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