CONTROL OF THE DOUBLY SALIENT PERMANENT MAGNET SWITCHED RELUCTANCE MOTOR. David Bruce Merrifield. Masters of Science In Electrical Engineering

CONTROL OF THE DOUBLY SALIENT PERMANENT MAGNET SWITCHED RELUCTANCE MOTOR David Bruce Merrifield Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Masters of Science In Electrical Engineering Dr. Krishnan Ramu, Chair Dr. Douglas Lindner Dr. William Baumann May 4, 21 Blacksburg, Virginia Keywords: Permanent Magnet Switched Reluctance Motor, PMSRM, SRM, Firing Angle Selection Efficiency Based Control, Current Control, Speed Control, Average Torque Control

CONTROL OF THE DOUBLY SALIENT PERMANENT MAGNET SWITCHED RELUCTANCE MOTOR David Bruce Merrifield ABSTRACT The permanent magnet switched reluctance motor (PMSRM) is hybrid dc motor which has the potential to be more effect than the switched reluctance (SRM) and permanent magnet (PM) motors. The PMSRM has a both a salient rotor and stator with permanent magnets placed directly onto the face of common pole stators. The PMSRM is wound like the SRM and can be controlled by the same family of converters. The addition of permanent magnets creates nonlinearities in both the governing electrical and mechanical equations which differentiate the PMSRM from all other classes of electric motors. The primary goal of this thesis is to develop a cohesive and comprehensive control strategy for the PMSRM so as to demonstrate its operation and highlight its efficiency. The control of the PMSRM starts with understanding its region of operation and the underlying torque production of the motor. The selection of operating region is followed by a both linear and nonlinear electrical modeling of the motor and the design of current controllers for the PMSRM. The electromechanical model of the motor is dynamically simulated with the addition of a closed loop speed controller. The speed controller is extended to add an efficiency searching algorithm which finds the operating condition with the highest efficiency online.

Acknowledgements Firstly, I would like to thank my advisor Dr. Krishnan Ramu for his continued support and guidance. He has introduced me to the field of electric motors and drives and has allowed me to work on controls problems which are both challenging and exciting. I would also like to thank Nimal Lobo who designed the PMSRM which this thesis is based upon, and whose assistance has been not only appreciated but also essential to my research. Thanks to Ramu Inc. for their funding and support of my research. This includes Dr. Gray Roberson and Ethan Swint who have helped me immensely over the past year on both this research plus much more. Finally, I would like to thank my family for their unconditional support in all aspects of my life. iii

Table of Contents List of Figures......v List of Tables..vi 1 Introduction..1 1.1 Introduction to the SRM and PMSRM..1 1.2 Thesis Proposal and Contributions...4 2 SRM and PMSRM Background...6 2.1 Operation of the SRM...6 2.2 Operation of the PMSRM 8 2.3 Converter Topologies for the SRM and PMSRM 11 2.3.1 The Asymmetric Converter...11 2.3.2 One Switch per Phase Converters.12 2.3.3 Pulse Width Modulation... 13 2.4 Control of the SRM..15 3 Control Principle for the PMSRM..17 3.1 Startup..17 3.2 Region of Operation.18 3.2.1 Effect of the Advance Angle.2 3.2.2 Effect of the Dwell Angle.21 3.2.3 Effect of Speed on the Firing Angles 23 3.2.4 Selection of Firing Angles....26 3.2.5 Sensitivity Analysis..31 3.3 Control Overview...33 4 Current Control Design and Simulation.37 4.1 Hysteresis Current Control..37 4.2 PI Current Control...39 4.2.1 Linearization of the PMSRM Current Model...4 4.2.2 PI Control Design.43 4.2.3 Anti-windup PI Control 46 iv

4.2.4 PI Controller Simulation...48 4.3 Adaptive Current Control 5 4.3.1 PMSRM System Model with Structured Non-Linearites.51 4.3.2 MRAC Current Control 52 4.3.3 Adaptive Current Control Simulation...54 5 Speed Control Design and Simulation 6 5.1 Speed Loop Linearization 6 5.2 Design of a Speed Feedback Filter..63 5.3 PI Speed Controller..65 5.4 PI Torque Controller....7 5.5 Efficiency Searching Algorithm.. 71 5.6 Comparison of Speed Control Designs....77 6 Conclusions....79 6.1 Summary......79 6.2 Future Research...8 References. 81 Appendix A: 8/1 4ecore PMSRM Specifications.....83 Appendix B: Deadbeat Current Controller Design.......84 v

List of Figures Figure 1.1: Two-phase 8-1 4ecore SRM 2 Figure 1.2: Two-phase 8-1 PMSRM..3 Figure 2.1: Flux Path of a 4ecore SRM...6 Figure 2.2: Operation of the SRM...7 Figure 2.3: Flux Path of the PMSRM..8 Figure 2.4: Operation of the PMSRM. 9 Figure 2.5: The Asymmetric Bridge Converter.12 Figure 2.6: The Split-dc Converter 13 Figure 2.7: PWM Chopping of the SRM...14 Figure 3.1: PMSRM Cogging Torque..18 Figure 3.2: Torque Profile of 4ecore PMSRM...19 Figure 3.3: Simulated Current and Torque with Selected Advance Angles. 2 Figure 3.4: Simulated Average Torque as a Function of Advance Angle and Reference Current.21 Figure 3.5: Simulated Current and Torque with Selected Dwell Angles..22 Figure 3.6: Simulated Average Torque as a Function of Dwell Angle and Reference Current...23 Figure 3.7: Simulated Current and Torque for Varying Speeds... 24 Figure 3.8: Simulated Average Torque as a Function of Dwell Angle and Speed 24 Figure 3.9: Simulated Average Torque as a Function of Advance Angle and Speed...25 Figure 3.1: Simulated Average Torque with Set Speed and Current Command 26 Figure 3.11: Maximum Average Torque...28 Figure 3.12: Transformation of the Torque Table.29 Figure 3.13: Lookup Tables for the Dwell and Advance Angles.....3 Figure 3.14: Current Waveform with Varying Dc-link Voltage 31 Figure 3.15: Average Torque as a Function Dwell Angle with Variable Dc-link Voltage...32 Figure 3.16: Average Torque as a Function Advance Angle with Variable Dc-link Voltage..33 Figure 3.17: Hardware Overview..34 Figure 3.18: General Two Phase PMSRM Control Block Diagram..35 Figure 3.19: Average Torque Control Block Diagram..36 Figure 4.1: Hysteresis Controller with Asymmetric Converter.38 vi

Figure 4.2: Simulation of the Hysteresis Current Controller 39 Figure 4.3: Self Inductance of the PMSRM..41 Figure 4.4: Block Diagram of the Linear 2-phase PMSRM Current Controller.43 Figure 4.5: Root Locus of the Linear Electrical Model of the PMSRM 44 Figure 4.6: Small Signal Step Response of the PI Controller...45 Figure 4.7: Anti-windup PI Current Controller Block Diagram 47 Figure 4.8: Simulated Current Response with Anti-Windup PI Current Control.47 Figure 4.9: Simulated phase current and voltage with 6A command 49 Figure 4.1: Simulated Phase Current at 36rpm...5 Figure 4.11: Block Diagram of One Phase of the PMSRM Model Reference Adaptive Controller.. 51 Figure 4.12: Comparison of Actual Nonlinearities to Matched Nonlinearities 55 Figure 4.13: Adaptive Current Control Simulations at 16rpm with i = 7A....56 Figure 4.14: Simulated Adaptive Gain Convergence at 16rpm.....57 Figure 4.15: Adaptive Current Control Simulations at 36rpm with i = 7A....58 Figure 5.1: Open-loop Small Signal PMSRM Mechanical Model... 62 Figure 5.2: IIR Speed Feedback Filter...64 Figure 5.3: Speed Filter Frequency Response... 63 Figure 5.4: Closed-loop Speed Control Block Diagram.... 64 Figure 5.5: Step Response and Root Locus of the PI Compensated Mechanical System....67 Figure 5.6: Dynamic Closed Loop Speed Control Simulation with Disturbance Inputs..68 Figure 5.7: Dynamic Closed Loop Speed Control....69 Figure 5.8: Closed-loop Torque Control Block Diagram with Firing Angle Lookup...7 Figure 5.9: Dynamic Closed Loop Torque Control Simulation with Disturbance Inputs....71 Figure 5.1: Efficiency Searching Algorithm Flow Chart....73 Figure 5.11: Frequency Response of the Power Averaging Filter. 75 Figure 5.12: Simulation of the Efficiency Searching Algorithm... 76 vii

List of Tables Table 4.1: Nominal Inductance values for the 2Hp PMSRM 42 Table 4.2: Nominal R eq Values for the 2Hp PMSRM...43 Table 4.3 Adaptive Parameters for simulation at 16rpm with i = 7A.56 Table 4.4: Control Parameters for Adaptive Control at 36rpm with i = 7A...58 Table 5.1: Selected Values of the EMF Constant..63 Table 5.2: Average Cogging Flux. 63 Table 5.3: Speed Filter Parameters 64 Table 5.4: Motor Efficiency of the Different Control Schemes 77 Table A.1: Dimensions of the 4ecore PMSRM...83 viii

1 Introduction 1.1 Introduction to PMSRM The switched reluctance motor (SRM) has seen growing interest in high volume commercial and industrial markets for variable speed motors. The basis for the SRM originated in the 185s but was not implemented until 1969 when S.A Nasar proposed that a dc SRM is practicable to develop [1]. His proposals have become more viable due to drastic advances in both power switching devices and electronic controllers. Since then SRM development has steadily improved; they can now be found commercially in drives as small as printer servos all the way to 4-kW compressor drives [2]. Commercially, there are many benefits of the SRM compared to other variable speed motors. The simple shape of the rotor and stator as well as the simple windings applied lends itself to inexpensive mass production. By design, SRMs require fewer raw materials, due to shorter stack lengths and more compact windings while still offering comparable power density to induction and permanent magnet motors. As with other variable speed motors the SRM requires an electronic power converter but can require as little as one IGBT and one diode per phase, compared to a variable speed induction or permanent magnet motor, which need at least twice as many switches. The focus of past and present research of SRMs deals with some inherent disadvantages which must be solved to insure the commercial success of the SRM. Firstly, control of an SRM is a non-trivial task due to the non-linear inductance and torque profiles from the varying air gap between the rotor and stator poles. Also, the absolute position of the rotor is necessary for phase excitation and commutation. In most cases this requires additional hardware in the form of a position sensor, which can either be magnetic or an optical encoder. Additionally, the SRM inherently produces a large amount of acoustic noise which can be of great concern in many commercial and industrial applications. The SRM has salient poles on both the stator and rotor, with dc windings on the stator but no magnets or windings on the rotor. The numbers of rotor and stator poles, as well as the number of phases of the machine are central criterion of the motor design process. The motor is operated by exciting a phase of the stator, which causes the rotor to come to an aligned position. 1

At that point, the other phase(s) are out of alignment. By then commutating the first phase and exciting a subsequent phase the rotor will move to an aligned position with the new phase. When this sequence is properly orchestrated the rotor will spin at a continuous rate generating torque. [2] A two-phase SRM with a common pole e-core structure is presented in [3] and [4], and has been shown to have reduced amounts of steel and copper compared to other SRMs while increasing the power density and overall efficiency. The e-core SRM stator is comprised of sections with three poles. The two outer poles have the windings for each of the phases, while the middle pole (common pole) has no windings and is shared between both phases. This structure can be used to create a segmental stator with two independent e-cores. Another structure would be to have a single stator comprised of poles with phase windings alternated with larger common poles. The e-core design allows for shorter flux paths which allows for reduction in copper wire and core loss which in turn yield higher efficiency. A two phase e-core SRM with 8 stator poles and 1 rotor poles is shown in figure 1.1. Figure 1.1: Two-phase 8-1 4-eore SRM ( Krishnan Ramu) An ac motor called the duel stator doubly salient permanent magnet motor is presented in [5] and [6], which has a similar structure to an SRM with segmental stator separated by a pair of magnets. This machine is shown to have higher torque density and efficiency than other ac 2

motors. It is controlled as an ac machine; therefore it only gets positive torque from the magnets because the net reluctance torque contribution is zero. The doubly salient permanent magnet switched reluctance motor (PMSRM), has been proposed [7], [8] to allow for the torque production of a SRM with the addition of PM torque. This machine has the same mechanical structure as the segmental e-core SRM with magnets placed along the face of the common stator poles. The PMSRM is a dc motor with torque and inductance properties similar to that of the SRM which allow the motor to be controlled with any drive used to power an SRM. Although, there have been no attempts at controlling the PMSRM, its similar structure and characteristics allow many of the control techniques used for the SRM to be adapted to its control. Figure 1.2: Two-phase 8-1 PMSRM ( Krishnan Ramu) The increased efficiency given by these magnets comes with additional difficulties in construction, modeling and control of the motor. The rotor must be carefully inserted as to not touch the brittle on the surface of the stator poles which could easily shatter. As with the SRM, the torque of the PMSRM is a nonlinear function of rotor position and current but with the additional problem of having a non-uniform zero crossing. That is, for each current there is a unique point at which the positive torquing region begins and ends. The flux of the machine is the sum of the inductive flux and the magnetic flux which adds additional non-linearity to the 3

machine model. Starting up the PMSRM also can also be difficult since the rotor will be fixed in one of four possible places due to the magnetic attraction while the machine is at rest. 1.2 Thesis Proposal and Contributions The central goal of this thesis is to design, simulate and implement a control strategy for the PMSRM. The region of operation, specified by the motors firing angles, will be analyzed for torque, speed, and efficiency, all of which are critical measures of performance for a variable speed motor. Linear and non-linear techniques will be used to analyze the current-voltage relationship and will be used to design three current controllers for the PMSRM. A linear mechanical model of the PMSRM will be used to design a speed filter and controller to allow for variable speed operation. The closed loop speed controller will be augmented with a self-tuning efficiency controller that will optimizing the firing angles of the motor to find the most efficient operation at any particular load and speed. The contributions of this thesis are: Analysis, simulation and verification of the operating region of the PMSRM based on its relationship to average torque, speed and efficiency of the motor. A control strategy is developed to maximize average torque. Design of a current controller for the PMSRM using both linear and structured non-linear modeling of the motors electromagnetic equations. This includes the design of a gain scheduled anti-windup PI controller and model reference adaptive controller. Design of a speed controller for the PMSRM based on an original linear mechanical model of a doubly salient permanent magnet motor which accounts for reluctance plus magnetic torque contributions. A self-tuning efficiency algorithm for any SR motor which finds the most efficient combination of firing angles online based on real time feedback in a speed controlled system. The simulations in this thesis are based on a 2-HP two phase 4e-core PMSRM motor designed in N. Lobo s PhD dissertation [11]. The torque and inductance data was extracted from 4

FEA simulations. The rated operating point of this motor is 36rpm with a 3.8 Nm load therefore this operating point will be the primary concern in simulation. The relevant parameters of this motor are attached in appendix A. 5

2 SRM and PMSRM Background From a mechanical and a control standpoint the SRM and the PMSRM have many similarities. Although they have different electrical and mechanical models, both have similar overall structures, torque production and use the same electronic power converters. Two common SRM converter topologies are presented for the PMSRM, each of which has specific advantages. In addition, control methods for the SRM which may be applicable to the PMSRM are investigated. 2.1 Operation of the SRM The basic magnetic structure of a 4ecore SRM is shown in figure 2.1. When a phase is excited, the magnetic flux moves through the rotor pole into the stator pole around to the common pole and back into the rotor. With an e-core structure the common poles of the SRM are used to shorten the length of the majority of the flux path. The shorter path results in lower core losses compared to a traditional SRM structure in which the flux path would travel through the rotor from one excited stator pole to the other. Figure 2.1: Flux Path of a 4ecore SRM (Adapted from [4]) When the rotor pole and stator pole are unaligned almost all of the flux is through the air gap resulting in a minimum inductance value. As the poles overlap the flux path is through the 6

rotor pole into the stator and back through the common pole. In this region the inductance increases as the two poles move closer to alignment. As the inductance reaches its maximum the torque production becomes zero. The current through the winding is then turned off during this region so that negative torque will not be created once the rotor continues to move. As the rotor pole moves past the stator pole, the slope of the inductance becomes negative and if current is applied then negative torque is produced. L a Phase A i a Phase B L b i b Λ a Λ b Flux Linkage Torque Rotor Position Figure 2.2: Operation of the SRM From [2], the electrical model of one phase of an SRM is given by: v = R s i + 7 dλ θ, i dt Where v is the voltage applied across the windings, R s is the resistance of the windings, i is the current through the windings and λ is the flux linkage of each phase, which is equivalent to the product of the current and the inductance. (2.1)

λ = L θ, i i where L is inductance. The mechanical model of the SRM is defined by its torque production, which is given by: T e = 1 dl θ, i i2 2 dθ The torque is a function of the current squared which allows for positive torque regardless of the current polarity allowing for simple converter design. Negative torque occurs when the inductance has a negative slope. (2.2) (2.3) 2.2 Operation of the PMSRM The stator of the PMSRM is the same as that of the SRM with the only difference being a small amount of steel removed from the face of the common poles which is replaced by a permanent magnet. All four of the magnets are placed with the same magnetic direction opposite to that of the windings. Placing the magnets on the stator poles allows for the construction of a solid back-iron which is much easier for manufacturing than other doubly salient permanent magnet designs. The windings of the PMSRM produce unipolar current, thus even under a winding fault the magnet will not be demagnetized. (a) (b) Figure 2.3: Flux Path of the PMSRM (a) Phase A aligned (b) Phase B aligned (Adapted from [11]) 8

The when a winding is excited the flux path of the PMSRM goes through the stator pole, through the aligned rotor pole. The flux then splits to the two adjacent rotor poles and moves into each of the adjacent common poles, returning through the active pole. As the rotor pole passes an excited stator pole the winding flux and inductance begins to rise, as does the flux contribution from the PM. The inductance reaches its maximum at the aligned position, and then begins to decrease. λ a Phase A i a λ pm a Phase B λ b λ pm b i b Self Inductance L b L a Λ a Λ b Flux Linkage Torque Rotor Position Figure 2.4: Operation of the PMSRM As with an SRM, the PMSRM is designed to have as little mutual inductance between phases as possible. With this assumption, the instantaneous torque produced by a two phase PMSRM is: 9

T = 1 2 i L 2 a,b a,b θ + i L pm a,b a,bi pm θ + 1 2 i L pm 2 pm θ The first term is the torque produced by the self inductance of each winding. This is represented as either phase a or phase b which assumes that only one phase is producing torque at a time. The second term is the magnet torque, which is a function of the winding current, the equivalent current through the permanent magnet, i pm, and the change in inductance of the magnet with respect to the active phase. The third term is the cogging toque, and is a function of only the rotor position. The equivalent current of the PM times the position derivative of the PM with respect to each phase is equal to the position derivative of the PM s flux, shown as: i pm dl pm a,b dθ = dλ pm a,b dθ The cogging torque of one electrical cycle must be zero; therefore, when considering the actual torque of the machine this term can be ignored. Considering the zero effect of torque and substituting equation 2.5 into 2.4 results in the following equation for torque of the DSPSRM: T = 1 2 i dl 2 a,b a,b dθ + i dλ pm a,b dθ The general electrical model for one phase of the PMSRM is the same as the SRM s electrical model given in 2.1. However the flux of the PMSRM is the sum of the reluctance flux and the flux of the PM: λ θ, i = il θ, i + λ pm θ The result of combining equations 2.1 and 2.7 is the complete electrical model for one phase of the PMSRM, which is: il θ, i v = Ri + d dt + dλ pm θ dt The electrical and mechanical models of the PMSRM are very similar to their SRM counterparts; however in both cases they have an additional effect from the PM. In the (2.4) (2.5) (2.6) (2.7) (2.8) 1

mechanical equation the additional nonlinearity comes from the EMF of the PM, which is a function of the change in inductance times the phase current. For the electrical model, the voltage term is a function of the change in PM flux with respect to time. Overall control strategies as well as linear techniques used for SRM control design must be modified to account for these additional terms. 2.3 Converter Topologies for the SRM and PMSRM As opposed to induction and other ac motors, the SRM and the PMSRM have unidirectional voltage from a dc voltage source, typically coming from a rectified ac source. Any converter used for the SRM can be used for the PMSRM. The converters can range in cost and functionality, and should be selected based on the application. This section presents two converters, the asymmetric bridge and the split-dc, both of which are well suited for the control of the PMSRM. The pulse width modulation (PWM) technique is also presented for sustaining average current with current control and can be used with either converter type. 2.3.1 The Asymmetric Converter Shown in figure 2.5, the asymmetric bridge converter is a specialized controller designed for the SRM. The asymmetric converter has independent phase control which can be implemented with any number of phases. This converter also allows freewheeling operation, or the ability to apply zero volts, and can recover mechanical energy with regeneration. Since the asymmetric converter has two switches and two diodes per phase, it is typically limited to high power and high performance applications. However it is a suitable choice for the initial development of the PMSRM. 11

T1 T3 + Vdc - D2 + Va - Phase A D1 D4 + Vb - Phase B D3 T2 T4 Figure 2.5: The Asymmetric Bridge Converter When switches T1 and T2 are turned on the full dc bus voltage is applied to phase A causing the current in the windings to rise. When the current in phase A is to be commutated, both switches are then turned off. Since the current remains in the same direction both diodes D1 and D2 become forward biased and the negative bus voltage is seen across the winding, creating a rapid decrease in current. While the switches are turned off, the current is circulating back through the voltage source, which is called regeneration. In addition, while the current is high, switch T2 can be turned off, causing D1 to be forward biased and giving zero volts across the windings. Thus the asymmetric converter has three degrees of freedom, and can command a voltage of ±V dc or V. For most applications the PMSRM will need an ac input, necessitating the addition of a full bridge rectifier to the converter. 2.3.2 One Switch per Phase Converters From an electronics standpoint, one of the largest advantages of the PMSRM and the SRM over other variable speed drives is one switch per phase converters. While two switch perphase converters, such as the asymmetric converter, are good for high performance applications as well as laboratory research and testing, high volume commercial markets demand configurations with less switches. For mass produced, low cost motor drives the number of switches can make a significant difference in the overall cost of the system. There are a few 12

possible implementations for one switch per phase, one choice being the split dc supply converter. Vdc/2 T1 D1 Phase A D2 Phase B Vdc/2 T2 Figure 2.6: The Split-dc Converter When switch T1 is on, diode D2 is reverse biased and current flows through the winds of phase A with half of the dc bus voltage applied across its windings. When T1 is off current flows through the phase A windings, through D2 and back into the second capacitor, regenerating it. This converter configuration can be augmented with a split leg rectifier to allow for an ac input source (split-ac converter) which has even less devices than a full bridge rectifier; however, the capacitor size must be larger for this converter to maintain equal voltages across both phases. 2.3.2 Pulse Width Modulation One of the most common methods for current regulation is using the PWM method to apply an average voltage to each phase, which is maintained by the duty cycle of the power device. PWM control allows the implementation of controllers which command a voltage between +V dc with a duty cycle of one and V dc with a duty cycle of zero. A typical control scheme for the SRM would be to apply full voltage when the inductance is rising to reach the 13

desired current level. The current is regulated at the commanded current for the torque producing region using PWM chopping. When the inductance begins to decrease, full negative voltage is applied to bring the current to zero as quickly as possible. I i* 2 i d 1.5 V dc V -V dc T s Figure 2.7: PWM Chopping of the SRM For each cycle the PWM scheme is defined as: v = V dc t dt s V dc dt s t T s (2.9) Where d is the duty cycle and T s is the PWM period. The performance and efficiency of the PWM switching scheme are both directly related to the PWM frequency. As the frequency increases the output current ripple, Δi, decreases as does the efficiency of the converter since the total number of turn-on and turn-off losses increase. PWM chopping is vital for the implementation of current regulators, such as the PI controller presented later, in both the SRM and the PMSRM. In addition, PWM chopping allows for increased efficiency and control compared to other methods of current control, such as hysteresis (presented in section 4). 14

2.4 Control of the SRM With the success of SRMs over the past 3 years their control has become a well established area. The control can be broken into three general subcategories which are not necessarily separate but highlight distinct areas within the general control structure. Current or voltage control directly manages the power applied to the motor and is based on the electrical model of the motor. The time in which voltage is applied to the windings is determined by the firing angles which control the power available to the machine as well as its efficiency. Torque or speed control is used to create a closed loop control to regulate the speed of the motor. The combination of when each phase is turned on or off plus the current reference and the speed of the motor determine how much torque can be produced by the motor. Any given load or speed may have any number of combinations of firing angles. Therefore, selection of these angles can be used to additionally control the efficiency of the motor. In [9] exhaustive simulations are used to map the optimal firing angles to maximize the efficiency and torque of an 8/6 SRM. In [1], Gribble develops specific formulas to calculate firing angles so as to conserve energy by maximizing the torque output while minimizing input power through current and voltage control. Using only the position of mechanical overlap, and the aligned and unaligned inductances, a general equation for optimal firing angle selection is given. This method uses general inductance relationships as opposed to exhaustive simulation of exact parameters. The work of Gribble is expanded in [11] where a firing angle calculator is presented to select angles based only on speed, dc-link voltage, reference current and the aligned and unaligned inductance. The firing angle calculator is augmented by an efficiency optimizing algorithm which varies the turn-off angle to obtain the optimal efficiency. In [12] the turn-off angle is optimized to a curve-fit model while the turn-on angle is adjusted to place the peak of the phase current at the position where the inductance begins to rise. In all cases, efficiency is maximized by finding critical points in the relationship between when, how long and how much current is fired into each phase and their respective output torque and speed. Current control is an integral part of the overall SRM control system, and developing high performance current regulators is critical to operation as well the implementation of other more complicated control systems such as efficiency based control. Hysteresis control is a simple and effective low performance control which is accepted in industry do to its ease of 15

implementation. Linear PI control design is a well established industrial norm for current control. However linear models of the SRM will vary with any change in firing angle, speed and current reference itself which indicates decreased performance during variable speed operation. In [13] a modified anti-windup PI controller was introduced which used linear gain scheduling based on current and speeds to improve the overall performance. Also, nonlinear control strategies, such as model reference adaptive control, have been demonstrated as effective current control strategies [14]. Similarly, adaptive control has been implemented on ac drives including the SRMs ac brother, variable reluctance motor [15] and the synchronous permanent magnet motor. [16]. The torque and speed control methods presented in [2] keep set firing angles while using current reference as the speed controller s command which is a single-input single-output system. For most applications of SRMs the speed control can be relatively low performance in which case a PI controller will provide sufficient transient response without steady state error. In any cases, the performance of the speed controller is directly linked to the response of the current controller. 16

3 Control Principle of the PMSRM The operating performance of the PMSRM is highly dependent on when the phase currents are turned on and off which means an absolute knowledge of the rotor position is required. In addition, the amount of current allowed into each of the windings has a significant effect on the operation of the motor, and it s control is also required. The choice of these three control variables affect the speed, torque, efficiency and the acoustic noise produced by the motor. Generally, a given operating point with a set load and speed is desired to operate the PMSRM as efficiently as possible with the least amount of acoustic noise; however, the exact relationship between the control variables and the efficiency is not very straightforward. In addition, the acoustic noise is greatly affected by the motor design leaving only so much room for improvement from a controls standpoint. The relationships between the controls and the performance of the motor can be quantified through extensive simulation. In this thesis, two types of dynamic simulations are used to model the PMSRM. In this chapter and the following chapter, the speed of the motor is set to a constant rate and the torque produced is measured through a torque lookup table generated from FEA simulations, which is a function of rotor position and current. The current is calculated from the governing electrical equation 2.8 where the flux and inductance are looked up from a separate FEA generated table. This type of simulation allows measuring what the average torque output of the motor will be while easily varying the control parameters as well as the speed. The second type of simulation, used in chapter 5, is a dynamic speed simulation. In this case, the speed is a function of the mechanical model of the motor, which includes the torque computed from the lookup. 3.1 Startup For an SRM startup is an important aspect of operation. If the position is known, the current controller with PWM chopping can apply enough torque to initially spin the rotor. If the starting position is unknown, firing and holding one phase current will move the rotor to a known aligned position. From there, the mechanical sensor or position observer can then compute the position of the rotor with respect to each electrical cycle and the startup can continue as normal. 17

Torque (Nm) 2 1.5 1.5 -.5-1 5 1 15 2 25 3 35 Rotor Position (deg) Figure 3.1: PMSRM Cogging Torque Unlike the SRM, the PMSRM has cogging torque as a result of the PMs on the stator. Under no load the rotor will be locked in any one of four positions. However with any sort of external cogging forces the position will not necessarily be one of the four cogging positions. For the sake of simplicity the position at anytime will be considered known by a position sensor, but will not be assumed to be in one of the cogging locations. The reluctance torque of the PMSRM is significantly larger than the torque generated from the PM allowing for startup in the same fashion of the SRM. 3.2 Region of Operation In the most basic sense, the operation of the PMSRM is determined by when, how long, and how much current is applied to each phase and the resulting torque created by the motor. Torque of a PMSRM is a nonlinear function of current and rotor position which can be represented as three distinct control inputs. The first is the advance angle, θ a, which designates how many mechanical degrees prior to a set point the current is excited. The second control variable is the dwell angle, θ d, which indicates how long excitation lasts before the current is commutated. Together, the advance and dwell angle are referred to as the machines firing 18

Torque (Nm) angles. The third input is the current reference, i, which must be controlled with an added current regulator. 2 15 a d 22A 1 5-5 -1 A -15-2 5 1 15 2 25 3 35 Rotor Position (deg) Figure 3.2: Torque Profile of 4ecore PMSRM All three control inputs have significant contributions to the operation of the motor and often under-constrain operating points. An under-constrained system means that multiple combinations of inputs can result in the same torque and speed output. In order to select the best control inputs additional constraints must be considered; including the efficiency and acoustic noise produced. In certain cases the current controller may be omitted to allow for single pulsing operation, and controlled solely by the firing angles. However for a speed regulator the current command is usually used as the single control output variable. This allows for the firing angles to be determined offline based on the maximum torque production. Another option, which is shown later in this thesis, is to use a real time algorithm to select the firing angles to maximize efficiency. 3.2.1 Effect of the Advance Angle 19

The advance angle determines when the phase is excited relative to the rotor position, which in turn affects the current rise as well as the torque produced by motor. The ratio between torque produced and amp seconds of excitation is directly related to the efficiency of the machine. The resulting tradeoff between torque produced and efficiency must be maximized for ideal operation. The advance angle should be chosen to produce the necessary torque while minimizing the total time of excitation however as the advance angle is minimized the current reference must be increased 1 θ a = 2 1 θ a = 5 1 θ a = 1 i a 5 5 5 1 1 1 5 T a 5 5-5 -5 1 2 3 4 θ (deg) (a) -5 1 2 3 4 θ (deg) (b) Figure 3.3: Simulated Current and Torque with Selected Advance Angles (θ d = 1, i = 6A, ω r = 36rpm) -1 1 2 3 4 θ (deg) (c) Figure 3.3 shows the simulated effect of different advance angles on the torque production of the PMSRM. For this simulation the speed, dwell angle and current reference were kept constant. The current was regulated by an ideal current controller and the torque was measured as the average over one mechanical revolution of the rotor. A small advance angle prohibits the current from reaching the current reference. The larger advance results in excitation through a region of increased inductance which impedes the current rise. With the increased advanced angle the current rise is accelerated in addition to the time in the positive torque region increased. However with too large of an advance, as in 3.3(c), negative torque occurs at the beginning of excitation which negates any additional torque produced by a faster rise and decreases the average torque production. 2

Average Torque (Nm) 6 5.5 5 4.5 4 3.5 3 12A 11A 1A 9A 8A 7A 2.5 2 1.5 1 2 3 4 5 6 7 8 9 1 Advance Angle,, (deg) a 6A 5A Figure 3.4: Simulated Average Torque as a Function of Advance Angle and Reference Current(θ d = 1, ω r = 36rpm) The effects of an increasing advance angle with different reference currents are shown in figure 3.4. The peak torque production happens with an advance angle similar to figure 3.3(b) in which the excitation begins prior to any negative torquing. In addition, the advance angle must increase to maintain maximum average torque as the current reference rises or else the reference is not achieved indicated in figure 3.4 when the same average torque is produced regardless of the current references until the advance angle is increased. 3.2.2 Effect of the Dwell Angle The dwell angle controls how long the current is maintained at the reference level. As the rotor leaves the primary torque production region the current needs to be commutated in order to prevent excitation in the negative torque region. As the dwell angle increases the inductance also increases. In all cases, it is important to have the phase current completely commutated at a position no further than the peak inductance. If this is not done, then the motor will create negative torque. 21

1 θ d =2 θ d =5 θ d =1 θ d =15 i a 5 1 T a 5-5 1 2 3 4 1 2 3 4 Figure 3.5: Simulated Current and Torque with Selected Dwell Angles(θ a = 5, i = 6A, ω r = 36rpm) 1 2 3 4 1 2 3 4 θ (deg) θ (deg) θ (deg) θ (deg) (a) (b) (c) (d) Simulated results of increasing dwell angles are shown in figure 3.5. For small dwell angles, negative torque is produced from the magnet as the current comes to zero, then it rises back as the magnet provides its positive torque. Keeping the dwell on for too long will enter the negative torquing region as in figure 3.5 (d). Therefore, the dwell should be long enough to prevent negative torque contribution from the PM but not so long as to produce negative reluctance torque. In addition, the positive contribution of the PM is at the end of the commutation period. Leaving current during this time increases the number of amp seconds into the windings while getting only a minimal increase in torque production. 22

Average Torque (Nm) 6 5.5 5 4.5 4 3.5 3 2.5 12A 11A 1A 9A 8A 7A 6A 5A 2 1.5 5 6 7 8 9 1 11 12 13 14 15 Dwell Angle,, (deg) d Figure 3.6: Simulated Average Torque as a Function of Dwell Angle and Reference Current(θ a = 5, ω r = 36rpm) As the dwell angle is increased there is a noticeable peak in torque production which varies with current reference. For larger current references maximum torque output is achieved with smaller dwell angles because it will take a longer amount of time to decrease the current. On the other hand, small current references maintain maximum output from larger dwell angles which allow for longer positive torque production. The relationship between the dwell and the average torque appears to be linear for a constant speed. 3.2.3 Effect of Speed on the Firing Angles The speed of the motor is not an implicit control variable; however, like the firing angles, it has an impact on the rate of change of the current which in turn has an effect on the torque produced each phase. Since the dynamic operation of the motor will occur with a functioning closed loop speed controller, it will be assumed that the reference speed is the same as the actual speed. Generally, higher speeds limit the rise of current and require a larger advance angle to produce torque equivalent to that of lower speeds. Figure 3.7 shows the simulated current and torque of the PMSRM for set firing angles and a constant current reference. 23

Average Torque (Nm) 1 ω r = 12 rpm ω r = 24 rpm ω r = 36 rpm i a 5 1 5 T a -5 1 2 3 4 1 2 3 4 1 2 3 4 θ (deg) θ (deg) θ (deg) Figure 3.7: Simulated Current and Torque for Varying Speeds (θ a = 5, θ d = 1, i = 6A, ) At low speeds the current waveform has a sharp rise and is maintained at the reference level for the positive torque region. As the speed increases, the rise is not as sharp, indicating that the advance angle must be increased for higher speeds. The decline of current after commutation in the 12rpm simulation is so rapid that it begins to produce negative torque. Thus a longer dwell is required at lower speeds to ensure that no negative cogging torque is produced. 3.5 3 2.5 2 1.5 1.5 12rpm 16rpm 24rpm 3rpm 36rpm 2 4 6 8 1 12 14 16 18 2 Dwell Angle (deg) 24

Average Torque (Nm) Figure 3.8: Simulated Average Torque as a Function of Dwell Angle and Speed(θ a = 5, i = 6A, When the current reference and the advance angle are held constant the maximum average output torque is nearly constant for all speeds with a decreasing dwell angle. The trend of increasing speeds correlates to figure 3.7 where it was noted that at lower speeds the sharp decrease in current created negative torque before the positive torque contribution from the PM. Therefore at lower speeds the dwell angle must be increased to avoid negative torque. If the dwell is increased past a certain point there is a sharp drop-off in average torque for all speeds. 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 1 2 3 4 5 6 7 8 9 1 Advance Angle (deg) Figure 3.8: Simulated Average Torque as a Function of Advance Angle and Speed(θ d = 1, i = 6A, 36rpm 3rpm 24rpm 18rpm 12rpm Holding the current reference and dwell angle constant shows the maximum torque for increasing speed with an increased advance angle. In this case, a dwell angle of 1 degrees was used which, as seen in figure 3.7, indicates that there will be larger torque production for higher speeds. The larger values of average torque for the higher speeds are slightly misleading since different combinations of dwell angles could create more torque for any speed. What can be taken from this simulation is that larger advance angles need to be used to maximize torque at higher speeds. At higher speeds the excitation period is shorter in time, which means that by 25

Torque (Nm) increasing the advance a longer amount of time is allowed for the current to rise and reach its maximum. 3.2.4 Selection of Firing Angles Analysis of the effects of the firing angles gives a clear picture on how these variables affect steady state operation of the motor. With current control in mind, choosing firing angles that produce the maximum torque at the highest current will allow for the greatest range of torque control. The selection of firing angles can neglect the maximum torque level at smaller current levels since the current level can be raised for additional torque. For instance, choosing a dwell angle of 8.5 will give a current controller the ability to generate anywhere from to 5.75 Nm of torque while a dwell of 14 would only allow for to 4.75 Nm. Offline selection of the firing angles are possible and can be done so as to maximize the torque output, however the angles will only be optimum for one operating condition. For more precise control of the PMSRM the performance of the motor must be characterized based on its operating conditions as well as its control variables. In addition, the control structure must change in order to allow the firing angles to be dynamically updated. 4 3 2 1 15 1 5 1 d (deg) 5 a (deg) 26

Figure 3.9: Simulated Average Torque with Set Speed and Current Command Each current reference and speed combination has a map similar to the one shown in figure 3.9, with increasing torque output for increased current reference and decreased torque output for increased speed. Likewise, the firing angle combination that yields the peak will vary as a function of speed and current. Visualizing all four dimensions of this performance map is difficult, however it can be thought of in a numeric method. The complete map of average torque can be expanded to a four dimensional matrix with each element referring to a specific advance and dwell angle with a set speed and current reference. This torque matrix is represented as: T ω, i, θ a, θ d = T ω, i, θ a, θ d T ω, i, θ a, θ dm T ω, i, θ an, θ d T ω, i, θ an, θ dm T ω k, i, θ a, θ d T ω k, i, θ a, θ dm T ω k, i, θ an, θ d T ω k, i, θ an, θ dm 27 T ω, i j, θ a, θ d T ω k, i j, θ a, θ dm T ω, i j, θ an, θ d T ω k, i j, θ an, θ dm T ω k, i j, θ a, θ d T ω k, i j, θ a, θ dm T ω k, i j, θ an, θ d T ω k, i j, θ an, θ dm In order to obtain a complete performance map of a motor the elements of the matrix in 3.1 can be obtained through an extensive simulation. For simplicity, the simulations used in this thesis were performance using an ideal current regulator to remove additional disturbances from overshoot or slow rise time which may come with actual control implementation. In addition, the simulations need to be run at the different intervals of speed which requires the speed to be held constant. The simulation of the current is dynamic but the mechanical model is considered constant, which means that the speed is not affected by the torque production. When the simulation is completed, the resulting performance map has figures similar to 3.9 for each of the reference current and speed combination. For a given operating point maximum efficiency will occur when the largest amount of torque is produced with the smallest amount of current, or at the peak of the graph. Hence, the torque can be maximized for each unique reference current and speed combination. The point of maximum torque also corresponds to a pair of firing angles, which can be thought of as two separate sets of dependent variables (3.1)

Maximum Torque (Nm) matched to each point of maximum torque. The set of maximum torque and the corresponding firing angles as a function of speed and reference current are represented as: T max, θ amax, θ dmax = T ω,i, θ a ω,i, θ d ω,i T ω,i j, θ a ω,i j, θ d ω,i j T ωk,i, θ a ω k,i, θ d ω k,i T ωk,i j, θ a ω k,i j, θ d ω k,i j (3.2) 7 6 5 4 3 2 1 1 2 Speed (rpm) 3 4 2 4 6 8 1 12 Current Reference (A) Figure 3.1: Maximum Average Torque The maximum average torque output of the motor is linear for low speeds and smaller current reference levels. As the speed and current reference increase, the maximum torque declines. When simulating or running the motor the torque is an output variable determined by the speed, current and firing angles. However, in closed loop control design it is necessary to have reference current as a function of torque since the output of the system is torque, and the input is reference current. Current can be represented as a function of torque performing a transformation of the maximum torque matrix. T max i, ω r i T, w r (3.3) This transformation is accomplished through a relatively simple linear interpolation of the simulated torque matrix. The first step is to determine the range of the torque inputs. On the 28

Torque Command (Nm) upper side range of torque is limited by the average torque output of the motor and on the lower side it is limited by the smallest average torque of the lowest current reference. The step size of the torque lookup is user defined. A larger step size will give a poorer resolution but a smaller table that is easier to implement in a microcontroller. For each speed, the corresponding torque is interpolated for every point in the range. The fraction of the each torque relative to a known torque is proportional to the current. The following figure shows the transform process with a step size of 1Nm. A horizontal line is brought from the torque command to the line. This projection onto the current axis provides the corresponding current. 6 5 4 3 2 1 4 3 ω r (rpm) 2 1 2 4 6 8 1 12 Reference Current (A) Figure 3.11: Transformation of the Torque Table The advance and dwell angle matricides do not need to undergo the same transformation because they can be looked up based on the speed, which is the control input, and the current reference, which is found with the previous current lookup table. When the maximization of the torque map occurs, each index from for the dwell and advance angle need to be preserved so that there is no offset. 29

18 θ d 16 14 12 1 1 8 2 4 6 8 Reference Current (A) 1 12 4 3 2 ω r (rpm) (a) 6.5 6 θ a 5.5 5 4.5 4 4 12 1 8 6 Reference Current (A) 4 2 1 2 3 ω r (rpm) Figure 3.12: Lookup Tables for, (a) the Dwell Angle and, (b) the Advance Angle (b) The data in figure 3.12 shows the firing angle lookup tables which was acquired through simulation. The dwell angle is at a maximum when the current and the speed are at their lowest which may seem counterintuitive at first. However with slower speeds and current levels the current will drop to zero very quickly. If the current reaches zero too soon then negative torque 3

Current (A) will occur before the magnet torque comes into effect. At higher current levels the dwell needs to be shorter so that there is enough time for the current to reach zero before the negative torque region. Generally, the dwell angle seems to have a negative linear relationship between both the current and the speed. The advance angle lookup table shows a peak advance angle when the speed and the current are at a maximum. The lowest advance angle comes at lower speeds and high current reference due to the fact that the current will have the additional time to rise at lower speeds, rendering additional advance angle unnecessary. The advance is constant for the lowest current level; however it drops off for increased current. 3.2.5 Sensitivity Analysis The previous simulations have all been performed assuming a constant dc link voltage. While this is acceptable to demonstrate trends and the general operation of the motor it fails to account for the voltage ripple caused by rectification. The converters presented in chapter two are dc converters, which in almost all cases will come from a rectified ac signal which will have an oscillating ac component, or ripple. The size of the ripple will be determined mainly by the size of the capacitors. In most applications total system cost is a priority, meaning that expensive electronics with higher performance will be substituted by the lowest cost working replacement. Finding the limits of the system s electronics are important to determining the most cost effective solution. 9 8 7 6 5 4 3 34V 2 31V 1 28V 5 1 15 2 25 3 35 4 Rotor Position (deg) 31

Average Torque (Nm) Figure 3. 13: Current Waveform with Varying Dc-link Voltage The magnitude of the voltage on the dc link has its primary impact on the rise and fall rate of the current. This in turn affects how much torque is produced for a given set of firing angles. For example, a smaller dc link voltage will account for a slower rise time which would require a larger advance angle to match the performance of a larger voltage. The accompanying slower turn-off time will require a shorter dwell angle so as to avoid negative torque production. 5 4.5 8A 4 3.5 6A 3 2.5 4A 2 1.5 1 6 8 1 12 14 16 Dwell Angle (deg) Figure 3.14: Average Torque as a Function Dwell Angle with Variable Dc-link Voltage The simulation above shows the average torque as a function of dwell and current with a dc-link voltage that is 31V ±1%. As the current reference increases, the discrepancy between the maximum and minimum voltages becomes larger. In addition, the location of peak torque occurs at increasing dwell angles for increasing bus voltages since with a quicker turn-off time the dwell can be larger without producing negative torque. With an increased dwell angle the overall average torque increases. 32

Average Torque (Nm) 4.5 4 8A 3.5 3 6A 2.5 2 1.5 4A 1 1 2 3 4 5 6 7 8 9 1 Advance Angle (deg) Figure 3.15: Average Torque as a Function Advance Angle with Variable Dc-link Voltage 3.3 Control Overview Given the performance benefits coupled with relative low cost, microcontrollers and digital signal processors (DSP) are the most effective and efficient way to implement the control scheme. The absolute position feedback can come from an encoder wheel, a magnetic Hall Effect sensor, or potentially a sensor-less position observer. For closed loop current regulation the controller also needs current feedback from analog current sensors discretized using an analog to digital converter (ADC). In addition, dc-link voltage of the converter is useful for control and can be scaled down with a voltage divider and then fed through the ADC. The PWM output of the controller can be directly connected to the gate driver on the converter. 33

DSP Control Logic PWM Timer 2-Phase Asymmetric Converter i a,b PM- SRM θ ADC ADC ADC v dc Voltage Sensor Current Sensor Position Sensor Figure 3.16: Hardware Overview For variable speed operation of the PMSRM an outer-loop speed regulator will be implemented in software to control the current regulator. With this type of operation the dwell and advance angles will remain constant in order to reduce torque ripple; one of the primary contributors to acoustic noise in a motor. In turn, the software defined current regulator will produce a reference voltage output. For PWM current control is used the reference will be a duty cycle with 1 representing the dc link voltage and indicating the negative dc link voltage. The PWM will use a DTA converter to send the duty cycle to each phase. In addition, the PWM will control when each phase should be on or off based on the rotor position and the firing angles. 34

Phase A H c ω + - ω f G ω i - + + - G c a G c b d a d b Switching Signals PWM Asymmetric Converter i a θ PMSRM H c i b Phase B H ω ω m Figure 3.17: General Two Phase PMSRM Control Block Diagram The self-tuning efficiency based algorithm presented in chapter 5 will retain the same structure as the above control diagram as the above controller with the only difference being the switching signals come from an additional controller. Once the algorithm has completed then operation of the motor will continue with fixed angles. The previous control method uses set firing angles (computed either online or off) with a variable current. However, as seen in section 3.2.4, the average torque output of the PMSRM can only be controlled for efficiency by using all three variables. The following control scheme can be implemented to maximize the efficiency by maximizing the average torque. ω r θ a i, ω r θ d i, ω r θ a θ d + G T T e i T, ω r i - ω r Figure 3.18: Average Torque Control Block Diagram 35

The speed error is placed through a torque controller which generates a torque command. Given the torque command and the speed reference a current command value is determined with a two dimensional interpolation from a lookup table generated from the simulation in section 3.2.4. This value will then be used to calculate the firing angles that match the current and speed reference based on another lookup table. For any combination of current command and speed reference there is one unique set of firing angles which guarantees that torque will be the maximal possible for the current, and therefore the most efficient combination. In this control scheme the current regulator remains the same; and is provided a reference current from the lookup table. The PWM functionality remains the same even though the firing angles are time varying. 36

4 Current Control Design and Simulation As stated in [2], the heart of any motor drive s control system is current control. The electromagnetic nonlinearities of both the SRM and the PMSRM make current control a nontrivial task. Although they are slightly different, both machines have inductances that vary with position and current with similar values. In addition, both machines use the same electronics and have similar operational regions; therefore the approach to controlling the PMSRM will closely follow the control of the SRM. Hysteresis controller has been proven to be a simple and effective approach; however, its lack of sophistication is evident in its large current ripple and significant switching losses. Pulse width modulation is a more efficient means of current control which can also provide a large performance increase. Of the available types of linear control, proportional plus integral (PI) is the most common controller for SRM current control, and is considered the benchmark of all controllers. A drawback for the PI controller is that the controller must be specifically designed for set operating point. During variable speed operation the speed and current reference will change considerably, rendering the gains of the controller ineffective. In addition, based on manufacturing methods and materials, the actual motor may vary up to 3 percent from the model. Gain scheduling uses gains selected for linearized points within the operating region to improve performance at variable speeds and loads. With a hysteresis band to avoid excessive changing of gains the speed and current feedback are used to select gains from a lookup table of gains designed from either linear or experimental design. Another approach for current control design is to use a nonlinear adaptive control algorithm that recognizes both parameter uncertainty and model nonlinearity. The controller is designed so that the known linear model is separated from the parameter uncertainties and nonlinearities. From this form, the controller is designed to drive the system to a reference current while adapting for the unknown. Thus, the more that is known about the plant model the better the performance. If the nonlinearities are only partially known and there is significant uncertainty stable control can still be achieved. The downside of adaptive control is that it can be difficult to implement the sophisticated adaptive algorithm on a DSP or MCU that has significant computational and numerical limitations. 37

4.1 Hysteresis Current Control Due to its simplicity and ease of implementation, hysteresis current control is a viable option for low performance applications and general operation. The most basic hysteresis control strategy is implemented by applying full positive voltage to the phase whenever the current feedback is less than the reference current value. Likewise, whenever the current feedback is greater than the reference then full negative voltage is applied. The resulting current ripple is directly related to the controller frequency. When using the asymmetric converter the hysteresis controller can also output zero volts by turning on the switch T1 and turning off T2. With the asymmetric converter positive voltage is applied when both switches are on, and negative voltage is applied when both switches are off. A slightly higher performance control strategy can be used given this extra degree of freedom by adding a boundary around the current command. When the current feedback is within the boundary then zero volts are applied. When the feedback is out of bounds, then the controller behaves like a typical hysteresis controller. Figure 4.1: Hysteresis Controller with Asymmetric Converter The switching behavior for the controller is summed up as: T 1 = ON, i i + i OFF, else T 2 = ON, i i i OFF, else 38

Current (A) (4.1) Correlating to figure 2.5 T 1 is the switch connected to the positive dc link and T 2 is the switch connected to the negative terminal of the dc link. The error boundary, i, can either be a set value or, for more accuracy, a percentage of the current command. The controller was simulated for the PMSRM as shown in figure 4.2. In this simulation the boundary was chosen to be 1% of the current command. 15 i a i b i * 1 5.5.1.15.2.25.3.35.4.45 Time (s) Figure 4.2: Simulation of the Hysteresis Current Controller From the simulation it is clear that the hysteresis controller can accurately track the current reference. As the reference is increased the magnitude of the current ripple becomes significant. The simplicity of implementing the hysteresis controller makes it an attractive option in certain low performance settings; however the large current ripple is a cause for concern in higher performance applications. 4.2 PI Current Control PI control is a proven method of current control in SRM s as well as in the control of other motors. The integral term is vital to eliminating the steady state error that will 39

undoubtedly be present due to the systems nonlinearities combined with general model uncertainties. The linear transfer function will give a transfer function with current as the output signal and voltage as the input. To apply this, a PWM chopping scheme described in chapter 2 must be implemented. While the PI controller may be a simple solution it is a proven and robust method for current control which is very easily implemented. 4.2.1 Linearization of the PMSRM Current Model The relationship between voltage and current for the PMSRM is: v t = R s i t + 4 dλ(θ, i) dt The voltage is considered to be the control input to the system and the current is the state and the output of the system. When calculating the effect of the current for the PMSRM the flux is given by λ = L θ, i i + λ pm θ. Substituting the flux equation and taking the partial derivative results in: v t = R s i t + L θ, i di t dt + i t dl θ, i dt + dλ pm θ dt The derivative of the cogging flux is only a function of speed and can be assumed to be a constant value since the change in flux will be near constant for the excitation range. When considering the linear system, this term will be absorbed by the large signal voltage term since it is a time invariant constant. The linear system can be found by substituting the following smallsignal perturbations at the following operating points: i = i + δi v = v + δv ω m = ω m + δω m For current control it is desired to have current as the output signal and voltage as the input signal. Combining the perturbations with the voltage equation results in: dδi dt L θ, i = R s + dl(θ, i) ω dt δi + δv (4.2) (4.3) (4.4) (4.5)

Inductance (H) Taking the Laplace transform of the above equation the transfer function is found to be: G p s = δi s δv s = 1 sl + R eq The equivalent resistance has been substituted into the equation as: (4.6) R eq = R s + ΔL ω m (4.7) Also the nominal inductance, L, and the nominal change in inductance, ΔL, have been introduced. To complete the linearization these nominal points must be calculated for each operating point of a particular speed and current. The speed can be chosen based upon whatever the desired operations of the motor. However the current will be varying over the operating region, an operating current is chosen for based on what the average current will be at each specific operating point..3.25 2A 4A 6A 8A 1A.2.15.1 12A 14A 16A 18A 2A 22A.5 5 1 15 2 25 3 35 Rotor Position (deg) Figure 4.3: Self Inductance of the PMSRM The inductance of the PMSRM is a nonlinear function of position and current. When the rotor pole is aligned with the stator pole the inductance is at a maximum. Likewise, when the rotor and stator are unaligned the inductance is at a minimum. As the current increases the change in inductance becomes smaller. The nominal inductance is found by taking the arithmetic mean of inductance for the current specified by the operating region. 41

L = L a + L u 2 i=i (4.8) Where L a is the inductance at the aligned position (the maximum inductance), and L u is the inductance at the unaligned position (the minimum inductance). Likewise, the nominal change in inductance is found by taking the difference in inductance divided by the difference in position. ΔL = dl dθ i=i = L a L u θ a θ u i=i (4.9) The aligned position is θ a, and θ u is the unaligned position. Variable speed operation will require a range of operating currents, all of which will have unique nominal inductance values, as seen in table 4.1. i 2A 4A 6A 8A 1A 12A L.194.187.182.175.171.161 ΔL.15.15.14.12.9.8 Table 4.1: Nominal Inductance values for the 2Hp PMSRM The nominal values for the equivalent resistance are a function of both current and the speed of the motor. ω rpm ω (rad/sec) 2A 4A 6A 8A 1A 12A 36 377. 6.25 6.25 5.87 5.12 3.99 3.61 33 345.6 5.78 5.78 5.43 4.74 3.71 3.36 3 314.1 5.31 5.31 4.99 4.36 3.42 3.11 27 282.7 4.84 4.84 4.55 3.99 3.14 2.86 24 251.3 4.36 4.36 4.11 3.61 2.86 2.61 21 219.9 3.89 3.89 3.67 3.23 2.57 2.35 18 188.5 3.42 3.42 3.23 2.86 2.29 2.1 R eq 42

15 157.1 2.95 2.95 2.79 2.48 2.1 1.85 12 125.7 2.48 2.48 2.35 2.1 1.73 1.6 Table 4.2: Nominal R eq Values for the 2Hp PMSRM 4.2.2 PI Control Design The next step in linear control design is the addition of a PI controller. The following block diagram shows the closed loop current control strategy. Each phase will need a spate controller since the error signals will be different, although the model, controller, and reference will be the same + e ia K P s + K i s 1 sl + R eq i a i + e ib K P s + K i s 1 sl + R eq i b Figure 4.4: Block Diagram of the Linear 2-phase PMSRM Current Controller The plant transfer function is represented by, G p s and the PI controller is represented by: G c s = K P s + K i s It is desirable to place the controller zero as close to the systems real pole as possible in order to cancel it out. This means that the integral gain should be selected to have the same value which is the equivalent resistance divided by the nominal inductance. That is: (4.1) K i = R eq L (4.11) In the real system the controller s zero will not cancel the system s pole out since the plant model will not be identical to the actual system. If the system pole is smaller than the controller pole then the root locus will have a part on the real axis from the pole at the origin to the controller zero and a second 43

Imaginary Axis part moving into the left hand plane away from the system pole. If the system pole is larger, then the locus will split before the zero, and then rejoin the real axis somewhere past the zero. In both cases one of the poles will go towards the zero and the other will go to negative infinity. Selecting a large proportional gain will have the same effect in either case. Root Locus 2 15 1 5-5 -1-15 -2-35 -3-25 -2-15 -1-5 Real Axis Figure 4.5: Root Locus of the Linear Electrical Model of the PMSRM The proportional gain must be large enough to ensure that the full dc voltage is commanded as soon as the initial excitation command is seen. If full voltage is not seen immediately, the rise time of the system will be limited. Inaccuracies in the linear model can be seen by simulating the linear and the nonlinear models side by side as shown in figure 4.6. 44

Phase Voltage (V) (V) Phase Current (A) (A) Phase Voltage (V) (V) Phase Current (A) (A) 2.5 2 1.5 1 i a i * i a lin.5 4 2 v a v a lin -2-4.5 1 1.5 2 2.5 3 3.5 4 4.5 Time (ms) 8 (a) 6 4 i a i * i a lin 2 4 2 v a v a lin -2-4.5 1 1.5 2 2.5 3 3.5 4 4.5 Time (ms) Figure 4.6: Small Signal Step Response of the PI Controller for Current Reference of (a) 2A and (b) (b) 6A for the linear and nonlinear model The accuracy of the linear design can be evaluated by simultaneous simulation of the linear and nonlinear models with the same PI controller. For the case in figure 4.5(a), when the current reference is 2A, the two models behave slightly different. First of all, the rise time of the 45

linear model is slower, meaning that the actual system pole is further in the left half plane. The slower rise time is also a result of the nonlinear inductance being lower than the nominal inductance at the time of excitation. The command of the current regulator is limited in the nonlinear model since the voltage cannot rise above the dc-link voltage, yet in the linear model the voltage does rise above the dc-link threshold. For the 6A current command, the transient response of the two models are very similar, however the nonlinear model looks more like a 2 nd order response and has a faster response despite having a limited voltage command. In both cases the linear approximation does not exactly match the nonlinear system response although it does provide a good general guideline for control design. 4.2.3 Anti-windup PI Control For any converter topology there will always be a limited dc bus voltage which limits the control signal which means that the control signal must be limited to V dc V V dc. Adding a saturation function to the control signal is a necessary step in implementation of the controller however it leads to a large build up of integral term (windup) as error continues to compound. As the current approaches the reference the large magnitude of the integral term produces an excessively large contribution resulting in overshoot. A solution to avoid the negative effects of saturation is to introduce an anti-windup PI controller. The anti-windup can be implemented in different ways, but the goal is to limit when the integration occurs and at what rate. One way to do this is to stop or reset the integration process once the controller is saturated. Another way to avoid windup is to initially apply a full command signal. Then when the output is within a certain threshold of the reference begin the control algorithm. Within typical operating conditions the PMSRM s PI current controller will saturate as soon as excitation begins. In addition, the inductance is increasing through the excitation range, resulting in a further increase from integrator windup. Therefore the anti-windup action can be chosen to set the integral value to zero every time the output of the controller reaches saturation. The block diagram for such a controller is shown in figure 4.7. 46

Average Voltage (V) (V) Phase Current (A) (A) Figure 4.7: Anti-windup PI Current Controller The addition of the integral limitation from the anti-windup adds additional non-linearity to the system. When the controller is in saturation the controller behaves strictly as a proportional controller. As the current waveform reaches the reference the control comes out of saturation and the integral term begins to function. After each phase excitation is over, the error signal for the integral term must be reset so as to not affect the next excitation period. 12 1 PI w/antiwindup PI i a 8 6 i b i* 4 2 4 2 v a v b -2-4 1 2 3 4 5 6 7 8 9 Time (sec) x 1-3 Figure 4.8: Simulated Current Response with Anti-Windup PI Current Control on Phase A and PI Current Control of Phase B 47