topic 13: vector control of ac induction motors spring 2004 ece 8830 - electric drives

Topic 13: Vector Control of AC Induction

Motors

Spring 2004

ECE 8830 - Electric Drives

Introduction

Scalar control of ac drives produces good steady state performance but poor dynamic response. This manifests itself in the deviation of air gap flux linkages from their set values. This variation occurs in both magnitude and phase.

Vector control (or field oriented control) offers more precise control of ac motors compared to scalar control. They are therefore used in high performance drives where oscillations in air gap flux linkages are intolerable, e.g. robotic actuators, centrifuges, servos, etc.

Introduction (cont’d)

Why does vector control provide superior dynamic performance of ac motors compared to scalar control ?

In scalar control there is an inherent coupling effect because both torque and flux are functions of voltage or current and frequency. This results in sluggish response and is prone to instability because of 5th order harmonics. Vector control decouples these effects.

Torque Control of DC Motors

There is a close parallel between torque control of a dc motor and vector control of an ac motor. It is therefore useful to review torque control of a dc motor before studying vector control of an ac motor.

Torque Control of DC Motors (cont’d)

A dc motor has a stationary field structure (windings or permanent magnets) and a rotating armature winding supplied by a commutator and brushes. The basic structure and field flux and armature MMF are shown below:


The field flux f (f) produced by field current If is orthogonal to the armature flux a (a) produced by the armature current Ia. The developed torque Te can be written as:

Because the vectors are orthogonal, they are decoupled, i.e. the field current only controls the field flux and the armature current only controls the armature flux.

'e t a fT K I I


DC motor-like performance can be achieved with an induction motor if the motor control is considered in the synchronously rotating reference frame (de-qe) where the sinusoidal variables appear as dc quantities in steady state.

Two control inputs ids and iqs can be used for

a vector controlled inverter as shown on the next slide.


With vector control:

ids (induction motor) If (dc motor)

iqs (induction motor) Ia (dc motor)

Thus torque is given by:

where is peak value of sinusoidal space vector. 'e t qs t ds qsrT K i K i i r r


This dc motor-like performance is only possible if iqs

* only controls iqs and does not affect the flux , i.e. iqs and ids are orthogonal under all operating conditions of the vector-controlled drive.

Thus, vector control should ensure the correct orientation and equality of the command and actual currents.

r

Equivalent Circuit of Induction Motor

The complex de-qe equivalent circuit of an induction motor is shown in the below figure (neglecting rotor leakage inductance).

Equivalent Circuit of Induction Motor (cont’d)

Since the rotor leakage inductance has been neglected, the rotor flux = , the air gap flux.

The stator current vector Is is the sum of the ids and iqs vectors. Thus, the stator current magnitude, is related to ids and iqs by:

r

m

sI

2 2s ds qsI i i

Phasor Diagrams for Induction Motor

The steady state phasor (or vector) diagrams for an induction motor in the de-qe (synchronously rotating) reference frame are shown below:

Phasor Diagrams for Induction Motor (cont’d)

The rotor flux vector is aligned with the de axis and the air gap voltage is aligned with the qe axis. The terminal voltage Vs slightly leads the air gap voltage because of the voltage drop across the stator impedance. iqs contributes real power across the air gap but ids only contributes reactive power across the air gap.

( )r m mV

Phasor Diagrams for Induction Motor (cont’d)

The first figure shows an increase in the torque component of current iqs and the second figure shows an increase in the flux component of current, ids. Because of the orthogonal orientation of these components, the torque and flux can be controlled independently. However, it is necessary to maintain these vector orientations under all operating conditions.

How can we control the iqs and ids components of the stator current Is independently with the desired orientation ?

Principles of Vector Control

The basic conceptual implementation of vector control is illustrated in the below block diagram:

Note: The inverter is omitted from this diagram.

Principles of Vector Control (cont’d)

The motor phase currents, ia, ib and ic are converted to ids

s and iqss in the stationary

reference frame. These are then converted to the synchronously rotating reference frame d-q currents, ids and iqs.

In the controller two inverse transforms are performed:

1) From the synchronous d-q to the stationary d-q reference frame; 2) From d*-q* to a*, b*, c*.

Principles of Vector Control (cont’d)

There are two approaches to vector control:

1) Direct field oriented current control - here the rotation angle of the iqs

e vector with respect to the stator flux qr’s is being directly determined (e.g. by measuring air gap flux)

2) Indirect field oriented current control - here the rotor angle is being measured

indirectly, such as by measuring slip speed.

Direct Vector Control

In direct vector control the field angle is calculated by using terminal voltages and current or Hall sensors or flux sense windings.

A block diagram of a direct vector control method using a PWM voltage-fed inverter is shown on the next slide.

Direct Vector Control (cont’d)


The principal vector control parameters, ids* and

iqs*, which are dc values in the synchronously

rotating reference frame, are converted to the stationary reference frame (using the vector rotation (VR) block) by using the unit vector cose and sine. These stationary reference frame control parameters ids

s* and iqss* are then

changed to the phase current command signals, ia*, ib*, and ic* which are fed to the PWM inverter.


A flux control loop is used to precisely control the flux. Torque control is achieved through the current iqs

* which is generated from the speed control loop (which includes a bipolar limiter that is not shown). The torque can be negative which will result in a negative phase orientation for iqs in the phasor diagram.

How do we maintain idsand iqs orthogonality? This is explained in the next slide.


Here the de-qe frame is rotating at synchronous speed e with respect to the stationary reference frame ds-qs, and at any point in time, the angular position of the de axis with respect to the ds axis is e (=et).

From this phasor diagram we can write:

and coss

dr er sinsqr er


Thus,

, , and

The cose and sine signals in correct

phase position are shown below:

cossdr

e

r

sinsqr

e

r

2 2s s

dr qrr


These unit vector signals, when used in the vector rotation block, cause ids to maintain orientation along the de-axis and the iqs orientation along the qe-axis.

Summary of Salient Features of Vector Control

A few of the salient features of vector control are:

The frequency e of the drive is not controlled (as in scalar control). The motor is “self-controlled” by using the unit vector to help control the frequency and phase.

There is no concern about instability because limiting within the safe limit automatically limits operation to the stable region.

sI

Summary of Salient Features of

Vector Control (cont’d)

Transient response will be fast because torque control by iqs does not affect flux.

Vector control allows for speed control in all four quadrants (without additional control elements) since negative torque is directly taken care of in vector control.

Flux Vector Estimation

The air gap flux can be directly measured in a machine using specially fitted search coils or Hall effect sensors. However, the drift in the integrator with a search coil is problematic at very low frequencies. Hall effect sensors tend to be temperature-sensitive and fragile.

An alternative approach is to measure the terminal voltage and phase currents of the machine and use these to estimate the flux. These techniques are discussed on pp. 363-368 of the Bose text.

Indirect Vector Control

Indirect vector control is similar to direct vector control except the unit vector signals (cose and sine) are generated in a feedforward manner.

The phasor diagram on the next slide can be used to explain the basic concept of indirect vector control.

Indirect Vector Control (cont’d)


The ds-qs axes are fixed on the stator and the dr-qr axes are fixed on the rotor. The de-qe axes are rotating at synchronous speed and so there is a slip difference between the rotor speed and the synchronous speed given by:

Since, , we can write:

e r sl

e edt e r sl


In order to ensure decoupling between the stator flux and the torque, the torque component of the current, iqs, should be aligned with the qe axis and the stator flux component of current, ids, should be aligned with the de axis.

We can use the de-axis and qe-axis equivalent circuits of the motor (shown on the next slide) to derive control expressions.


The rotor circuit equations may be written as:

( ) 0drr dr e r qr

dR i

dt

( ) 0qrr qr e r dr

dR i

dt


The rotor flux linkage equations may be written as:

These equations may be rewritten as:

dr r dr m dsL i L i

qr r qr m qsL i L i

1 mdr dr ds

r r

Li i

L L

1 mqr qr qs

r r

Li i

L L


Combining these with the earlier equations allows us to eliminate the rotor currents which cannot be directly obtained. The resulting equations are:

where .

0dr mrdr r ds sl qr

r r

d LRR i

dt L L

0qr mrqr r qs sl dr

r r

d LRR i

dt L L

sl e r


For decoupling control the total rotor flux needs to be aligned with the de-axis and so we want: qr=0 => dqr/dt =0

If we now substitute into the previous equations, we get:

and

where has been substituted for dr .

r

r r

m dsrr

dLL i

R dt

m r

sl qs

rr

L Ri

L

r


For implementing the indirect vector control strategy, we need to take these equations into consideration as well as the equation:

Note: A constant rotor flux results in the equation:

so that the rotor flux is directly proportional to ids in steady state.

m dsr L i

e r sl


An implementation of indirect vector control for 4-quadrant operation is shown below:


Features of this implementation: Diode rectifier front-end with a PWM inverter with

a dynamic brake in the dc link. Hysteresis-band current control. Speed control loop generates the torque

component of current, iqs*.

Constant rotor flux is maintained by using the desired ids

*.

The slip frequency sl* is generated from the

desired iqs*.

Indirect Vector Control (cont’d) Slip gain Ks is given by:

e and e are given by:

and

The incremental encoder is necessary for indirect vector control because the slip signal locates the rotor pole position with respect to the dr axis in a feedforward manner.

*

*sl m r

sqs r r

L RK

i L

*e sl r edt

Indirect Vector Control (cont’d) If iqs

*<0 for negative torque, phasor iqs is reversed and sl (and sl) will be negative.

The speed control range can be extended into the field weakening region by incorporating the dotted line part of the implementation (see figure below). Note: Closed loop flux control is now required.


Harmonic content of hysteresis-band current control is not optimum. Also, at higher speeds the current controller will saturate in part of the cycle because of the high back emf.

Synchronous current control can be used to overcome these problems. See Bose text, pp. 372-374 for details.


A dc motor-like electromechanical model can be derived for an ideal vector-controlled drive using the following equations:

3

2 2m

e qsrr

LPT i

L

r r

m dsrr

dLL i

R dt

2 re L

dT T J

P dt


A transfer function block diagram is shown below:

Note: The torque Te responds instantly but the flux has first order delay (with time constant =Lr/Rr).


The physical principle of vector control can be explained more clearly with the help of the below de-qe

equivalent circuits:


Since ids and iqs are being controlled, we can ideally ignore the stator-side parameters. With qr=0 under all conditions, the emf source on the rotor side de-circuit slqr=0. This means that in steady state ids flows only through the magnetizing inductance, Lm, but in the transient case, is shared by the rotor circuit whose time constant = Llr/Rr.


In the qe-circuit when torque is controlled by iqs the emf sldr changes instantaneously (because ). Since qr=0, this emf causes a current (Lm/Lr)iqs to flow through the rotor resistor Rr. If Llr is neglected and flux is constant, ids is seen to only flow through Lm and iqs only flows through the rotor side, as desired.

/sl dr m r qs rL R i L

r


A serious issue with respect to indirect vector control is that of slip gain detuning. This is due primarily to variation in rotor resistance. This effect is illustrated below where Rr=actual rotor resistance and

= estimated rotor resistance.

rR


Continuous on-line tuning of Ks is very complex and computationally intensive. However, two methods, one based on extended Kalman filtering (EKF) for parameter estimation and a second one based on a model referencing adaptive controller (MRAC) approach are good options. The EKF method will be considered later when studying sensorless vector control but the MRAC method is described next.


In the MRAC approach a reference model output signal X* that satisfies the tuned vector control condition is usually a function of ids

* and iqs

*, motor inductances, and operating frequency. The adaptive model X is estimated based on motor feedback voltages and currents as shown in the next slide. X is compared to X* an the resulting error used to estimate the slip gain through a P-I compensator. Slip gain tuning is achieved when X=X*. sK


Suppose we decide to use torque as the model parameter X. Thus,

Substituting Lmids* for gives:

The actual torque can be estimated from the stator frame variables using the equation:

* * *3

2 2m

e qsrr

LPX T i

L

r

2* * * *3

2 2m

e ds qsr

LPX T i i

L

3

2 2s s s s

e ds qs qs ds

PX T i i


Note: Lm and Lr parameter variations affect the estimation accuracy of X* and at low speeds, the stator resistance Rs affects the estimation accuracy of X.

Stator Flux-Oriented Vector Control

Until now we have only considered rotor flux-oriented vector control. Airgap flux or stator flux-oriented vector control is also possible but at a cost of a coupling effect that requires decoupling compensation. See Bose text pp. 381-384 for details.

Vector Control of Current-Fed Inverter Drive

Vector control can also be extended to current-fed drives as illustrated below:

Vector Control of Current-Fed Inverter Drive (cont’d)

Drive operates with regulated rotor flux and the speed control loop is the outer loop. The speed loop generates the torque command Te

* which is then divided by K to generate iqs

*. The flux loop generates ids

*.

is used to control the firing angle of the phase controlled rectifier through a feedback loop. The inverter frequency is controlled by a phase-locked loop (PLL) so that the stator current, , is maintained at the desired torque angle with respect to the rotor flux.

r

sI

sI

Vector Control of Cycloconverter Drive

Vector control can also be used with a Scherbius drive with cycloconverter as shown:

Vector Control of Cycloconverter Drive (cont’d)

Recall in the Scherbius drive, +sPg is sent to the line in subsynchronous motoring and -sPg is sent to the line in supersynchronous motoring, where sPg is the slip energy. Currents Ip and IQ are the in-phase and quadrature current components, with respect to the slip voltage, Vr. The error from the speed control loop generates the desired current, Ip

*’ and IQ* may be set to zero (as shown).


The unit vector signals are obtained from the following equations:

where and ,

and is the amplitude of the line voltage.

cos cos cos cos sin sinsl e r e r e r

sin sin sin cos cos sinsl e r e r e r

cossds

es

v

V sin

sqs

es

v

V

sV


To illustrate how the drive works, consider a drive that is accelerating from a subsynchronous speed with a command supersynchronous speed.

At subsynchronous speed: IP >0, sl >0, and sPg>0.

At synchronous speed, sl=0 and IP is dc.

At supersynchronous speed: IP <0, => sl <0, and sPg<0.

topic 13: vector control of ac induction motors spring 2004 ece 8830 - electric drives

Documents

motor control

control inputs

field oriented control

precise control of ac

ds induction motor

armature flux

field current i f

dc quantities