永磁电机转矩常数的深度课件
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 NTRODUCTIONHE torque constant is defined as the ratio of the torquedelivered 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 changeThe 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 isb a s V I R E V ++=(1)where V s is the applied DC voltage source, E is the back EMF, V bis the voltage drop of one-pair brushes, I is the input DCcurrent, and R a is the armature resistance. Equation (1) can be coupled with load mechanical equations by introducing⎩⎨⎧==I k T k E T mmE ω (2)where ωm is the angular velocity in mechanical rad/s, T m is theelectromagnetic (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 , orcan 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 forPermanent Magnet MachinesD. Lin, P. Zhou and Z. J. CendesT©2008 IEEE.III. T ORQUE C ONSTANT IN BLDC M OTORSEven 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 repeatableoperating states with the state period of 60 electrical degrees.Fig. 1. Y-connected three-phase windings with the bridge-type inverterA . Voltage equation (1) is no longer applicable The voltage equation (1) is no longer applicable in BLDCmotors 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 constantIn 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 viabranches 3 and 2. When branch 1 is off, the phase-A currentfreewheels through branch 4, which makes winding A to connect in parallel with winding C. If the voltage drop acrossthe conducting transistor in branch 2 is the same as that acrossthe freewheeling diode in branch 4, the average back EMFduring the current operating state is])(21[10∫∫++=sf 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 inducedvoltages, T s is the state period in second (corresponding to 60electric degrees), and T f is the current freewheeling duration,as shown in Fig. 2. It is obvious from (3) that the average backEMF varies with the current freewheeling duration, andtherefore k Eis not constant for various operations.Fig. 2. Rectified back EMF from trapezoidal line-to-line induced voltagesFor 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 casewhere 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 andpulse width.Fig. 3. Variation of k T with the rotor speed C . k T is no longer equal to k EIn 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 thecurrent freewheeling even though the line-to-line induced voltage may have a flat waveform in 60 electric degrees by aspecial design (see the solid lines inside T s in Fig. 2). Theinput current also contains significant ripples because thefreewheeling current is in nature of “generator” current. Byexamining the power conversion, one gets∫⋅⋅=s T s m m dt i e T T 01ω(4)∫⋅∆⋅∆+=sT sdt i e T EI 01where, ∆e and ∆i are the ripples of the DC back EMF and theinput current, respectively. From (4), one concludes that atload conditions k T ≠ k E because T m ωm ≠ EI .D . kE 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 OTORSThe torque constant in PM AC motors can be defined as the ratio of the torque to the peak value of the input AC phasecurrents I peak , and the back-EMF constant is the ratio of thepeak value of the induced phase voltages E peak to the speed of the rotor, as expressed below [2] ⎩⎨⎧==peakT m mE 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 EMF 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 /⋅=ωi 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 phasecurrent and back EMF, and m is the number of phases. For thepower conversion, the mechanical power must be equal to theelectric power, that ism m T ωi rms rms E mI ϕcos =. (7) As a resulti E T k mk 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 bythe permanent magnets and the phase currents are ignored. Inorder to show the effects of the spatial field harmonics on thetorque constant, a three-phase 4-pole PM synchronousmachine, as show as in Fig. 4, is analyzed using 2D transientfinite element method (FEM). To focus on observing thevariation of the torque constant with the internal power factorangle, the change in saturation caused by armature currents isignored, and thus linear materials are used for all components.Fig. 4. The one-pole geometry layout of the three-phase 4-pole PM synchronousmachine Three-phase windings are applied with DC currents as follows⎪⎩⎪⎨⎧−=−==IAm I IAm I IAmI CB A *5.0*5.0 (9) where IAm is set to be 0 and 1A via parametric analysis. Therotor 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 producedby the phase currents, and the other is the cogging torque component which is produced by the permanent magnets at 0phase currents. Because linear materials are used, the torque component produced by the phase currents can be directlyderived from the result of the torque at IAm = 1A minus thetorque at IAm = 0, as shown in Fig. 6. By definition, the curvein Fig. 6 shows the torque constant because the torque isproduced by unit phase currents. One notes that the torqueconstant is not a constant as had been anticipated and istherefore 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 ONCLUSIONThe 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 PermanentMagnet 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.。
《永磁同步电机》幻灯片PPT
3 2
N3(iB
iC)
iiN N32
1 0
1 2 3 2
1 2
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PMSM电机的FOC控制策略
考虑变换前后总功率不变，可得匝数比应为 N 3 2
N2 3
可得
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21 30
1 2 3 2
1 2
3 2
iiiC BA
坐标系变换矩阵：
C3/2
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1
3 0
1 2 3 2
1 2
3 2
C 2/3
1
2 3
1 2
1 2
0
3
2
3 2
PMSM电机的FOC控制策略
如果三相绕组是Y形联结不带零线，那么有
iAiBiC0
于是
3
i i
2 1
2
0 2
iA iB
2
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PMSM电机的FOC控制策略
〔2〕Park〔2s/2r〕变换
U1
VF1
VF3
VF5
H1
译
A
码
H2
电
B
H3
路
VF4
VF6
VF2
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Y联结三三通电方式的控制原理图
PMSM和BLDC电机的工作原理
vab
0
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Y联结三三通电方式相电压和线电压波形
t
a)
VF6VF1VF2导通时合成转矩
Tc 2
b) VF1VF2VF3导通是合成转矩
《永磁电机设计》PPT模板课件
表1-3 铁氧体永磁材料牌号及其主要磁性能
牌号
剩余磁感应强 度 Br
T kGs
磁感应强度 矫顽力 H c
kA/ m
kOe
内禀矫顽力
H cJ
kA/ m
kOe
最大磁能积
(BH)max
kJ/m3
MG·O e
Y8T Y10T Y15 Y20 Y23 Y25 Y28 Y32
0.2~0.235 ≥0.2
0.28~0.36 0.32~0.38 0.32~0.37 0.36~0.40 0.37~0.40 0.40~0.42
大部分稀土永磁的退磁曲
线全部为直线，回复线与退磁 曲线相重合，可以使永磁电机 的磁性能在运行过程中保持稳 定，这是在电机中使用是最理 想的退磁曲线。
图1－4 （b） 回复线
3、内禀退磁曲线
磁性材料在外磁场作用下被磁化后产生的内在磁感应强度，称为 内禀磁感应强度 B i ，又称为磁极化强度 J 。
J 0M
式中，M为磁化强度（A/m）
（1－3）
由铁磁学理论可知，在磁性材料中 B = 0M+ 0H
在均匀的磁性材料中，上式的矢量和可改成代数和
（1－4）
B i 0MB0H
若取绝对值，则式（2-5）可改写成
Bi B0H
（1－5） （1－6）
描述内禀磁感应强度Bi (J )与磁场强度 H关系的曲线 Bi f(H)是表征
B rt1 B rt(0 11IL 0 ) 10 1 B0(rt10 t0)
（1－11）
式中，IL和 Br 取绝对值。
（2）磁稳定性是指在施加外磁场条件下永磁体磁性能发 生变化的情况。
理论分析和实践证明，一种永磁材料在工作温度时的 内禀矫顽力 H cJt 越大，内禀退磁曲线的矩形越好（或者说 H K 越大），则这种永磁材料的磁稳定性越高，即抗外磁 场干扰能力越强。
永磁电机转矩常数的深度
In Depth Study of the Torque Constant for Permanent-Magnet Machines
指導老師：黃昌圳 學生：陳育俊
摘要
介紹 PMDC馬達的轉矩常數回顧 BLDC馬達的轉矩常數 PM交流馬達的轉矩常數 轉矩常數的其他定義方式 結論
輸入電流因為也包含顯著的漣波。藉由功率轉 換的定理，可以得到
1 Tmm Ts
0
Ts
1 e i dt EI Ts
0
Ts
e i dt
(4)
Δe和Δi是直流反電動勢和輸入電流的漣波，相 對地，從由上式，得知因為Tmωm≠EI所以在負 載情況kT≠kE。
D. kE在負載下不再被精確測量
交流永磁馬達的轉矩常數被定義為轉矩與交流 輸入相電流Ipeak的峰值的比例，並且反電動勢 常數為感應相電壓Epeak與動子轉速的比例
E peak k E m T k I m T peak
(5)
大部分交流永磁馬達操作如同同步馬達。在永 磁同步馬達，內部的功率因素角ψi是反電動勢 相量和電流相量之夾角，從機械負載的觀點， 夾角會隨著負載自動被調整，因此一般不是零。 在這些情況，傳送的機械功率為
(13)
R1是相繞組電阻，Ld和Lq是在dq軸上的繞組同步 電感，p代表d/dt，並且 . eq -md -(nppmd) m (14)
λmd是由永磁轉換成d軸的繞組磁通交鏈，ωm是 rad/s的機械角速度，npp是馬達的極對數。 以N· m的轉矩為 Tm n pp (Lq - Ld )idiq - mdiq (15) 大部分的表面型永磁馬達 Lq=Ld，因此轉矩為
永磁电机 ppt课件
12
i
电流入
A× Z
×
Y
0
ωt
C×
B
电流出
X
ωt=0时电流和磁场情况
A、C两相电流t=0时为正，因此首端流入、末端流出。 B相电流t=0时为负，末端流入、首端流出。 相邻线圈电流流向一致，在气隙中生成合成磁场。
PPT课件
13
i ωt =120°
0
ωt
ωt =120°时电流和磁场情况
A× Z×
Y
×B CX
11
实际三相电动机
的旋转磁场是如 何产生的呢？
三相异步电动机的三相定子绕组以互隔1200的方式嵌放在定子铁 芯中。当三个绕组分别接入三相交流电后，便可以产生旋转磁场。
i
A
Z
Y
0
ωt
B
C
X
规定：电流为正值时，电流从绕组首端流入，从 末端流出；电流为负值时，电流从绕组末端流入， 从首端流出。
PPT课件
1）变压器电动势：线圈与磁场相对静止，只有磁通变化。
2）运动电动势：线圈与磁场相对运动，引起磁通变化。 e=Blv 符合右手定则。
PPT课件
8
PPT课件
9
PPT课件
10
前面，我们回顾了电磁感应方面的一些基础知识，下面我们在此基础上开始了解电机 方面的理论知识，首先，来了解下右手定则。
PPT课件
1、直线电流 产生的磁场
2、环形电流 产生的磁 场
PPT课件
4
三、磁场对电流的作用
1、通电导体在磁场中会受到力的作用，电 磁力的方向符合左手定则。
左手定则 伸开左手，四指并拢，拇指与四指垂直，并 且在同一平面里，让磁感线垂直穿过手心，使 四指指向电流方向，这时大拇指所指的方向就 是通电导线在磁场中所受磁场力的方向。
永磁电机 ppt课件
BLDC电机
图1-5 如此不断改变两头螺线管的电流方向，内转子就会不停转起来了。改变 电流方向的这一动作，就叫做换相（commutation）。注意：何时换相只与 转子的位置有关，而与转速无关。 以上是两相两级无刷电机的工作原理，，下面我们来看三相两极无刷电 机的构造。
直流无刷电机 基本原理
BLDC电机
在了解永磁电机之前，为便于理解永磁电机 工作的基本原理，我们先简要回顾一下电磁感应 方面的一些基本理论及我们比较熟悉的普通三相 异步电动机和单相异步电动机与直流电机的基本 工作原理。通过对比分析，有助于我们快速的掌 握永磁电机的基本工作原理。首先，来了解一下 电磁感应方面的一些基础知识。
即：B=
F IL
B：均匀磁场的磁感强度（T） F：通电导体受到的电磁力（N） I：导体中的电流强度（A） L：导体在磁场中的有效长度（m）
9
3、一匝匝数为N的线圈在磁场中，若与线圈交链
的磁通Φ发生变化，则线圈上会感应出电动势e,称为 电磁感应, e的正方向与Φ符合右手螺旋定则。
E=-N△φ/△t
上式的含义是指，电磁感应的电动势与线圈匝数和磁通的变化率成正比。 负号是指在感应电动势作用下而在线圈里产生的感应电流所产生的Φ′将 逆着Φ变化。
空间对称嵌放。起动绕阻与电容C串联，使起动绕组
电流i2和工作绕组电流i1产生90°的相位差，即：
C i2
i i1 i2 i1
A
A
A
ωt Y
BY
BY
B
M
0 45°90°
360°
1～
X
X
X
(a) 电路图
(b) 波形图
t 0 t90 t180
加入起动绕组后，和工作绕组并联连接于单相交流电源上。
永磁电机转矩常数的深度共41页文档
31、别人笑我太疯癫，我笑他人看不 穿。(名 言网) 32、我不想听失意者的哭泣，抱怨者 的牢骚 ，这是 羊群中 的瘟疫 ，我不 能被它 传染。 我要尽 量避免 绝望， 辛勤耕 耘，忍 受苦楚 。我一 试再试 ，争取 每天的 成功， 避免以 失败收 常在别 人停滞 不前时 ，我继 续拼搏 。
33、如果惧怕前面跌宕的山岩，生命 就永远 只能是 死水一 潭。 34、当你眼泪忍不住要流出来的时候 ，睁大 眼睛， 千万别 刻变得 清澈明 晰。盐 。注定 要融化 的，也 许是用 眼泪的 方式。
35、不要以为自己成功一次就可以了 ，也不 要以为 过去的 光荣可 以被永 远肯定 。
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71、既然我已经踏上这条道路，那么，任何东西都不应妨碍我沿着这条路走下去。——康德 72、家庭成为快乐的种子在外也不致成为障碍物但在旅行之际却是夜间的伴侣。——西塞罗 73、坚持意志伟大的事业需要始终不渝的精神。——伏尔泰 74、路漫漫其修道远，吾将上下而求索。——屈原 75、内外相应，言行相称。——韩非
永磁同步电机的模型和方法课件
电流方程
电流方程描述了PMSM的定子 电流与转子位置之间的关系。
电流方程通常表示为：I = Iq×sin(θr) + Id×cos(θr)，其中 I是电流矢量，Iq是定子电流矢 量，Id是直轴电流矢量，θr是转
子位置角。
该方程反映了随着转子位置的变 化，定子电流矢量的变化情况。
磁链方程
磁链方程通常表示为：Ψ = L0×I + L1×(θr)，其中Ψ 是磁通链数，L0和L1是与电机结构有关的常数，θr 是转子位置角。
06 参考文献
参考文献
01
总结词
详细描述了PMSM的数学模型和等效电路模型，并给出了仿真结果和实
验结果。
02 03
详细描述
本文介绍了永磁同步电机的数学模型和等效电路模型，通过仿真和实验 验证了模型的准确性和有效性。该文还对PMSM的控制器设计进行了详 细讨论，为PMSM的控制提供了理论依据。
总结词
磁链方程描述了PMSM的磁通链数与转子位置角之间 的关系。
该方程反映了随着转子位置的变化，磁通链数的变化 情况。
转矩方程
转矩方程描述了PMSM的输出转矩与定子电流之间的关系。
转矩方程通常表示为：T = (P/2π)×(θr×Iq)，其中T是输出转矩，P是电机极对数，θr是转 子位置角，Iq是定子电流矢量中的直交分量。
永磁同步电机的发展趋势和挑战
发展趋势
随着技术的不断发展，永磁同步电机将朝着更高效率、更高可靠性、更小体积和更低成本的方向发展 。同时，随着智能制造和物联网技术的快速发展，永磁同步电机的智能化和网络化也将成为未来的发 展趋势。
挑战
尽管永磁同步电机具有许多优点，但在高温、高湿、高海拔等恶劣环境下运行时，仍存在一些挑战。 例如，高温会导致永磁材料性能下降，高湿会使电机腐蚀生锈，高海拔会使电机功率下降等。因此， 提高永磁同步电机的环境适应性是当前面临的重要问题之一。
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電樞反應影響的飽和度變化可以被忽略 。
b)在運作時電刷位置被機械固定即使他可以被調整 。 c)在電刷寬度的角度內每個線圈的電流完成換相，並
且換流持續時間不受轉子速度的影響。
d)即使電樞阻抗並沒有排列在q軸，也沒有磁阻轉矩。
由特性3)的觀點，反電動勢kE可以在發電機模 式無載情況下被測量，轉矩常數kT可以直接從 kE獲得，或者從負載操作下獲得。
A. 固定電樞磁場旋轉轉子
三相繞組是用於直流由下表示
IIba
Im -0.5
Im
(9)
Ic -0.5 Im
Im經過參數分析設定為0和1A。轉子轉速設定 為1500rpm，轉子的起初位置設定在a相繞組在
t=0時有正的最大感應電壓的位置。
在Im為0和1A時，轉矩顯示下圖。
我們可以看到Im=1A的轉矩包括兩個成分： 其中一個成分由相電流產生，另一個頓轉轉矩
永磁电机转矩常数的深度
摘要
介紹 PMDC馬達的轉矩常數回顧 BLDC馬達的轉矩常數 PM交流馬達的轉矩常數 轉矩常數的其他定義方式 結論
介紹
Tm=kTI。
E=kEωm。 kT和kE 將電路方程式與機械方程式結合一起，
並且廣泛使用在馬達運動控制。 兩個常數使用在PM馬達必須討論以下：
馬達控制可以很簡單的從上面兩式預測永磁直
流馬達的特性。
BLDC馬達的轉矩常數
1
3
5
6
2
4
6
ab ac bc ba ca cb
A. 電壓方程式再也不能應用
電壓方程式在無刷直流馬達因為自感壓降再也 不被應用。在永磁直流馬達，由電流換相造成 的自感感應電壓並非有助於電刷兩端的電壓降。 然而，在無刷直流馬達，自感壓降可以比擬成 電阻壓降。
輸入電流因為也包含顯著的漣波。藉由功率轉 換的定理，可以得到
Tmm
1 Ts
Ts e i dt EI 1
0
Ts
Ts e i dt
0
(4)
Δe和Δi是直流反電動勢和輸入電流的漣波，相 對地，從由上式，得知因為Tmωm≠EI所以在負 載情況kT≠kE。
D. kE在負載下不再被精確測量
交流永磁馬達的轉矩常數被定義為轉矩與交流
輸入相電流Ipeak的峰值的比例，並且反電動勢 常數為感應相電壓Epeak與動子轉速的比例
Epeak kEm
Tm
k T I peak
(5)
大部分交流永磁馬達操作如同同步馬達。在永
磁同步馬達，內部的功率因素角ψi是反電動勢 相量和電流相量之夾角，從機械負載的觀點，
電流飛輪二極體、任意的反電動勢波形、凸極、 任意的脈波寬度和觸發角和內部功率因素角度。
PMDC馬達的轉矩常數回顧
在PMDC馬達，電路方程式為 Vs=E+RaI+Vb (1)
Vs是供應直流電壓源，E是反電動勢，Vb是一 對電刷的電壓降，I是輸入直流電流，並且Ra是 電樞電阻。
可以將機械方程式結合在一起。
成分由永磁的0相電流產生。
因為線性材料被使用，由相電流產生的轉矩成 分可以從Im=1A的轉矩減去Im=0A的轉矩直接獲 得，如下圖所示。根據定義，下圖的曲線表示 轉矩常數因為轉矩由每單位相電流產生。指出 轉矩常數由此可以得知並不是常數，因此不適 合使用在交流永磁電機。
夾角會隨著負載自動被調整，因此一般不是零。
在這些情況，傳送的機械功率為
Tmm kTIpeak Epeak /kE
2k T
mk Ecos i
mI rmsErmscos i
(6)
Tmm kTIpeak Epeak /kE
2k T
mk Ecos i
mI rmsErmscos i
Irms 和Erms代表弦波相電流和反電動勢的根均方 值(rms)，m是相數。為了能量轉換，機械功率
E
Tm
k Em
kTI
(2)
ωm是在rad/s的機械角速度，Tm是在N·m的電磁 (氣隙)轉矩，kE是在V·s/rad的反電動勢常數，kT 是在N·m/A的轉矩常數。
kE和kT 有以下的特性： 1)在度量單位下， kE=kT 。 2) kE和kT 是常數。 3) kE和kT 可以測量。
特性1)可以從 EI= Tm ωm得證。
下圖顯示表面型無刷直流馬達(固定不變的觸發 角和脈波寬度)隨著速度改變的kT。
C. kT再也不等於kE
在無刷直流馬達，跨於直流端的反電動勢一般 包含與任意線對線反電動勢波形相關的漣波。 即使線對線感應電壓可能藉由一個特殊的設計， 在60°電機角有一個平坦的波形，電流飛輪二 極體漣波仍需要被考慮。
一定等於電機功率，也就是說
Tmm mI rmsErmscos i (7)
其中
kT
m 2
kEcos i
(8)
kT
m 2
kEcos i
由上式可以得到一個結論，即使當飽和效應
可以被忽略時，kE可能是常數。但是永磁同 步馬達的kT不是常數。kT隨著內部功率角而 改變，功率角隨著機械負載不同而改變。
一個3相4極的永磁同步電 機，如右圖所示，使用2D 暫態有限元素方法(FEM) 分析可以指出轉矩常數隨 著內部功率因素角度而改 變。為了觀察隨著內部功 率因素角度而改變的轉矩 常數，電樞電流造成的飽 和變化可以被忽略，如此 線性材料被使用於所有元 件。
B. kT和kE再也不是常數
在無刷直流馬達，E是跨於直流端的平均反電 動勢，他的值將會隨著電流飛輪二極體的導通 時間而改變。
E 1
Ts
Tf 0
1 2
eba
ebc dt
Ts Tf
e
bc
dt
(3)
很明顯的知道平均反電動勢隨著電流飛輪二極
體導通時間而改變，因此kE在不同的操作並不 是定值。
只要Tf<Ts，飛輪二極體的電流總是會減少輸入 的直流和減少轉矩，因此kT隨著電流飛輪二極 體的導通時間而改變，導通時間隨著轉子速度 改變。
另外一個kT並不是定值情況，在內藏型永磁馬 達(IPM)，磁阻轉矩成分也會貢獻氣隙轉矩作 為突極效應的結果，並且磁阻轉矩成分並不會 線性正比於直流。此外，直流無刷馬達的觸發 角和脈波寬度通常須控制的。這也是一個kT不 是常數的情況。