CHAPTER 2 MODELLING OF INTERIOR PERMANENT MAGNET SYNCHRONOUS MOTOR

21 CHAPTER 2 MODELLING OF INTERIOR PERMANENT MAGNET SYNCHRONOUS MOTOR 2.1 INTRODUCTION The need for adjustable speed drives in industrial applications has been increasing progressively. The variable speed drives loads are ranging from pumps and fans to complex drives in paper machines, rolling machines and cranes. Hence, more demanding control schemes are required to achieve speed and torque control. It is now recognized that the control strategies for PMSMs are FOC and Direct Torque Control (DTC).Both the control strategies have their own advantages and disadvantages and they are implemented successfully in the industrial applications. This chapter presents the design of dynamic mathematical model of inverter fed IPMSM drive. The various blocks involved in this drive system, the principle and performance of FOC and DTC techniques for PMSM drive are also discussed. 2.2 PERMANENT MAGNET SYNCHRONOUS MOTOR DRIVE SYSTEM The basiccomponents of the IPMSM drive area speed controller, PWM current controller, position sensor, the motor and an IGBT based voltage source inverter. The reference current for the inner current control loop is generated by the outer speed loop. The modules are connected as shown in Figure. 2.1. The speed of the motor is compared with its reference

22 value and the speed error is processed in the speed controller. The output of this controller is considered as the reference torque.a limit is put on the speed controller output depending on the permissible maximum winding currents. The reference current generator block generates the three phase reference currents (ia*, ib*& ic*) using the limited peak current magnitude as decided by the controller and the position sensor. Figure2.1 Schematic Block Diagram for Drive System 2.3 MODELLING OF IPMSM Detailed modelling of IPMSM system is required for proper simulation of the system. A mathematical model for the vector control of the PMSM can be derived from its equivalent circuit. The synchronously rotating rotor reference frame is chosen, to enable the transformation of winding quantities to the synchronously rotating reference frame that is revolves at rotor speed. This is represented in Figure 2.2.

23 Rotor q-axis Iq Stator mmf a C b d q r Id b c a Figure 2.2 IPM machine Synchronously Rotating d-q reference frame The rotor reference frame is preferred as the position of rotor magnets determines the stator voltages and currents, the instantaneously induced EMF and subsequently the stator currents and torque of the machine. The consequences are that the speed difference between the stator and rotor magnetic fields is zero and the stator q and d axis winding currents have a fixed phase relationship with the rotor magnet axis. The model of PMSM has been developed with respect to rotor reference frame using the following assumptions: 1) Saturation is neglected. 2) The induced EMF is sinusoidal. 3) Core losses are negligible. 4) There are no field current dynamics. 5) Damper windings effects are not present in IPMSM.

24 An IPMSM is described by the following set of general equations. Direct and quadrature axis voltage equations are given by Equations(2.1) and (2.2) Vd d d Rsid rq (2.1) dt Vq dq Rsiq rd (2.2) dt Flux linkages can be expressed as given in Equations(2.3) and (2.4) q Lqiq (2.3) (2.4) d Ldid f Substituting Equations(2.3) & (2.4) into (2.1) & (2.2),the following equations are obtained. Vq V d d R s i q r L d i d f L q i q (2.5) dt d dt Rsid rlqiq Ldid f (2.6) Equations (2.5) and (2.6) in State space form is represented by dlq V q Rs dt Vd rlq rld Rs dld dt r i f q d f id dt (2.7)

25 Equation(2.8) gives the electrical torque developed by the motor Te 3 P diq 2 2 qid (2.8) Substituting Equations(2.3) and (2.4) inequation(2.8),it becomes T e 3 P f i 4 q L d L q i q i d (2.9) The torque balance equation for the motor is given in Equation (2.10) Te d B J m TL m (2.10) dt Solving for rotor mechanical speed from (2.10),it is defined Te TL Bm m dt (2.11) J and rotor electrical speed is P r m (2.12) 2 where vq and vd - the q,d axis voltages Lq and Ld - q, d axisselfinductances iq and id - q, d axis stator currents R s - stator resistance per phase λ f - the constant flux linkage due to rotor permanent magnet λ d - flux linkage in d axis

26 λ q - flux linkage in q axis ω r - the angular rotor speed P - the number of pole pairs of motor T L - the load torque T e - electrical torque B - the rotor damping coefficient J - the inertia constant θ - position of the rotor The mathematical Simulink model for IPMSM and model with coordinate transformation control unit are shown in Figures 2.3 and 2.4, respectively. Figure 2.3 IPMSM Mathematical model

27 Figure 2.4 Matlab/Simulink Model for IPMSM with coordinate transformation control unit 2.3.1 Equivalent Circuit of IPMSM For the purpose of analysis, equivalent circuits of the motors are used for study and simulation of motors. From the d-q modellingof the motor using the stator voltage equations, the equivalent circuit of the motor can be derived. Assuming rotor d axis flux from the PMs, itis represented by a constant current source as described by Equation(2.13),we get (2.13) where λ f is the constant flux linkage due to rotor permanent magnet L dm is mutual inductance and i f is PM field current The equivalent circuit and the corresponding mathematical Simulink model of quadrature axis current are shown in Figures 2.5 and 2.6 respectively. Similarly, direct axis parameters are presented in Figure 2.7 and Figure 2.8.

28 The equivalent circuits are: 1. Dynamic stator q-axis equivalent circuit 2. Dynamic stator d-axis equivalent circuit R s + - ω r λ d L Is =L q -L m V qs L qm Figure2.5 Stator q-axis equivalent circuit Figure 2.6Matlab/Simulink model for quadrature axis electricalcurrent R s + - ω r λ q L Is =L d -L m V qs L dm I f Figure 2.7 Stator d-axis equivalent circuit

29 Figure2.8 Matlab/Simulink model for direct axis electrical current 2.3.2 Park Transformation Park(1920) has presented a new approach to electric machine analysis. Stator and rotor rotating variables can be realized as constants with reference to a fixed frame on the rotor. Park s transformation converts vectors in a balanced two-phase orthogonal stationary system into an orthogonal rotary reference frame. It is a familiar three-phase to two-phase transformation in electrical machines study. The Park transformation is representedin Figure 2.9. The dq0 variables are converted into ABC dq0 parameters by Equation(2.14) and in reverse by Equation (2.15) (Andresen et al1994). q f ABC fdq0 T (2.14) q 1 fdq0 fabc T (2.15) where f can be either voltage or current of the stator parameters and the transformation matrix T is represented by Equation (2.16) 2 2 cos( qm) cos( qm ) cos( qm ) 3 3 2 2 2 T ( q ) sin( qm) sin( qm ) sin( qm ) (2.16) 3 3 3 1 1 1 2 2 2

30 With reference to the rotor frame, the time-varying inductance is presented in the voltage equation due to rotor spinning is thus eliminated. Figure 2.9 Park Transformation ABC to dq By applying Park s transformation, the stator voltage can be transformed to rotor reference frame. Transformation can be expressed as in Equations(2.17) and (2.18). Vqr = Vqs cos θ Vds sin θ (2.17) Vdr = Vds cos θ + Vqs sin θ (2.18) where, Vqs, Vds - Vqr, Vdr - q- and d axis voltages of the stationary system q- and d axis voltages of rotating reference frame The simulation diagram shown in Figure 2.10 explains the implementation of Park transformation.

31 Figure 2.10 Simulation Diagram of Co-ordinate Transformation 2.3.3 Vector Rotator The difficulty involved in machine calculations can be reduced by transforming the machine parameters. Generally,in three phase motor models, the park transformations are employed. It transforms three phase quantities into two phase quantities.inverse Park transformation, i.e., dq0_to_ abc transforms the two-phase variables denoted as d and q axis into three quantities (direct axis, quadratic axis and zero-sequence components).depending upon the rotor position, the motor reference currents (ia, ib and ic) are generated according to the Equations (2.19) to (2.21). Ia* = Iq cos θ + Id sin θ (2.19) Ib* = Iq cos (θ-120) + Id sin (θ-120) (2.20) Ic* = Iq cos (θ+120) + Id sin (θ+120) (2.21)

32 where, Ia, Ib and Ic - stator currents θ - position of the rotor The vector rotator is modelled based on the above three equations as shown in Figure 2.11.Stator three phase currents are calculated using the Simulink block of vector rotator. Figure 2.11Simulation diagram of Vector rotator 2.3.4 Current Transformation The d-and q- axis command currents are obtained by using the motor voltage equations as given in Equations (2.1) and(2.2).the simulation diagram of the current transformation is explained in Figure 2.12.

33 Figure 2.12 Simulation diagram of Current Transformation 2.4 CURRENT CONTROLLERS IPMSMs are fed by Voltage Source Inverters(VSI). Appropriate inverter and control technique have to be unified for the desired response of the motor. DC voltage is convertedto AC voltage by the inverter. Gate signal for the inverters is generated by the current controllers. Hysteresis and PWM are the two different types of current control techniques for inverters.pwm current controller is used in this research. 2.4.1 Pulse Width Modulation Current Controller PWM controllers are based on the principle of comparing a triangular carrier wave with the error of the controlled signal. If the triangle waveform is greater than the error, the rectangular wave output is in the low state, else in the high state. Required PWM signal is generated and shown in Figure2.13.

34 Figure 2.13 PWM Current Controller 2.5 POSITION SENSOR The operation of PMSM requires the knowledge of the position of the rotor,when operated without damper winding.there are four main devices for the measurement of position and are the potentiometer, linear variable differential transformer, optical encoder and resolvers. A position sensor with the required accuracy can be selected on the basis of application. 2.5.1 Optical Encoder Optical encoder is the most commonly used encoder as shown in Figure 2.14. It consists of a rotating disk, a light source and a photo detector.the disk,whichis mounted on the rotary shaft, has coded patterns of opaque and transparent sectors. As the disk rotates, these patterns interrupt the light emitted onto the photo detector by generating a digital pulse or output signal. Figure 2.14 Optical Encoder

35 Optical encoders offer the advantages of digital interface. There are two types of optical encoders, these are incremental encoder and absolute encoder. Incremental encoders have excellent precision and are simple to implement. The absolute encoder captures the exact position of a rotor with a precision directly related to the number of bits of the encoder. 2.6 CONTROL METHODS Synchronous motors used in Variable Frequency Drive (VFD)can run at different speeds. Control methods of IPMSM depend on what quantities they control. The scalar control algorithm controls only magnitudes, whereas both magnitude and angles are controlled in the vector control. Vector control is the method preferred for controlling a IPMSM. The current vector is controlled by FOC and the torque fluctuation is controlled by DTC method. An overview of different control methods can be seen in Figure 2.15. Figure 2.15 Overview of Control Strategies of AC drives

36 2.6.1 Scalar Control It is a simple control technique, in which the association between voltage or current and frequency are kept constant through the entire speed range of the motor. The frequency is set according to the required synchronous speed and the magnitude of the voltage/current is adjusted to maintain the ratio between them as constant. The name scalar control states absence of control over the angle. The rotor position is not considered and acts like an open loop, Hence,it is easy to implement. System instability results when the applied frequency exceeds a certain limit. To overcome this, synchronization of the rotor to the electrical frequency is to be assured by constructing the rotor with damper windings (Chandana Perera et al2003). 2.6.2 Vector Control A higher dynamic performance of the drive system is achieved with the control of both magnitude and the angleof the flux. FOC and DTC are the two different types of techniques exist in vector control. 2.6.2.1 Field oriented control (FOC) FOC was developed prominently in the1980s to meet the challenges of transient condition analysis and oscillating the flux with torque responses in inverter fed induction and synchronous motor drives. 2.6.2.2 Derivation of vector control IPMSM drive Vector control separates the torque and flux in the machine through its stator excitation inputs. Vector control of the three-phase PMSM is derived from its dynamic model. Considering the currents as inputs, the three-phase currents are represented as per Equations from (2.22) to (2.24).

37 i a is sinr t d (2.22) 2 ib is sinrt d (2.23) 3 2 ic is sinrt d (2.24) 3 where δ is the angle between the rotor field and stator current phasors. Using Park s transformation, the stator currents are referred to the rotor reference frame with the rotor speed ω r. The q and d axis currents are constants in the rotor reference frames, since δ is constant for a given load torque. As these constants are similar to the armature and field currents in the separately excited dc machine,the q axis current is distinctly equivalent to the armature current of the DC machine; the d axis current is field current, but not in its entirety. It is only a partial field current.the other part is contributed by the equivalent current source representing the permanent magnet field. For this reason, the q axis current is referred as the torque producing component of the stator current and the d axis current is referred to as the flux producing component of the stator current. Using park s transformation, this stator current should be transformed to rotor reference frame as per Equation (2.25). iq cos qr 2 id sin qr 3 i0 1/ 2 cos sin qr 120 cosqr 120 q 120 sinq 120 r 1/ 2 r 1/ 2 ia ib ic (2.25) Substituting the Equations (2.22), (2.23) and (2.24) in (2.25) and solving, the quadrature axis and direct axis currents can be calculated in terms of stator current i s which is represented in Equation(2.26).

38 i i q d i s sin d cos d (2.26) Using the Equations (2.9) and (2.26), the electromagnetic torque is obtained as per Equation (2.27) L L i 2 sin 2d i sin d 3 P 1 Te 2 2 d q s f s (2.27) 2 In order to achieve the behaviour like DC motor,the control needs knowledge of the position of the instantaneous rotor flux or rotor position of PM motor. Determination of the three phasecurrent is possible through knowledge of the position of the rotor. desired. Its calculation using the current matrix depends on the control a. Constant Torque Operation. b. Flux weakening Operation. These options are basedon the physical limitation of the motor and the inverter. The limit is established by the rated speed of the motor.figure 2.16 shows at which speed, the constant torque operation finishes and flux weakening starts.

39 2.6.3 Constant Torque Operation In this control strategy, the d-axis current is kept zero, while the vector current is aligned with the q-axis in order to maintain the torque angle equal with 90ᵒ. The torque Equation (2.27) becomes Te 3 P fiq 2 2 (2.28) Equation(2.28) and it can be rearranged as Equation (2.29) 3 P Te ktiq where, kt f (2.29) 2 2 T e Constant Torque I m Flux-Weakening ω rated ω m Figure 2.16 Torque and Power Curve in wide speed range A point to note is that the torque Equation(2.28)resembles that of the DC machine where the torque depends only on quadrature axis current, when the field flux is considered as constant and hence, its equivalent operation is provided.

40 2.6.4 Flux Weakening Operation The speed of an IPMSM going beyond base speed ω b can be controlled through field weakening control. However, the field weakening speed range is small due to a weak armature reaction effect. Appropriate control techniques are necessary for improving the performance of the machine. Figures2.17 and 2.18 show the phasor diagram of FW control and torque-speed curve, respectively. As the stator voltage tends to saturate at the edge of the constant torque region and V s =ω e λ s, the stator flux λ s must be weakened beyond the base speed ω b. q e I ds X s V f ' =ω e λ f V s =ω e λ s V s ' =ω e λ s ' î s -i ds δ a i qs β α θ λ s λ s ' δ A' λ a A λ a λ f d e e -i qs d s Figure 2.17 Phasor diagram of flux weakening control

41 Hence, the stator current control remains possible. This means that a demagnetizing current i ds must be injected into the stator side. In view of low armature reaction flux ψ a, this demagnetizing demands largeids, within the specified machine stator current rating. Therefore, it appears that the weakening of λ s is small by giving a small range of field weakening speed control. It is also evident that with constant λ f, V f increases proportionally with ω e and the overexcited machine gives a leading power factor at the machine terminal. V s T e I s λ r L s ' > L s L s L s ' Speed (ω r ) ω b ω r1 Figure 2.18 Torque Speed curve of flux weakening control above base speed ω b In FW region with rated Î s =i qs, Î s can now be rotated anticlockwise in the a - a locus so that the stator current Î s and d-axis current id are calculated as per Equations (2.30) and (2.31) Î s = 2 2 iqs i ds (2.30) id ={ ψ/[2(lq-ld)]- [ψ2/[2(lq-ld)2]+ iq2]} (2.31) In other words, ids helps to weaken the stator flux. With a constant magnitude of λ a, the λ s Phasor is reduced and rotated in the locus A - A. At point A, Î s = -i ds, which corresponds to zero developed torque at speed ω ri. The machine power factor is zero leading to this orientation of

42 voltages V s and V f. Obviously, within the rated stator current, the field weakening range can be increased, if the synchronous inductance L s is increased to L s. Constant torque mode i ds =0, but in flux weakening mode, flux is controlled inversely with speed, with ids control generated by the flux. 2.7 SUMMARY In this chapter, a mathematical model of PMSM has been derived. By using the Park s transformation, all time-varying inductances in the voltage equations are eliminated and in turn, the models are simplified and vector control algorithms are implemented. Dynamic stator d and q-axis equivalent circuit of motor are derived using stator voltage equations.finally, Constant-torque operation and FW control are derivedfor an IPMSM drive system. Implementation of PI and AWPI controllers for IPMSM drive are discussed in chapter3.