ACCURATE TORQUE PULSATION ESTIMATION AND REDUCTION
OF IPMSMS VALID FOR ANY OPERATING POINT AND CURRENT

WAVEFORM

By

Tiraruek Ruekamnuaychok

A THESIS

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Electrical Engineering—Master of Science

2019

ABSTRACT

ACCURATE TORQUE PULSATION ESTIMATION AND REDUCTION OF

IPMSMS VALID FOR ANY OPERATING POINT AND CURRENT

WAVEFORM

By

Tiraruek Ruekamnuaychok

In this thesis, a method to estimate the torque pulsation of IPMSMs is proposed. The

method is based on an analytical torque equation of IPMSMs derived by using the concepts

of energy and co-energy of a magnetic circuit. This method is valid and accurate for any

operating point, including that under heavy magnetic saturation, and for any arbitrary

current waveform. Also, by using the proposed equation, a method to reduce the torque

pulsation is proposed. Simulation and experimental results show that the proposed methods,

for both estimation and reduction, are valid, accurate and effective.

Copyright by
TIRARUEK RUEKAMNUAYCHOK
2019

ACKNOWLEDGMENTS

I would like to thank Prof. Elias G. Strangas, my advisor, for giving me knowledge and

understanding regarding electrical machines and drives and all the support he made. Also, I

would like to thank Shaopo Huang, a visiting scholar from Hebei University of Technology,

for helping me in the experimental setup.

iv

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Calculation of Stored Field Energy, Stored Field Co-Energy
and Torque of a Magnetic Circuit . . . . . . . . . . . . . . . . . .
2.1 Calculation of Stored Field Energy . . . . . . . . . . . . . . . . . . . . . . .
2.2 Calculation of Torque from Stored Field Energy . . . . . . . . . . . . . . . .
2.3 Calculation of Stored Field Co-Energy . . . . . . . . . . . . . . . . . . . . .
2.4 Calculation of Torque from Stored Field Co-Energy . . . . . . . . . . . . . .

Chapter 3 The Complete Torque Equation of IPMSMs . . . . . . . . . . .
3.1 Energy Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Instantaneous Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Stored Field Energy Torque . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Total Torque
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4 Torque Pulsation Estimation Algorithm . . . . . . . . . . . . . .
4.1 Stored Field Energy Torque . . . . . . . . . . . . . . . . . . . . . . . . . . .
Instantaneous Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
4.3 Total Torque
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
4
8
9
11

12
12
16
16
22

23
24
30
31

Chapter 5 Torque Pulsation Reduction of IPMSMs

. . . . . . . . . . . . .

32

Chapter 6 Total Flux Linkage Estimation . . . . . . . . . . . . . . . . . . . .

35

Chapter 7

Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
7.1 Torque Pulsation Estimation — Constant Idq
7.2 Torque Pulsation Estimation — Oscillating Idq
. . . . . . . . . . . . . . . .
7.3 Torque Pulsation Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

36
36
41
45

Chapter 8 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Chapter 9 Conclusion and Possible Future Work . . . . . . . . . . . . . . .

51

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

v

LIST OF FIGURES

Figure 2.1: Magnetic circuit with one electrical input and one mechanical output. . .

Figure 2.2:

Integration path for the calculation of the stored field energy at any final
state (λf , θf ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

8

Figure 2.3:

Integration path for the calculation of the stored field co-energy at any
final state (if , θf ).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Figure 3.1: Some specific initial and final state of an IPMSM and its path. . . . . . .

14

Figure 3.2: One possible integration path. . . . . . . . . . . . . . . . . . . . . . . . .

15

Figure 3.3: Relation between λd and id.

. . . . . . . . . . . . . . . . . . . . . . . . .

18

Figure 3.4: Relation between λq and iq.

. . . . . . . . . . . . . . . . . . . . . . . . .

19

Figure 4.1: 2-D cross section for FEA. . . . . . . . . . . . . . . . . . . . . . . . . . .

24

Figure 4.2: Operating point and some possible paths of injection. . . . . . . . . . . .

25

Figure 4.3: λd at each step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.4: λq at each step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

26

Figure 4.5: Relation between λd and id at a specific electrical position in the example. 27

Figure 4.6: Relation between λq and iq at a specific electrical position in the example. 27

Figure 4.7:

Integration in d Axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Figure 4.8:

Integration in q Axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Figure 4.9: Co-energy function in d Axis.

. . . . . . . . . . . . . . . . . . . . . . . .

28

Figure 4.10: Co-energy function in q Axis.

. . . . . . . . . . . . . . . . . . . . . . . .

29

Figure 4.11: Stored field energy torque at the operating point.

. . . . . . . . . . . . .

29

Figure 4.12: λdq at the operating point. . . . . . . . . . . . . . . . . . . . . . . . . . .

30

Figure 4.13: Instantaneous torque at the operating point. . . . . . . . . . . . . . . . .

30

vi

Figure 4.14: Comparison between the torque calculated by FEA and the estimated
. . . . . . . . . . . . . . . . . . . . . . . . .

torque using the algorithm.

31

Figure 5.1: Torque at neighboring operating points.

. . . . . . . . . . . . . . . . . .

34

Figure 5.2: Reducing torque pulsation current at 108Nm obtained by inverting the
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

table.

34

Figure 7.1: Cogging Torque.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

Figure 7.2: Comparison of torque estimations based on eq. 7.1, 7.2 and 7.3 at Is = 2A. 38

Figure 7.3: Comparison of torque estimations based on eq. 7.1, 7.2 and 7.3 at Is = 12A. 38

Figure 7.4: Comparison of torque estimations based on eq. 7.1, 7.2 and 7.3 at Is = 22A. 39

Figure 7.5: Torque comparison at Is = 2, 4, 6 and 8A.

. . . . . . . . . . . . . . . . .

40

Figure 7.6: Torque comparison at Is = 10, 12, 14 and 16A. . . . . . . . . . . . . . . .

40

Figure 7.7: Torque comparison at Is = 18, 20, 22, 24 and 26A.

. . . . . . . . . . . . .

40

Figure 7.8:

id at each step.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Figure 7.9:

iq at each step.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Figure 7.10: λd at each step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 7.11: λq at each step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

42

Figure 7.12: Torque comparison at Is = 2, 4, 6 and 8A.

. . . . . . . . . . . . . . . . .

42

Figure 7.13: Torque comparison at Is = 10, 12, 14 and 16A. . . . . . . . . . . . . . . .

43

Figure 7.14: Torque comparison at Is = 18, 20, 22, 24 and 26A.

. . . . . . . . . . . . .

43

Figure 7.15: Arbitrary id.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Figure 7.16: Arbitrary iq.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Figure 7.17: Torque comparison under arbitrary current waveforms.

. . . . . . . . . .

44

Figure 7.18: Reducing torque pulsation current in d axis.

. . . . . . . . . . . . . . . .

45

Figure 7.19: Reducing torque pulsation current in q axis.

. . . . . . . . . . . . . . . .

45

vii

Figure 7.20: Torque without and with the torque pulsation reduction at Is = 22A.

. .

46

Figure 8.1: Experiment Setup.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Figure 8.2: Is = 18A.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Figure 8.3: Is = 20A.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Figure 8.4: Is = 22A.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Figure 8.5: Is = 24A.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Figure 8.6: Reducing torque pulsation current in d axis.

. . . . . . . . . . . . . . . .

50

Figure 8.7: Reducing torque pulsation current in q axis.

. . . . . . . . . . . . . . . .

50

Figure 8.8: Torque without and with the torque pulsation reduction from the trans-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ducer at Is = 22.5A.

50

viii

Chapter 1

Introduction

Interior Permanent Magnet Synchronous Motors (IPMSMs) are used widely nowadays

due to many of their advantages such as high power density and high efficiency. Nevertheless,

one of their main disadvantages is that they may have high torque pulsation.

The term “torque pulsation” refers to any oscillation that exists in the developed torque

of a motor. This oscillation can lead to many negative effects such as audible noise and rotor

speed variation. More importantly, it creates vibrations which can seriously damage a motor

or any other thing connected to it.

For IPMSMs, the torque pulsation can have many components that are from many dif-

ferent sources such as control scheme, power converter and noise. However, its dominating

component is usually from the construction of the motor itself. This work focuses solely on

this component and assumes that all other components are zero. Note that throughout this

work, whenever the term torque pulsation is used, it refers to this component.

Ideally, the construction of IPMSMs should be such that the stator winding and the

radial magnetic flux density produced by the magnets are sinusoidally distributed in space.

In this case, the developed torque will be constant as long as the phase current is purely

sinusoidal. Unfortunately, such construction is not possible in practical situations. As a

result, there will be the torque pulsation even if the phase current does not have harmonics.

1

Over the years, there have been many attempts in reducing the torque pulsation of

IPMSMs. The most commonly used control method in doing so is called “Current Harmonic

Injection.” Its concept is to obtain the torque pulsation first. Then, inject current with some

harmonics in order to have them compensate for the pulsation. In this case, the difficulty

lies mainly in how the torque pulsation itself is obtained.

Generally, there are two ways to obtain it. The first way is to use a torque transducer.

However, its installation is not convenient, and it is likely to have some error. Moreover,

what it actually measures is not the developed torque of a motor. Rather, it measures

the shaft torque which is the developed torque passed through the transfer function of the

shaft. Therefore, injecting current harmonics based on this torque may not be as effective

as desired.

The other way is to obtain by estimation. In literature, there have been many attempts

in deriving an analytical torque equation of IPMSMs that takes the torque pulsation into

consideration [3–17]. Some of them derived the torque equation by using the product of back-

emfs and currents [2–5]. Some of them analyzed harmonics contained in the general torque

equation [6–8]. Alternatively, [9, 10] derived the torque equation with focus on derivative

of inductances with respect to the rotor position. However, these works usually considered

only some components of the torque pulsation, not the total one.

In recent years, the trend in deriving the analytical torque equation is based on the

concepts of energy and co-energy of a magnetic circuit [11–17]. In [13], N. Nakao and K.

Akatsu proposed an analytical equation for the torque of IPMSMs, including the torque

pulsation, based on the concept of co-energy. Nevertheless, the co-energy in this work was

not thoroughly defined, and the structure of the equation itself was based on an assumption

that the magnetic saturation was linear.

2

In short, until now, there has been no analytical torque equation of IPMSMs that takes

all components of the torque pulsation into consideration, is complete in derivation and is

valid for any operating point, including that under heavy magnetic saturation. This work

proposes such an equation together with offline algorithms that make use of it. Then, based

on the proposed equation, a method to reduce the torque pulsation based on the current

harmonic injection is developed. Simulation and experimental results show that the equation,

algorithms and reduction method, are valid, accurate and effective.

3

Chapter 2

Calculation of Stored Field Energy,

Stored Field Co-Energy and Torque of

a Magnetic Circuit

The calculation of the stored field energy, stored field co-energy and torque of a magnetic

circuit are very important in this work. They serve as fundamental in deriving the total

torque equation of IPMSMs. In this chapter, they will be reviewed in detail.

2.1 Calculation of Stored Field Energy

For a magnetic circuit, the term “stored field energy” refers to the energy that is stored

inside the circuit in the form of magnetic field. Neglecting losses, it can be identified by

taking difference between the input and output energy.

Consider a magnetic circuit with one electrical input and one mechanical output in Fig.

2.1 where e is induced voltage, i is injected current, φ is magnetic flux, λ is total magnetic

flux linkage and θ is the angle of rotation. The electrical input power, Pe, is given by (2.1).

The total input energy from initial time t0 to final time tf , denoted by ∆W e

t0→tf
calculated by integrating the electrical input power over the time period as (2.2).

, can be

4

Figure 2.1: Magnetic circuit with one electrical input and one mechanical output.

Pe = ie

= i

dλ
dt

t0

tf(cid:90)
tf(cid:90)
λf(cid:90)

t0

λ0

Pedt

i

dλ
dt

dt

idλ

∆W e

t0→tf

=

=

=

5

(2.1)

(2.2)

The output mechanical power, denoted by Pm, can be calculated as (2.3) where T is

electromagnetic torque in Nm. Consequently, the total output energy from initial time t0 to

final time tf , denoted by ∆W m

t0→tf
power over the time period as (2.4).

, can be calculated by integrating the mechanical output

Pm = T

dθ
dt

t0

tf(cid:90)
tf(cid:90)
θf(cid:90)

t0

Pmdt

T

dθ
dt

dt

T dθ

∆W m

t0→tf

=

=

=

(2.3)

(2.4)

θ0

By taking the difference between the input and output energy, the stored field energy

from initial time t0 to final time tf , denoted by ∆W f

t0→tf

can be expressed as (2.5).

∆W f

t0→tf

= ∆W e

t0→tf

− ∆W m

t0→tf

λf(cid:90)

=

idλ −

θf(cid:90)

T dθ

(2.5)

λ0

θ0

Note that (2.5) can be interpreted as a line integral scalar function defined on a 2-dimensional

space (λ, θ), and its alternative meaning is the difference of the stored field energy between

the initial state (λ0, θ0) and the final state (λf , θf ).

6

Taking the initial state to be (λ0 = 0, θ0 = 0) the stored field energy at the final state

(λf , θf ), denoted by W f

f , can be calculated by (2.6).

λf(cid:90)

0

idλ −

θf(cid:90)

0

T dθ

W f

f =

(2.6)

By neglecting the hysteresis loss, the stored field energy becomes conversative and the

path of integration from the initial state (0, 0) to the final state (λf , θf ) in (2.6) can be

arbitrarily chosen. The simplest path is to integrate along the direction of θ from θ = 0 to

θ = θf first while keeping λ fixed at zero, i.e., to integrate from (0, 0) to (0, θf ). In this

integration, (2.6) is zero. This is because when λ is kept zero, dλ will also be zero. Moreover,

there will be no magnetization which results in zero T . After that, integrate from (0, θf )

to (λf , θf ). Using this integration path, the stored field energy at any final state (λf , θf )

can be calculated as (2.7) where the vertical line represents the fact that θ is kept fixed at

θf . Fig. 2.2 illustrates this calculation. Note that to use (2.7), the function i(λ, θ) must be

known. Theoretically, if the exact geometry of the magnetic circuit is known, the function

can be analytically obtained. In Fig. 2.2, the blue dot line represents this function at θf .

(2.7)

λf(cid:90)

0

(cid:12)(cid:12)(cid:12)(cid:12)θf

idλ

W f

f =

7

Figure 2.2: Integration path for the calculation of the stored field energy at any final state
(λf , θf ).

2.2 Calculation of Torque from Stored Field Energy

By using the fact that the stored field energy (2.6) is a function of two variables, which

are λ and θ, the differential of the stored field energy can be expressed as (2.8).

dW f =

∂W f
∂λ

dλ +

∂W f
∂θ

dθ

(2.8)

Moreover, according to (2.5) and (2.6), the differential of the stored field energy can also be

expressed as (2.9).

dW f = idλ − T dθ

(2.9)

By comparing the last terms of (2.8) and (2.9), the torque of the magnetic circuit can be

expressed as (2.10).

(2.10)

T = −∂W f
∂θ

8

2.3 Calculation of Stored Field Co-Energy

Calculating the torque from the stored field energy is somewhat inconvenient. This is due

to the fact that it requires integration of the current with respect to the flux linkage, which

is not intuitive. A more convenient way of calculating the torque is to calculate through

the “stored field co-energy.” Note that the stored field co-energy is not a physical quantity.

Rather, it is a concept that is defined and developed just to simplify the calculation.

In this section, calculation of the stored field co-energy for the given magnetic circuit

will be presented. Then, in the following section, calculation of the torque by using this

co-energy concept will be discussed.

For the magnetic circuit shown in Fig. 2.1 with the stored field energy equation of (2.6),

the stored field co-energy at the state (λf , θf ), denoted by W c

f is defined as (2.11).

(2.11)

(2.12)

f = if λf − W f
W c

f

By substituting the stored field energy (2.6) into (2.11), it yields

Note that

f = if λf −
W c

λf(cid:90)

0

idλ +

θf(cid:90)

0

T dθ

λf(cid:90)

0

idλ =

if(cid:90)

0

λdi

if λf −

9

Hence, (2.12) is changed to (2.13), with the 2-dimensional space of (i, θ).

if(cid:90)

0

W c

f =

λdi +

θf(cid:90)

0

T dθ

(2.13)

Similar to the stored field energy, (2.13) is conservative under the assumption that the

hysteresis loss is neglected. Following the same path of integration but with respect to i,

the stored field co-energy at the final state (λf , θf ) is given by (2.14). For virtualization, see

Fig. 2.3.

if(cid:90)

0

(cid:12)(cid:12)(cid:12)(cid:12)θf

λdi

W c

f =

(2.14)

Figure 2.3: Integration path for the calculation of the stored field co-energy at any final state
(if , θf ).

10

2.4 Calculation of Torque from Stored Field Co-Energy

Following the same logic from the stored field energy, the differential of the stored field

co-energy can be expressed as (2.15) and (2.16).

dW c =

∂W c
∂i

di +

∂W c
∂θ

dθ

dW c = λdi + T dθ

(2.15)

(2.16)

By comparing each term of these two equations, the torque of the magnetic circuit can be

expressed by (2.17).

T =

∂W c
∂θ

(2.17)

It can be seen in this chapter that the torque of the given magnetic circuit can be

expressed using either the stored field energy or the stored field co-energy. Moreover, as long

as the function i(λ, θ), or equivalently λ(i, θ), is known, its stored field energy, stored field

co-energy and torque can be calculated immediately.

Note that even though this chapter focuses only on the magnetic circuit with one electrical

input and one mechanical output, the concept itself is generally true of any other magnetic

circuits, and it will serve as fundamental in deriving the total torque equation of IPMSMs

in the next chapter.

11

Chapter 3

The Complete Torque Equation of

IPMSMs

In this chapter, the complete torque equation of IPMSMs is derived. The derivation is

based on the following concept. When an IPMSM is considered under the rotor frame of

reference, its energy can be considered as having two components. The first component is

the energy that is not stored in the form of magnetic field. The second component, on the

other hand, is the energy that is stored in the form of magnetic field. To derive the torque

equation, these two components will be identified first. Then, their corresponding torques

will be derived separately. Finally, adding the two torques gives the total torque of IPMSMs.

3.1 Energy Decomposition

For IPMSMs, the input power can be expressed as (3.1) where eabc and iabc are phase

back-emfs and currents respectively. Consequently, transforming (3.1) into the rotor frame

of reference yields (3.2) where edq and idq are d-q axis back-emfs and currents.

Pin = eaia + ebib + ecic

(cid:0)edid + eqiq

(cid:1)

=

3
2

12

(3.1)

(3.2)

Note that the d-q axis back-emfs edq can be expressed as (3.3) and (3.4) where ωe is electrical

speed in rad/s and λdq are d-q axis total flux linkages.

ed = −ωeλq +

eq = ωeλd +

dλd
dt
dλq
dt

(3.3)

(3.4)

By substituting (3.3) and (3.4) into (3.2), the final expression for the input power of IPMSMs

can be obtained as (3.5).

Pin =

3
2

ωe

(cid:18)
λdiq − λqid

(cid:19)

(cid:18)

+

3
2

id

dλd
dt

+ iq

dλq
dt

(cid:19)

(3.5)

The output power of IPMSMs can be expressed as (3.6) where T is the electromagnetic

torque in Nm and θm is the mechanical rotor position in radians.

Pout = T

dθm
dt

(3.6)

The amount of energy being stored in the form of magnetic field from initial time t0 to

final time tf , denoted by ∆W f

t0→tf

, can be identified by taking integral of the difference

between the input and output power as (3.7).

tf(cid:90)

t0

tf(cid:90)

t0

Pindt −

∆W f

t0→tf

=

Poutdt

(3.7)

By substituting (3.5) and (3.6) in (3.7), the equation becomes (3.8), where P is the number

of pole pairs.

13

(cid:1)dθm

3
2

P(cid:0)λdiq − λqid
λqf(cid:90)

iddλd +

iqdλq

3
2

θm0

θmf(cid:90)
λdf(cid:90)
θmf(cid:90)

λd0

3
2

∆W f

0→f =

+

−

λq0

(3.8)

T dθm

θm0

Note that (3.8) can be considered as a line integral scalar function defined on a 3-

dimensional space (λd, λq, θm) and its alternative physical interpretation is the difference of

the stored energy between the initial state (λd0, λq0, θm0) and the final state (λdf , λqf , θmf ).

In terms of usage, (3.8) is to be applied on the path on which the state is moved. For exam-

ple, consider Fig. 3.1. Suppose at some initial time, the state is at (λd0, λq0, θm0) and at the

next time instance, the state is moved along the blue line to the final state (λdf , λqf , θmf ).

Then, this blue line is the path on which (3.8) is to be applied.

Figure 3.1: Some specific initial and final state of an IPMSM and its path.

14

However, by neglecting the hysteresis loss, the stored energy becomes conservative and

hence, the integration path can be arbitrarily chosen. By taking the path as shown in Fig.

3.2, where the line (λ0, 0, θm) represents any state that has zero idq, i.e., no current injected,

it can be shown that the first term of (3.8) is always zero while the other terms are not. This

concludes that for the given input power of (3.5), the input energy due to the first will not

be stored in the form of magnetic field while the input energy due to the second term will.

Figure 3.2: One possible integration path.

Following this logic, the energy of IPMSMs can be considered as having two components.

The first component is called “instantaenous energy.” It is the energy that is not stored in

the form of magnetic field, and it corresponds to the input energy due to the first term of

(3.5). The second component, on the other hand, is the energy that is stored in the form of

magnetic field, and it corresponds to the input energy of the second term in (3.5) combined

with the existing energy due to the magnets. In this work, this component is called “stored

field energy.”

15

3.2

Instantaneous Torque

Since the input energy due to the first term in (3.5), which is the first term in (3.8), is not

stored in the form of magnetic field, it will be transferred directly to the output in the form

of torque. This torque at any final state (λdf , λqf , θmf ) can be derived by simply taking

derivative of the energy itself with respect to the rotor position as (3.9) and will be called

“instantaneous torque.” Note that this torque is generally the main torque of the motor and

it contains both constant and oscillating components.

P(cid:0)λdf iqf − λqf idf

(cid:1)

Tins =

3
2

(3.9)

3.3 Stored Field Energy Torque

As mentioned earlier, in addition to the instantaneous energy, the motors also have the

stored field energy, and the existence of this energy also develops torque. This torque is

called in this work “stored field energy torque”, and it can be derived as follows.

By removing the first term in (3.8), the amount of energy being stored in the form of

magnetic field from any initial state to any final state can be expressed as (3.10).

λdf(cid:90)

λd0

λqf(cid:90)

iqdλq −

θmf(cid:90)

λq0

θm0

iddλd +

3
2

∆W f

0→f =

3
2

T dθm

(3.10)

By taking the initial state to be that when there is no idq with zero θm, i.e., (λd0, 0, 0), with

an assumption that at this state, the existing energy due to the magnets is K, the stored

field energy of IPMSMs at any final state (λdf , λqf , θmf ) can be expressed as (3.11).

16

(cid:34)

3
2

λdf(cid:90)

λd0

W f

f =

λqf(cid:90)

0

iqdλq −

θmf(cid:90)

0

iddλd +

3
2

(cid:35)

T dθm

+ K

(3.11)

By taking the differential of (3.11), the equation becomes (3.12).

dW f

f =

3
2

iddλd +

iqdλq − T dθm

3
2

(3.12)

Furthermore, by using the fact that (3.11) is a function of three variables, its differential can

also be expressed as (3.13).

dW f

f =

∂W f
f
∂λd

dλd +

∂W f
f
∂λq

dλq +

∂W f
f
∂θm

dθm

(3.13)

By comparing the terms of (3.12) and (3.13), the stored field energy torque can be expressed

as (3.14).

Tsf e = −∂W f

f
∂θm

.

(3.14)

Alternatively, dividing (3.12) by θm yields (3.15). Consequently, by rearranging (3.15), the

stored field energy torque can also be expressed as (3.16).

dW f
f
dθm

=

Tsf e =

3
2

3
2

id

id

dλd
dθm

dλd
dθm

+

+

3
2

3
2

iq

iq

dλq
dθm

dλq
dθm

− T
− dW f

f
dθm

(3.15)

(3.16)

As mentioned earlier, using the stored field energy is not convenient. In the following

derivation, expression of the stored field energy torque in terms of the stored field co-energy

will be presented.

17

For IPMSMs, the stored field co-energy at any final state (λdf , λqf , θmf ) is defined as

(3.17). Then, by substituting (3.11) into (3.17), the equation becomes (3.18). Note that

the stored field co-energy is not a physical quantity. Rather, it is just a tool developed to

simplify the calculation.

W c

f =

3
2

(cid:34)

=

iqf λqf +

idf λdf − W f

f

3
2

3
2

iqf λqf +

3
2

idf λdf − 3
2

λdf(cid:90)

iddλd − 3
2

λd0

λqf(cid:90)

0

iqdλq +

(cid:35)

T dθm

θmf(cid:90)

0

(3.17)

− K

(3.18)

In general, the relations between λd and id and between λq and iq can be expressed as

(3.19) and (3.20). For virtualization, see Fig. 3.3 and 3.4.

(3.19)

(3.20)

λdf(cid:90)
λqf(cid:90)

λd0

iddλd +

iqdλq +

idf λdf =

iqf λqf =

idf(cid:90)
iqf(cid:90)

0

λddid

λqdiq

0

0

Figure 3.3: Relation between λd and id.

18

Figure 3.4: Relation between λq and iq.

By using (3.19) and (3.20), the stored field co-energy (3.18) is reduced to (3.21), where

the second term is negative since the integration is applied on a positive function but in

negative direction.

(cid:34)

3
2

iqf(cid:90)

0

W c

f =

λqdiq +

idf(cid:90)

0

3
2

λddid +

(cid:35)

− K

T dθm

θmf(cid:90)

0

(3.21)

For convenience in applying (3.21), the limit of integration of the second term is reverted

so that the integration becomes positive. As a result, (3.21) becomes (3.22), which is the final

expression for the stored field co-energy of IPMSMs at any final state (λdf , λqf , θmf ). Note

that (3.22) is still a line integral scalar function, but it is now defined on a 3-dimensional space

of (id, iq, θm). Also, by neglecting hysteresis loss, the conservative property still remains.

− K

(3.22)

(cid:34)

W c

f =

iqf(cid:90)

0

3
2

λqdiq − 3
2

0(cid:90)

idf

θmf(cid:90)

0

T dθm

λddid +

(cid:35)

Taking the differential of stored field co-energy yields (3.23). Note that the differential

must be taken on the original expression (3.21), not the one modified for convenience (3.22).

19

(cid:34)

− 3
2

idf(cid:90)

0

λqdiq − d

(cid:35)

λddid

+ T dθm

λqdiq +

3
2

λddid + T dθm

dW c

f =

=

3
2

3
2

(3.23)

In addition, since the stored field co-energy is a function of three variables, its differential

can also be expressed as

dW c

f =

∂W c
f
∂iq

diq +

∂W c
f
∂id

did +

∂W c
f
∂θm

dθm

(3.24)

Similar to the stored field energy, by comparing the terms of (3.23) and (3.24), the torque

can be expressed as (3.25)

Tsf e =

∂W c
f
∂θm

(3.25)

Alternatively, dividing (3.23) by θm yields (3.26). Consequently, by rearranging (3.26), the

torque can be expressed as (3.27).

dW c
f
dθm

=

Tsf e =

+

3
λq
2
dW c
f
dθm

diq
dθm
− 3
2

λd

3
2
diq
dθm

did
dθm
− 3
2

λq

+ T

λd

did
dθm

(3.26)

(3.27)

Up to this point, it can be seen that the stored field energy torque can be expressed in

terms of the stored field energy as (3.14) and (3.16) or in terms of the stored field co-energy

as (3.25) and (3.27).

20

The easiest and most practical expression is the one in the stored field co-energy term with

normal derivative (3.27). The reason is that this equation explicitly includes the oscillation

in the currents, i.e.,

did
dt and

diq
dt , while (3.25) does not. This makes it easier to use in

practical situations since most of the time, idq are not constant due to the fact that the

current controllers are not ideal. Note that after this point, the derivation of the torque

expression will be based on (3.27) only.

By integrating (3.22) from θm = 0 to θm = θmf with idq kept fixed at zero, and then

integrating from (id, iq) = (0, 0) to (id, iq) = (idf , iqf ) with θm kept fixed at θmf , the stored

field co-energy at any final state (idf , iqf , θmf ) can be calculated as (3.28), where the vertical

lines represent that the integrations are evaluated at θmf . Note that (3.28) now represents

a numerical value, not a function.

(cid:34)

W c

f =

iqf(cid:90)

0

3
2

λqdiq

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)θmf

0(cid:90)

idf

− 3
2

(cid:35)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)θmf

+

θmf(cid:90)

0

λddid

Tcoggingdθm − K

(3.28)

By substituting (3.28) into (3.27) and using the relation θe = P θm, where θe is the

electrical position, the stored field energy torque at any final state (idf , iqf , θef ) can be

expressed as (3.29), where (cid:102)W c

id and (cid:102)W c

iq are defined as (3.30) and (3.31). Note that even

though in terms of derivation, (3.29) is the stored field energy torque at any final state

(idf , iqf , θmf ), but since the expression holds true for any final state, the 3-dimensional

space of (idf , iqf , θmf ) can be changed to (idf , iqf , θef ) without loss of generality. Also note

that, according to (3.29), the stored field energy torque is expressed in terms of the derivative,

therefore, this torque will have only oscillating components.

21

Tsf e =

P

iqf ,θef

3
2

(cid:34)
(cid:32) 0(cid:90)
(cid:32) iqf(cid:90)

idf

0

d
dθe

λddid

(cid:18)(cid:102)W c
(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)θef
(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)θef

=

=

idf ,θef

(cid:102)W c
(cid:102)W c

iqf ,θef

λqdiq

(cid:19)

−(cid:102)W c

idf ,θef

− λqf

diqf
dθe

− λdf

didf
dθe

(cid:35)

+ Tcogging

(3.29)

(3.30)

(3.31)

3.4 Total Torque

Finally, adding the instantaneous and stored field energy together, the final expression

for the torque of IPMSMs at any final state (idf , iqf , θef ) can be expressed as (3.32).

P(cid:0)λdf iqf−λqf idf

(cid:1)+

Ttotal =

3
2

(cid:34)

(cid:18)(cid:102)W c

3
2

P

d
dθe

iqf ,θef

(cid:19)
−λqf

−(cid:102)W c

idf ,θef

(cid:35)

diqf
dθe

−λdf

didf
dθe

+Tcogging

(3.32)

22

Chapter 4

Torque Pulsation Estimation

Algorithm

It can be seen that to use the final equation (3.32), the relation between idq, λdq and θe

must be known. Unfortunately, it is almost impossible to obtain the relation analytically due

to many reasons such as the complexity of magnetic field calculation. Moreover, the equation

itself is not straightforward; it requires the calculation of the stored field co-energy. Hence, in

this chapter, an algorithm to identify the relation experimentally and consequently calculate

the torque is proposed. Note that this algorithm contains an offline motor characterization

which requires a motor to be connected to a dynamometer or a variable mechanical load.

In this entire work, the cogging torque Tcogging is assumed to be known. In addition, to

use (3.32), the exact waveforms of λdq are required. In this section, they are assumed to be

known first. In a later section, a method of how to obtain them will be presented.

For simplicity, let the algorithm be explained along with an example. The motor used in

the example is a 10-pole IPMSM with concentrated winding and fractional slot per pole per
phase. Its rated current is 25A and the operating point is chosen at id = −4 and iq = 20.

Its 2-D cross section for FEA is shown in Fig. 4.1. Note that although the operating point

in this example is assumed to be constant, the algorithm is actually valid for any arbitrary

waveforms of idq.

23

Figure 4.1: 2-D cross section for FEA.

4.1 Stored Field Energy Torque

In the algorithm, the stored field energy torque will be calculated first followed by the

calculation of the instantaneous torque. Adding these two torques together gives the total

torque of IPMSMs.

The following steps explain in detail of how to apply (3.29) to calculate the stored field

energy torque for a given operating point over a range of electrical position.

1. Suppose that the given operating point is at (idop, iqop). First, rotate the motor at

some arbitrary rotor speed. Then, inject idq step by step from the origin (0, 0) to the

operating point. During the injection, the rotor speed is to be maintained. This is why

a dynamometer or a variable mechanical load is required. The path for the injection

can be arbitrarily chosen as long as the current at each step is increasing. Fig. 4.2

shows some possible paths of injection. The path can be from the origin to point a or

b and then to the operating point. Alternatively, it can be directly from the origin to

the operating point as the curve or the straight line. Then, at each step, collect idq and

λdq for at least one electrical cycle. Note that at the origin, λdq are the flux linkages

due to the magnets. Also note that after collecting the data at the operating point, the

dynamometer or the variable mechanical load is no longer needed. As for the example,

24

the desired operating point is given as id = −4, iq = 20. The injection path is chosen
as (id, iq) = (−0.5, 2.5), (−1, 5), (−1.5, 7.5), (−2, 10), (−2.5, 12.5), (−3, 15), (−3.5, 17.5)
and (−4, 20).

Figure 4.2: Operating point and some possible paths of injection.

Figure 4.3: λd at each step.

25

Figure 4.4: λq at each step.

2. Extract the data from all steps of injection but at one and the same electrical position,

and plot them with idq as the x-axis and λdq as the y-axis (2 plots, one for the d axis

and the another one is for the q axis). Note that for the q axis, starting from the

iq = 0 to iq at the operating point, but for the d axis, starting from id at the operating

point to id=0. Then, for better accuracy, apply interpolation between each step. The

results are the relations between the current and flux linkage in d and q axes at this

(cid:12)(cid:12)(cid:12)(cid:12)θef

(cid:12)(cid:12)(cid:12)(cid:12)θef

chosen electrical position, i.e., λd(id, iq)

and λq(id, iq)

which are to be used in

(3.30) and (3.31). Note that different paths of injection may give different relations,

but ultimately, they all will give the same stored field energy and torque. For the

example, this step is illustrated in Fig. 4.5 and 4.6.

3. Apply integral with respect to the current to the relations obtained from step 2. Note

that the integration here is the area under the curve and its sign is assigned to be

positive. The results of the integrations will be the d and q axis current co-energy at

the chosen electrical position, i.e.,(cid:102)W c

and(cid:102)W c

idf ,θef

iqf ,θef

of (3.30) and (3.31). Fig.4.7

and 4.8 illustrate this process according to the example.

26

Figure 4.5: Relation between λd and id at a specific electrical position in the example.

Figure 4.6: Relation between λq and iq at a specific electrical position in the example.

Figure 4.7: Integration in d Axis.

27

Figure 4.8: Integration in q Axis.

4. Repeat steps 2 and 3 for every electrical position of the collected data. Then, by

stacking the resulting current co-energy at all electrical positions together, the complete

current co-energy functions over an entire range of electrical positions in the d and q

(θe), can be obtained. In the example, the complete current

axis, (cid:102)W c

(θe) and (cid:102)W c

idf

iqf

co-energy functions are shown in Fig. 4.9 and 4.10.

Figure 4.9: Co-energy function in d Axis.

28

Figure 4.10: Co-energy function in q Axis.

5. The final step is to substitute all terms into (3.29). The terms(cid:102)W c

idf ,θef

and(cid:102)W c

iqf ,θef

are obtained from the previous step. The terms λdqf and idqf are from the collected

data at the step of the operating point. As for the cogging torque, it is assumed to be

known. In the example, the stored field energy torque is shown in Fig. 4.11.

Figure 4.11: Stored field energy torque at the operating point.

29

4.2

Instantaneous Torque

Calculation of the instantaneous torque is comparatively easier than that of the stored

field energy torque. It can be done simply by applying (3.9) directly at the operating point.

The important thing here is that all harmonics must be included. Fig. 4.12 shows λdq at

the operating point and Fig. 4.13 shows the corresponding instantaneous torque.

Figure 4.12: λdq at the operating point.

Figure 4.13: Instantaneous torque at the operating point.

30

4.3 Total Torque

Finally, adding the two torques gives the total torque of IPMSMs (3.32). In the example,

the torque calculated by FEA and the torque from the estimation are compared in Fig. 4.14.

Figure 4.14: Comparison between the torque calculated by FEA and the estimated torque
using the algorithm.

One of the most important things is that by considering each step of injection as another

operating point and repeating the calculation presented above, the torque at any point on

the path of injection can be obtained. For example, suppose the path of injection is chosen

to be the curve line in Fig. 4.2, after injecting the current step by step to the operating

point (idop, iqop), the ultimate result is the torque at any point over the entire curve.

31

Chapter 5

Torque Pulsation Reduction of

IPMSMs

For a given desired level of torque, the idea in torque pulsation reduction by means of

control is to inject less current when the torque is higher than the desired level. On the

other hand, when the torque is lower than the desired level, inject more current. By doing

so, it is expected that the injected current will compensate the torque, making it constant.

In this work, the current is called “reducing torque pulsation current.”

For IPMSMs, provided that the torque pulsation is known, the problem is reduced to

how to calculate the reducing torque pulsation current. The analytical calculation is not

simple. To illustrate the reason, suppose the total torque, (3.32), had only the instantaneous

term, it seems that once λdq are known, the reducing torque pulsation current could be

calculated without any difficulties. However, that is definitely not the case. This is because

due to the saturation, the waveforms of λdq depend heavily on the level of current. Once the

normal current is replaced by the reducing torque pulsation current, the waveforms of λdq

will deviate. Since the reducing torque pulsation current was calculated based on the first

waveforms of λdq, this current would not reduce the torque pulsation as much as it should.

Moreover, if the stored field energy torque is taken back into consideration, the complexity

will become much higher.

32

One possible analytical solution is to first identify the relation between the level of current,

total flux linkage and the co-energy and then calculate the reducing torque pulsation current

accordingly. However, this is equivalent to creating a look-up table.

Hence, it would be much simpler to create a look-up table for the torque directly, instead

of this relation. The idea in using a look-up table to reduce the torque pulsation is the

following. Given a desired operating point, first identify the torque pulsation at neighboring

operating points for at least one electrical cycle. This would allow creation of a look-up table

with current and electrical position as inputs and torque as an output. Then, by inverting

the table at some desired constant torque, the reducing torque pulsation current can be

obtained.

The next question is what should be the neighboring operating points mentioned above.

The answer depends on which current will be used to reduce the torque pulsation. For

IPMSMs, this could be only id, only iq or both. The important thing is that the look-up

table can be inverted only when the desired constant torque is below the minimum of the

torque from the highest current and above the maximum of the torque from the lowest

current in the table.

For example, suppose that some IPMSM is desired to be operated at id = −13.7A and

iq = 100A. The task is to reduce the torque pulsation by using only iq. The neighboring
operating points are chosen at iq = 80, 120 and 140A while id is kept fixed at −13.7A. The

corresponding torque of each level, including that of the desired operating point, are shown in

Fig. 5.1. Note that these torques will be used to create the look-up table and its acceptable

range is between the two black lines.

33

Figure 5.1: Torque at neighboring operating points.

Suppose that the desired operating point has the average torque of 108Nm. By inverting

the table at 108Nm, the reducing torque pulsation current can then be obtained, as shown

in Fig 5.2.

Figure 5.2: Reducing torque pulsation current at 108Nm obtained by inverting the table.

34

Chapter 6

Total Flux Linkage Estimation

In the torque estimation algorithm, the waveforms of the total flux linkage in d and q

axis are needed. They can be obtained by integrating the phase back-emfs and transforming

them to d-q quantities as (6.1).

λdq = Tabc→dq

(cid:20)(cid:90) (cid:0)vabc − iabcrs

(cid:21)
(cid:1)dt

(6.1)

where rs is the resistance of the stator winding, vabc is the phase terminal voltages and

Tabc→dq is the transformation matrix.

However, the phase terminal voltages vabc are usually inaccessible since most of the time,

the motor is driven by PWM. Due to this reason, vabc are replaced by the reference voltages

of the PWM ˆvabc.

35

Chapter 7

Simulation Results

The motor used in this chapter is the same motor used in the examples previously.

For convenience, its specification is repeated here. The motor is a 10-pole IPMSM with

concentrated windings and fractional slot per pole per phase. The rated current is Is = 25A
and its MTPA is approximately at 110◦, for all Is. Its cogging torque is shown in Fig 7.1.

Figure 7.1: Cogging Torque.

7.1 Torque Pulsation Estimation — Constant Idq

To illustrate the accuracy of the proposed estimation method among the other methods

in literature, three important methods are chosen from the literature, and their estimates

are compared to the estimate from the proposed method and the torque from FEA. The
comparison will be made at Is = 2, 12 and 22A, all at 110◦, which represent no, moderate

and heavy magnetic saturations.

36

The path of injection in the proposed method is Is = 2A− 22A with a step size of 2A, all
at 110◦, and the three methods are based on the following equations (7.1), (7.2) and (7.3).

3
2

T =

T = (cid:101)eaia +(cid:101)ebib +(cid:101)ecic

P (λdiq − λqid)
(cid:34)

ωm
[idq]T d[φdq]
dθe

1
2

T =

3
2

P

− 1
2

[φdq]T d[idq]
dθe

+ [idq]T d[ψdq]
dθe

+ [λF dq]T × [idq]

(7.1)

(7.2)

(cid:35)

+ Tcogging (7.3)

where (cid:101)eabc are no-load phase back-emfs, iabc are phase currents, λdq, ψdq and idq are d-q

axis total flux linkages, flux linkages from the magnets and currents respectively. ωm is the

mechanical speed in (rad/s), θe is the electrical position in radians, P is the number of pole

pairs, Tcogging is the cogging torque and all the matrices are defined as below with [λF dq] is
the average value of [λdq] and [φdq] = [λdq] − [ψdq].

λq


λd
ψd


id

ψq

iq

[λdq] =

[ψdq] =

[idq] =

Equation (7.1) is the general torque equation for IPMSMs that is normally used in control

applications. Equation (7.2) is one of the traditional torque equations for the torque pulsation

[2]. Equation (7.3) is one of the recent and promising analytical torque equations of IPMSMs

that, similar to this work, was derived based on the concept of co-energy [13].

37

The comparisons are shown in Fig. 7.2, 7.3 and 7.4. In each figure, the red plot is the

torque obtained from FEA. The pink, black and green plots are the torque obtained by

using (7.1), (7.2) and (7.3) respectively, and the blue plot is the torque obtained by using

the proposed estimation method.

Figure 7.2: Comparison of torque estimations based on eq. 7.1, 7.2 and 7.3 at Is = 2A.

Figure 7.3: Comparison of torque estimations based on eq. 7.1, 7.2 and 7.3 at Is = 12A.

38

Figure 7.4: Comparison of torque estimations based on eq. 7.1, 7.2 and 7.3 at Is = 22A.

It can be seen that (7.2) completely fails to estimate the torque in all cases. Equation

(7.1) can estimate the average torque correctly in all cases but fails to estimate the pulsation.

This is because the equation considers only the instantaneous torque. Equation (7.3) can

estimate the average torque correctly but can estimate the pulsation only at Is = 2A, where

there is no magnetic saturation. The reason is that the co-energy used in the derivation of

the equation was not thoroughly defined and the structure of the equation itself was based

on linear magnetic saturation. As for the proposed method, it can estimate the torque

accurately in all cases.

To further express that the proposed estimation method is valid and accurate for any

operating point, the path of injection is extended to Is = 26A, with a step size of 2A, all at
110◦, and the estimation method is then applied to each step. Fig. 7.5, 7.6 and 7.7 show

comparisons of the torques at all steps between those obtained by FEA (Red) and those

obtained by using the proposed estimation method (Blue). It can be seen that the method

can estimate the torque pulsation very accurately and for any operating point.

39

Figure 7.5: Torque comparison at Is = 2, 4, 6 and 8A.

Figure 7.6: Torque comparison at Is = 10, 12, 14 and 16A.

Figure 7.7: Torque comparison at Is = 18, 20, 22, 24 and 26A.

40

7.2 Torque Pulsation Estimation — Oscillating Idq

To show that the estimation does not require idq to be constant, this section repeats the

previous section but with 0.5sin(6ωst) and 0.5cos(6ωst) added into id and iq respectively at

each step of injection. The waveforms of the idq and their corresponding λdq at each step

of injection are shown in Fig. 7.8 to 7.11, and the comparisons of the torques are shown in

Fig. 7.12, 7.13 and 7.14.

Figure 7.8: id at each step.

Figure 7.9: iq at each step.

41

Figure 7.10: λd at each step.

Figure 7.11: λq at each step.

Figure 7.12: Torque comparison at Is = 2, 4, 6 and 8A.

42

Figure 7.13: Torque comparison at Is = 10, 12, 14 and 16A.

Figure 7.14: Torque comparison at Is = 18, 20, 22, 24 and 26A.

In order to verify the fact that the estimation algorithm is accurate and valid for any

current waveform, arbitrary waveforms of current are injected into the motor. Then, the

estimation method is applied. The arbitrary waveforms of current are shown in Fig. 7.15

and 7.16 and the comparison of the torques is shown in Fig. 7.17.

43

Figure 7.15: Arbitrary id.

Figure 7.16: Arbitrary iq.

Figure 7.17: Torque comparison under arbitrary current waveforms.

44

7.3 Torque Pulsation Reduction

By using the estimated torques of Is = 2A to Is = 26A, the look-up table is created. It

is assumed here that the task is to reduce the torque pulsation at Is = 22A. By inverting

the table at the torque value of this point, the reducing torque pulsation currents, in both

d and q axes, can be obtained and are shown in Fig. 7.18 and 7.19. The comparison of the

torque without and with the torque pulsation reduction is shown in Fig. 7.20.

Figure 7.18: Reducing torque pulsation current in d axis.

Figure 7.19: Reducing torque pulsation current in q axis.

45

Figure 7.20: Torque without and with the torque pulsation reduction at Is = 22A.

46

Chapter 8

Experimental Results

The motor used in this experiment is the motor of the 2-D cross section that has been

used throughout this work. The setup is shown in Fig. 8.1. In the figure, the dark blue

motor is the test motor. It is connected to a dynamometer (light blue motor) which helps

maintaining constant rotor speed. The test motor is also connected to a torque transducer.

Note that the torque obtained from this transducer is used only for comparison. As for the

cogging torque, it is obtained from FEA simulation.

Figure 8.1: Experiment Setup.

The desired operating point in this experiment is assumed to be Is = 22.5A with 110◦

and the task here is to reduce the torque pulsation at this point. The path of injection is
chosen to be from Is = 2A to 30A with a step size of 2A, all at 110◦.

47

Fig. 8.2-8.5 show the comparisons of the torque obtained from the transducer (Red) and

the torque obtained by the estimation (Blue) of Is = 18, 20, 22 and 24A at 110◦.

Figure 8.2: Is = 18A.

Figure 8.3: Is = 20A.

48

Figure 8.4: Is = 22A.

Figure 8.5: Is = 24A.

Using the estimated torques from each step of injection, the look-up table is created.
Then, by inverting the table at the torque level of Is = 22.5A at 110◦, the reducing torque

pulsation current in d and q axis can be obtained. They are shown in Fig. 8.6 to 8.7. The

comparison of the torque without and with the torque pulsation reduction by injecting these

currents is shown in Fig. 8.8.

49

Figure 8.6: Reducing torque pulsation current in d axis.

Figure 8.7: Reducing torque pulsation current in q axis.

Figure 8.8: Torque without and with the torque pulsation reduction from the transducer at
Is = 22.5A.

50

Chapter 9

Conclusion and Possible Future Work

In this work, a novel method in torque pulsation estimation and reduction of IPMSMs

are proposed. This work significantly differs from other similar works in literature in the fact

that the proposed estimation method is valid and accurate for any operating point, including

that under heavy magnetic saturation region, and for any current waveform. Simulation and

experimental results show that the proposed method is valid, accurate and effective.

Nevertheless, the proposed estimation method has two requirements. First, it requires the

cogging torque to be known in advanced. In the case that the cogging torque is not known,

the torque pulsation can still be estimated and reduced. However, this torque pulsation

is the total torque pulsation with the subtraction of the cogging torque, not the total one.

Second, it requires a motor to be connected to a dynamometer or a variable mechanical load.

However, by doing so only one time, not only can the torque at a specific operating point be

estimated, but the torque at any point within the path of injection can also be estimated.

One possible future work is to have the method online.

It is expected that by doing

so, the accuracy of the method may decrease.

In this case, the problem may lie in how

to maintain the accuracy within a given acceptable level. Other possible future works may

include cogging torque estimation of IPMSMs and analysis of influence of the eddy and

hysteresis loss on the estimation method.

51

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52

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