The idea of field orientation, or vector control as it is called as well, can be briefly stated as a ‘control method that converts an AC machine into its DC machine equivalent from the c
Trang 12 MATHEMATICAL MODELLING OF AN INDUCTION MACHINE
AND THE SUPPLY
2.1 INTRODUCTION
As far as the AC machines are concerned, simple speed control systems are not capable of providingdecoupled (independent) flux and torque control All the so called scalar speed control methods(constant volts/hertz control, slip frequency control, voltage control etc.) are able of controlling thesteady-state behaviour of the machine only All these methods rely on controlling the rms values of ACvoltage and/or current while instantaneous torque depends on instantaneous values of currents.Therefore torque developed by the machine exactly corresponds to the commanded torque in steady stateonly, while the dynamic response is generally sluggish and slow Transition from one steady-state toanother is not controllable and follows internal dynamics of the machine The idea of field orientation,
or vector control as it is called as well, can be briefly stated as a ‘control method that converts an AC machine into its DC machine equivalent from the control point of view and thus enables instantaneous decoupled control of flux and torque’ Instantaneous decoupled flux and torque control
is made possible by control of instantaneous current values rather than rms values Extremely fastresponse, that fully corresponds to the one obtainable from a DC machine, is enabled by this method ofspeed control However, the control system capable of realising such a good quality speed control is,due to AC nature of all the variables in the machine, much more complicated Due to significantly morecomplex structure of AC machines, compared to DC machines, application of field orientation as apractical speed control method has become possible only by microprocessors Field oriented control isnowadays applied in variety of manners in conjunction with both induction and synchronous machines(sinusoidal and trapezoidal permanent magnet synchronous machines, wound rotor synchronousmachines, synchronous reluctance machines) The emphasis here is on the two most frequent types of
AC machines that are utilised in vector controlled drives, namely three phase squirrel cage (singly fed)induction machine (IM) and three phase sinusoidal permanent magnet synchronous machine (SPMSM)
As shown shortly, electro-magnetic torque of a three-phase induction motor can be expressed in terms ofphase currents of stator and rotor as
where θ denotes instantaneous position of the rotating rotor phase ‘A’ magnetic axis with respect to
stationary stator phase ‘a’ magnetic axis and L aA is the peak value of the mutual inductance betweenstator and rotor windings of the machine This torque expression holds true in both steady-state andtransient operation of the induction machine The angleθis determined with the speed of rotation, thatis
Note that rotor currents are induced in rotor windings and they are thus governed by feeding conditions
at stator side (and load) Hence both flux and torque component of the current stem from stator (thereare no independent windings for separate flux current and torque current control, in contrast to DCmachines) The question then arises: is it possible somehow to express the torque of the inductionmachine in terms of some other, fictitious currents in such a way that it resembles torque expression for
a separately excited DC machine? In other words, can the torque be somehow transformed into the form
Trang 2where flux ψ may be stator flux, air gap flux or rotor flux linkage, and i qs is a certain fictitiouscomponent of the stator current If such a transformation is possible, then induction machine may bemade to behave from the control point of view as separately excited DC machine FIELD ORIENTED CONTROL (VECTOR CONTROL) is a theory which enables achievement of the stated goal, not only withrespect to induction machines but for all the other listed types of AC machines.
Field oriented control may be therefore shortly defined as a set of control methods which, with respect
to control of the machine, enable conversion of an ac machine into an equivalent separately excited
DC machine Thus field oriented control enables decoupled (independent) control of flux and torque in
an AC machine by means of two independently controlled (fictitious) currents, as the case is in aseparately excited DC machine
It has to be noted that, as instantaneous time-domain variables are under consideration at all times andthe subject of analysis is dynamic (transient) behaviour of an AC machine, it is not possible to use inanalysis approach with phasor representation of sinusoidal quantities The variables are not sinusoidal(except in steady-state) nor are the regimes under consideration steady-states The whole theory of fieldoriented control relies on machine modelling in time-domain
Vector control requires existence of the current control, in very much the same way as it was explained
in conjunction with a separately excited DC machine However, in the case of a DC machineinstantaneous change of torque requires only instantaneous change of the current amplitude, since thearmature current is a DC current In the case of an AC machine requirement of instantaneous change ofcurrent is much more involved To illustrate this, consider a steady state operation of an inductionmachine Let us assume that the supply source is capable of providing purely sinusoidal currents of anyamplitude and any frequency Instantaneous stator phase currents are then given with:
such, that a transient-free torque response is obtained In other words, it is necessary to provide control
of stator current amplitude, frequency and phase in an appropriate manner Field-oriented controlactually explains how these parameters have to be changed in order to obtain a transient-free torqueresponse
In order to further examine behaviour of an induction machine torque response, Fig 2.1 illustrates load acceleration from standstill, with 50 Hz, rated sinusoidal voltage supply Note that this is not thecase of a variable speed drive and that there is not any control of the motor It is simply an accelerationtransient with mains supply At time instant zero the motor is connected to the mains There is no loadconnected to the shaft The motor accelerates from standstill to the steady-state no-load speed In finalsteady state operation the motor torque is zero, while the speed is constant no-load speed As witnessed
no-by the torque trace in Fig 2.1, torque developed no-by the motor is highly oscillatory during the transient
It even takes negative values in some instants, during which the speed reduces rather than increases.Recall that in a high performance drive torque response is required to be instantaneous and equal to themaximum allowed torque during the transient Complex nature of an induction machine makes such atorque response rather difficult to achieve and that is why vector control is widely used
Trang 3In a standard induction motor drive with open loop or closed loop V/f speed control torque transientduring transition from one operating speed to the other behaves similarly to the trace of Fig 2.1 Hencemore dedicated control has to be used if high performance is to be achieved.
-10010203040
• What the commutator does in a DC machine physically (i.e enables decoupled flux and torque
control), has to be done in an AC machine mathematically (theory of vector control or field oriented
control);
• Instantaneous flux and torque control require that the machine windings are fed from current
controlled AC sources;
• Current and speed sensing is necessary in order to obtain the feedback signals for real time control
(current and speed are controlled in closed loop manner, with current control loop embedded withinthe speed control loop)
Compared to the statements given at the end of discussion of high performance DC drives, one notesthat the first and the last two are the same However, the second statement replaces the second and thethird in the list for DC drives and is the subject that will be discussed shortly
2.2 HISTORY AND APPLICATIONS OF FIELD ORIENTED CONTROL
Rapid development in industry automation asks for permanent improvement of different types of electricdrives The imposed requirements are increased reliability, decrease in electric energy consumption,minimisation of the maintenance costs and improved capability of dealing with complicated and precisetasks required by the given technological process About 50% of the generated electric energy indeveloped countries is converted into mechanical energy by means of electrical drives, and about 20 %
Trang 4of drives are variable speed drives Variable speed drives which were for an extended period of timebased on standard DC machines are more and more being substituted with appropriate variable speed
AC drives The annual rate of substitution varies in different areas of applications, but attains evensuch value as 15 % per year in the field of servo drives The main reason for such a widespreadutilisation of DC drives in the past is the capability of decoupled flux and torque control in DCmachines, which asks for just a moderate investment in appropriate power electronics source It wasnot until the fundamentals of field oriented control were set forth, that such a decoupled control of fluxand torque in AC machines became feasible
Basic principles of field oriented control show that it is possible to realise theoretically perfectlydecoupled control of flux and torque in AC machines The idea of field oriented control requires thatinstantaneous values of magnitude and position of the stator current space vector with respect to theappropriate flux space vector in the machine, in relation to which the orientation is performed, can becontrolled The way of obtaining field oriented control is to orientate stator current space vector withrespect to rotor flux space vector The notion of "space vector" stems from the general theory ofelectric machines and the other popular name for field oriented control which is widely used, namely
"vector control", has its origin in the fact that field oriented control is frequently dealt with in terms ofspace vectors, which are commonly applied in analysis and modelling of AC machines Realisation ofdecoupled control of flux and torque is possible with both induction and synchronous machines and theycan be fed from a converter which is either of the voltage or current source type
The original realisations of rotor flux oriented control from early seventies employ analogue techniques.Due to the complexity of the control part of the system, which is caused mainly by necessity to performco-ordinate transformation, analogue versions of the field oriented control did not find widerapplication
Development in microprocessors in the late seventies made however realisation of vector controlledinduction motor drives both attractive and achievable During the last fifteen years, research in the area
of field oriented control has become subject of wide interest in the whole world Superiority ofdynamics of vector controlled induction machines in relation to classic control algorithms represents thefundamental reason for such a trend in development of controlled AC drives On the other hand, thecomplexity of the control system inevitably forces researchers to look for simpler control schemes whichshould still be able to retain dynamic behaviour comparable to vector controlled drives However, forhigh performance drives, where the most severe constraints are imposed on dynamics, simplified controlmethods can not be expected to replace field oriented control due to poorer dynamic behaviour As aconclusion to this discussion it can be stated that field oriented control remains the best available choicefor the applications where decoupled control of flux and torque is an absolute "must" in order to obtainthe highest possible accuracy and speed of the drive response
Application areas of vector controlled induction machines in industry are numerous One of the mostfrequent applications is in machine tools, were usually induction machines with rated speed of 1500 rpmare utilised and field weakening feature extends the range of operating speeds up to 4500-6000 rpm.Completely digital versions of field oriented controllers for machine tools, manufactured by Bosch, arecapable of operating at as high speeds as 10.000 rpm, thus providing for speed range in the fieldweakening region of up to 1:6 The advantage of vector controlled induction machines in relation tofield oriented permanent magnet synchronous machines, in the domain of machine tools, is simpleprovision for field weakening feature Another important area of application are servo drives whereeither permanent magnet synchronous machines or induction machines are used, depending on operatingrequirements
If operation in the field weakening region is needed, induction machines are advantageous and they areused in servo drives for positioning The applications discussed so far comprise low and medium powerrange Induction motor drives with field oriented control are however used in high power range as well.This type of application was initiated in Japan in late seventies The complete automation of aproduction line in paper industry, where requirement on speed control accuracy is 0.02% and speed has
to be varied in the range 180:1, is performed by five induction machines with power ratings 340-500
Trang 5kW in 1979 A number of vector controlled induction machines with power ratings of the order
100-300 kW have been installed in the period 1980-1983 in steel industry Two complete production lineswere introduced in 1979 in Japan in steel rolling mills, each containing 40 vector controlled inductionmachines in the power range 5.5-11 kW
The research in the area of field oriented control, due to the complexity of the overall system, runs inparallel in a number of different sub-areas, namely VLSI design, power electronic converters, moderncontrol techniques and parameter variation effects and parameter identification (as will be shown later,vector control schemes require accurate knowledge of induction machine parameters) New laboratoryprototypes utilise single chip for all the control functions, or are alternatively based on applicationspecific integrated circuits Topologies of power electronic converters which ask for semiconductorswitches with bi-directional current flow and bi-directional voltage blocking capabilities are gainingmore and more attention recently, because it is expected that such switches will become available in thenear future At this stage, instead of bi-directional switches, appropriate combinations of unidirectionalswitches are utilised for experimental purposes As the bi-directional switches are still not available,converter topology with resonant DC link at the moment seems to be more prosperous solution
Development of high-speed low-cost microprocessors and signal processors enables implementations ofmore and more sophisticated control algorithms in vector controlled drives Different methods based onmodern control theory are being proposed with ultimate goal to further improve the drive performance.Among the large variety of the methods, the most important seem to be application of state observers,model reference adaptive control and state-space controllers The need for application of moderncontrol theory stems from the complexity of an AC machine as a control object, whose parameters arevariable The ideal decoupled control of flux and torque can be obtained by means of standard vectorcontrol approach only if the parameters of an AC machine are exactly known and constant This isunfortunately not the case in reality The parameters of machines are subject to variation due to theirdependency on operating state of the machine A discrepancy between parameter values assumed at thestage of the control system design and actual parameter values in the machine results, causing loss ofdecoupled torque and flux control and deterioration in quality of dynamic response
TRANSFORMATION
2.3.1 Model of the machine in terms of physical phase variables
As the field oriented control asks for instantaneous control of machine flux and torque via instantaneouscurrent control, it is not possible to deal with induction machine representation in terms of equivalentcircuit and phasors Instead, time domain mathematical model in the original phase reference frame has
to be utilised as a starting point Furthermore, this model has to be mathematically transformed intonew fictitious reference frame by suitably chosen mathematical transformation It becomes obviouseven from this short discussion that the process of designing and achieving decoupled flux and torquecontrol in an induction machine is much more tedious than with DC machines
The procedure of mathematical modelling of an induction machine is subject to a number of differentcommon assumptions and idealisations More specifically, it is assumed that stator phase windings areidentical with mutual space displacement of exactly 120 degrees, that magneto-motive force of awinding is sinusoidally distributed along the air gap circumference, that air gap is constant, that rotorcage winding can be substituted with a balanced three-phase winding, that winding resistances andleakage inductances are constant parameters, eddy-currents and iron losses are neglected as well as allthe parasitic capacitances, and finally, it is assumed that magnetising curve can be treated as a linearfunction, i.e that the main flux saturation can be neglected
Voltage equilibrium equations of a three-phase induction machine in original phase domain, if the abovelisted assumptions are adopted, are given with the following expressions (underlined symbols denotematrices) in terms of time domain instantaneous variables:
Trang 6v R i d
dt
v R i d
dt
abc s abc abc
ABC s ABC ABC
ç öø÷ æèç − öø÷
−
æ è
+
æ è
ù
û
ú ú ú ú ú ú ú
23
232
3
232
3
23
ù
û
ú ú
whereω is once more electrical speed of rotation of rotor, mechanical power is taken as positive when it
leaves the machine (for motoring) and electromagnetic torque can be expressed in terms ofinstantaneous phase currents as
Schematic representation of a three-phase induction machine in original phase domain is given in Fig.2.2 Model described with (2.1), (2.9) is very inconvenient for any type of analysis and it has to betransformed by applying an appropriate mathematical transformation The main shortcomings of thismodel are time dependent coefficients of differential equations (all the mutual inductances betweenstator and rotor phases are indirectly time dependent through dependence on rotor angular positionθ),
Trang 7and full inductance matrix with 36 non-zero inductance terms The system of differential equations thatdescribe the machine is said to be non-linear, with time-varying coefficients There are all together sevenfirst-order differential equations, six for the electrical sub-system (voltage equilibrium equations) andone for the mechanical sub-system (equation of rotor motion).
Fig 2.2 - Schematic representation of a three-phase induction machine: all the windings are placed
on magnetic axes (windings are illustrated for phases a and A); rotor windings A,B,C rotate with
rotor, while stator windings a,b,c are stationary.
2.3.2 Transformation of the model into a common reference frame, rotating at an arbitrary angular speed
Mathematical model of an induction motor, expressed in terms of phase variables and parameters, can
be transformed into a corresponding model in the so-called common reference frame, by means ofappropriate mathematical transformations In general, two approaches are possible The first oneutilises the model given in the preceding section as the starting point and relies on the use of matrixtransformations The second approach at first defines so-called space vectors and then appliesappropriate transformation without resorting to the use of matrices The approaches lead to the samefinal result In what follows, the matrix transformation approach is used Space vectors are defined inthe following sub-section
Regardless of which approach is used, the idea is to replace the physically existing machine with itsthree-phase stator and rotor windings with a fictitious machine whose all windings are in the commonreference frame This means that stationary stator windings and rotor windings that rotate at rotor speedare all replaced with new fictitious stator and rotor windings that all have the same speed This speed ofthe common reference frame can be arbitrarily selected for an induction machines, due to the uniformair gap
In order to transform the model of the machine from the original phase variables into new variables, it isnecessary to apply appropriate transformation matrices on stator and rotor variables If the statorequations and rotor equations are transformed by means of As and Ar transformation matricesrespectively,
Trang 83
23
232
3
231
2
12
12
2
3
23
232
3
231
2
12
12
equations of an induction machine in arbitrary common reference frame result Note that thetransformation matrices for stator and rotor windings differ in the sense that different angles are met in
sin and cos terms in these two matrices The procedure of transforming equations of an induction
machine from original phase domain into so called arbitrary reference frame may be viewed, as alreadypointed out, as substitution of actual phase windings with new fictitious windings These new windingsare all, in general, rotating; it is important to realise that both original stator (stationary) and rotor(rotating) windings are substituted with new windings that have the same arbitrary speed of rotation(hence the name "common reference frame") The model obtained after the application of thetransformation may be given as follows (indices "s" and "r" denote stator and rotor variables andparameters, respectively):
Trang 9According to (2.11)-(2.12) each set of three-phase windings is substituted with a new set of three
windings These are labelled d,q and o It turns out however that if the machine is star connected without connected neutral, or if the machine is fed from a symmetrical three-phase source, the o components cannot exist It is for this reason that corresponding o fictitious windings are completely
omitted from further considerations
One obvious benefit of being able to omit a pair of windings is that from now on the machine with sixwindings can be described with only four equivalent windings The total number of voltage equilibriumequations thus reduces from six to four
A graphical illustration of transformation of original phase domain windings into equivalent rotatingwindings in arbitrary d,q frame of reference is given in Fig 2.3 Mutual correlation between differentangles defined in equations (2.10) is self-explanatory from Fig 2.3 Correlation between original phasedomain and new d,q,o domain is described with (superscripts “s” and “r” refer to stator and rotorvariables, respectively)
s abc s
dqo
s
s abc s
r abc r
dqo
r
r abc r
The angles introduced in the transformation matrices and the instantaneous rotor angular position angle
θ are defined and mutually correlated through the following expressions:
θr =θ θs− ; θ θ= +òωdt ; θs=θs +òωa dt
0 t
T e= 3P ds qs i − qs ds i
Trang 10Note that the transformation expressions (2.17) are always applied in one direction for voltages and in
the opposite direction for currents (say a,b,c to d,q for voltages and d,q to a,b,c for currents, or the
other way round)
q
stationary axis a
v R i d
dt
v R i
d dt
ω ψ
00
2
32
Trang 11where f stands for currents or voltages.
The model (2.20)-(2.23) in an arbitrary frame of reference yields corresponding model for any specifiedvalue of the common reference frame angular velocity Typical values of the angular velocity of thecommon reference frame fall into one of the two categories: constant ones or variable ones The constantangular velocity of the common reference frame is usually selected for simulation of the motordynamics Typical choices are the stationary frame of reference (ωa = 0; normally selected forsimulation of a mains fed induction motor) and synchronous reference frame (ωa= 2π50; often selected
for analysis of an inverter fed induction machine) If the machine is a wound rotor one and there is apower electronic converter on the rotor side, a variable speed reference frame firmly attached to therotor ‘A’ axis (ωa = ω) is usually selected in simulations However, from the point of view of the
control, quite different choices are made As discussed shortly, common reference frame is selected asfirmly attached to one of the flux linkage space vectors (variable speed common reference frames) andthe whole concept of vector control is based on such a selection The issue is deferred for section 3 andonly one specific reference frame is looked at in more detail next In the special case when ωa = 0,equations in stationaryα,β frame of reference result:
d dt
r r
The equations (2.21) remain valid, provided that appropriate index substitution is done, d→α and
q→β, and provided that it is recognised that due to ωa = 0, it follows that θs= 0 and θr = −θ.Transformation expressions (2.23) take the form