Tài liệu High Performance Driver P2 pdf

HIGH PERFORMANCE DRIVES - 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 providing decoupled (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 the steady-state behaviour of the machine only All these methods rely on controlling the rms values of AC voltage 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 state only, while the dynamic response is generally sluggish and slow Transition from one steady-state to another 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 fast response, that fully corresponds to the one obtainable from a DC machine, is enabled by this method of speed 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 more complex structure of AC machines, compared to DC machines, application of field orientation as a practical speed control method has become possible only by microprocessors Field oriented control is nowadays applied in variety of manners in conjunction with both induction and synchronous machines (sinusoidal and trapezoidal permanent magnet synchronous machines, wound rotor synchronous machines, 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 of phase currents of stator and rotor as ø ÷ ö ( iaiC + ibi A + iciB ) + sin + ố ỗ ổ 2 ( ia iB + ibiC + ici A ) ứ ữ ố ỗ ổ ) ỵ ý ỹ ( ỵ í ì Te = − LaA sin θ ia i A + ib i B + ic iC + sin θ − (2.1) where θ denotes instantaneous position of the rotating rotor phase ‘A’ magnetic axis with respect to stationary stator phase ‘a’ magnetic axis and LaA is the peak value of the mutual inductance between stator and rotor windings of the machine This torque expression holds true in both steady-state and transient operation of the induction machine The angle θ is determined with the speed of rotation, that is θ = ωdt (2.2) ò 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 (there are no independent windings for separate flux current and torque current control, in contrast to DC machines) The question then arises: is it possible somehow to express the torque of the induction machine 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 Te = Cψ d iqs  E Levi, 2001 (2.3) HIGH PERFORMANCE DRIVES - where flux ψ may be stator flux, air gap flux or rotor flux linkage, and iqs is a certain fictitious component of the stator current If such a transformation is possible, then induction machine may be made 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 with respect 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 a separately excited DC machine It has to be noted that, as instantaneous time-domain variables are under consideration at all times and the subject of analysis is dynamic (transient) behaviour of an AC machine, it is not possible to use in analysis 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 field oriented 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 machine instantaneous change of torque requires only instantaneous change of the current amplitude, since the armature current is a DC current In the case of an AC machine requirement of instantaneous change of current is much more involved To illustrate this, consider a steady state operation of an induction machine Let us assume that the supply source is capable of providing purely sinusoidal currents of any amplitude and any frequency Instantaneous stator phase currents are then given with: ( ) i b = I sin( ω e t − ϕ − 2π / 3) i c = I sin( ω e t − ϕ − 4π / 3) i a = I sin ω e t − ϕ (2.4) Suppose that a step speed command increase takes place, that asks for instantaneous stepping of the torque The problem of stepping the torque in the machine from the appropriate steady state value to the maximum permissible value in order to achieve the fastest possible acceleration of the drive may be understood in terms of three-phase stator currents as a problem of providing new set of currents: ( ) i b1 = I sin( ω e1 t − ϕ − 2π / 3) i c1 = I sin(ω e1 t − ϕ − 4π / 3) i a1 = I sin ω e1 t − ϕ (2.5) 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 control actually explains how these parameters have to be changed in order to obtain a transient-free torque response In order to further examine behaviour of an induction machine torque response, Fig 2.1 illustrates noload acceleration from standstill, with 50 Hz, rated sinusoidal voltage supply Note that this is not the case of a variable speed drive and that there is not any control of the motor It is simply an acceleration transient with mains supply At time instant zero the motor is connected to the mains There is no load connected to the shaft The motor accelerates from standstill to the steady-state no-load speed In final steady state operation the motor torque is zero, while the speed is constant no-load speed As witnessed by the torque trace in Fig 2.1, torque developed 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 the maximum allowed torque during the transient Complex nature of an induction machine makes such a torque response rather difficult to achieve and that is why vector control is widely used  E Levi, 2001 HIGH PERFORMANCE DRIVES - In a standard induction motor drive with open loop or closed loop V/f speed control torque transient during transition from one operating speed to the other behaves similarly to the trace of Fig 2.1 Hence more dedicated control has to be used if high performance is to be achieved 40 Sinusoidal, 50 Hz supply Torque (Nm) 30 20 10 -10 0.05 0.1 0.15 0.2 Time (s) 0.25 0.3 Torque for no-load acceleration Fig 2.1 - Variation of the induction motor torque during no-load acceleration from standstill with 50 Hz sinusoidal supply On the basis of the considerations of this sub-section and discussion of high performance DC motor drives, the following statements can be made: • High performance operation requires that the electro-magnetic torque of the motor is controllable in real time; • 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 within the speed control loop) Compared to the statements given at the end of discussion of high performance DC drives, one notes that the first and the last two are the same However, the second statement replaces the second and the third 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 electric drives The imposed requirements are increased reliability, decrease in electric energy consumption, minimisation of the maintenance costs and improved capability of dealing with complicated and precise tasks required by the given technological process About 50% of the generated electric energy in developed countries is converted into mechanical energy by means of electrical drives, and about 20 %  E Levi, 2001 HIGH PERFORMANCE DRIVES 10 - of drives are variable speed drives Variable speed drives which were for an extended period of time based 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 even such value as 15 % per year in the field of servo drives The main reason for such a widespread utilisation of DC drives in the past is the capability of decoupled flux and torque control in DC machines, which asks for just a moderate investment in appropriate power electronics source It was not until the fundamentals of field oriented control were set forth, that such a decoupled control of flux and torque in AC machines became feasible Basic principles of field oriented control show that it is possible to realise theoretically perfectly decoupled control of flux and torque in AC machines The idea of field oriented control requires that instantaneous values of magnitude and position of the stator current space vector with respect to the appropriate flux space vector in the machine, in relation to which the orientation is performed, can be controlled The way of obtaining field oriented control is to orientate stator current space vector with respect to rotor flux space vector The notion of "space vector" stems from the general theory of electric 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 of space vectors, which are commonly applied in analysis and modelling of AC machines Realisation of decoupled control of flux and torque is possible with both induction and synchronous machines and they can 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 perform co-ordinate transformation, analogue versions of the field oriented control did not find wider application Development in microprocessors in the late seventies made however realisation of vector controlled induction 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 of dynamics of vector controlled induction machines in relation to classic control algorithms represents the fundamental reason for such a trend in development of controlled AC drives On the other hand, the complexity of the control system inevitably forces researchers to look for simpler control schemes which should still be able to retain dynamic behaviour comparable to vector controlled drives However, for high performance drives, where the most severe constraints are imposed on dynamics, simplified control methods can not be expected to replace field oriented control due to poorer dynamic behaviour As a conclusion to this discussion it can be stated that field oriented control remains the best available choice for the applications where decoupled control of flux and torque is an absolute "must" in order to obtain the highest possible accuracy and speed of the drive response Application areas of vector controlled induction machines in industry are numerous One of the most frequent applications is in machine tools, were usually induction machines with rated speed of 1500 rpm are 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, are capable of operating at as high speeds as 10.000 rpm, thus providing for speed range in the field weakening region of up to 1:6 The advantage of vector controlled induction machines in relation to field oriented permanent magnet synchronous machines, in the domain of machine tools, is simple provision for field weakening feature Another important area of application are servo drives where either permanent magnet synchronous machines or induction machines are used, depending on operating requirements If operation in the field weakening region is needed, induction machines are advantageous and they are used in servo drives for positioning The applications discussed so far comprise low and medium power range 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 a production 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  E Levi, 2001 HIGH PERFORMANCE DRIVES 11 - kW in 1979 A number of vector controlled induction machines with power ratings of the order 100300 kW have been installed in the period 1980-1983 in steel industry Two complete production lines were introduced in 1979 in Japan in steel rolling mills, each containing 40 vector controlled induction machines 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 in parallel in a number of different sub-areas, namely VLSI design, power electronic converters, modern control techniques and parameter variation effects and parameter identification (as will be shown later, vector control schemes require accurate knowledge of induction machine parameters) New laboratory prototypes utilise single chip for all the control functions, or are alternatively based on application specific integrated circuits Topologies of power electronic converters which ask for semiconductor switches with bi-directional current flow and bi-directional voltage blocking capabilities are gaining more and more attention recently, because it is expected that such switches will become available in the near future At this stage, instead of bi-directional switches, appropriate combinations of unidirectional switches 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 of more and more sophisticated control algorithms in vector controlled drives Different methods based on modern 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 modern control theory stems from the complexity of an AC machine as a control object, whose parameters are variable The ideal decoupled control of flux and torque can be obtained by means of standard vector control approach only if the parameters of an AC machine are exactly known and constant This is unfortunately not the case in reality The parameters of machines are subject to variation due to their dependency on operating state of the machine A discrepancy between parameter values assumed at the stage of the control system design and actual parameter values in the machine results, causing loss of decoupled torque and flux control and deterioration in quality of dynamic response 2.3 PHASE-DOMAIN MODEL TRANSFORMATION OF AN INDUCTION MACHINE AND ITS 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 instantaneous current control, it is not possible to deal with induction machine representation in terms of equivalent circuit 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 into new fictitious reference frame by suitably chosen mathematical transformation It becomes obvious even from this short discussion that the process of designing and achieving decoupled flux and torque control 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 different common assumptions and idealisations More specifically, it is assumed that stator phase windings are identical with mutual space displacement of exactly 120 degrees, that magneto-motive force of a winding is sinusoidally distributed along the air gap circumference, that air gap is constant, that rotor cage winding can be substituted with a balanced three-phase winding, that winding resistances and leakage inductances are constant parameters, eddy-currents and iron losses are neglected as well as all the parasitic capacitances, and finally, it is assumed that magnetising curve can be treated as a linear function, 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 above listed assumptions are adopted, are given with the following expressions (underlined symbols denote matrices) in terms of time domain instantaneous variables:  E Levi, 2001 HIGH PERFORMANCE DRIVES 12 v abc = Rs i abc + v ABC = Rs i ABC ψ ψ abc dψ abc dt dψ ABC + dt (2.6) = L s i abc + L sr i ABC ABC (2.7) = Lr i ABC + Ltsr i abc where lower case indices apply to stator quantities, while upper case indices denote rotor quantities, and i ABC = i A iB cosθ Lab Lac L s = Lba Lbb Lbc Lca Lcb Lcc cos θ − cos θ + 2π (2.8.a) t 2π 2π cosθ (2.8.b) L AA L AB L AC L r = LBA û ú ú ú ù è ç æ Laa cos θ − LBB LBC LCA LCB LCC ë ê ê ê é 2π cosθ ứ ữ ố ỗ ổ ố ỗ ỉ cos θ + 2π ø ÷ cos θ − cos θ + iC û ú ú ú ù t ứ ữ vC t ib ic ố ỗ æ ë ê ê ê ê ê ê ê é L sr = LaA i abc = ia ố ỗ ỉ vB t ø ÷ v ABC = v A vc ố ỗ ổ vb ỷ ỳ ỳ ỳ ø ú÷ úư ú ø ú÷ ùư v abc = va (2.8.c) ë ê ê ê é Each matrix equation in (2.6)-(2.7) is therefore an abbreviated way of writing three equations, one per phase The angle θ, as already discussed, denotes instantaneous position of magnetic axis of rotor phase ‘A’ winding with respect to stationary magnetic axis of stator phase ‘a’ winding and is correlated with rotor (electrical) speed of rotation through dependence θ = ω dt (2.2) ò The mechanical equation of motion is the same as for a DC or any other motor (a torque that describes mechanical losses is now included and the equation is given in terms of electrical speed of rotor rotation) J dω + kω P dt P Te − TL = (2.9) where ω 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 of instantaneous phase currents as ø ÷ ( iaiC + ibi A + iciB ) + sin + ố ỗ ổ 2 ( ia iB + ibiC + ici A ) ø ÷ ố ỗ ổ ) ỵ ý ỹ ( ợ í ì Te = − LaA sin θ ia i A + ib i B + ic iC + sin θ − (2.1) Equation (2.1) for electromagnetic torque is, as already discussed, significantly more complicated than the corresponding equation for a DC machine (2.3) The main reasons are that the machine under consideration is an AC, three phase machine, and, additionally, currents in rotor windings are induced current (i.e rotor and stator windings are not fed from separate supplies, as in a DC machine) Hence the ‘excitation’ current and ‘armature’ current in the case of an induction machine stem from the same supply 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 be transformed by applying an appropriate mathematical transformation The main shortcomings of this model are time dependent coefficients of differential equations (all the mutual inductances between stator and rotor phases are indirectly time dependent through dependence on rotor angular position θ),  E Levi, 2001 HIGH PERFORMANCE DRIVES 13 - and full inductance matrix with 36 non-zero inductance terms The system of differential equations that describe the machine is said to be non-linear, with time-varying coefficients There are all together seven first-order differential equations, six for the electrical sub-system (voltage equilibrium equations) and one for the mechanical sub-system (equation of rotor motion) b ω C B a θ A c 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 of appropriate mathematical transformations In general, two approaches are possible The first one utilises the model given in the preceding section as the starting point and relies on the use of matrix transformations The second approach at first defines so-called space vectors and then applies appropriate transformation without resorting to the use of matrices The approaches lead to the same final result In what follows, the matrix transformation approach is used Space vectors are defined in the following sub-section Regardless of which approach is used, the idea is to replace the physically existing machine with its three-phase stator and rotor windings with a fictitious machine whose all windings are in the common reference frame This means that stationary stator windings and rotor windings that rotate at rotor speed are all replaced with new fictitious stator and rotor windings that all have the same speed This speed of the common reference frame can be arbitrarily selected for an induction machines, due to the uniform air gap In order to transform the model of the machine from the original phase variables into new variables, it is necessary to apply appropriate transformation matrices on stator and rotor variables If the stator equations and rotor equations are transformed by means of As and Ar transformation matrices respectively,  E Levi, 2001 HIGH PERFORMANCE DRIVES 14 - − sin θ s + ø ÷ cos θ r + è ç æ è ç æ − sin θ r − 2π ø ÷ − sin θ r è ç æ è ç æ 2π cos θ r − 2π cosθ r 2π − sin θ r + 2 2π 2π û ú ú ú ø ú÷ úư ỳ ứ ỳữ ựử ố ỗ ổ ứ ữ cos s + ố ỗ ổ ố ỗ ổ ờ ờ ê ê é ë ê ê ê ê ê ê ê é − sin θ s − 2π ø ÷ − sin θ s 2π è ç æ Ar = cos θ s − û ú ú ú ø ú÷ ú ú ø ú÷ ùö As = cosθ s (2.10.a) (2.10.b) equations of an induction machine in arbitrary common reference frame result Note that the transformation 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 already pointed out, as substitution of actual phase windings with new fictitious windings These new windings are 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 the transformation may be given as follows (indices "s" and "r" denote stator and rotor variables and parameters, respectively): dψ ds − ω a ψ qs dt dψ qs vqs = Rsiqs + + ω a ψ ds dt dψ os vos = Rsios + dt (2.11) dψ dr − ( ω a − ω ) ψ qr dt dψ qr vqr = Rriqr + + ( ω a − ω ) ψ dr dt dψ or vor = Rrior + dt (2.12) vds = Rsids + vdr = Rridr + where d-q-o axis flux linkages are given with ψ ds = Lsids + Lmidr ψ qs = Lsiqs + Lmiqr (2.13) ψ os = Losios ψ dr = Lridr + Lmids ψ qr = Lriqr + Lmiqs (2.14) ψ or = Lorior for stator and for rotor equivalent windings, respectively In equations (2.13)-(2.14) the inductance terms are correlated with phase domain inductances through the following expressions Ls = Laa − Lab = Lγs + Lm Lr = LAA − LAB = Lγr + Lm Lσs = Laa + Lab Lm = ( / 2) LaA Lσr = LAA + LAB  E Levi, 2001 (2.15) HIGH PERFORMANCE DRIVES 15 - According 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 six windings can be described with only four equivalent windings The total number of voltage equilibrium equations thus reduces from six to four A graphical illustration of transformation of original phase domain windings into equivalent rotating windings in arbitrary d,q frame of reference is given in Fig 2.3 Mutual correlation between different angles defined in equations (2.10) is self-explanatory from Fig 2.3 Correlation between original phase domain and new d,q,o domain is described with (superscripts “s” and “r” refer to stator and rotor variables, respectively) s i s = A s i abc dqo s v dqo = A s v s abc ψs dqo = As ψ s ir dqo vr dqo ψr dqo = Ar ψ r = Ar i r abc = Ar v r abc abc (2.16) abc Note however that the rotor windings are not accessible and therefore phase values of rotor currents cannot be measured This means that the transformation expressions that are relevant are only those that apply to stator The same transformation applies to voltages, current and flux linkages Hence, the transformation and inverse transformation for stator variables, described with (2.16), is governed with the following algebraic equations: i ds = (2 / 3)(i a cosθ s + i b cos(θ s − 2π / 3) + i c cos(θ s − 4π / 3)) i qs = −( / 3)(i a sinθ s + i b sin(θ s − 2π / 3) + i c sin(θ s − 4π / 3)) v a = v ds cosθ s − v qs sinθ s v b = v ds cos(θ s − 2π / 3) − v qs sin(θ s − 2π / 3) v c = v ds cos(θ s − 4π / 3) − v qs sin(θ s − 4π / 3) (2.17) i a = i ds cosθ s − i qs sin θ s i b = i ds cos(θ s − 2π / 3) − i qs sin(θ s − 2π / 3) i c = i ds cos(θ s − 4π / 3) − i qs sin(θ s − 4π / 3) v ds = ( / 3)(v a cosθ s + v b cos(θ s − 2π / 3) + v c cos(θ s − 4π / 3)) v qs = − ( / 3)( v a sinθ s + v b sin(θ s − 2π / 3) + v c sin(θ s − 4π / 3)) The angles introduced in the transformation matrices and the instantaneous rotor angular position angle θ are defined and mutually correlated through the following expressions: θ = θ ( 0) + ω dt ; ò t θ s = θ s ( 0) + ωa dt ò t θr = θ s −θ ; (2.18) The equation of mechanical equilibrium remains unchanged, Te − TL = J dω + kω P dt P (2.9) while electromagnetic torque may be expressed in terms of new variables as Te =  E Levi, 2001 ( P ψ dsiqs − ψ qsids ) (2.19) HIGH PERFORMANCE DRIVES 16 - Note 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 d-axis A-axis stationary axis θ qs θr ωa a ω ω ωa θs A θs θr qr ωa 90° d dr θ a-axis rotor ds q-axis stator Fig 2.3 - Illustration of winding transformation from phase domain to arbitrary common frame of reference As already noted, zero-sequence components can be omitted from further consideration, so that four instead of six voltage equilibrium equations have to be considered The second important result of the applied transformation is that time dependence in coefficients in the model has been eliminated and the number of non-zero inductance terms has been greatly reduced, to only eight in the absence of zerosequence components Last but not least, the model in an arbitrary frame of reference enables easy derivation of vector control principles Note that the torque equation (2.19) is significantly simplified already, compared with (2.1) Note as well that the new d-q axes in Fig 2.3 are mutually perpendicular Summarising the derivation so far, the equations of the new model relevant for further considerations are the following: v ds = R s i ds + v qs = R s i qs + dψ ds dt dψ qs dt − ω a ψ qs = R r i dr + + ω a ψ ds = R r i qr + ψ ds = Lsids + Lmidr ψ qs = Lsiqs + Lmiqr Te = Te = 3 ( dt dψ qr dt − (ω a − ω )ψ qr (2.20) + (ω a − ω )ψ dr ψ dr = Lr i dr + Lm i ds ψ qr = Lr i qr + Lm i qs ) P ψ dsiqs − ψ qsids = ( dψ dr ) P P ψ dmiqs − ψ qmids = Lm Lr (ψ ( dr iqs − ψ qr ids PLm idr iqs − iqr ids ) (2.21) ) (2.22) where alternative form of the torque equation is obtained by simple algebraic manipulation, using (2.21) As a singly fed induction machine is discussed here, rotor phase voltages and hence d-q axis voltages as well are equal to zero The model is completed with transformation expressions (2.17):  E Levi, 2001 HIGH PERFORMANCE DRIVES 17 f ds = ( / 3)( f a cosθ s + f b cos(θ s − 2π / 3) + f c cos(θ s − 4π / 3)) f qs = − (2 / 3)( f a sin θ s + f b sin(θ s − 2π / 3) + f c sin(θ s − 4π / 3)) (2.23) f a = f ds cosθ s − f qs sin θ s f b = f ds cos(θ s − 2π / 3) − f qs sin(θ s − 2π / 3) f c = f ds cos(θ s − 4π / 3) − f qs sin(θ s − 4π / 3) where f stands for currents or voltages The model (2.20)-(2.23) in an arbitrary frame of reference yields corresponding model for any specified value of the common reference frame angular velocity Typical values of the angular velocity of the common reference frame fall into one of the two categories: constant ones or variable ones The constant angular velocity of the common reference frame is usually selected for simulation of the motor dynamics Typical choices are the stationary frame of reference (ωa = 0; normally selected for simulation 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 a power electronic converter on the rotor side, a variable speed reference frame firmly attached to the rotor ‘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 as firmly attached to one of the flux linkage space vectors (variable speed common reference frames) and the whole concept of vector control is based on such a selection The issue is deferred for section and only one specific reference frame is looked at in more detail next In the special case when ωa = 0, equations in stationary α,β frame of reference result: vαs = Rsiαs + dψ αs dt vβs = Rsiβs + vαr = Rsiαr + dψ αr + ωψ βr dt vβr = Rsiβr + Te = ( P ψαs i βs − ψ βsiαs ) dψ β s (2.24) dt d ψ βr dt − ωψ αr (2.25) (2.26) 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 ω a = , it follows that θ s = and θr = − θ Transformation expressions (2.23) take the form f αs = (2 / 3)( f a − 0.5 f b − 0.5 f c ) f βs = − (2 / 3)( − f b / + f c / 2) f a = f αs (2.27) f b = −0.5 f αs + f βs / f c = −0.5 f αs − f βs / By performing the mathematical transformation described above, actual stator and rotor three phase windings, which are stationary with respect to stator and rotor respectively, have all been substituted, as already mentioned, with new equivalent windings that rotate at an arbitrary speed ωa Consequently, relative motion between stator and rotor windings has been eliminated by this transformation as now both stator and rotor equivalent windings have the same angular velocity It has to be noted that both stator and rotor voltages, currents and flux linkages have been transformed, so that correlation between phase and d-q axis variables has to be accounted for in any application of the model  E Levi, 2001 HIGH PERFORMANCE DRIVES 18 - 2.3.3 Space vectors and space vector models of an induction machine Stator voltage space vector and stator current space vector are defined in stationary ωa = common frame of reference as (symbols for space vectors are underlined while superscript "s" denotes stationary reference frame): vs = s is = s 3 ( va + avb + a vc ) ( ia + aib + a 2ic ) , a=e j (2.28) 2π Each of the two defined space vectors is a simultaneous representation of all the three three-phase quantities This means that the complete three-phase system may be treated using a single quantity for any of the variables Note that space vectors, as defined in (2.28) are complex variables, and are, in contrast to phasors, functions of time Note that, in contrast to phasors, no requirement has been imposed that phase quantities must be sinusoidal Space vectors are therefore applicable to both sinusoidal and non-sinusoidal supply Note that no requirement on steady-state conditions has been imposed, as the case is with phasors Space vectors are therefore applicable to both steady-state and transient operation Since space vectors are complex numbers, they can be represented in the complex plane As complex plane is orthogonal system of axes, real and imaginary part of a space vector are essentially the components in the stationary reference frame defined in (2.27) Hence v s = vαs + jvβs = vse jβ s s (2.29) i s = iαs + jiβs = ise jε s s An illustration of stator voltage and current space vectors is given in Fig 2.4 Note that, in contrast to phasors, space vectors are not stationary in this complex plane They move as time goes by and therefore illustration in Fig 2.4 applies to only one specific instant in time Im (β axis) vβs Im (β axis) vss vss vs βs iss iβs is s Re (α axis) iαs εs Re (α axis) vαs is Fig 2.4 - Illustration of stator voltage and stator current space vectors in one specific instant in time In order to transform space vectors from stationary reference frame to a common reference frame rotating at an arbitrary speed, it is only necessary to change the argument (phase) of the space vectors by an angle of transformation θs Hence in an arbitrary reference frame stator voltage and current space vectors can be given as v s = v se − jθ s s i s = i se− jθ s s  E Levi, 2001 (2.30a) HIGH PERFORMANCE DRIVES 19 - The term e − jθ s is often called vector rotator as it rotates the space vector in stationary reference frame by an angle θs Substituting (2.29) into (2.30a) yields v s = v ds + jvqs = v s e i s = ids + jiqs = is e The scaling factor ( j β s −θ s ( j ε s −θ s ) (2.30b) ) present in equations (2.28) and in transformation matrices (2.10) is correlated with ratio of powers in original three phase domain and in the new reference frame By application of (2.29) or (2.10) per-phase power in original and new reference frame is invariant (but the total power is not, as the new reference frame is essentially two-phase) Space vectors of rotor quantities are defined in exactly the same way as for stator voltage and current From (2.30b) one observes that a complex stator voltage space vector is defined as real d-axis component plus j times imaginary q-axis component Thus the real d-q axis model given with (2.20)(2.21) can be converted into a corresponding complex model From (2.30b) and (2.20) one has = R s (i ds + ji qs ) + = Rs i s + dψ s dt − ω a ψ qs + j R s i qs + dt d (ψ ds + jψ qs ) dt d (ψ ds + jψ qs ) dt + j a ố ỗ ổ = R s (i ds + ji qs ) + dψ ds dψ qs ø ÷ v s = v ds + jv qs = R s i ds + + ω a ψ ds = dt − ω a (ψ qs − jψ ds ) = (2.31) + jω a (ψ ds + jψ qs ) = s The procedure for derivation of the other equations of the complex space vector model is identical to the one applied in (2.31) Hence the complete space vector model, that results by manipulating d and q axis equations of (2.20)-(2.21) as real and imaginary parts of complex variables can be written as: v s = Rs i s + = Rr i r + dψ dt dψ + jω a ψ s s + j (ωa − ω ) ψ r dt (2.32) r ψ = Ls i s + Lm i r s (2.33) ψ = Lr i r + Lm i s r where the remaining space vectors are, analogously to (2.30b), given with ψ = ψ se ( j φ s −θ s ) ψ = ψr e s ψ s = ψ se ( j φ r −θ s r jφ s ψ s = ψr e r s jφ r ) i r = ir e s i r = ir e ( j ε r −θ s ) jε r (2.34) Electromagnetic torque can be expressed in terms of space vectors as P Lm Lr Im i s ψ ỵ í ì Te = r } (2.35) where line above the symbol denotes complex conjugation Illustration of stator voltage and current space vectors in the reference frame defined with θs = βs is given in Fig 2.5 Note that, compared to Fig 2.4, the only thing that has been changed is the common reference frame, that is, the axes of the complex plane Since the angle of vector rotation (or matrix transformation) has been selected as equal to the stator voltage space vector instantaneous position (θs = βs), d-axis (real axis) is at all times aligned with the voltage space vector This means that the voltage space vector in this specific rotating  E Levi, 2001 HIGH PERFORMANCE DRIVES 20 - reference frame is a pure real quantity, since its imaginary part is zero As will be shown shortly, this idea of having certain quantity as a pure real number in a conveniently chosen reference frame constitutes the basis of vector control β axis Re (d axis) vs Im (q axis) βs = θs ωe iss 90° εs α axis Fig 2.5 - Stator voltage and current space vectors in the common reference frame firmly attached to the stator voltage space vector (ω a = ω e , ω e = stator voltage angular frequency) Equations (2.32)-(2.33) enable construction of an equivalent circuit, similarly as it is done with phasor equations Space vector equivalent circuit of an induction machine is shown in Fig 2.6 Note that although the appearance of the circuit resembles the steady state phasor equivalent circuit of an induction machine, the principle difference is in the variables present in the circuit The other difference of equally the same importance is the fact that the circuit shown in Fig 2.6 describes both transient and steady state operation of the machine Here the variables are space vectors which will become equal to (scaled) phasors only in steady-state operation in synchronous reference frame, provided that the supply voltages are sinusoidal (this will be shown in one of the examples shortly) It should be noted once more that the d-q axis variables are simply real (d) and imaginary (q) components of space vectors Therefore space vectors are complex variables of time and are sometimes termed instantaneous space phasors All the voltage sources shown in Fig 2.6 represent rotational electromotive forces Note that their values (except for the one in rotor circuit) are dependent on selected speed of the reference frame, ωa Rs Lγ s is vs j ω a Lγ s i s + Lγ r Rr j ω a Lγ r i r + ir Lm + im jωψr + j ω a Lm i m Fig 2.6 - Dynamic equivalent circuit of an induction machine in arbitrary frame of reference in terms of space vectors  E Levi, 2001 HIGH PERFORMANCE DRIVES 21 - Summarising this sub-section, an induction machine can be described with the complex space vector model v s = Rs i s + dψ s dt + jωa ψ = Rr i r + s ψ = Ls i s + Lm i r P Lm Lr v s = vds + jv qs = v s e dt + j (ωa − ω )ψ r r Im i s ψ ỵ í ì r ψ = Lr i r + Lm i s s Te = dψ r } ( j β s −θ s (2.36) ) i s = ids + jiqs = i s e ( j ε s −θ s ) This complex model now contains only two differential equations for voltage equilibrium Note however that the reduction from four equations in real d-q model to only two equations in complex space vector model is only apparent; each complex equation contains real and imaginary part Hence the total number of differential equations that have to be solved remains five (one for mechanical motion and four for voltage equilibrium) 2.4 MODELLING OF THE SUPPLY 2.4.1 Sinusoidal three-phase supply As already noted, an induction motor used in a high performance drive will have to be supplied from a source of variable voltage, variable frequency, such as an inverter Modelling of such a supply is considered in the following sub-section, while this one examines standard mains sinusoidal supply of fixed voltage and fixed frequency There are two reasons for this The first one is that the induction machine models, described in the previous section, were originally developed for the analysis of dynamic behaviour of the machine under sinusoidal supply conditions, rather than for the control purposes The second reason is that, as will be shown in Chapter 3, control part of the system in any vector controlled drive generates sinusoidal references for currents or voltages The fact that a power electronic converter has to be used for supply in a high performance drive is a necessity which stems from non-availability of an ideal sinusoidal voltage source generator, which could deliver purely sinusoidal voltages of any required voltage and frequency Let the induction motor be supplied from a three-phase voltage source with the following phase to neutral voltages (angular frequency of the voltages is denoted just with ω, for simplicity; note that, when the motor model is under consideration, this symbol stands for rotor speed of rotation): va = 2V cosωt vb = 2V cos(ωt − 2π / 3) (2.37) vc = 2V cos(ωt − 4π / 3) Angular frequency and the rms value of the voltage are constants in these expressions In what follows different reference frames will be looked at and the expressions for transformed stator voltage will be derived Next, corresponding space vectors will be found and the so obtained stator voltage space vectors will then be used to examine the operation of the machine by means of the dynamic equivalent circuit of Fig 2.6 under steady-state sinusoidal operating conditions A correlation between phasors and space vectors will thus be established for steady-state sinusoidal operation and it will be shown that for these specific conditions dynamic equivalent circuit reduces to well-known steady-state phasor equivalent circuit  E Levi, 2001 HIGH PERFORMANCE DRIVES 22 - Let us at first determine components of the stator voltages in the stationary reference frame From equation (2.27) on has vαs = ( / 3)(v a − 0.5vb − 0.5v c ) (2.38) v βs = − ( / 3)(− vb / + vc / 2) Substitution of phase to neutral voltages (2.37) into (2.38) enables calculation of the phase voltage components in the stationary reference frame: vαs = ( / 3)(v a − 0.5vb − 0.5v c ) = ( / 3) 2V ( cosωt − 0.5 cos( ωt − 2π / 3) − 0.5 cos( ωt − 4π / 3) ) = = (2 / 3) 2V ( cosωt − 0.5 cosωt cos 2π / − 0.5 sin ωt sin 2π / − 0.5 cosωt cos 4π / − 0.5 sin ωt sin 4π / 3) = ( ) = (2 / 3) 2V cosωt + 0.25 cosωt + 0.25 cosωt − 0.25 sin ωt + 0.25 sin ωt = = (2 / 3) 2V ( / 2) cosωt vαs = 2V cosωt ≡ v a ( )( ) v βs = − ( / 3)(− vb / + vc / ) = / vb − vc = 3V ( cos( ωt − 2π / 3) − cos(ωt − 4π / 3) ) = = 3V ( cosωt cos 2π / + sin ωt sin 2π / − cosωt cos 4π / − sin ωt sin 4π / 3) = ( = 3V sin ωt / ) v βs = 2V sin ωt Hence vαs = 2V cosωt ≡ va (2.39) v βs = 2V sin ωt Alfa component is thus identically equal to the phase ‘a’ voltage This is so since the alfa axis and the phase ‘a’ axis are aligned (i.e they coincide) Beta component is 90 degrees displaced from alfa component - this is so since alfa-beta co-ordinate system is an orthogonal co-ordinate system Note that both components have the peak value equal to the peak value of the phase voltages and that the both components are of the same frequency as phase voltages Hence these two components continuously vary in time Space vector of stator phase voltages is determined with (2.28) in terms of phase voltages, or with (2.29) in terms of alfa-beta components of the phase voltages From (2.29) v s = v αs + jv βs = v s e s jβ s v s = 2V ( cos ωt + j sin ωt ) = 2V exp( jωt ) s 2 v s = v αs + v βs = 2V (2.40) β s = ωt The space vector of stator phase voltages continuously travels in the complex plane as time goes by, since its phase is a function of time The amplitude of the space vector is constant Hence the trajectory that the space vector describes in a complex plane is a circle The space vector will make one full revolution in the complex plane for the time interval equal to the period of the phase voltages Projections of the space vector on real and imaginary axes of the complex plane are alfa-beta components; since the argument of the space vector continuously changes, its projections on the real and imaginary axes vary continuously in time as well Figure 2.7 illustrates space vector of stator phase voltages in the complex plane for a couple of characteristic instants At time instant zero space vector is aligned with the real axis and therefore alfa component equals peak voltage, while beta component is zero In time instant equal to one quarter of the period, space vector is along imaginary axis; hence beta component equals peak voltage, while real part - alfa component - is zero  E Levi, 2001 HIGH PERFORMANCE DRIVES 23 - Thus we conclude that the voltage space vector for sinusoidal supply conditions continuously travel along the circle, of radius equal to the peak value of the phase voltages The speed at which the space vector moves along the circle is equal to the angular frequency of the phase voltages, ω Im (β) ωt=90° ωt=135° ωt=45° vss ωt=0° Re (α) √2V Fig 2.7 - Space vector of stator phase voltages for sinusoidal supply conditions As already noted, space vector can be calculated directly from (2.28) using phase voltages defined in (2.37) This is illustrated next, although the result is, of course, the same expression already given with (2.40) From (2.28) one has vs s = vs = s ( va + avb + a vc ) a=e j 2π 2V ( cosωt + a cos(ωt − 2π / 3) + a cos( ωt − 4π / 3) ) The expression for the space vector is now most easily found if one recalls the well-known correlation cosδ = 0.5( exp( jδ ) + exp(− jδ )) Hence vs = s = 2V 2V (e jωt +e − jωt (e ω + e j t − jωt + ae j ( ωt − π / ) + aa *e jωt + ae + aae − j ( ωt − 2π / 3) − jωt + a 2e + a a 2* e jωt j ( ωt − π / ) + a a 2e + a 2e − jωt − j ( ωt − π / ) )= ) 1+ a + a2 = a* = a vs = s = vs = s 3 a 2* = a 2V 2V 2V v s = 2Ve s ( e ω (1 + a j t a3 = ( e ω ( 3) + e 2 j t e jωt a + aa ) + e − jωt ( 0) a4 = a − jωt (1 + a + a) ) = ) jωt The symbol * denotes complex conjugate Consider next an arbitrary common reference frame, with the speed of rotation equal to ωa The instantaneous position of the real axis (d-axis) of this reference frame  E Levi, 2001 HIGH PERFORMANCE DRIVES 24 - with respect to the stationary phase ‘a’ axis (that coincides with alfa axis) is then determined with the angle θ s = ω a dt In all the cases when speed ωa is constant, the angle simply becomes θ s = ωa t Let us ò now investigate correlation between space vectors and d-q axis components in the arbitrary reference frame and the corresponding quantities in the stationary reference frame The speed of the common reference frame will now be selected as equal to the angular frequency of the stator sinusoidal voltages, ωa = ω Hence θ s = ωt Such a reference frame is usually called synchronously rotating reference frame Space vector of stator voltages in this new reference frame can be most easily found using the concept of vector rotator, introduced in (2.30) From (2.30) and (2.40) one gets v s = vds + jv qs = v s e s β s = ωt = vse jβ s − jθ s e = vse ( j β s −θ s ) θ s = ωt v s = vds + jv qs = v s e v s = 2V = vds − jθ s j ( ωt − ωt ) (2.41) = v s = 2V v qs = In this new common reference frame d-axis is selected as aligned with the voltage space vector Real component of the space vector is therefore at all times equal to the space vector itself, while the imaginary component is zero at all times Moreover, real component and the space vector itself are pure constant DC quantities This is an important result It shows that the AC property of the space vector in stationary reference frame has been eliminated by selecting the reference frame firmly attached to the space vector itself This idea will be used later on, in derivation of vector control principles Figure 2.8 illustrates voltage space vector in the common reference frame aligned with the space vector Note that this is the same vector as the one of Fig 2.7 The only thing that has changed is the coordinate system Instead of the stationary complex plane, as in Fig 2.7, we have now a complex plane that rotates in synchronism with the space vector (d-q axes) The voltage space vector and the reference frame are both rotating now, at the same speed equal to the angular frequency of the phase voltages Moreover, d-axis of the complex plane is aligned with the voltage space vector at all times β Im (q) ωa = ω Re (d) ωa = ω vs βs=θs = ωt =45° √2V α Fig 2.8 - Voltage space vector in the synchronously rotating reference frame aligned with the vector itself, for one specific instant in time In a more general case the common reference frame can be selected as rotating at the same speed as does the space vector of stator voltages, but with certain constant angular displacement between the space vector of the stator voltages and the d-axis of the reference frame In other words, ωa = ω and θ s = ωt + θ o , θ o = const In such a case d-q axis components still remain constant DC quantities but now both of them exist (that is, q-axis component is not equal to zero) Derivation is identical to the one of (2.41):  E Levi, 2001 HIGH PERFORMANCE DRIVES 25 v s = vds + jv qs = v s e s β s = ωt − jθ s = vse jβ s − jθ s e = vse ( j β s −θ s ) θ s = ωt + θ o v s = vds + jv qs = v s e ( j ωt − ωt −θ o ( v s = 2V cosθ o − j sinθ o ) = vse − jθ o = 2Ve − jθ o (2.42) ) vds = 2V cosθ o = const vqs = − 2V sin θ o = const The reference frame and the space vector of stator voltages are illustrated in Fig 2.9 β Im (q) ωa = ω Re (d) ωa = ω vds ω θs=ωt+25°=45° vs βs = ωt =20° α √2V vqs Fig 2.9 - Voltage space vector in the synchronously rotating reference frame that is not aligned with the voltage space vector Space vector of stator voltages in ωa = ω, θ s = ωt reference frame (or indeed in any other rotating reference frame) can be found using at first transformation of stator phase voltages into d-q axis reference frame From (2.23) and (2.30) one has vds = ( / 3)( va cosθ s + vb cos(θ s − 2π / 3) + v c cos(θ s − 4π / 3)) vqs = − ( / 3)(v a sin θ s + vb sin(θ s − 2π / 3) + vc sin(θ s − 4π / 3)) v s = vds + jvqs Substituting (2.37) and using θ s = ωt one gets: vds = ( / 3) 2V (cosωt cosωt + cos(ωt − 2π / 3) cos(ωt − 2π / 3) + cos(ωt − 4π / 3) cos(ωt − 4π / 3)) vqs = − ( / 3) 2V (cosωt sin ωt + cos(ωt − 2π / 3) sin(ωt − 2π / 3) + cos(ωt − 4π / 3) sin(ωt − 4π / 3)) ( 2 2 vds = ( / 3) 2V cos ωt + 0.25 cos ωt + 0.75 sin ωt + 0.25 cos ωt + 0.75 sin ωt ( 2 vds = ( / 3) 2V 15 cos ωt + sin ωt ) ) vds = 2V vqs = − ( / 3) 2V ( 0.5 sin 2ωt + 0.5 sin (ωt − 2π / 3) + 0.5 sin 2(ωt − 4π / 3) ) vqs = v s = 2V = vds vqs = The result is of course the same as before and this derivation illustrates the possibility of obtaining the same result using alternative defining expressions  E Levi, 2001 HIGH PERFORMANCE DRIVES 26 - Only the space vector of stator voltages has been considered so far Stator phase currents will be in the case of steady-state operation with sinusoidal supply sinusoidal as well and will lag the phase voltages by an angle φ Thus ia = I cos( ωt − φ ) ib = I cos( ωt − 2π / − φ ) (2.43) ic = I cos( ωt − 4π / − φ ) The procedure of finding stator current components in stationary reference frame and the corresponding stator current space vector in stationary reference frame is identical to the one applied for voltages The result is iαs = I cos( ωt − φ ) i βs = I sin(ωt − φ ) is s (2.44) = iαs + ji βs = I ( cos( ωt − φ ) + j sin( ωt − φ ) ) = I exp[ j ( ωt − φ ) ] = I exp( jε s ) Similarly, the procedure of finding the stator current space vector and the corresponding d-q axes components in a common reference frame that rotates at a specified speed is identical to the one used for voltages In the already introduced synchronous reference frame firmly attached to the voltage space vector (ωa = ω, θ s = ωt ) one obtains i s = i ds + ji qs = Ie − jφ i ds = I cos φ (2.45) i qs = I sin( −φ ) Space vector of stator currents therefore lags the voltage space vector by angle φ (power factor angle) regardless of the reference frame Figure 2.10 illustrates mutual relationship between the voltage and the stator current space vectors in the stationary reference frame and in the synchronously rotating reference frame firmly attached to the voltage space vector β Im (β) vss ω Im (q) ω iss vs ω βs = ωt φ Re (d) βs = ωt = θs φ is α Re (α) εs = ωt − φ=βs − φ Fig 2.10 - Illustration of stator voltage space vector and stator current space vector in the stationary reference frame and in the reference frame firmly attached to and aligned with voltage space vector, for one specific instant in time Next, it is interesting to examine the dynamic equivalent circuit of Fig 2.6 for steady-state operation of an induction machine under sinusoidal conditions The circuit and the corresponding equations (2.32)(2.33) are given for the arbitrary reference frame Let us at first look at the synchronously rotating reference frame, ωa = ω, θs = ωt From the previous consideration we now know that in this specific reference frame v s = 2V i s = Ie  E Levi, 2001 − jφ (2.46) HIGH PERFORMANCE DRIVES 27 - Since in the motor model the symbol ‘ω’ stands for rotor speed of rotation, let the synchronous angular frequency (i.e frequency of the supply voltages) be denoted as ωe instead of just ω, while ω retains the same meaning as before (rotor speed of rotation) Equation (2.46) shows that both the stator voltage and the stator current space vectors are pure DC quantities, of constant magnitude In any steady-state operation time derivative of such DC quantities equals zero and the equations (2.32)-(2.33) become, by setting d/dt = 0, equal to: v s = Rs i s + jωe ψ ψ = Ls i s + Lm i r s s = Rr i r + j (ωe − ω )ψ (2.47) ψ = Lr i r + Lm i s r r Substitution of flux linkage equations into voltage equations yields ( ) ( v s = R s i s + jωe L s i s + Lm i r = R s i s + jω e Lγs i s + Lm (i s + i r ) )( ( ) )( ( ) = Rr i r + j ω e − ω Lr i r + Lm i s = Rr i r + j ω e − ω Lγr i r + Lm (i s + i r ) ) (2.48) Introducing reactances and slip ‘s’ one further gets ( v s = Rs i s + jX γs i s + jX m i s + i r = Rr i r + ωe − ω ωe [ j X γr i r + X m ( (i ) (i s + ir )] ) + i )] v s = Rs i s + jX γs i s + jX m i s + i r [ = Rr i r + sj X γr i r + X m s (2.49) r or finally ( v s = Rs i s + jX γs i s + jX m i s + i r ( ) [ ( ) = Rr / s i r + j X γr i r + X m i s + i r )] These equations describe the well-known phasor equivalent circuit of an induction machine for steadystate operation with sinusoidal supply, Fig 2.11 The only difference is that now all the variables are space vectors rather than phasors Procedure for analysing the steady-state operation of an induction motor using space vectors is therefore identical to the one used when phasors are applied In other words, given the space vector of the supply sinusoidal voltages, current space vector can be simply calculated using is = vs Z Z = R s + jX γs + ( jX m Rr / s + jX γr ) (2.50) Rr / s + jX γr + jX m Rs jXγs jXγr is vs = √2 V ir jXm Rr/s im Fig 2.11 - Equivalent circuit for steady-state operation with sinusoidal supply, synchronously rotating reference frame It has to be noted once more that this is valid only for steady-state operation with sinusoidal supply  E Levi, 2001 HIGH PERFORMANCE DRIVES 28 - It is interesting to further examine dynamic equivalent circuit for the stationary reference frame, again assuming sinusoidal supply conditions and the steady-state operation The general space vector model in stationary reference frame is directly obtainable from (2.32)-(2.33) by setting the speed of the reference frame to zero Hence v s = Rs i s + s s dψ s s s = Rr i r + dt s ψ s = Ls i s + Lm i r s dψ s r dt − jωψ s (2.51) r s ψ s = Lr i r + Lm i s s s r From the consideration of the stator voltage and current space vectors, we already know that in the stationary reference frame v s = 2V exp( jω e t ) s (2.52) i s = I exp( jω e t ) exp(− jφ ) s Thus in steady-state operation with sinusoidal supply all the quantities in the stationary reference frame are AC, of angular frequency ωe Time derivative d/dt therefore becomes in stationary reference frame equal to jωe Then it follows from (2.51) that ( ) ( s s v s = Rs i s + jωe Ls i s + Lm i r = Rs i s + jω e Lγs i s + Lm (i s + i r ) s s s s s s ( )( ) )( ( ) s s s s = Rr i r + j ωe − ω Lr i s + Lm i s = Rr i r + j ω e − ω Lγr i r + Lm (i s + i r ) s s r ) which is identical to (2.48) Hence the resulting equivalent circuit is again described with (2.49) and (2.50) remains valid The circuit is illustrated in Fig 2.12 and the only difference compared to Fig 2.11 is that all the variables now have a frequency dependent term, exp(jωet) Rs jXγs jXγr iss irs vss = √2 V exp(jωet) jXm Rr/s ims Fig 2.12 - Equivalent circuit for steady-state operation with sinusoidal supply, stationary reference frame It is important to underline once more the differences between phasors and space vectors Phasors can be used only for sinusoidal quantities and for steady-state operation In contrast to this, space vectors are applicable to non-sinusoidal supply conditions and describe both the steady-state and dynamic operation Space vectors are, in general, functions of time, in contrast to phasors For sinusoidal operation space vectors have amplitude equal to peak values (while phasors have magnitude equal to rms values) and space vectors contain a time varying exponential term in all the reference frames except for the synchronous As already pointed out, any high performance drive will be fed from a power electronic converter since an ideal sinusoidal voltage source of variable frequency and variable magnitude is not available The most frequently used supply is the so-called voltage source inverter (VSI) Operation and modelling of the VSI are therefore examined next  E Levi, 2001 ... the considerations of this sub-section and discussion of high performance DC motor drives, the following statements can be made: • High performance operation requires that the electro-magnetic... 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... response rather difficult to achieve and that is why vector control is widely used  E Levi, 2001 HIGH PERFORMANCE DRIVES - In a