Vector Control Of Im

Vector Control of Induction Machines us

dq αβ

θ

2

3

IM

3

2

αβ dq

θ

is

Introduction • The traditional way to control the speed of induction motors is the V/Hz-control • Low dynamic performance • In applications like servo drives and rolling mills quick torque response is required. • Desire to replace dc drives led to vector control • Braunschweig, Leonhard, Blaschke, Hasse,

What is vector control? • Vector control implies that an ac motor is forced to behave dynamically as a dc motor by the use of feedback control. • Always consider the stator frequency to be a variable quantity. • Think in synchronous coordinates.

Basic blocks of a vector controlled drive

us

dq αβ

θ

2

3

IM

3

2

αβ dq

θ

is

Addition of a block for calculation of the transformation angle us

dq αβ

2

3

3

IM

θr

θ T ra n s fo r m a tio n a n g le c a lc u la tio n

2

αβ dq

θ

is

The current is controlled in the d- and q-directions ref s

i

 i

ref sd

 ji

ref sq

magnetization torque production

Vector controller isref +

-

C u rre n t c o n tro lle r

us

dq αβ

2

3

3

IM

θr

θ T ra n s fo rm a tio n a n g le c a lc u la tio n

2

αβ dq

θ

is

Stator and rotor of an induction machine

Magnetization current from the stator

The flux

The rotation

ω1 ωr

View from the rotor

ω2

Induced voltage and current e = v × B dl

B

ω2

v

v

Torque production F ω2

Ampere-turn balance

ω2

Rotor flux orientation • Difficult to find the transformation angle since the direction of the flux must be known • Flux measurement is required • Flux sensors (and fitting) are expensive and unreliable • Rotor position measurement does not tell the flux position • The solution is flux estimation

Rotor flux orientation using measured flux Original method suggested by Blaschke •Requires flux sensors •Flux coordinates: aligned with the rotor flux linkage   r    arctan     r 

Rotor flux orientation q

β

d

ρ

ψr

y e f

α

 j

  y      s

s r s r



*



 y s 

From Chapter 4 iss R s

Lsl

+

ims

s s

u

d ss  uss  R s iss dt

L rl i s r

Lm

R ′r +

jω r ψ rs

(stator) d rs  j r  rs  R r irs dt

(rotor)

Transformation to flux coordinates d j f j f j f j & e  j   s e  us e  R s is e dt f s

d rf j  e  j & rf e j   j r  rf e j   R r irf e j  dt

d sf  usf  j1 sf  R s isf dt

d f f   j 2  r  R r ir dt f r

 2  1   r The flux coordinate system is ”synchronous” only at steady-state. During transients the speed of the rotor flux and the stator voltage may differ considerably.

The rotor equation (5.9) d f f   j 2  r  R r ir dt f r

1 f Lm f i  r  is Lr Lr f r

d R r f Lm R r f f   j 2  r   r  is dt Lr Lr f r

Split into real and imaginary parts d

 0 f rq

f rq

dt

0

d R r f Lm R r f    rd  isd dt Lr Lr f rd

Lm R r f 0   2   isq Lr f rd

Rotor flux dynamics are slow Lr Tr  Rr

ψ

f rd0

L i

f m sd0

Torque control



3 Lm f * f T  p  Im   r  is 2 Lr

3 Lm f f T  p   rd isq 2 Lr

i

ref sd

  Lm

ref r



Rotor flux orientation using estimated flux • The rotor flux vector cannot be measured, only the airgap flux. • Flux sensors reduce the reliability • Flux sensors increase the cost • Therefore, it is better to estimate the rotor flux.

The "current model" in the stator reference frame (Direct Field Orientation) d s s  j r  r  R r ir dt 1 s Lm s s ir   r  is Lr Lr s r

s ˆ  d r  Lm s 1 s     jr ˆ r  is dt Tr  Tr 

The current model T ref

ψ rdref

C u rre n t c o n tro l

usf

s

ρˆ

isf

s

f

f

ρˆ

C u rre n t m odel

IM d riv e

uss

ωr

iss

The "current model" in synchronous coordinates (Indirect Field Orientation) f sq f rd

Lm Rr f Lm i 0  2   isq  2   Lr Tr  f rd



f rd0

L i

f m sd0



1 isq 2   Tr isd

Transformation angle    1 d t 1 isq 1   r   2   r   Tr isd

Remarks on indirect field orientation • Does not directly involve flux estimation (superscript f dropped) • Not ”flux coordinates” but ”synchronous coordinates” • Since the slip relation is used instead of flux estimation, the method is called indirect field orientation

Indirect field orientation based on the current model T ref

ψ rdref

C u rre n t c o n tro l f s

i

usf

sy

s

IM d riv e

uss

θ s lip re la tio n

θ sy

s

ωr

iss

Feedforward rotor flux orientation ref sq ref sd

1 i 1  r   Tr i

•Significantly reduced noise in the transformation angle •Fast current control is assumed (ref.value=measured value) •No state feedback => completely linear

The voltage model •The current model needs accurate values of the rotor time constant and rotor speed •The trend is to remove sensors for cost and reliability reasons •Simulate the stator voltage equation instead of the rotor voltage equation

dˆ s s  us  Rs is dt s s

 s  Ls is  Lm ir Solve for the rotor current and insert in

 r  Lr ir  Lm is Lr s s s    s  Ls is   Lm is  Lm s r

Multiplication by yields

Lm /Lr

 Lm s L  s s  r   s   Ls  is Lr Lr   2 m

2 m

L Lsσ L Lr

Solve for



s r

Lr s s ˆ   ˆ s  L is  Lm s r

Direct field orientation using the voltage model T ref

ψ rdref

C u rre n t c o n tro l

f

s

ρˆ

isf

s

usf

ψˆ rdf f

ρˆ

V o lta g e m odel

uss

IM d riv e iss

Stator flux orientation "Direct self-control" (DSC) schemes first suggested by Depenbrock, Takahashi, and Noguchi in the 1980s.

dˆ s  us dt s s

1 nominal

At low frequencies the current model can be used together with:

Lm s s ˆ  ˆ r  L is Lr s s

Field weakening isdref

M a x im u m to rq u e ra n g e

ω r max

F ie ld w e a k e n in g ra n g e = > R e d u c e d to rq u e

ω rref

Current control i u

s

+

s

R

L +

e

s

s

di s s s L  u  R i  e dt d j j j j L  i e   u e  R i e  e e dt  d i j j  j j j L e + i jω e   u e  R i e  e e  dt   di  L + i jω   u  R i  e  dt 

di L  u   R + j L  i  e dt d id L  ud  R id   L iq  ed dt

L

d iq dt

 uq  R iq   L id  eq

Transfer function and block diagram of a three-phase load

1 G (s)   s  j  L  R e u

+

G (s)

i

Review of methods for current control • Hysteresis control • Stator frame PI control • Synchronous frame PI control

Hysteresis control (Tolerance band control) • Measure each line current and subtract from the reference. The result is fed to a comparator with hysteresis. • Pulse width modulation is achieved directly by the current control • The switching frequency is chosen by means of the width of the tolerance band. • No tuning is required. • Very quick response

Drawbacks of hysteresis control • The switching frequency is not constant. • The actual tolerance band is twice the chosen one. • Sometimes a series of fast switchings occur. • Suitable for analog implementation. Digital implementation requires a very high sampling frequency.

Stator frame PI control • Two controllers: one for the real axis and one for the imaginary axis • Cannot achieve zero steady-state error • Tracking a sinusoid means that steady-state is never reached in a true sense • Integral action is useless except at zero frequency

Synchronous frame PI control • In a synchronous reference frame the current is a dc quantity at steadystate. • Zero steay state error is possible. • Coordinate transformations necessary • Easily implemented on a DSP • Usually the best choice!

Design of synchronous frame PI controllers di L  u   R + j L  i  e dt Remove cross-coupling

u  u  j L i di L  u  R i  e dt

1 G ( s )  sLR e

iref

+

F (s) -

u′

G ′( s )

+

G (s) + jω L

i

Desired closed-loop system  Gc ( s )  s 

ki F (s)  k p  s

 tr  ln(9) F ( s )G ( s ) Gc ( s )  1  F ( s )G ( s )

Choice of controller parameters   R -1 F ( s )   G ( s )    s L  R    L+ s s s

kp   L ki   R

Speed control • Applications: pumps and fans in the process industry, paper and steel mills, robotics and packaging, electric vehicles • Very different dynamic requirements • Most drives have low to medium high requirements on dynamics. These drives are considered here. • Cascade control is sufficient

Block diagram of a speedcontrolled drive system Inverter

+

ω

ref m

−

εω

+ Speed Current controller i ref − ε I controller u ref

i

Electric motor

ωm

The mechanical system

d m Te  Tl  J  b m dt T − l Te +

1/ ( J s + b )

ωm

The speed controller • The task of the speed controller is to provide a reference value for the torque (or current) which makes the mechanical system respond to the speed reference with a specified rise time.

 tr  ln(9)

Block diagram with speed controller Speed controller

+

ω

ref m

− εω

Fω ⋅ cT

Inner loop ref sq

i

1/ cT

− Tl Te +

1 cT  3 Lm ref p   rd 2 Lr

1/ ( J s + b ) ω m



ref m

1  m  F   m J sb 1 F   Go J sb

Go  Gc   Go  1 s  

 Go  s

Choice of controller parameters 1  F   J sb s

  b F   J s  b   { J  s s { P I

Realistic choice of bandwidth • Care must be taken that the bandwidth of the speed controller is not unnecessarily high. • In fact this should be decided during the first steps in the design process of a drive system • The bandwidth is directly connected to the current rating of the inverter.

A change in the speed reference

iqp     J How large steps should be foreseen?

  max  Cmax mbase

With

iqp  I nom

and

     max

I nom  Cmax mbase  J I nom   Cmax mbase J Check if the current controller is sufficiently fast.