Minimum Copper Loss Flux-Weakening
Control of Surface Mounted Permanent
Magnet Synchronous Motors
Jiunn-Jiang Chen and Kan-Ping Chin, Member, IEEE
Abstract—This study presents a novel current loop design
method capable of automatic flux-weakening control with minimum copper loss for surface mounted permanent magnet synchronous motors (SPMSM). The proposed current controller can automatically re-compute the d-axis current command to defer output voltage saturation. Consequently, the motor op-erations in the flux-weakening region are also contained in the stable operating region. Analysis results indicate that since the output voltage vector in the flux-weakening region produced by this controller is consistently on the boundary of the maximum output voltage vector allowed by the inverter, the corresponding flux-weakening current is the optimal value in the sense of min-imum copper loss. This minmin-imum copper loss design differs from the maximum output torque design and the constant power design of the flux-weakening control methods found in the literature. Experimental results further demonstrate the feasibility of the proposed current controller and its ability to maximize the speed range of the motor drive for a given inverter capacity.
Index Terms—Current control, flux-weakening control, PMSM,
torque.
I. INTRODUCTION
P
ERMANENT magnet synchronous motors (PMSM) have recently been widely adopted for servo drives in industrial applications, because of their numerous attractive features such as high efficiency, superior power density, and high torque to inertia ratio. However, the servo controlled PMSMs have a limited operating range, owing to the control strategies applied and the inverter capacities installed. An inverter’s capacity is fixed during motor operations. As a result, how to apply control strategies to effectively expand the operating range under a fixed inverter capacity is an interesting topic in the field of PMSM controls. The output voltage limit of the inverter is generally a major limitation to expanding the operating range of PMSM controls. As the back emf rises with an increasing speed, the back emf cancels a portion of the input voltage, which is bound by the inverter capacity, and limits the current injected to the motor, thus limiting the maximum speed of the PMSM. To extend the operating speed range of PMSMs, numerous algorithms were proposed to accomplish flux-weak-ening control [1]–[7]. Employing a negative d-axis current toManuscript received May 21, 2001; revised March 1, 2003. Recommended by Associate Editor P. C. Luk.
J.-J. Chen is with Aetas System Incorporated Company, Hsinchu, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]).
K.-P. Chin, deceased, was with the Department of Mechanical Engineering, National Chiao-Tung University, Hsinchu 300, Taiwan, R.O.C.
Digital Object Identifier 10.1109/TPEL.2003.813751
suppress the rising back emf is a common characteristic of these flux-weakening control strategies for PMSMs. The negative d-axis current is the so-called flux-weakening current, while the extended operating region achieved by flux-weakening control is the so-called flux-weakening region. References [1]–[4] proposed several flux-weakening controls of various PMSMs, including surface mounted permanent magnet synchronous motor (SPMSM) and interior permanent magnet synchronous motor (IPMSM). These methods design flux-weakening current command schemes from the perspective of maximum torque output or constant power output under steady-state analyzes. However, these methods are extremely sensitive to the motor parameters used to compute the flux-weakening current command and the switching velocity. Moreover, even if the flux-weakening current command computed from im-precise motor parameters can still perform flux-weakening control, the original intention is violated and the operating range is reduced. Existing investigations [5]–[7] have presented flux-weakening controls of PMSMs that do not require prior knowledge of motor parameters. One of the methods proposed that the flux-weakening current command should be propor-tional to the filtered q-axis current error [5]. This method is inherently instable, because the existence of the q-axis current error in the flux-weakening region is required to maintain the generation of the flux-weakening current command. Another method indicated that the flux-weakening current command be adjusted through a proportional-integral (P-I) computation of the voltage error between the maximum output voltage vector and the output voltage vector of the current controller [6], [7]. This method can theoretically achieve effective flux-wakening control because it attempts to stop voltage saturation and expand the operating range. However, the voltage commands produced by this method must pass through the saturated voltage region. Consequently, the original output commands produced by the current control loop must be replaced by commands modified by the overmodulation strategy. This result effectively changes the current controller to a new form with unknown stability and performance characteristics, and may not achieve the original design objectives. Therefore, PI gains must be selected appropriately to ensure stability in the voltage saturation region.
Flux-weakening control should aim primarily to defer voltage saturation and expand the operating range. This study proposes a novel flux-weakening control method that differs from ex-isting flux-weakening control methods in that it is integrated into the current loop design. The proposed current controller can 0885-8993/03$17.00 © 2003 IEEE
automatically re-compute the d-axis current command to defer output voltage saturation, so that the current control loops in the normal operation region and in the flux-weakening op-eration region are coherent. Accordingly, control sta-bility and performance can be guaranteed even in the flux-weak-ening region. Furthermore, because the output voltage vector of the current controller is always on the boundary of the maximum output voltage vector during operations in the flux-weakening region, the corresponding flux-weakening current is insensitive to the motor parameters used in the controller and will be the op-timal value for minimizing copper loss [8]. This design feature differs from the maximum output torque design and the constant power design [1]–[4].
The rest of this paper is organized as follows. Section II presents the mathematical model of a SPMSM. Section III then reviews and discusses why flux-weakening control is needed to extend the operating range of motor control under a fixed inverter capacity, and also discusses how flux-weakening control can be achieved while minimizing copper loss. Mean-while, Section IV describes the proposed current controller and its implementation technique, and Section V summarizes the experimental results. Finally, conclusions are made in Section VI.
II. MOTORMODEL
The mathematical model of the electrical subsystem of a SPMSM in a synchronous frame, or the so-called frame, can be described as
(1) (2) where denotes the electrical state vector whose com-ponents are the stator currents in the frame; de-notes the input vector whose components are input voltages in the frame; is the rotor velocity; is the phase inductance; denotes the phase resistance; is the number of pole pairs;
represents the torque (permanent magnet) constant; and denotes the electromagnetic output torque.
III. REVIEW OFFLUX-WEAKENINGCONTROL
This section discusses the theoretical steady-state operable region of motor drives with a fixed inverter capacity and the essentials of flux-weakening controls. Moreover, the condi-tions required to accomplish flux-weakening control while minimizing copper loss are also revealed.
A. Theoretical Steady-State Operable Region
Inverter capacity is generally fixed during motor controls. In-verted capacity, including voltage and current constraints, can be expressed as
(3) (4)
Fig. 1. Theoretical steady-state operable region expressed on thei 0i plane.
The current constraint (3) stipulates that the magnitude of the inverter output current vector can not exceed the maximum cur-rent, , while the voltage constraint (4) stipulates that the magnitude of the inverter output voltage vector can not exceed the maximum voltage, . Generally, is determined by the inverter current rating and is constant. Meanwhile, is determined by the dc link voltage, , and forms a hexagon in the voltage vector space if a space vector pulse width mod-ulation (SVPWM) strategy is applied. For convenience, the in-side tangent circle of the hexagon is commonly used to replace the hexagon as the voltage limit. Consequently, the maximum
voltage can be expressed as .
The theoretical steady-state operable region of motor control under a fixed inverter capacity can be completely expressed by and . Assuming that the motor has reached its steady-state, the voltage equations of the motor model can be expressed as
(5) (6) By substituting (5) and (6) into (4), the voltage constraint can be re-expressed as
(7) Given a designated operating speed, , and a fixed inverter ca-pacity, and , the two constraints (3) and (7) can be plotted together on an plane, as displayed in Fig. 1. Be-cause (7) contains the variable , the voltage constraint area varies with operating speed. Actually, as illustrated in Fig. 1, the voltage constraint circle shrinks and moves to the left as the designated operating speed increases. On the other hand, the current constraint is not dependent on operating speed and the current constraint circle remains unchanged in the plane. Notably, unlike the figures of general flux weakening analyses presented in [1]–[3], Fig. 1 does not ignore the effect of stator
Fig. 2. Minimum copper loss design.
resistance. The area within the intersection of the current and voltage circles corresponds to the solutions that satisfy both in-equalities (3) and (7). This area is the theoretical steady-state operable region expressed on the plane for the desig-nated operating speed and the inverter capacity. For example, the shadowed area in Fig. 1 denotes the theoretical steady-state operable region for the operating speed, .
In addition to clearly indicating the operable region, Fig. 1 also shows the necessity of performing flux-weakening con-trol to extend the operating range of the motor concon-trol. For SPMSMs, because the d-axis current does not generate an elec-tromagnetic torque, as expressed in (2), it is usually controlled to zero under normal operations to reduce copper loss from winding resistance. However, according to Fig. 1, the regions where are not always within the operable region, espe-cially when the motor is operating with a high load torque, large , or/and high speed. Under these circumstances, the d-axis cur-rent, , must be maintained at a negative value to shift the oper-ating point laterally into the operable region (shadowed area in Fig. 1). The negative d-axis current is the so-called flux-weak-ening current, and the flux-weakflux-weak-ening control is responsible for driving the flux-weakening current such that the motor always operates inside the operable region even when operating condi-tions vary.
B. Minimum Copper Loss
To simplify the following analyzes, Fig. 1 is re-plotted to Fig. 2, which only contains the current constraint circle and voltage constraint circle for a single speed . Furthermore, the shadowed area of the operable region in Fig. 2 represents motor operation when . In Fig. 2, line segment and arc denote the minimum copper loss regions of the operable range, because operating points in these regions have the smallest mag-nitude of for a given . Line segment belongs to the normal operating region , while arc , which is on the voltage constraint boundary, belongs to the flux-weakening operating region . Consequently, a necessary condi-tion to minimizing copper loss in a flux-weakening operacondi-tion is to keep the output voltage vector of the current controller on the voltage constraint boundary in the flux-weakening region. However, if voltage constraint is considered alone, two corre-sponding values exist on the voltage constraint boundary for any given value in the flux-weakening region, corresponding to points and in Fig. 2. Obviously, reducing the magnitude of , corresponding to point in Fig. 2, will minimize copper loss. The above observations provide the foundation for
de-veloping flux-weakening current control with minimum copper loss in the following section. Moreover, if the motor is driven to remain on the intersection point between the current con-straint boundary and the voltage concon-straint boundary, for ex-ample point in Fig. 2, then the output torque generated by the motor is continuously maximized, as is the operating range for the given inverter capacity.
IV. CURRENTLOOPDESIGN
This section develops a current controller whose operating ranges can cover normal and flux-weakening operating regions. This outstanding feature comes from the ability of the proposed controller to allow re-selection of the d-axis current command to avoid overmodulation and to defer voltage saturation. Moreover, selecting d-axis current commands appropriately according to the suggestions outlined in Sections IV-C and IV-D, the current controller can drive the motor with minimum copper loss when q-axis current command is inside the operable region, and with maximum output torque when q-axis current command exceeds the operable region.
A. Current Controller Design
The design objective of the proposed current controller is to defer output voltage saturation to as late as possible. The ana-lyzes in Section III confirmed that the negative d-axis current can prevent early output voltage saturation and expand oper-ating range to the flux-weakening region. Consequently, output voltage saturation can be prevented while preserving control sta-bility, by including a term which contains the negative d-axis current command in the control laws represented by and . By applying feedback linearization method [9], the current con-trol laws embodied with the aforementioned idea are designed as
(8) (9) where and are current command inputs; and
are current control errors; and are the integration of current control errors; , , and are control gains. When the magnitude of the original computed output voltage vector exceeds that of the maximum output voltage vector, the term in (9) makes it possible to directly re-select a negative to reduce the magnitude of the
and to avoid output voltage saturation.
To analyze the control stability, we substitute control laws (8) and (9) into the electrical subsystem (1). As a result, the closed-loop error dynamics becomes
According to the singular perturbation method [10], [11], owing to the two-time-scale property possessed by small electrical ma-chines, the mechanical variable in (10) can be treated as a frozen value. Consequently, system (10) becomes a linear time-invariant system, and numerous linear analysis techniques can be applied to analyze the system. Using the Laplace
transfor-mation method and choosing ,
, , and allow us
to express the relationships between command inputs and con-trol errors as
(11)
(12) Clearly, stable poles of (11) and (12) can be assigned via properly chosen positive control gains, , , , and . Consequently, the control stability is guaranteed. Additionally, by applying the final value theorem, the steady-state error
even if is a ramp input, and even if is a parabolic input and is a ramp input. B. Digital Implementation
Before introducing the methods to select , the fundamental implementation methods of the proposed current controller should be introduced first.
Because the proposed controller is implemented digitally, the integral terms of control laws (8), (9) can be digitized as
(13) (14) where denotes the sequence index, and represents the sam-pling period of the current loop. Rewrite control laws (8), (9) by isolating the terms, we get the digitized control laws as
(15)
(16) where
In the next two subsections, methods to select are intro-duced. Section IV-C describes the method to select to achieve minimum copper loss control when is inside the operable re-gion. Section IV-D presents a method for selecting to maxi-mize output torque control when exceeds the operable region. C. Inside the Operable Region
Two rules can be applied for selecting , depending on whether the output voltage vector calculated from the control laws exceeds the maximum output voltage vector.
Situation 1: : When the magnitude of
the output voltage vector computed from (15) and (16) exceeds that of the maximum output voltage vector, is chosen to
sat-isfy the maximum voltage equation .
Substituting (15) and (16) into this maximum voltage equation, can be obtained by solving the following quadratic equation:
(17) If the solutions of (17) are complex values with nonzero imagi-nary parts, the desired exceeds both the voltage constraint re-gion and the operable rere-gion. Section IV-D suggests selections for , while exceeds the operable region. Meanwhile, if (17) has two real solutions of , the one with a smaller magnitude is chosen for minimum copper loss design as seen in (18), shown at the bottom of the page. Similarly, if the current vector com-bined by components and derived from (18) exceeds the current constraints and the operable region, the selection of needs to be dealt with according to the suggestions presented as follows in Section IV-D.
Situation 2: : When the magnitude of
the output voltage vector is smaller than or equal to that of the
maximum output voltage vector, , output
voltage saturation does not occur, and should return to a value that keeps the motor running while minimizing copper loss. is computed as
if (19)
where denotes a positive integral gain. The tuning rule of in (19) depicts that at the next cycle is modified by the
in-tegration of the voltage difference, ,
and is limited to an upper bound at 0. Basically, the tuning rule
Fig. 3. Flow chart of the proposed current controller wheni remains within the operable region.
(19) will not affect the stability of the current control. How-ever, during digital implementation the integral gain, , must be upper bounded.
Fig. 3 illustrates a flow chart of the aforementioned current controller. The methods employed above for selecting focus on running a motor inside the operable region. Under such cir-cumstances, the requested can always be achieved and the obtained by the controller will minimize copper loss. The fol-lowing subsection discusses how to select when the requested
exceeds the operable region. D. Outside the Operable Region
The requested is unobtainable if no can move the op-erating point into the operable region. In other words, the re-quested is outside the operable region. Under such circum-stances, a compromise approach is to select so as to maximize torque output in the operable region. Since the operating point that maximizes output torque can be limited by either
1) the current condition;
2) both current and voltage conditions together;
3) the voltage condition, the following three maximum torque control designs exist.
Situation 1: Limited by the Current Condition: If the oper-ating point of the maximum torque control is only limited by the
current condition, should clearly equal and should be zero.
Situation 2: Simultaneously Limited by the Current Condi-tion and the Voltage CondiCondi-tion: If the operating point of the maximum torque control is simultaneously limited by the cur-rent condition and the voltage condition, for example point
in Fig. 2, can be substituted into (17),
and (17) can then be solved combined with the constraint
equa-tion to obtain the precise values of and
that maximize output torque. However, due to the complex computation involved in solving the bi-quadratic equation, this approach is unsuitable for implementation in real-time control systems. An alternative method is suggested herein, in which the operating point gradually approaches the maximum torque point along the voltage constraint boundary. An tuning rule, similar to (19), is proposed to replace the precise value ob-tained from the bi-quadratic equation
(20) where denotes a positive integral gain; and and represent the output voltages for the initial and at the be-ginning of that cycle. Moreover, must be re-calculated each cycle to satisfy the output voltage equation, . Al-though this method cannot always maximize torque output, it is capable of maximizing torque output when the system reaches its steady state.
Situation 3: Limited by the Voltage Condition: When the op-erating point of the maximum torque control is only limited by the voltage condition, the precise values of and for maxi-mizing output torque can be directly obtained from (7) as
sgn
However, to simplify the computation, similar rules for selecting and as in Situation 2 are adopted, except that a minimum value of is added to tuning rule (20). From (7), the minimum value of for maximizing output torque is
(21) Similarly, once is selected for a given cycle, must also be re-computed for that cycle to satisfy the output voltage equation,
.
To summarize the discussions in Sections IV-C and IV-D, a complete flow chart for realizing the proposed current controller is displayed in Fig. 4. Because and are bounded inputs, the proposed current laws (8), (9) can achieve stable current control. Additionally, following the proposed current command selection rules, the motor operation always minimizes copper loss.
V. EXPERIMENTALRESULTS
Experiments are performed to verify the proposed current controller. Because flux-weakening control can expand the op-erating range of the motor, high-speed experiments under a
des-Fig. 4. Complete flow chart of the proposed current controller.
Fig. 5. Block diagram of the motor drive.
ignated inverter capacity are helpful for examining the perfor-mance of the automatic flux-weakening control of the proposed current controller. A simple PI velocity controller with 1 kHz sampling rate is implemented in these experiments. Meanwhile, the proposed current controller is implemented based on the flow chart in Fig. 4 with a sampling rate of 5 kHz. The block diagram of the motor drive is illustrated in Fig. 5, where de-notes the rotor position.
Fig. 6 illustrates the experimental setup, which includes a Sinano #7CB30-2SE6F permanent magnet synchronous motor with ratings and identified parameters as listed in Table I, the inverter of a Micro Trend UT90 Drive, a proprietary control card made in-house, and a PC. The control card converts the analog phase current measurements into digital signals, decodes the encoder signals, and generates SVPWM switching signals to control the inverter. The PC is used to compute the control
al-Fig. 6. Experimental setup.
TABLE I
RATINGS ANDIDENTIFIED PARAMETERS OF THESINANO#7CB30-2SE6F MOTOR
gorithms and the coordinate transformations among the vector space, the stator reference frame, and the rotor reference frame. Three experiments are performed, in which the DC link voltage of the inverter, , is set at 140 V, such that the motor drive will enter into the flux-weakening region when the motor is operated at high speeds. According to the motor parameters listed in Table I and the given , the minimum speed that needs flux weakening can be calculated at 3311 rpm. Thus, flux-weakening control becomes necessary when the operating speed exceeds 3311 rpm. Additionally, the maximum current limit of inverter, , is set at 2A.
First, a smooth velocity tracking experiment is performed, and the experimental results are shown in Fig. 7. In this experi-ment, velocity command, as illustrated in Fig. 7(a), is a smooth trapezoidal trajectory that ranges between 3000 rpm and 4000 rpm. Under this velocity command, according to the experi-mental results, is always inside the operable region. Conse-quently, the computation of the current controller follows the flow chart illustrated in Fig. 3. As exhibited in Fig. 7(c), the response of is negative in the high-speed range and returns to zero in the low speed range. Moreover, the output voltage
ratio, , closely approaches one in the
flux-weakening region, as shown in Fig. 7(b). The analysis in Section III indicates that the response of displayed in Fig. 7(c) always minimizes copper loss.
Further experiments are conducted by changing the initial ve-locity command into a step input, and the experimental results
Fig. 7. Results of the flux-weakening operation experiment for the smooth velocity tracking command. (a) velocity command and velocity; (b) output voltage ratio = v + v =V ; (c) d-axis current; (d) q-axis current; (e) current ratio = i + i =I .
Fig. 8. Results of the flux-weakening operation experiment for the velocity tracking command whose initial command is a step. (a) velocity command and velocity; (b) output voltage ratio = v + v =V ; (c) d-axis current; (d) q-axis current; (e) current ratio = i + i =I .
are presented in Fig. 8. Maximum output torque control limited only by the current condition, as discussed in Section IV-D, Sit-uation 1, present in the initial accelerated region. As shown in Fig. 8(d), (e), is close to 2A in the initial accelerated region, and the current ratio, , also closely ap-proaches one in that region.
An additional experiment is conducted to determine whether the proposed current controller can efficiently employ the max-imum inverter capacity, , to drive the motor. Fig. 9 shows the experimental results. The initial step velocity com-mand is up to 5000 rpm, as exhibited in Fig. 9(a), which is outside the operable region of the motor for the given inverter capacity. Under such circumstances, according to previous
an-Fig. 9. Results of the flux-weakening operation experiment for the velocity tracking command whose high speed part exceeds the operable range. (a) velocity command and velocity; (b) output voltage ratio
= ( v + v =V ); (c) d-axis current; (d) q-axis current; (e) current
ratio = ( i + i =I ).
alyzes, the proposed current controller should perform a max-imum output torque control. Fig. 9(b), (e) clearly displays that the maximum output torque control in the high-speed region is simultaneously limited by both the current and the voltage conditions, as discussed in Section IV-D, Situation 2, where the output voltage ratio, , and the current ratio, , are both close to one. Because the flux-weakening current always minimizes copper loss in the motor, the experimental results imply that the current controller can efficiently employ the maximum inverter capacity to drive the motor even in the flux-weakening region. Consequently, the highest speed in Fig. 9(a) is the highest oper-ating speed for the motor driver for the given inverter capacity.
VI. CONCLUSION
This study presents a novel current loop design method that can achieve automatic flux-weakening control with minimum copper loss for SPMSM. The outstanding characteristics of the proposed current controller come from the control law (9), which contains a term. This term allows to be directly reselected to avoid output voltage saturation while preserving control stability. As a result, the proposed current controller can achieve an active and stable flux-weakening control. Moreover, the proposed current controller always keeps the output voltage vector on the voltage constraint boundary in the flux-weakening region, indicating that the flux-weakening current is constantly optimized for minimizing copper loss. Additionally, when the requested exceeds the operable region, the proposed current controller can drive the motor to achieve maximum output torque control. Experimental results demonstrated the feasibility of the proposed current controller, and verified that the proposed controller can efficiently employ the inverter capacity to drive the motor to the maximum operating speed possible for a given inverter capacity.
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Jiunn-Jiang Chen was born in Kaohsiung, Taiwan,
R.O.C., in 1972. He received the B.S. and the Ph.D. degrees from National Chiao-Tung University, Hsinchu, Taiwan, in 1995 and 2000, respectively.
He then joined the Aetas System Incorporated Company to work in the field of motor driving technique and imaging technique on imaging apparatus. His current research interests include power electronics, identification, signal processing, and electrical machine control.
Kan-Ping Chin (S’89–M’92) was born in Taipei,
Taiwan, R.O.C., on June 30, 1960. He received the B.S. degree in mechanical engineering from National Taiwan University, Taiwan, R.O.C., in 1982 and the M.S. and Ph.D. degrees from the Massachusetts Institute of Technology, Cambridge, in 1988 and 1991, respectively.
He was a faculty member at the National Chiao-Tung University, Taiwan, from 1992 to 2002. His premature and unanticipated death on February 8, 2002 was a loss not only to those who loved and admired him but to his burgeoning research on control of servo motors and microelectromechanical systems (micromotors).
