永磁电机转矩常数深度分析

Abstract —The torque con stan t, together with the back-EMF co n sta n t, was origi n ally used i n perma n e n t mag n et DC commutator motors (PMDC motors) to couple the electric circuit equation s with mechan ical equation s. But it is still an open question whether the con cept of the torque con stan t an d the back-EMF con stan t can be applied to brushless DC (BLDC) motors an d perman en t magn et (PM) AC machin es. This paper presen ts an in -depth study of the two con stan ts un der various real conditions in PM machines. The torque constant at various load con dition s is computed usin g tran sien t 2D fin ite elemen t an alysis (FEA). It is shown that the torque con stan t is n ot a constant for BLDC motors and PM AC machines.
Index Terms —Back-EMF constant, brushless DC motors, DC commutator machi n es, fi n ite eleme n t a n alysis, perma n e n t magnets, synchronous machines, torque constant.
I. I NTRODUCTION
HE torque constant is defined as the ratio of the torque
delivered by a motor to the current supplied to it, and the back-EMF constant is the ratio of voltage generated in the winding to the speed of the rotor. In PMDC motors, they are almost constant at various load conditions. The torque constant and back-EMF constant couple the electric circuit equations with mechanical equations, and are widely used in motor control.
It is of great interest to see whether the concept of the torque constant and the back-EMF constant can be applied to BLDC motors and PM AC motors. Some effort have been made in this regard [1][2]. Reference [2] indicates that an ideal BLDC motor (also called a square-wave motor), under the condition that the line-to-line back EMF waveform is trapezoidal and that the winding current waveform is ideally square, is electrically identical to a PMDC motor. The author also applies the concept of the torque constant and the back-EMF constant to sine-wave PM AC motors under the assumption that the internal power-factor angle between the back EMF and the current is fixed to zero.
However, in real cases, the winding current waveform is far from the ideal square-wave in BLDC motors due to current freewheeling. And, in PM synchronous motors, the internal power-factor angle is normally not zero because the torque angle is automatically adjusted according to the change
The authors are with Ansoft Corporation, Pittsburgh, PA 15219 USA
(phone: 412-261-3200; e-mail: dlin@, ping@,
zol@). in load. To this end, this paper presents an in-depth study of the torque constant and the back-EMF constant for BLDC motors and PM AC motors. The suitability of the use of the two constants in PM motors is discussed considering the following: current freewheeling, arbitrary back-EMF waveforms, salient pole, variable pulse width and trigger angle, and internal power factor angle.
II. R EVIEW OF THE T ORQUE C ONSTANT IN PMDC M OTORS In PMDC motors, the electric circuit equation is
b a s V I R E V ++=
(1)
where V s is the applied DC voltage source, E is the back EMF, V b
is the voltage drop of one-pair brushes, I is the input DC
current, and R a is the armature resistance. Equation (1) can be coupled with load mechanical equations by introducing
⎩⎨
⎧==I k T k E T m
m
E ω (2)
where ωm is the angular velocity in mechanical rad/s, T m is the
electromagnetic (air-gap) torque in Nm, k E is the back-EMF constant in Vs/rad, and k T is the torque constant in Nm/A. The torque constant and the back-EMF constant have the following properties:
i. k T = k E in the metric unit system; ii. k T and k E are constant; iii. k T and k E are measurable.
Property (i) is obvious from the fact that the electric power (EI ) is equal to the mechanical power (T m ωm ) during power conversion.
Property (ii) follows since: (1) PMDC motors have large air gaps due to surface mounted magnets, thus the saturation change caused by the armature reaction is negligible; (2) the brush position is mechanically fixed during operation even if it is adjustable; (3) the current in each coil completes commutating within the angle of the brush width, and the commutating duration is independent of the rotor speed; and (4) there is no reluctance torque even if the armature reaction is not aligned with the q-axis.
Based on property (ii), the back-EMF constant k E can be measured at no-load condition operating in generator mode. The torque constant k T can be obtained directly from k E , or
can be measured at load operation. It is straightforward to predict the performance of PMDC motors from (1) and (2) in motor control. In-Depth Study of the Torque Constant for
Permanent Magnet Machines
D. Lin, P. Zhou and Z. J. Cendes
T
©2008 IEEE.
III. T ORQUE C ONSTANT IN BLDC M OTORS
Even though the torque constant and the back-EMF constant in BLDC motors are defined in the same way as those in PMDC motors as shown in (2), there are some essential differences regarding the torque constant and the back-EMF constant between BLDC motors and PMDC motors. For the sake of easy discussion, take a Y-connected three phase winding with bridge-type inverter as an example, as shown in Fig. 1. The trigger pulse width for each branch is 120 electrical degrees in turn and the inverter has 6 repeatable
operating states with the state period of 60 electrical degrees.
Fig. 1. Y-connected three-phase windings with the bridge-type inverter
A . Voltage equation (1) is no longer applicable The voltage equation (1) is no longer applicable in BLDC
motors due to the inductance voltage drop. In PMDC motors, the inductance induced voltage caused by the current commutating will not contribute to the voltage drop across the brush terminals. However, in BLDC motors, the inductance voltage drop becomes comparable with the resistance voltage drop.
B. k T and k E are no longer constant
In BLDC motors, E used in (2) is the average back EMF across the DC link, and its value will vary with the current freewheeling duration. In Fig. 1, assume at the previous operating state, the source voltage V s is applied to winding terminals AC via branches 1 and 2, and at the current operating state, V s is applied to winding terminals BC via
branches 3 and 2. When branch 1 is off, the phase-A current
freewheels through branch 4, which makes winding A to connect in parallel with winding C. If the voltage drop across
the conducting transistor in branch 2 is the same as that across
the freewheeling diode in branch 4, the average back EMF
during the current operating state is
])(21[10∫∫++=s
f f T T BC T BC BA s dt e dt e e T E
(3) where, e BC and e BA are instantaneous line-to-line induced
voltages, T s is the state period in second (corresponding to 60
electric degrees), and T f is the current freewheeling duration,
as shown in Fig. 2. It is obvious from (3) that the average back
EMF varies with the current freewheeling duration, and
therefore k E
is not constant for various operations.
Fig. 2. Rectified back EMF from trapezoidal line-to-line induced voltages
For the circuit of Fig. 1, as long as T f < T s , the freewheeling currents always reduce the input DC current and increase the delivered torque, and therefore, k T varies with the current freewheeling duration which in turn varies with the rotor speed.
Another case in which k T is not constant is, in interior permanent magnet (IPM) motors, the reluctance torque component also contributes to the air-gap torque due to the salient-pole effects, and the reluctance torque component is not linearly proportional to the DC current. Furthermore, the trigger angle and the pulse width of the controlling signals in BLDC motors are usually controllable. This is also a case
where k T is not constant.
Fig. 3 shows the variation of k T with the speed of a typical surface mounted BLDC motor with fixed trigger angle and
pulse width.
Fig. 3. Variation of k T with the rotor speed C . k T is no longer equal to k E
In BLDC motors, the back EMF across DC link normally includes ripples associated with arbitrary line-to-line back-EMF waveforms. The ripples become considerable due to the
current freewheeling even though the line-to-line induced voltage may have a flat waveform in 60 electric degrees by a
special design (see the solid lines inside T s in Fig. 2). The
input current also contains significant ripples because the
freewheeling current is in nature of “generator” current. By
examining the power conversion, one gets
∫⋅⋅=s T s m m dt i e T T 0
1ω
(4)
∫⋅∆⋅∆+
=s
T s
dt i e T EI 01
where, ∆e and ∆i are the ripples of the DC back EMF and the
input current, respectively. From (4), one concludes that at
load conditions k T ≠ k E because T m ωm ≠ EI .
D . k
E is no longer measurable By measuring the air-gap torque (which is obtained from the load torque and the mechanical loss) and the DC component of the input current at load operation, k T can be determined. However, k E is no longer measurable at load conditions for BLDC motors. It cannot be measured by driving the motor as a generator and rectifying the line voltage with a rectifier as described in [2] because k E at load conditions is different from that at the no-load condition. Also it cannot directly be obtained from k T because k E ≠ k T at load conditions.
IV. T ORQUE C ONSTANT IN PM AC M OTORS
The torque constant in PM AC motors can be defined as the ratio of the torque to the peak value of the input AC phase
currents I peak , and the back-EMF constant is the ratio of the
peak value of the induced phase voltages E peak to the speed of the rotor, as expressed below [2] ⎩⎨⎧==peak
T m m
E peak I k T k E ω. (5) Most PM AC motors operate as synchronous motors. In PM synchronous motors, the internal power factor angle ϕ i , the angle between the back EM
F phasor and the current phasor, is automatically adjusted based on the mechanical load and is normally not zero. In these cases, the delivered mechanical power is E peak peak T m m k E I k T /⋅=ω
i rms rms i E T E mI mk k ϕϕcos cos 2⋅= (6) where I rms and E rms denote RMS values of sine-wave phase
current and back EMF, and m is the number of phases. For the
power conversion, the mechanical power must be equal to the
electric power, that is
m m T ωi rms rms E mI ϕcos =. (7) As a result
i E T k m
k ϕcos 2
=
. (8)
One concludes from (8) that k T is not constant for PM synchronous motors even though K E may be constant when the saturation effects can be ignored. It varies with the internal power angle which in turn varies with the mechanical load.
Equation (8) is derived under the assumption that the spatial harmonics of the air-gap magnetic fields produced by
the permanent magnets and the phase currents are ignored. In
order to show the effects of the spatial field harmonics on the
torque constant, a three-phase 4-pole PM synchronous
machine, as show as in Fig. 4, is analyzed using 2D transient
finite element method (FEM). To focus on observing the
variation of the torque constant with the internal power factor
angle, the change in saturation caused by armature currents is
ignored, and thus linear materials are used for all components.
Fig. 4. The one-pole geometry layout of the three-phase 4-pole PM synchronous
machine Three-phase windings are applied with DC currents as follows
⎪⎩⎪⎨⎧−=−==IAm I IAm I IAm
I C
B A *5.0*5.0 (9) where IAm is set to be 0 and 1A via parametric analysis. The
rotor speed is set to be 1500rpm, and the rotor initial position is set to such a position that the phase-A winding has positive maximum induced voltage at time = 0. The computed torques at IAm = 0 and 1A are shown in Fig. 5. It can be seen from Fig. 5 that the torque at IAm = 1A consists of two components: one is the component produced
by the phase currents, and the other is the cogging torque component which is produced by the permanent magnets at 0
phase currents. Because linear materials are used, the torque component produced by the phase currents can be directly
derived from the result of the torque at IAm = 1A minus the
torque at IAm = 0, as shown in Fig. 6. By definition, the curve
in Fig. 6 shows the torque constant because the torque is
produced by unit phase currents. One notes that the torque
constant is not a constant as had been anticipated and is
therefore not suitable for use with PM AC machines.
Fig. 5. Torques at different phase currents varying with the internal power factor angle ϕ i (time=20ms corresponds to ϕ i =360 electric degrees)
Fig. 6. Torque produced by unit phase current varying with the internal power factor angle ϕ i (time=20ms corresponds to ϕ i =360 electric degrees)
V. C ONCLUSION
The torque constant and the back-EMF constant which were originally used in PMDC motors are generally not suitable for BLDC motors and PM synchronous motor analysis. Detailed computations of both constants with real motors reveal that they are no longer constant but, instead, vary significantly with load conditions.
R EFERENCES
[1]Electro-Craft Handbook, Fifth Edition, August 1980, ISBN 0-960-1914-0-
2.
[2]J.R. Hendershot Jr, and T. J. E. Miller, Design of Brushless Permanent
Magnet Motors, Magna Physics Publishing and Clarendon Press, Oxford, 1994.
Din gshen g Lin received his B.S. and M.S. degrees in Electrical Engineering from Shanghai University, Shanghai, China, in 1982 and 1987, respectively. He is currently a Senior Research and Development Engineer at Ansoft Corporation, Pittsburgh, PA. Before he joined Ansoft in 1999, he was an Associate Professor of electrical engineering at Shanghai University. His research interests include design and optimization techniques of electrical machines and electromagnetic field computation. He received the third prize of the Chinese National Award of Science and Technology, in 1987, and two second prizes of the Shanghai City Award of Science and Technology, in 1986 and 1989.
Ping Zhou received his M.S. degree from Shanghai University, China in 1987 and his Ph.D. degree from Memorial University of Newfoundland, Canada in 1994. He was with Shanghai University as a lecturer after his undergraduate study in the same university in 1977. He was a Visiting Scholar of Memory University of Newfoundland from 1989 to 1991. Since 1994, he jointed Ansoft Corporation in the R&D department. Currently, he is the manager of Electromechanical R&D group at Ansoft. His research interests include finite element numerical field computation, circuit coupling, multi-physics coupling and electrical machine modeling.
Zoltan Cendes is Founder and Chairman of Ansoft Corporation, Pittsburgh, PA, and is an Adjunct Professor at Carnegie Mellon University, Pittsburgh, PA. In addition to his role at Ansoft, Dr. Cendes has served as a Professor of Electrical and Computer Engineering at Carnegie Mellon University, as an Associate Professor of Electrical Engineering at McGill University, Montreal, Canada, and as an Engineer with the Corporate Research and Development Center of the General Electric Company in Schenectady, NY. Dr. Cendes received his M.S. and Ph.D. degrees in Electrical Engineering from McGill University and his B.S.E. degree from the University of Michigan.。