Summary Dynamical Operation, Vector Control, DTC and Encoder-less Operation

©Copyright Ned Mohan 2005, by

Summary

Dynamical Operation, Vector Control, DTC and Encoder-less Operation

t

t 0

0 TLoad

m

Seamless discussion of dynamic

control and encoder-less operation

©Copyright Ned Mohan 2005, by ©Copyright Ned Mohan 2005, by

What is an Electric-Motor Drive?

Power Processing Unit (PPU) fixed

form

measured speed/ position

speed / position Motor

Electric Drive

Load

input command (speed / position)

Power Signal adjustable

Electric Source form (utility)

Sensors

Controller

- Harnessing of Wind Energy

-Hybrid Electric Vehicles

Power Processing Unit

V





a b

c

a

( )

q t q t

( ) q t

( )

,

( )

control a

v t

,

( )

control b

v t

,

( )

control c

v t

tri

( ) v t

V





a b

c

a

( )

q t q t

( ) q t

( )

,

( )

control a

v t

,

( )

control b

v t

,

( )

control c

v t

tri

( ) v t

acmotor



c

( ) d t

b

( ) d t

a

( ) d t V





a b

c acmotor



c

( ) d t

b

( ) d t

a

( ) d t V





a b

c

Reference Book: First Course on Power Electronics by Ned Mohan, published

by MNPERE (See www.MNPERE.com for details).

©Copyright Ned Mohan 2005, by

a axis b axis

c axis

_m i_a i_b

i_c i_A i_C i_B

A axis B axis

C axis

_m

( )a

i_a i_b

i_c

  





v_a v_b

v_c

R_s









v_A 0 v_B 0

v_C 0

R_r i_A i_B

i_C

Stator circuit Rotor circuit

( )b

 

Figure 2-5 Rotor circuit represented by three-phase windings.

(b)

Equivalent Windings in A Squirrel-Cage Rotor

3

m

2

L  L

s s m

L  L

 L

( ) ( ) ( ) 0

A B C

i t  i t  i t  ( ) ( ) ( ) 0

a b c

i t  i t  i t 

3

m

2

L  L

r r m

L  L

 L

,1

cos

aA m phase m

L  L

 

Stator

Rotor

Mutual

Dynamic Analysis of Induction Machines in Terms of dq-

Windings

©Copyright Ned Mohan 2005, by

Representation of Stator MMF by Equivalent dq Windings

2 / 3 4 / 3

( ) ( ) ( ) ( )

a j j

s a b c

i  t  i t  i t e

 i t e

( ) ( )

2

a s a

s

N

F t   i  t

 

3 2

2 2

s s d

sd sq s

N N

i  ji  i 

 i

 ji

  ² ₃ i 

,1

winding magnetizing inductance =( 3/ 2) (3/ 2)

m phase m phase m

dq L

L L







axis a axis

b

axis c

is



at t

axis d axis

q

isd

isq ³₂^Ns

da

axis a axis

b

axis c

is

at t

axis d axis

q

projection projection

sq 2

i   3

projection projection

sd

i 2

  3

da

Mutual Inductance between dq Windings on the Stator and the Rotor

a axis stator A axis

rotor d axis

_m i_sd i_sq

q axis at t

i_rq

i_rd

_m

 i_s

 i_r

3 2isd

3 2isq

3

2irq 3

2ird

d

d

da

dA 3

2N^s

3 2N^s 3

2N^s

Figure 3-3 Stator and rotor mmf representation by equivalent dq winding currents.

a axis a axis stator A axis A axis

rotor d axis d axis

_m

_m i_sd i_sd i_sq

i_sq q axis

q axis at tat t

i_rq i_rq

i_rd i_rd

_m

_m

 i_s

 i_s

 i_r

 i_r

3 2isd

3 2isd

3 2isq

3 2isq

3 23irq

2irq 3

2ird

3 2ird

d

d

da

dA 3

2N^s

3 2N^s 3

2N^s

Figure 3-3 Stator and rotor mmf representation by equivalent dq winding currents.

Figure 3-3 Stator and rotor representation by equivalent dq winding currents. The dq winding voltages are defined as positive at the dotted terminals.

Note that the relative positions of the stator and the rotor current space vectors are not actual, rather only for definition purposes.

sd

L i

L i

    

L i

 L i

©Copyright Ned Mohan 2005, by

Figure 3-4 Transformation of phase quantities into dqwinding quantities.

[ ]T_{sabc dq} ia

ib

ic

sd

i

sq

i

(a) stator

[ ]T_rABCdq iA

iB

iC

rd

i

rq

i

(b) rotor

da

 _dA

Figure 3-4 Transformation of phase quantities into dqwinding quantities.

[ ]T_{sabc dq} ia

ib

ic

sd

i

sq

i

(a) stator

[ ]T_rABCdq iA

iB

iC

rd

i

rq

i

(b) rotor

da

 _dA

[ ]T_{sabc dq} ia

ib

ic

sd

i

sq

i

(a) stator

[ ]T_rABCdq iA

iB

iC

rd

i

rq

i

(b) rotor

da

 _dA

[ ]

2 4 ( )

cos( ) cos( ) cos( )

( ) 2 3 3 ( )

( ) 3 sin( ) sin( 2 ) sin( 4 ) ( )

3 3

da da da a

sd b

sq da da da c

Ts abc dq i t i t

i t i t

i t

 

  

 

  



     

 

    

    

        

 

2 4 ( )

cos( ) cos( ) cos( )

( ) 2 3 3 ( )

( ) 3 sin( ) sin( 2 ) sin( 4 ) ( )

3 3

dA dA dA A

rd B

rq dA dA dA C

Tr ABC dq i t i t

i t i t

i t

 

  

 

  



     

 

    

    

        

Mathematical Relationship between dq and phase

Winding Variables (abc to dq)

10

Mathematical Relationship between phase Winding Variables and dq (dq to abc)

 

cos( ) sin( )

( ) 2 4 4

( ) cos( ) sin( )

3 3 3

( ) 2 2

cos( ) sin( )

3 3

da da

a sd

b da da

c sq

da da

Ts dq abc

i t i

i t i

i t

 

 

 

 

 



 

  

     

       

     

   

 

    

 

 

  

cos( ) sin( )

( ) 2 4 4

( ) cos( ) sin( )

3 3 3

( ) 2 2

cos( ) sin( )

3 3

dA dA

A rd

B dA dA

C rq

dA dA

Ts dq ABC

i t i

i t i

i t

 

 

 

 

 



 

  

     

       

     

   

 

  

 

 



11

©Copyright Ned Mohan 2005, by

sd s sd

d

v R i

dt   

  

sq s sq

d

v R i

dt   

  

Derivation of Voltages in dq Windings

 

 

 

v R i d

dt 

 

     

[ ]

[ ]

[ ]

d

T v R T i T

dt 







  ^v

^{ R i}

 

^{ [ ] T}

_dt ^d  ^{[ ] T}

  ^



        ^{ }

1 0 0 1

0 1 1 0

[ ] [ ] [ ] [ ]

s s abc dq s dq abc s abc dq s dq abc

dq dq dq dq

d

d d

v R i T T T T

dt dt



 

   

    

   

   

  

     

rd r rd

d

v R i

dt   

  

rq r rq

d

v R i

dt   

  

12

a  a x i s d  a x i s q  a x i s

ir q

ir d is q

d

d a d d

s u b t r a c t

d u e t o i_{s q} a n d i_{r q}

d u e t o i_{r q} l e a k a g e f l u x



 

F i g u r e 3 - 8 T o r q u e o n t h e r o t o r - a x i s .d

a  a x i s d  a x i s q  a x i s

ir q

ir d is q

d

d a d d

s u b t r a c t

d u e t o i_{s q} a n d i_{r q}

d u e t o i_{r q} l e a k a g e f l u x



 

a  a x i s d  a x i s q  a x i s

ir q

ir d is q

d

d a d d

s u b t r a c t

d u e t o i_{s q} a n d i_{r q}

d u e t o i_{r q} l e a k a g e f l u x



 

F i g u r e 3 - 8 T o r q u e o n t h e r o t o r - a x i s .d

Electromagnetic Torque on the Rotor d-Axis

0

3/ 2

ˆ ( )

2

s r

rq sq rq

g m

mmf

N L

B i i

L

  

   

 

 

,

3/ 2 ˆ

2

d rotor

N

T _{ } ^  r B ^ _ i

  

2

, 0

3/ 2

( )

2

s r

d rotor sq rq rd

g m

N L

T r i i i

L

 

  

          



0 2

,

3

( )

2 2

s r

d rotor sq rq rd

g m

Lm

N L

T r i i i

L

 

   

 

          

 

,

( )

d rotor m sq r rq rd rq rd rq

T L i L i i i





  



13

©Copyright Ned Mohan 2005, by

ir q ir d

q a d d

s u b t r a c t d u e t o i_{s d} a n d i_{r d}

d u e t o l e a k a g e f l u xi_{r d} q  a x i s

d  a x i s

a  a x i s is d

q

 



F i g u r e 3 - 9 T o r q u e o n t h e r o t o r - a x i s .q

ir q ir d

q a d d

s u b t r a c t d u e t o i_{s d} a n d i_{r d}

d u e t o l e a k a g e f l u xi_{r d} q  a x i s

d  a x i s

a  a x i s is d

q

 



ir q ir d

q a d d

s u b t r a c t d u e t o i_{s d} a n d i_{r d}

d u e t o l e a k a g e f l u xi_{r d} q  a x i s

d  a x i s

a  a x i s is d

q

 



F i g u r e 3 - 9 T o r q u e o n t h e r o t o r - a x i s .q

Electromagnetic Torque on the Rotor q-Axis

,

( )

q rotor m sd r rd rq rd rq rd

T L i L i i i





    



, ,

em d rotor q rotor

T  T  T

( )

em

2 p

T   i   i

em L

mech eq

d T T

dt   J ^

Net Electromagnetic

Torque

Simlink-based dq-Axis Simulation of Induction Motor

15

©Copyright Ned Mohan 2005, by

Simulation Results

16

Mathematical Description of Vector

Control

17

©Copyright Ned Mohan 2005, by

a axis stator A axis

rotor d axis

_m i_sd i_sq

q axis att

i_rq

i_rd

_m

 i_s

i_r

3 2isd

3 2isq

3

2irq 3

2ird

d

d

da

dA 3

2N^s

3 2N^s 3

2N^s

Figure 3-3 Stator and rotor mmf representation by equivalent dq winding currents.

a axis a axis stator A axis A axis

rotor d axis d axis

_m

_m i_sd i_sd i_sq

i_sq q axis

q axis attatt

i_rq i_rq

i_rd i_rd

_m

_m

 i_s

 i_s

i_r

i_r

3 23isd

2isd

3 2isq

3 2isq

3 2irq

3

2irq 3

2ird

3 2ird

d

d

da

dA 3

2N^s

3 2N^s 3

2N^s

Figure 3-3 Stator and rotor mmf representation by equivalent dq winding currents.

Figure 5-1 Stator and rotor mmf representation by equivalent dq winding currents.

The d-axis is aligned with ^_r.

Motor Model with the d-Axis Aligned with the Rotor Flux Linkage Axis

( ) 0

rq

t

 

r

i L i

  L ( ) 0

d

t

dt  

Dynamic Circuits with the d-Axis Aligned with the

Rotor Flux Linkage Axis

Calculation of 

:

rq m

dA r sq

rd r rd

i L

R i

 ^{ }  ^  

Calculation of Torque

em

2 p

T    i

: T

rq r rq

d

v R i

dt   

  

rq m sq

r

i L i

  L

p  L 

19

©Copyright Ned Mohan 2005, by

D-Axis Rotor Flux Dynamics

( ) ( )

1

rd m sd

r

s L i s

 s

 



r r r

L

  R

rd m

rd sd

r r

d L

dt i

 

 

 





0

rd r rd

d

v R i

dt   

  

rd

L i

L i

  

Motor Model

 _{  }

  

Figure 5-4 Motor model with d-axis aligned with .  ^





1

m r

L s 





N/D N da

1/ s



T









L

i

i

2

m r

L p

 L

2 p







 

21

©Copyright Ned Mohan 2005, by

Speed and Position Loops for Vector Control

Figure 5-8 Vector controlled induction motor drive with a current-regulated PPU.

(measured) d dt/

Motor PPU

regulated current to

abc dq

to dq

abc

a*

i

b*

i

c*

i

ia

ib

ic

isd

isq

Tem

rd

da ^{Fig. 5-4}

mech

mech

(measured)

mech mech Tem

(calculated)

P PI PI



  





 







 PI

mech ^*

rd

*rd



mech

rd(calculated)

*sd

i

*sq

* i

mech ^*mech _*

Tem

(measured)

da

mech

Estimated Motor Model

22

Design of Speed Loop

rd

L i

 

2 *

2

em m sd sq

r k

L

T p i i

 L

 

 



mech

k

sq( )

i s

p ki

k  s 1

sJeq

*mech

 T_em

23

©Copyright Ned Mohan 2005, by

Simulation of CR-PWM Vector Controlled Drive

using Simulink

24

Figure 5-11 Simulation results of Example 5-2.

Simulation Results of a Vector

Controlled Induction Motor Drive

25

©Copyright Ned Mohan 2005, by

Space-Vector Pulse-Width-

Modulated (SV-PWM) Inverters

Advantages

• Full Utilization of the DC Bus Voltage

• Same simplicity as the Carrier-Modulated PWM

• Applicable in Vector Control, DTC and V/f Control

Synthesis of Stator Voltage Space Vector

Figure 7-1 Switch-mode inverter.

q

q

q





V

a b

c

N

 

va vb



vc 

i

i

i

0 2 / 3 4 / 3

( ) ( ) ( ) ( )

a j j j

s a b c

v t   v t e  v t e

 v t e

; ;

a aN N b bN N c cN N

v  v  v v  v  v v  v  v

0 2 / 3 4 / 3

0

j j j

e  e

 e



0 2 / 3 4 / 3

a

( )

s aN bN cN

v t   v e  v e

 v e

0 2 / 3 4 / 3

( ) ( )

a j j j

s d a b c

v t   V q e  q e

 q e

27

©Copyright Ned Mohan 2005, by

0 7

Figure 7-2 Basic voltage vectors ( and not shown).v v -axis a vs

1(001) v

3(011) 2(010) v

v

6(110) v

4(100)

v v5(101)

sector 1 sector 2

sector 3

sector 4

sector 5

sector 6

Basic Voltage Vectors

0 1 0

2 / 3 2

3 / 3

4 / 3 4

5 / 3 5

6 7

(000) 0

(001) (010) (011) (100) (101) (110)

(111) 0

sa

a j

s d

a j

s d

a j

s d

a j

s d

a j

s d

a j

s d

sa

v v

v v V e

v v V e

v v V e

v v V e

v v V e

v v V e

v v











 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Synthesis of Voltage Vector in Sector 1

1 j0

v   V e

v   V e

ˆ

s s

v   V e



yv3

Figure 7-3 Voltage vector in sector 1.

1 3

(1)

1 [

0]

s

v xT v yT v zT

 T   

  

1 3

(2) v 

 xv   yv  (3) x y z    1

0 / 3

(4) V e ˆ

 xV e

 yV e

29

©Copyright Ned Mohan 2005, by

Limit on the Amplitude of the Stator Voltage Space Vector

,max ,max

(4) ( ) 3 ˆ 0.707

2 2

phase d

LL d

V V

V rms    V

,max

3

(5) ( ) 0.612

LL

2 2

V rms  V  V

300

ˆ_s,max V

Vd Vd

Figure 7-7 Limit on amplitude .Vˆ_s

,max

ˆ

(1) v 

( ) t  V

e

0

,max

60 3

(2) ˆ cos( )

2 2

s d d

V  V  V

,max

2

ˆ ˆ

(3) 3 3

phase s

V

V  V 

Synthesis using Carrier-Modulated PWM

0 7

Figure 7-4 Waveforms in sector 1; z z   z .

Ts s/ 2

T

z7 0/ 2

z z₀/ 2

/ 2 / 2 y

y / 2

x x/ 2

vaN

vbN

vcN 0

0

0

Vd

Vd

Vd 0

, control a tri v

v

, control b v

, control c v

,

,

,

ˆ / 2

ˆ / 2

ˆ / 2

control a a k tri d

control b b k tri d

control c c k tri d

v v v

V V

v v v

V V

v v v

V V

 

 

 

max( , , ) min( , , ) 2

a b c a b c

k

v v v v v v

v  

( ) ( ) ( ) 0

a b c

v t  v t  v t 

31

©Copyright Ned Mohan 2005, by

Synthesis of Space Vector using Carrier-Modulated

PWM in Simulink

32

Figure 7-6 Simulation results of Example 7-1.

Control Waveforms for Carrier Pulse-Width-Modulation

33

©Copyright Ned Mohan 2005, by

Direct Torque Control (DTC) and Encoder-less Operation of

Induction Motors

DTC System Overview

Measured Inputs: Stator Voltages and Currents

Estimated Outputs: 1) Torque, 2) Mechanical Speed, 3) Stator Flux Amplitude and 4) its angle



 Vd

qa

qb

qc

ia

ib

ic

IM

 PI 

   





mech*



mech

em*

T

ˆ_s*



ˆ_s

 Tem

s



Estimator

Selection of

vs

35

©Copyright Ned Mohan 2005, by

Principle of DTC Operation

Figure 8-2 Changing the position of stator flux-linkage vector.







s ( ) t

 ^

( )

s t T

    v  s  T



( ) ( )

r t r t T

      

-axis a

Rotor -axisA

rA



2

ˆ ˆ sin 2

em

p L

T  L _   

sr s r

    

36

2

2 3 2 ˆ

slip r em

r

R T

 p



 

      

m r slip

    

( ) ( ) ^t ( ) ˆ ^{j s}

s s s s s s

t T

t t T v R i d e ^

   



       

   

( ) ˆ ^j

r r

r s s s r

m

L L i e

L

          

( ) ( )

r r

r d r t t T

dt T





 

    

 



Im( )

2

em p s conj s

T    i 

2

1

s r

L

   L L

Calculation of Stator Flux:

Calculation of Rotor Flux:

where

Estimating Torque:

Estimating Mechanical Speed:

s s s

d

v R i

dt 

   

 

37

©Copyright Ned Mohan 2005, by

Inverter Basic Vectors and Sectors

a-axis 1(001) v

3(011) v

2(010) v

6(110) v

4(100)

v v⁵(101)

b-axis

c-axis

1 3 2

4

5

6

Figure 8-3 Inverter basic vectors and sectors.

38

Stator Voltage Vector Selection in Sector 1

sector 1

 ^ s

v 1

v 3

v 2

v 4 v ⁵ v 6

Figure 8-4 Stator voltage vector selection in sector 1.

39

©Copyright Ned Mohan 2005, by

Selection of the Stator Voltage Space Vector

s e c t o r 1

 

v

v

v

v

v

v

F i g u r e 8 - 4 S t a t o r v o l t a g e v e c t o r s e l e c t i o n i n s e c t o r 1 .

s e c t o r 1

 

v

v

v

v

v

v

s e c t o r 1

 

v

v

v

v

v

v

F i g u r e 8 - 4 S t a t o r v o l t a g e v e c t o r s e l e c t i o n i n s e c t o r 1 .

Effect of Voltage Vector on the Stator Flux-Linkage Vector in Sector 1.

increase increase increase ^decrease

decrease decrease decrease increase

v s

v 3

v 2

v 4

v 5

T em  ^ˆ _s

a-axis 1(001) v

3(011) v

2(010) v

6(110) v

4(100)

v v⁵(101)

b-axis

c-axis

1 3 2

4

5

6

Figure 8-3 Inverter basic vectors and sectors.

Effect of Zero Stator Voltage Space Vector







( ) ( )

s t s t T

      

-axis a

Rotor -axis at t- t A 

rA

 Rotor -axis at t A

( )

r t T

  ( ) ^r t  

 ^

 m



sin 

 ( 

 

)

( )

em s r

T  k   

s

0



 

A

0



 

r m rA m

   

        T

   k ( 

)

41

©Copyright Ned Mohan 2005, by

DTC in Simulink

42

Fig. 2 Torque Waveforms.

43

©Copyright Ned Mohan 2005, by

Fig. 3 Speed Waveforms.

44

Fig. 4 Stator Flux.

45

©Copyright Ned Mohan 2005, by

Fig. 5 Stator and Rotor Fluxes.

46

Vector Control of Permanent-

Magnet Synchronous-Motor Drives

47

©Copyright Ned Mohan 2005, by

Non-Salient Permanent-Magnet Synchronous Motor

sd

L i

     

L i

axis c

axis a axis

b

N S

ib

ia

ic

m

( )a

axis c

axis a axis

b

N

S ib

ia ic

m Br



' a

a -axis

q

( )b

-axis d

Figure 9-1 Permanent-magnet synchronous machine (shown with =2). p

48

sd s sd

d

v R i

dt   

  

sq s sq

d

v R i

dt   

  

m

2 p

  

( )

em

2 p

T   i   i [( ) ]

2 2

em

p

p

T  L i   i  L i i   i

em L

mech eq

T T

d

dt   J ^

Non-Salient Permanent-Magnet Synchronous Motor

(Continued)

49

©Copyright Ned Mohan 2005, by

Controller in the dq Reference Frame

compd

( )

sd s sd s

d

v R i L i L i

dt 

   



compq

( )

sq s sq s

d

v R i L i L i

dt  

   



2 2

,

3

ˆ ( ˆ )

sd sq dq rated

2

i  i  I  I

Figure 9-3 Controller in the dq reference frame.

ia

ib

ic

Motor

 





*sq

i

*sd

i

( )

m 2p mech

  

sensor position

abc to dq

 



( )

mL is sd fd

  m s sqL i





*sd

v

*sq

v Inverter

PWM

p ki

k  s

m

isq

isd







decoupling terms

1 s

to abc

dq

*a

v

p ki

k  s

*b

v

c*

v

Vector Control of a Permanent-Magnet Synchronous-

Motor Drive

51

©Copyright Ned Mohan 2005, by

Figure 9-5 Simulation results of Example 9-1.

Simulation Results