Investigation of Force Generation in a Permanent Magnet
Synchronous Machine
W. Zhu, Student Member, IEEE, S. Pekarek, Member, IEEE, B. Fahimi, Senior Member, IEEE and B. Deken,
Student Member, IEEE
Traditional analysis of permanent magnet
synchronous machines has focused upon establishing a
relationship between the quadrature (q) and direct (d)
axis stator current (or voltage) and the electromagnetic
force created to establish rotation (torque). In this
paper, an alternative analysis of electromagnetic force
production is considered. Specifically, the influences of
q- and d-axis stator current on both the radial and
tangential components of the airgap flux densities are
first evaluated. Using a Maxwell Stress Tensor
approach, the fields are then used to evaluate both the
radial and tangential component of force density
created in the airgap of the machine. From this
perspective several interesting observations are made.
First, it is shown that the d-axis current has zero
influence on the average tangential force (torque), as
predicted using traditional analysis, but it has a
significant influence on the average radial component
of force. Second, it is shown that the q-axis current
contributes to both the average radial and average
tangential components of force. Interestingly, it is also
shown that under standard operating conditions, the
average radial force far exceeds that of the average
tangential component of force. Therefore, one can
conclude that the magnetic fields established create a
significant component of force in a direction that
cannot produce torque.
Abstract
Index Terms
Permanent Magnet Synchronous Motor,
Maxwell Stress Tensor, Force Density, Torque Generation.
I INTRODUCTION
Over the past half-century, advances in the design of
electric machinery have been relatively modest when
compared to quantum leaps made in the design of
controlled switches (semiconductors). Although advances
in electric machinery have been relatively modest, the
control of machines has changed dramatically due to
increased computing power and the power handling
capability of high power semiconductor devices. For each
class of electric machinery, controls now exist to achieve a
variety
of
desired
responses
(high-bandwidth
current/torque transduction, minimal torque ripple for a
desired average torque, maximum torque/amp, maximum
efficiency, etc). New materials, manufacturing techniques,
computer technologies, and semiconductor devices all
promise to ensure further progress is made along existing
drive system design guidelines.
The motivation for this research has started with a question
being asked by several researchers [1,2]–are there paths for new
machine design/excitation strategies that could lead to
significant changes in how electrical/mechanical energy
conversion is achieved? To begin to address this question, an
analysis of an existing surface-mount PM machine is performed.
However, in contrast to a traditional ‘macroscopic’ approach in
which the electromagnetic torque is the focus of the energy
conversion process, in this research an alternative view of force
generation in the PM machine is considered. Specifically, the
radial and tangential components of magnetic flux density in the
airgap are evaluated and used to calculate the electromagnetic
forces (radial and tangential components). This so-called
microscopic view of the force generation is used to view force
production from a different perspective and helps to further
understand how magnetic fields lead to force production.
Based upon this analysis, several observations have been
made. First, it is shown that the d-axis current has zero influence
on the average tangential force (torque), as predicted using
traditional analysis (in the absence of saturation). However, it
has a significant influence on the average radial component of
force. Specifically, it is shown that the relationship between daxis current and average radial force is a quadratic function.
Second, it is shown that the relationship between q-axis current
and average tangential force is linear (as predicted using
traditional analysis), while the relationship between the q-axis
current and average radial force is also a quadratic function –
although a different function than for the d-axis. Interestingly, it
is also shown that under standard operating conditions, the
radial force far exceeds that of the tangential component of
force. Therefore, one can conclude that the magnetic fields
established create a significant component of force in a direction
that does not produce motion.
In our opinion, the main contributions of this research are
two-fold. First, the analysis provides the community with a
more complete understanding of how typical excitation adjusts
the fields in the airgap and influences the overall force profile.
Second, the results point to a general direction for further
research in the area. Specifically, the fact that the majority of
force produced does not lead to motion raises a question as to
whether alternative geometries/excitation schemes can be
developed to yield a more productive force profile for electric
machines.
Prior to proceeding with the analysis, it is important to
highlight the works of researchers whose contributions are
closely related. Specifically, several researchers have used
analytical techniques to arrive at the magnetic field distribution
in the airgap of electric machines [3-12]. Among the early
efforts was Hague [3], who provided analytical solutions for the
magnetic fields in the airgap of a machine due to current
imbedded in iron or located inside the airgap of a machine.
1
φr
m
φs
m
More recently, analytical solutions for the fields in the
airgap of PM machines have been provided in [4-11].
Although these works are related, differences exist in the
methods used to account for stator slots, the co-ordinate
system applied (i.e. polar/rectangular), and the solution
method. Although very useful for understanding field
behavior and in design, none have explored the link
between q- and d-axis current excitation and the tangential
component of the field created by the stator windings.
Most recently, [12] has used analytical techniques to
explore the force profile under q- and d- excitation. The
primary goal of their effort was to consider the effect of
using d-axis field weakening on force ripple. Two key
assumptions in their analysis was 1) that the stator
windings produce a pure sinusoidally distributed field in
the airgrap, and 2) that this field contains a negligible
tangential component - i.e. only a radial flux density exists
from stator to rotor. Herein, to be more general, the fields
created by the stator windings are not assumed to be a pure
sinusoid – and in fact through FEA are confirmed to
contain harmonics. Moreover, it is shown that in general
one cannot assume that the stator windings produce a
unidirectional field. In fact the vector nature of the fields
created by the stator windings is critical to the production
of both the tangential and radial components of force.
C1
A2
A1'
θrm
S
B1
C2'
N
as-axis
N
B2
C1'
S
A2'
A1
C2
between the q- and as-axis (
rm
). Herein, so-called electrical
angles s , r , and r are defined by multiplying the mechanical
angles by the number of pole pairs. The relationship
(1)
s = r + r
exists between electrical angles: The following assumptions
have been made for the analysis provided:
• The stator teeth and permanent magnets are rigid; no
deformation due to radial and tangential force is
experienced by these components.
• The stator windings are concentrated and are wound at
a full-pitch.
• The permanent magnets are parallel-magnetized.
• The permanent magnets are not demagnetized by the
flux introduced by the phase currents.
• The flux density in the z-axis is assumed zero (no end
effects).
• Hysteresis and Eddy currents are neglected.
III MACROSCOPIC VIEW OF FORCE PRODUCTION
q-axis
B1'
purpose of analysis that is used in Section V, the angle sm is
defined as the position on the stator relative to the midpoint of
the phase-a stator slot shown and the angle rm is defined as the
position on the rotor relative to the midpoint of the permanent
magnet shown. Although not shown on the diagram, the
separation between angles rm and sm is the same as the angle
B2'
Prior to investigating force production under a so-called
‘microscopic’ view, it is useful to consider the analysis of the
machine from a more traditional perspective. In traditional
lumped-parameter-based machine analysis, a closed-form
expression for the electromagnetic torque is established using an
energy balance approach. This yields the expression [13]
P Wc
Te =
(2)
2
r
Where P is the number of magnetic poles and Wc is the coenergy of the coupling field that can be represented in a form
Wc = ias aspm + ibs bspm + ics cspm + Wcpm ( r )
(3)
In (3), Wcpm ( r ) represents energy that is due to the permanent
magnets being attracted to stator iron and the terms
Figure 1. A cross-sectional view of the PM machine and coordinate
system.
II MACHINE MODEL
The PM machine studied in this research is shown in
Fig. 1. It is a 1 Hp, 2000 rpm, 3-phase, 4-pole, 12-slot,
surface-mounted permanent magnet machine that is used
in home appliance applications. In Fig. 1, the phase-a
winding magnetic axis and the q-axis of the rotor are
shown, respectively. The mechanical rotor position ( rm )
is defined as the angle between the two axes. For the
asm
,
bsm
,
represent the influence of the flux of the permanent
magnets on the respective stator windings. For initial analysis,
the flux linkages are assumed of the form:
(4)
asm = mag sin ( r )
csm
bsm
=
mag
sin (
r
120 )
(5)
csm
=
mag
sin (
r
+ 120 )
(6)
Substituting (3) into (2) yields
2
Te =
P
ias
2
aspm
bspm
+ ibs
r
+ ics
r
cspm
+
Wcpm ( r )
r
r
(7)
Wcpm ( r )
where
represents the torque due to the
r
permanent magnets being attracted to the stator (cogging
torque).
When the equations of the PM machine are expressed
in terms of physical (abc) variables, the stator flux
linkages are functions of rotor position (due to (4)-(6)) and
thus the lumped-parameter model is time-varying. To
eliminate time-varying components a transformation of
variables is applied in which the dynamics of the machine
are represented in the rotor frame of reference [3]. The
resulting dynamic model of the machine expressed in the
rotor reference frame is of the form
vqsr = rs iqsr + r dsr + p qsr
(8)
vdsr = rs idsr
r
qs
r
r
qs
=L i
r
ds
= Lss idsr +
+p
r
ds
(9)
r
ss qs
(10)
(11)
m
3P
r
m iqs + Tcog
2 2
where rs represents the stator resistance,
Te =
r
qs
angular velocity, v
iqsr
voltages,
currents,
r
qs
and
and
r
ds
r
ds
and v
idsr
(12)
r
the rotor
the transformed stator
the
transformed
stator
the transformed stator flux linkages,
and Tcog the cogging torque. Equations (8)-(12) represent
the standard dynamic equations that are used to analyze
machine performance. Their derivation is described in
texts on PM machines and the response predicted using the
model is known to be accurate, provided that the
parameters are accurate. One item of interest is that from
(12) it can be observed that the electromechanical torque
generated by the machine is only associated with q-axis
current, the d-axis current has no contribution to average
torque. Based upon this observation, the d-axis current is
typically set to zero or a minimum value to achieve a
maximum torque per ampere ratio and minimum copper
loss. There are instances where the d-axis current is
introduced to have the effect of weakening the field
created by the magnet to facilitate operating at higher
machine speeds.
Although the standard equations offer the basic
principle to design the control of PMSMs, it is observed
that the information provided is limited from the sense that
there is little information on the overall force production.
Specifically, it is known that within the airgap there are
both radial and tangential components of force. To date,
there has been very little focus on how the q- and d-axis
components of current contribute to radial components of force,
with the exception of [12], wherein the focus was on force
ripple and assumed that the fields created by the stator winding
are unidirectional (i.e. radial from stator to rotor). Thus an
alternative microscopic investigation on torque and force
characteristics is described in the following section to address
both of these issues.
IV MICROSCOPIC VIEW OF FORCE PRODUCTION
An alternative to deriving the electromagnetic torque using
an energy balance approach is to derive the components of force
from the magnetic field using a Maxwell Stress Tensor method
[14]. Specifically, within the airgap of the machine the local
radial and tangential components of force density can be
expressed [14]
ft =
fr =
1
µ0
(13)
Br Bt
1
2µ0
(B
2
r
Bt2
)
(14)
where ft is the tangential component of force density (N/m^2),
f r is the radial component of force density (N/m^2), Br is the
radial component of the magnetic flux density and Bt is the
tangential component of magnetic flux density. µ0 is the
permeability of free space. Equations (13) and (14) provide the
basis for a microscopic investigation of the force production
within the machine.
As a first step in viewing the force under the light of (13)(14) it is useful to consider the magnetic flux densities in the
airgap of the machine subject to alternative stator excitation. To
facilitate this study, the machine was modeled using a finite
element approach wherein the radial and tangential components
of flux density were evaluated using a FEA model created using
the commercial package Maxwell [15]. A mesh with 21934
triangles was used in the calculations. To minimize error, the
contour of integration was established in the middle of the
airgap [16]. Hence the distribution of the flux density (and force
density) components was calculated in the middle of the airgap.
A convergence test was conducted to ensure accuracy.
Specifically, the number of triangles was increased to roughly
130000 and the difference between the forces calculated was
within 0.5%.
The tangential and radial flux density distribution along the
airgap for a machine in which the stator is de-energized and the
rotor is located at r = 0 is shown in Fig. 2. In Fig. 2, the
horizontal axis represents the position of an observer inside the
airgap of the machine. From the waveforms it can be seen that
the permanent magnet creates a significant radial component of
flux density (Brpm). There are changes in the radial component
around the regions directly below stator slots. In contrast, the
tangential distribution of the flux density (Btpm) has relatively
3
small amplitude and only occurs at the location
immediately around the stator slots. The tangential flux
density is attributed to flux from the magnet traveling to
the iron wall of the slots. Through visual inspection, it can
be seen that multiplication of Br and Bt will yield a
tangential component of force density that has equal
positive and negative components. Therefore, integration
of the tangential component of force density will yield a
value of 0 (confirmed numerically), meaning zero average
torque (as expected). In contrast, at zero stator current, the
radial component of force density will be nonzero and is
dominated by the radial component of the flux density.
0.5
Br
Bt
0.4
0.3
Flux density [T]
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
50
100
150
200
250
angular position in airgap (φsm)
300
350
Figure 2. Tangential and radial flux density in airgap generated by PMs.
As a second study, the overall tangential and radial
flux density distribution for a case in which current was
r
r
= 0 and iqs
= 4.6 A (rated), and the
applied in a manner ids
permanent magnets are maintained in the rotor, is shown in
Fig. 3. It is noted that the figure contains the distribution at
a single rotor position ( r = 0 ). Similar curves are
obtained at all positions of the rotor.
Br
Bt
0.5
Comparing the results shown in Fig. 3 with the flux
densities of the de-energized machine (Fig. 2), it can be seen
that the q-axis stator current influences both the radial and
tangential components of flux density. As one might expect, this
influence is most clear in the vicinity of stator slots. In contrast
to the de-energized machine, in each pole-pitch the tangential
component of flux density is positive over the region that the
radial component is positive. It is negative over the region that
the radial component of flux density is negative. Therefore, the
result is that the tangential component of force has a positive
average value (i.e. average torque is produced). It is also
interesting to consider that between stator slots, the radial
components of flux density are larger than the values observed
in the de-energized machine. This leads to an increase in the
average radial component of force.
The force densities computed at a single rotor position for
r
r
ids
= 0 and iqs
= 4.6 A are shown in Fig. 4. For clarity, the
negative value of the tangential component of force is plotted.
From these plots it can be observed that the force density
distributions along the airgap are far from uniform. This is due
to the existence of slots and a spatial current distribution that is
discontinuous. Considering the two waveforms of force density,
it is interesting to note that the peak values of both the radial and
tangential force densities are roughly the same. However, the
tangential component of force is created in the regions near the
stator slots. In contrast, the radial components are distributed
over an entire pole-pitch. Therefore, upon integration it is clear
that the radial component of force will be greater than the
tangential component. It is also interesting to note that a
relatively small percentage of the airgap space actively
participates in the torque generation process at a given rotor
position. It would appear that there is an opportunity to improve
torque density of a PMSM through design modifications
(geometries, winding configurations, excitation) that yield a
nonzero tangential force density over a wider region – similar to
the radial force density). Methods to search for such alternatives
using numerical techniques are being pursued in ongoing
research.
5
x 10
fr
-ft
0.4
1
0.3
Force density [N/(m*m)]
Flux density [T]
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0.5
0
-0.5
-0.5
-1
0
50
100
150
200
250
angular position in airgap (φsm)
300
350
Figure 3. Radial and tangential flux densities along the airgap when
r
r
ids
= 0 and iqs
= 4.6 A.
0
50
100
150
200
250
angular position in airgap (φsm)
300
350
Figure 4. Tangential and radial force density distribution along the airgap at a
single rotor position ( r = 0 ).
4
Br
Bt
0.6
0.4
Flux density [T]
A plot of the average component of radial and
tangential components of force/length of the machine
(N/m) as a function of q-axis current is shown in Fig. 5.
Comparing values of radial and tangential components of
force, it is clear that that radial component is much greater
than the tangential component. It is also observed that the
tangential component of force is linearly related to q-axis
current (as predicted from the macro-scopic model).
However, it can be seen that an increase in q-axis current
also leads to an increase in the radial component of force.
As will be shown in Section V, the relationship between
radial force and q-axis current is a quadratic function.
0.2
0
-0.2
-0.4
0
50
6000
100
150
200
250
angular position in airgap (φsm)
300
350
Figure 6. Radial and tangential flux densities along the airgap when
r
r
ids
= 4.0 A and ( iqs
= 4.6 A).
5000
4000
tangential force
radial force
5
x 10
3000
fr
-ft
1.5
2000
1
1000
0
0
1
2
3
4
5
6
q-axis current [A]
7
8
9
Figure 5. Average tangential and radial force with increasing q-axis
r
current ( ids
= 0 A).
The flux density distributions along the airgap for a
r
= 4.0 A
case in which the d-axis phase current is set to ids
r
= 4.6 A) are shown in Fig. 6. Comparing Fig. 6 to Fig.
( iqs
3, it can be observed that the introduction of the d-axis
current leads to a change in both the radial and tangential
components of flux density. If one compares values of the
radial component of flux density over a pole-pitch, there is
a general increase in value. In contrast, the tangential
component does not see a general increase in magnitude
over a pole pitch. Rather, as shown by the tangential
components that are encircled in the figure, there are
places where the tangential component of flux changes
r
= 0 A. Therefore, in
sign relative to their value when ids
r
contrast to the case in which ids
= 0 A, the multiplication
of Br and Bt leads to regions in which the tangential force
density distribution is negative. The radial and tangential
components of force density are shown in Fig. 7.
Force density [N/(m*m)]
Average value of radial and tangential force [N/m]
-0.6
7000
0.5
0
-0.5
-1
-1.5
0
50
100
150
200
250
angular position in airgap (φsm)
300
350
Figure 7. Tangential and radial force density distribution along the airgap at a
r
single rotor position ids
r
= 4.0 A ( iqs
= 4.6 A).
Shown in Fig. 8 are the flux density distributions along the
airgap for a case in which the d-axis phase current is set to
r
r
ids
= 4.0 A ( iqs
= 4.6 A). The radial and tangential components
of force density are shown in Fig. 9. Comparing the radial
component of flux density over a pole-pitch to those shown in
Fig. 3 and Fig. 6, there is a general decrease in value. In
contrast, the magnitude of the tangential component increases
slightly over a pole pitch. Similar to the case in which d-axis
current is positive, there are locations where the tangential
component of flux density changes sign relative to its value
r
= 0 A. Multiplication of Br and Bt leads to regions in
when ids
which the tangential force density distribution is negative.
Comparing radial force density, it is seen that the decrease in
radial flux density leads to a general decrease compared to the
r
r
= 0 A and ids
= 4.6 A.
cases in which ids
5
15000
B
Average value of radial and tangential force [N/m]
r
0.5
Bt
0.4
0.3
Flux Density [T)]
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
tangential force
radial force
10000
0
−0.5
-8
0
50
100
150
200
250
300
Angular Position in the Air Gap, fsm [degrees]
350
Figure 8. Radial and tangential flux densities along the airgap when
r
r
ids
= 4.0 A and ( iqs
= 4.6 A).
5
x 10
fr
1
Force Density [Nm−2]
5000
ft
0.5
0
−0.5
−1
0
50
100
150
200
250
300
Angular Position in the Air Gap, fsm [degrees]
350
Figure 9. Tangential and radial force density distribution along the airgap
r
at a single rotor position ids
=
r
4.0 A ( iqs
= 4.6 A).
Under both positive and negative d-axis current,
changes in radial and tangential flux densities are such that
the average value of the tangential component of force is
not related to d-axis current (as predicted by the
macroscopic model). The average value of the radial and
tangential components of the force/length are plotted as a
r
= 0 A) in Fig. 10. It can be
function of d-axis current ( iqs
seen in Fig. 10 is that the radial component of force
changes significantly with change in d-axis current.
-6
-4
-2
0
2
d-axis current [A]
4
6
8
Figure 10. Average tangential and radial force under different d-axis current
excitation (q-axis set to 0).
V ANALYSIS OF OBSERVATIONS
In order to provide some insight into the results observed in
the previous section, the contributions of the airgap flux
densities is explored analytically. For a PM machine, both Br
and Bt are created by two sources: the stator windings and the
permanent magnets. Since the PM machine usually has a
relatively large effective airgap, for the purposes of analysis,
saturation is neglected. Therefore, using superposition,
Br = Brpm + Brs
(15)
Bt = Btpm + Bts
(16)
where Brpm, Btpm, Brs and Bts denote the radial and tangential flux
density created by the PM and stator windings, respectively.
Substituting the field flux densities (15)-(16) into (13)-(14) the
tangential and radial force density can be expressed as:
1
ft =
Bts + Btpm Brpm + Brs
(17)
µ0
fr =
1
2 µ0
[( Brpm + Brs )2 ( Bts + Btpm )2 ]
(18)
Using (17) and (18), the tangential and radial force can be
derived from the design and excitation. Specifically, the overall
tangential and radial force/length at a single rotor position can
be expressed as:
P 2
Ft =
ft ( r ) Rd r
(19)
2 0
P 2
Fr =
f r ( r ) Rd r
(20)
2 0
and the electromagnetic torque expressed as
Te = Ft RLef
(21)
where R is the radius of the contour upon which the Maxwell
Stress Tensor is calculated and Lef is the effective stack length
(meaning the physical stack length multiplied by a stacking
factor). From (17), ft can be divided into four parts:
6
(i) Brpm Btpm
L
Brs =
(ii) B rs Bts
(iii) Brpm Bts
P
2 µ0
2
[Brpm Bts + Brs Btpm ]Rd
s
(22)
0
To consider the terms in (22) it is useful to expand each of
the flux densities in terms of a series. Specifically,
considering Fig. 2, one can see that Brpm can be expressed
as a Fourier series of the form
Brpm =
Brpmk cos(k r )
(23)
k =1
Similarly, it can be seen that Btpm can be expressed as a
Fourier series of the form
Btpm =
+ Br
cs
(25)
Btsk = Bt
+ Bt
as
bs
+ Bt
cs
(26)
Btpmk sin(k r )
where L is the number of stator slots, Br as , Br bs , Br cs , are
the radial components of flux density created by the phase
windings. Bt as , Bt bs , and Bt cs are the tangential components
of flux density created by the phase windings.
The tangential and radial flux density distribution along the
airgap created by the current in a single slot (obtained using the
FE model) is shown in Fig. 11. If both the stator and rotor iron
are unsaturated, the local flux densities created by current in
each slot are proportional to the current. Thus we can define:
Btsk ( s ) = islot b1 ( s )
(27)
Brsk ( s ) = islot b2 ( s )
(28)
where b1 and b2 are associated with the geometry of the
machine. From the numerical results it can be seen that for the
machine studied, b1 is an even function and b2 is an odd
function (with respect to sm ), respectively. This is consistent
with the analytical derivations made on simplified machines.
Specifically in [3] solutions of the Poisson equation in
cylindrical coordinates are obtained for 1) a machine assuming
stator currents embedded in stator iron and 2) a machine
assuming stator currents in the airgap with a smooth stator inner
surface. In both cases, the results indicate a solution with even
and odd symmetry, respectively.
Brsk [T]
contribution of (ii) on average tangential force is due to the
fact that Brs is an odd function and Bts is an even function,
which will become clear later in the analytical
development. Therefore, a conclusion is that only terms
(iii) and (iv) can generate an average torque. For the
analysis presented herein, the focus is on average values of
force and therefore using (iii) and (iv) one can express the
force/length due to these two terms as
=
bs
k =1
Terms (i) and (ii) produce cogging and reluctance torque
due to the existence of slots. However, both will yield zero
average force. A zero average value of term (i) can be seen
by considering Fig. 2. Specifically, integration of the
Brpm Btpm shown in Fig. 2 over a pole pitch is zero. The
iv )
+ Br
as
L
Bts =
(iv) Brs Btpm
Ft (iii
Brsk = Br
k =1
(24)
k =1
and therefore one might expect the series to only include
components that are multiples of the number of slots, it has
a fundamental period of 360 (electrical) degrees, i.e.
Btpm1 0 . This is due to the behavior of the flux around
the transition region between magnets and can be seen in
Fig. 2.
To evaluate the effect of the stator windings, it is
convenient to first consider that the net field created by
current in the stator windings is the sum of the fields
created by the current in each stator slot. Specifically, if
Brsk and Btsk represent the radial and tangential flux density
created by current in kth slot, then the net flux densities are
expressed
φsm
Btsk [T]
Prior to proceeding with the flux densities produced by the
stator windings, it is interesting to note that although
Btpm is due to interaction of the magnets with stator slots
φsm
Figure 11. Radial and tangential components of flux density introduced by
current in a single stator slot.
Based upon the result of current in a single stator slot, one
can express the tangential components of flux density
contributed by the individual phase windings in a form
Bt _ as ( s ) = ias B1 ( s )
(29)
Bt _ bs ( s ) = ibs B1 (
s
2 / 3)
(30)
Bt _ cs ( s ) = ics B1 (
s
+ 2 / 3)
(31)
7
the focus is on average tangential force which is represented by
the first term on the right hand side of the equal sign in (43).
Considering the expression, it is clear that the average tangential
force is only a function of the q-axis stator current.
The derivation of radial force follows a similar set of steps,
although the interaction of all components is a factor in the
average radial force. Plugging (25) and (26) into (18) and the
result into (20) yields
where B1 ( s ) is an even (zero average) periodic function
with respect to s . Specifically,
B1 ( s ) =
B1k cos(k s )
(32)
k =1
Similarly, one can express the radial components of flux
density due to individual phase windings in a form
Br _ as ( s ) = ias B2 ( s )
(33)
Br _ bs ( s ) = ibs B2 (
2 / 3)
s
Br _ cs ( s ) = ics B2 (
s
P
Fr =
4µ0
(34)
+ 2 / 3)
(35)
B2 k sin(k s )
Btpm (
+ i)
ibs = is cos(
r
+
Fr =
(37)
120)
i
(38)
+ B2 (
+
s
0
) Btpm (
s
2
3
s
2
+
2 /3
2
3
r
Btpm (
2 /3
s
is cos (
)
is cos
2 /3
P
2 µ0
+ B2
2
2 / 3+ 2
P
2 µ0
+ B2
+
P
2 µ0
=
iv )
i
)
s
+
i
2
3
)
Rd
s
r
r
s
+
Rd
r
is cos
Btpm (
r
+
r
i
)
+
B1 (
) Brpm (
B1
s
B1
s
s
r
+
)
2
3
Brpm (
s
r
)
2
3
Brpm (
s
r
)
s
r
iqs
= is cos( i )
(41)
r
ids
= is sin( i )
(42)
one can obtain a general form for the tangential
force/length due to Brpm Bts and Btpm Brs as
iv )
=
3P
8µ0
Br _ is
i = a ,b,c
(44)
r)+
Bt _ is
Rd
s
(B
2
2k
k , k 3,9, etc
B1k 2 ) ( iqsr 2 + idsr 2 ) +
$
Brpm1 + B11 Btpm1 ) idsr % R + Fr ( r )
k
&
(45)
where Fr ( r ) represents a component of radial force that is a
function of rotor position (force ripple). Force ripple is
evaluated in [12] for the special case in which the stator
windings yield only a pure sinusoidal radial flux density. A
value of average normal force is also provided in [12] based
upon the sinusoidal-radial approximation. Equation (45) will
simplify to the result provided in [12] if the same assumptions
are applied – although based upon the findings herein, these
cannot be made in general. Herein the focus is on the average
force which is represented by the first three terms on the right
hand side of the equal sign in (45). From (45) it is clear that the
average radial force is a quadratic function of both q- and d-axis
currents. Interestingly, the q- and d-axis expressions take
slightly different forms. Specifically the third term on the right
2
Btpmk
)+3
(B
21
r
(40)
Substituting (23), (24), (32), and (36) into (40), applying
trigonometric identities, and using the relationship that
Ft (iii
s
!9
"
!# 4
2
rpmk
s
2
3
Rd
s
P
4µ0
(B
ics = is cos( r + i 120)
(39)
and substituting (37)-(39), into (29)-(31) and (33)-(35) and
the net result into (22), one obtains the expression
Ft (iii
r)+
s
Substituting the Fourier series representation of each of the
components into (44) and using trigonomic identities to
simplify, one obtains a result for the radial force:
(36)
k =1
r
Brpm (
i = a ,b , c
Assuming phase current excitation of the form:
ias = is cos(
0
2
2
where B2 ( s ) is an odd (zero average) periodic function
with respect to s , i.e.
B2 ( s ) =
2
r
Btpmk B2 k iqs
R + Ft (
Brpmk B1k +
k
r
)
k
(43)
where Ft ( r ) represents a component of tangential force
that is a function of rotor position (i.e force ripple). Herein
hand side of the equal sign only includes ids . This accounts for
the more significant effect that the d-axis current has on the
radial component of force (seen in Fig. 8). Moreover, this same
term also shows how applying a negative d-axis current reduces
the radial component of force. In contrast, application of a qaxis current can only act to increase the radial component of
force. From (45) one can also observe how the different
components of the flux densities play a role in the production of
average radial force.
As noted in Section II, the effects of saturation, eddycurrents, and hysteresis have been neglected in the analysis of
force production. In saturation, the analysis starting with (25) is
invalid, since one cannot assume linearity. Initial research
indicates that in saturation the odd/even symmetry nature of the
radial and tangential components of flux density remain the
same. However, since the flux density components are nonlinear
8
functions of stator current, the tangential force cannot be
expressed as a linear function of q-axis current.
The effects of hysteresis and eddy currents on the flux
and force densities may also be of interest. However, since
machine models that include hysteresis and eddy current
effects are rarely necessary to predict the relationship
between torque and stator current, these effects may not be
significant (except possibly in high speed applications).
VI CONCLUSION
The magnetic flux density and force density
distribution along the airgap of a PM machine have been
investigated analytically by leveraging a field solution
obtained from finite element analysis. The effect of both qand d-axis phase current on the radial and tangential field
and force distribution has been analyzed. Based upon this
analysis, several observations have been made. First, it is
observed that under standard operation the majority of the
force produced in the machine is in the radial direction,
which cannot translate into rotor rotation. Second, it has
been found that the q- and d-axis components of current
both contribute to the average radial force – and that their
relationship can be expressed as a quadratic function. It is
also verified that the average component of tangential
force (torque) is linearly related to q-axis stator current,
while the d-axis current has no influence of average
torque.
Finally, through the analytical process an
evaluation of how stator excitation influences flux
densities to produce torque and radial force has been
provided which helps to provide a so-called microscopic
view of force production in the machine.
[11]
[12]
[13]
[14]
[15]
[16]
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38, No. 1, pp. 229-238, Jan. 2002.
Wang X., Li Q., Wang S., Li Q., “Analytical Calculation of Air-Gap
Magnetic Field Distribution and Instantaneous Characteristics of
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18, No. 3, pp. 424-432, Sept. 2003.
Jiao G. and Rahn C., “Field Weakening for Radial Force Reduction in
Brushless Permanent Magnet DC Motors,” IEEE Transactions on
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P. C. Krause, O. Wasynczuk, S. D. Sudhoff, Analysis of Electric
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Belahcen, A., “Overview of the Calculation Methods for Forces in
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
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9