 2 2 2 2cos cos cos cos cos cos cos cos cos

3 3 3 3 3 3

2 2 2 2 2sin 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 Ms

L

cos sin 1

2 2 2 2sin sin sin cos sin 1

3 3 3 3

1 1 1 2 2cos

1 1 1 1 1 1

2 2 2sin 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 2os cos cos cos

cos sin 13 3 3 3

2 2 2sin 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

21

3

2 2cos sin 1

3 3

2 2 2 2 2 2cos 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

s

L

2 2cos cos

3 3

2 2 2 2 2 2 2cos sin cos sin cos sin sin sin sin sin sin

3 3 3 3 3 32s

32 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 3cos cos cos sin sin sin 3

2 2 3 2 3 2 2 3 2 3 2

03

23

02

0

20

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 M

L

33

2

0 0

0

0 2

0

0

S

S

L M

L M

L MSMS

sL

L

0 0 0

0 0 0

0 0 0

0

L L LM lr ML 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 MqsL 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 lsMiqrL 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 04 4

0 0 0 02

r L L Ls ls M ML L r Lls M s M

L r L Lr M r r lr r M

L L L rr M r lr r ML 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 lsP

J

0 0 0

0 0 0

0

L L LM lr ML 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 MqsL 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 MqrL 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 02

vqs

vds

L L L LM ls lr lsL 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 lsP

J