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Unified Torque Expressions of AC Machines
Qian Wu
Outline
1. Review of torque calculation methods.
2. Interaction between two magnetic fields.
3. Unified torque expression for AC machines.
Permanent Magnet (PM) machine;
Synchronous Reluctance Machine (SynRM);
Induction Machine (IM);
4. Conclusion.
Torque Calculation Methods
Numerical method with FEA.
Maxwell stress tensor (𝐵𝑛∙𝐵𝑡
2𝜇0) in the air gap region.
Torque Calculation Methods
Based on the energy conversion theory
v
i dt
dRiu
idt
dRiui
2
Li
iddtRiuidt 2
Power balance
Energy balance
i
Enclosed area =
stored field energy id
Torque Calculation Methods
Similar for an electrical machine
,qs s qs qs r el ds
du R i
dt
qsdselrqsqsqssqsqsinq idt
diiRuiP ,
2
Input power of the q-axis winding
Copper loss Output mechanical power
For example, the q-axis, stator side winding analysis:
,ds s ds ds r el qs
du R i
dt
Rate change of the
stored field energy
dsqsqsdselrdqmec iiP ,,
Torque Calculation Methods
Another way to utilize the energy conversion theory
stator
rotor
Ni
Ni
1sF
• Uniform air gap • Sinusoidal MMF, B waveforms • Similar MMF on the stator and
rotor
Torque Calculation Methods
So finding the air gap average co-energy density air-gap total MMF (fundamental component)
srrsrssr FFFFF cos2 1121
211
air-gap actual field intensity
g
FH sr
peakag1
,
2
10
2,0
422
g
FHsrpeakag
Average of sin square function gives
stator
rotor
Average co-energy density assuming sinusoidal
air-gap field
Torque Calculation Methods
So the torque is obtained as
By accounting the volume of the air, the total co-energy
DLgg
FW sr
aveco
2
10,
4
By differentiation the co-energy, we obtain the torque
Torque expression for comparison
1. Some observations
Classical torque equation may involve different inductances and e.g.
PM flux linkage – direct comparison is not so obvious
We experience winding current excited magnetic field (MMF),
permanent magnet field (PMSM) and salient rotor magnetic field
modulation effects (sync. Reluctance motor).
2. An ideal torque expressions for AC machines.
Applying the same principle.
Intuitive understanding of torque production mechanism.
Torque expressions with the same geometrical parameters and
physical quantities.
Torque Calculation Methods
So steps to take
• Turn all other magnetic field into winding current
excited magnetic field (MMF)
• Using a uniform airgap
Magnet equivalent magnetic field
Consider a permanent magnet
Air gap B waveform or winding MMF waveform
It is possible to replace the magnet
with winding MMF for producing the
same air gap B waveform
Magnet equivalent magnetic field
For a magnet
J and B have the same unit: [T]
0J B H
The relationship
r pmB B H
Magnet equivalent magnetic field
0J B H r pmB B H
0r pmJ B H
0 0B M H
0J M M: moment
(not a constant
as well)
J is not a constant;
slightly decrease
when H increases
0
rBM
at H = 0
(Only is involved) 0
Convenient expression
Magnet equivalent magnetic field
The PM magnetic field
MMF Mh
N
S
g
h
N
S
g h
N
0g
MhB
h g
For example:
h = 1 mm
Br = 1.2 (T)
Mh = 955 (A.turns)
The MMF does not
decrease when the magnet
becomes narrower
(neglecting the leakage
flux)
0
rBM
Magnet equivalent magnetic field
Magnet MMF vs. winding MMF
2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7fundamental component of Bn
pole pair number
Bn (
T)
slot-opening stator
PM stator
Magnetic Coupling (MC) 1. Torque production of magnetic coupling could be typically explained with the
interaction between two magnetic fields.
2. Models:
4. Magnetic field from permanent magnets (fundamental component)
Inner PMs: 𝐵𝑛1_𝑖𝑛 = 𝑘𝐵𝜇0𝑀ℎ𝑖𝑛
𝑔+ℎ𝑖𝑛+ℎ𝑜𝑢𝑡sin(𝑃𝜃 − 𝜔𝑡 + 𝛽𝑖𝑛)
Outer PMs: 𝐵𝑛1_𝑜𝑢𝑡 = 𝑘𝐵𝜇0𝑀ℎ𝑜𝑢𝑡
𝑔+ℎ𝑖𝑛+ℎ𝑜𝑢𝑡sin(𝑃𝜃 − 𝜔𝑡 + 𝛽𝑜𝑢𝑡)
𝑀: magnetization intensity; 𝑘𝐵:𝐵𝑛1waveform factor considering the magnetizing direction of PMs.
Magnet equivalent magnetic field
The PM magnetic field
N
S N
N
S
N
S
Distributed current density layer
Current in the main air gap field
𝐹 = 𝑙𝑎𝐵𝑛 × 𝐼
Torque Evaluation of MC
2. Equivalent current 𝑘(𝜃) to replace the PMs on out side of MC:
satisfying 𝑘(𝜃) 𝑑𝜃 =𝐵𝑛1 𝜃
𝜇0(𝑔 + ℎ𝑖𝑛 + ℎ𝑜𝑢𝑡) (ensuring identical MMF)
yielding 𝑘 𝜃 = 𝑘𝐵𝑃𝑀ℎ𝑜𝑢𝑡cos(𝑃𝜃 − 𝜔𝑡 + 𝛽𝑜𝑢𝑡) (Peak value 𝑘𝑚 𝜃 = 𝑘𝐵𝑃𝑀ℎ𝑜𝑢𝑡)
Thus:
Equivalent current is sinusoidally distributed along the inner surface of outer iron.
Equivalent current is rotating at the same speed of PM in space.
Peak value of equivalent current is determined by the MMF of the magnetic field source and the pole pair number.
Torque Evaluation of MC 1. Equivalent model: current is located in the air gap magnetic field.
2. Electromagnetic force endured on the current at the position of 𝜃.
𝑓(𝜃) = 𝐿𝑎𝐵𝑛1(𝜃)𝑘(𝜃) (along the tangential direction)
3. Torque of MC:
𝑻 = 𝑹𝒇 𝜽 𝒅𝜽𝟐𝝅
𝟎= 𝑲𝒎𝑩𝒏𝟏𝒎𝑭𝒏𝟏𝒎 ∙ 𝐬𝐢𝐧(𝜷𝒓 − 𝜷𝒔)
𝐾𝑚 = 𝐿𝑎𝜋𝐷
2 (geometrical parameters)
𝐵𝑛1𝑚 = 𝑘𝐵𝜇0𝑀ℎ𝑖𝑛
𝑔+ℎ𝑖𝑛+ℎ𝑜𝑢𝑡 (peak value of fundamental magnetic field )
𝐹𝑛1𝑚 = 𝑘𝐵 𝑃𝑀ℎ𝑜𝑢𝑡 (peak value of fundamental MMF for P pole pairs )
𝑓 𝛿 = sin 𝛿 = sin 𝛽𝑟 − 𝛽𝑠 (relative position between rotor and stator magnetic field)
1. Torque comparison 2. Effect of pole pair number on torque
3. Observations:
A good agreement between the results from theoretical analysis and FE calculation.
Neglecting flux leakage between poles, the maximum torque value of MC is in a linear proportion to the pole number, which may be explained by torque expression of 𝑇 = 𝐾𝑚𝐵𝑛1𝑚𝐹𝑛1𝑚𝑓(𝛿).
Torque performance of MC
0 30 60 90 120 150 180 210 240 270 300 330 360-50
-40
-30
-20
-10
0
10
20
30
40
50
electrical degree
torq
ue
(N
m)
FE calculation
fundamental component
theoretical analysis
0 30 60 90 120 150 1800
1
2
3
4
5
6
7
8torque per pole
relative position (electrical degree)
torq
ue (
Nm
)
0 30 60 90 120 150 1800
10
20
30
40
50
60
70
P=1
P=2
P=3
P=4
P=5
𝑃 𝑇𝑚𝑎𝑥
1 11.5
2 29
3 44
4 57
5 67
Torque analysis of PM machine
1. Equivalent model:
Exactly the same with that of MC, thus the obtained torque expression may be applied for PM machine.
2. Specific expression of each term:
Stator MMF: 𝐹𝑠1_𝑝𝑜𝑙𝑒(𝜃) =𝑚
2
4
𝜋
1
2
𝐽𝑍𝑆𝑓𝑓𝑖𝑙𝑙𝑘𝑤
𝑚(2𝑃)sin(𝑃𝜃 − 𝜔𝑡 + 𝛽𝑠)
Thus total MMF: 𝐹𝑠1𝑚 = 𝑃𝐹𝑠1𝑚_𝑝𝑜𝑙𝑒
Rotor magnetic field: 𝐵𝑟1(𝜃) = 𝜇0𝑘𝐵𝑀ℎ
𝑔+ℎsin(𝑃𝜃 − 𝜔𝑡 + 𝛽𝑟)
Torque expression:
𝑇 = 𝐾𝑚𝐵𝑟1𝑚𝐹𝑠1𝑚𝑓 𝛿 = (𝐿𝑎𝜋𝐷
2) 𝜇0𝑘𝐵
𝑀ℎ
𝑔 + ℎ
𝐽𝑍𝑆𝑓𝑓𝑖𝑙𝑙𝑘𝑤
2𝜋sin(𝛽𝑟 − 𝛽𝑠)
1. Torque expression validation 2. Effect of pole pair number on torque
4. Observation:
Maximum torque value of PM is determined by the main dimension (𝐿𝑎 and 𝐷), the rotor magnetic field production capability of per pole (𝑀ℎand 𝑔), and stator total current . When neglecting flux leakage between poles, the pole pair number has no effect on the total torque.
Torque analysis of PM machine
0 30 60 90 120 150 180 210 240 270 300 330 360-100
-80
-60
-40
-20
0
20
40
60
80
100
eletrical degree
torq
ue
(N
m)
FE calculation
fundamental component
theoretical analysis
0 15 30 45 60 75 900
10
20
30
40
50
60torque of PM machines
rotor postion (mechanical degree)
torq
ue (
Nm
)
P=2
P=4
Torque analysis of SynRM 1. Model
Features: slot-opening stator with winding current; salient rotor without magnetic field source; non-uniform air gap;
2. Equivalent model
Features: identical stator configuration with that of original
model; uniform air gap (𝑔); rotor is assigned with current loading.
3. Conditions:
Keep the stator configuration unchanged, including structure and armature current. Keep the air gap magnetic field unchanged.
Torque analysis of SynRM
The derived torque equation
𝑇 = 𝐿𝑎𝜋𝐷
2𝜇0
𝐹𝑠1𝑚𝑝𝑜𝑙𝑒
2𝑔𝑞(1 −
𝑔𝑞𝑔𝑑) 𝐹𝑠1𝑚sin(2𝛽)
Equivalent rotor magnetic field
Stator total MMF (similar to PMSM)
𝐵𝑟𝑛1𝑚 = 𝜇0𝐹𝑠1𝑚_𝑝𝑜𝑙𝑒
2𝑔𝑞1 −
𝑔𝑞
𝑔𝑑
2 x angle between rotor and
stator magnetic field
Torque analysis of SynRM
0 10 20 30 40 50 60 70 80 90-40
-30
-20
-10
0
10
20
30
40
torque versus relative position betweenrotor and stator MMFs
relative position (mechnical degree)
torq
ue (
Nm
)
fundamental FE
FE calculation
theoretical analysis
Torque comparison
Torque analysis of IM
Expressions of the terms in torque expression
Rotor magnetic field 𝐵𝑟1𝑚 = 𝜇0𝐹𝑟1𝑚𝑔
Stator MMF 𝐹𝑠1𝑚 = 𝑃𝐹𝑠1𝑚_𝑝𝑜𝑙𝑒
Dimensional factor 𝐾𝑚 = 𝐿𝑎𝜋𝐷
2
Position function 𝑓 𝛿 = sin(𝛽𝑟 − 𝛽𝑠)
Torque calculation 𝑇 = 𝐾𝑚𝐵𝑟1𝑚𝐹𝑠1𝑚𝑓 𝛿 = 𝐿𝑎𝜋𝐷
2(𝜇0
𝐹𝑠1𝑚_𝑝𝑜𝑙𝑒
2𝑔)𝐹𝑠1𝑚sin2δ
Equivalent rotor magnetic field: 𝐵𝑟1𝑚 = 𝜇0𝐹𝑠1𝑚_𝑝𝑜𝑙𝑒
2𝑔
Conclusion
1. A unified torque expression is obtained for different AC machines, applying the same principle.
2. By FEM, the accuracy of this torque equation is validated;
3. Under the condition of the same stator configurations, only the peak value of rotor magnetic fields need to be compared for the torque comparison of AC machines .
PM machine: 𝐵𝑟1𝑚 = 𝜇0𝑘𝐵𝑀ℎ
𝑔+ℎ
IM: 𝐵𝑟1𝑚= 𝜇0𝐹𝑠1𝑚_𝑝𝑜𝑙𝑒
2𝑔
SynRM: 𝐵𝑟𝑛1𝑚= 𝜇0𝐹𝑠1𝑚_𝑝𝑜𝑙𝑒
2𝑔𝑞1 −
𝑔𝑞
𝑔𝑑