The correct analytical expression for the phase
inductance of salient pole machines
Gurakuq Dajaku, Dieter Gerling
Institute for Electrical Drives, University of Federal Defense Munich
Werner-Heisenberg-Weg 39, D-85577 Neubiberg, Germany
tel: +49 89 6004 3708, fax: +49 89 6004 3718,
e-mail: gurakuq.dajaku@unibw.de, dieter.gerling@unibw.de
Homepage: http://www.unibw.de/EAA
formulae for the total phase inductances for this type of
electrical machines shows that this parameter is constant
with rotor position, and doesn’t vary as is assumed in
many literatures. The analytical expression for the phase
inductances is verified and validated by comparing with
FE calculations.
Abstract- For the electrical machines with salient pole rotor
topology, due to the rotor saliency, the self- and mutual
inductance vary sinusoidally with the rotor position. In many
literatures analogous to the non-salient pole machines often
it is assumed that also for salient pole machines the phase
inductance is a function of rotor position, since the self
inductance of this type of machines depends on the rotor
position. The aim of this paper was to investigate further the
performances of this parameter. Deriving a correct formula
for the total phase inductance for this type of electric
machines shows that this parameter is constant with rotor
position, and doesn’t vary as is assumed in many literatures.
The new expression for the phase inductance is validated by
comparing with finite element method.
II. SELF- AND MUTUAL INDUCTANCES OF THE SALIENT
POLE MACHINES
Figure 1 shows cross section and the winding distribution
of a two-pole PM machine with inset magnets in the rotor
(salient pole machine topology).
I. INTRODUCTION
As is known, different mathematical models for PM
machines exist nowadays. Depending on the mathematical
model, self-, mutual- and phase inductances, or dqinductances appear in the main mathematical relations of
the PM machine. Therefore the winding inductance is an
important parameter required to model the performances
of electrical machines. In general, this parameter can be
determined analytically, or using finite element (FE)
method.
The self-, mutual-, and also dq-inductances are widely
studied in literature. For the electrical machines with
salient pole rotor topology, the self- and mutual
inductance vary with the rotor position due to the rotor
saliency. The dq-axis transformation provides the machine
inductances independent of rotor position by performing
the equations in a frame of reference that rotates in
synchronism with the rotor. Further, in analogy to
non-salient pole machines it is often assumed that also for
the salient pole machines the phase inductance can be
calculated according to: L phase = (3 / 2) ⋅ L self (e.g. in [2,
3]). Based on this hypothesis, the phase inductance is a
function of rotor position, since the self inductance of this
type of machines depends on the rotor position.
Therefore, the physical nature of this parameter is further
investigated and analysed in this paper. As a result, a
correct expression for the total phase inductances of the
salient pole electrical machines is derived. This expression
is valid even for non-salient pole electrical machines as a
special case of salient pole machines. Deriving the correct
1-4244-0743-5/07/$20.00 ©2007 IEEE
Fig. 1: Schematic diagram of a three phase synchronous machine.
Usually, during analysis of the electrical machines
different reference frames can be used. Figure 1 shows
two reference frames. For a three-phase electric machine,
the stator reference frame consists of three axes. The axes
u, v, and w denote the positive direction of the flux
produced by each winding (magnetic axes of the
respective phases). On the other side, for the salient pole
machines a two-axes (dq-axis) rotor reference frame is
useful. The d-axis is the axis of symmetry centered on one
rotor pole, and the q-axis is the axis of symmetry centered
between two rotor poles and leads the d-axis by 90
“electrical degree”.
Depending on the reference frame used, self-, mutual- and
phase inductances, or dq-inductances appear in the main
mathematical relations of the electrical machines.
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Self-inductance; In the case of salient pole synchronous
machines the effective air-gap length (reluctance of the
magnetizing path) isn’t constant along the circumferential
direction. As a result, the inductances of such machines
change with the rotor position. The variation of the stator
self-inductances with the rotor position for a salient pole
machine can be presented as [1]:
since the rotor q-axis is the path of lowest reluctance
(maximum permeance) for the air-gap flux.
III. PHASE INDUCTANCE OF THE SALIENT POLE MACHINES
The phase inductance (synchronous inductance) is the
effective inductance seen by one phase under the balanced
3-phase conditions of normal machine operation.
According to many literatures (e.g. in [2, 3]), the phase
inductance of the salient pole machines is calculated using
the following relation,
Luu = LA + LB ⋅ cos ( 2 ⋅ θ )
2π 

Lvv = LA + LB ⋅ cos  2 ⋅θ +

3 

2π 

Lww = LA + LB ⋅ cos  2 ⋅θ −

3 

(1)
LPhase =
where LA is the average value of the self-inductance, and
LB is the amplitude of the sinusoidal function in the
self-inductance curve. With θ is denoted the rotor
position (electric angle).
3
LSelf
2
(3)
Referring to the equ. (1) and (3), the phase inductance
would be a function of rotor position, since the
self-inductance of this type of machines depends on the
rotor position. Figure 3 shows the variation of the selfand phase (assumed, equ. (3)) inductance for the same
parameters ( LA , LB ) as before.
Mutual inductance; For a salient pole machine the
mutual inductance also changes with the rotor position.
Equation (2) describes the variation of the stator mutual
inductance with the rotor position [1],
1
2π 

Luv = Lvu = − LA + LB ⋅ cos  2 ⋅θ − 
2
3 

1
2π 

Luw = Lwu = − LA + LB ⋅ cos  2 ⋅θ + 
2
3 

1
Lvw = Lwv = − LA + LB ⋅ cos ( 2 ⋅θ + 2π )
2
(2)
The above equations show that due to the effect of the
rotor saliency, the self- and mutual-inductances vary with
“ 2θ ”. The following figure 2 shows the variation of the
self- and mutual inductances with rotor position for
LA = 0.266 mH and LB = 0.0848 mH .
Fig. 3. Variation of phase inductances with the rotor position [2, 3].
As should be shown in the 4-th section, the above
expression for the phase inductance is true only for the
non-salient pole machines. Deriving a correct formula for
the total phase inductance of salient pole electrical
machines shows that this parameter is constant with rotor
position, and doesn’t vary as is assumed in many
literatures.
III.1 THE
CORRECT EXPRESSION FOR THE PHASE
INDUCTANCE OF THE SALIENT POLE MACHINES
Under balanced three-phase conditions, the following
expressions for the current flux-linkages can be obtained
[4]:
Fig. 2. Variation of self- and mutual inductances with the rotor position.
3
3
ψu = LA ⋅ iˆ ⋅ cos (ωt +δ ) + LB ⋅ iˆ ⋅ cos (ωt −δ )
2
2
3
2
2π 
π

 3

ψv = LA ⋅ iˆ ⋅ cos ωt − + δ  + LB ⋅ iˆ ⋅ cos ωt − −δ 
2
3
3

 2


3
2π
2π 

 3

ψw = LA ⋅ iˆ ⋅ cos ωt + + δ  + LB ⋅ iˆ ⋅ cos ωt + −δ 
2
3
2
3




It is shown that the self-inductance of each stator phase is
a maximum when the rotor q-axis is aligned with the axis
of that phase, and the phase-phase mutual inductance is a
maximum when the rotor q-axis is aligned with the
midpoint of the two phases. This is the expected result
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(4)
where, with δ is denoted the electrical angle between
current iˆ and the rotor q-axis (load angle).
Let consider next the flux-linkage and the current in the
phase u,
iu = iˆ ⋅ cos (ωt + δ )
3
2
LU (δ ) =
3
2

where
3
2
3
3
ψ uB = LB ⋅ iˆ ⋅ cos (ωt − δ ) = LB ⋅ iˆ ⋅ cos (ωt + δ − 2δ )
2
2
ψ uA = LA ⋅ iˆ ⋅ cos (ωt + δ )
LU (δ ) = LV ( δ ) = LW ( δ ) = L (δ ) =
=
(9)
2
2
3
 LA + LB ⋅ cos ( 2δ )  +  LB ⋅ sin ( 2δ ) 
2 
Deriving a correct formula for the total phase inductance
of the salient pole machines, it is shown that this
parameter is constant with rotor position. Equation (9)
shows that the uvw phase inductances are always constant
and equal to each other; for the linear case they are
dependent only on the load angle δ .
For δ = 0° and δ = 90° we get from equ. (9):
The above expressions show that the first component of
the flux-linkage, ψ uA , is in phase with the current iu , and
the second component of the flux-linkage, ψ uB , is shifted
away from the current iu for the angle 2δ .
If with ψ̂ u , ψˆ uA = ( 3/ 2) ⋅ LA ⋅ iˆ , and ψˆ uB = ( 3/ 2) ⋅ LB ⋅ iˆ are
denoted the amplitudes of the ψ u , ψ uA , and ψ uB
respectively, the flux-linkage components can be
presented in phasor diagram as is shown in the figure 4.
3
( LA + LB ) = Lq
2
3
L ( δ = 90° ) = ( LA − LB ) = Ld
2
L ( δ = 0° ) =
(10)
that means that the total phase inductance is equal to the
q-, respectively d-inductance for the case when the total
flux due to currents flows through the q-, respectively drotor axis.
Otherwise, according to the equ. (9) the phase inductance
versus load angle δ (between q- and d-position) changes
as is shown in the figure 5. The following simulations are
made for the same LA and LB parameters as in section 2.
Fig. 4. Corresponding phasor diagram for the flux-linkage of phase u.
The phasor diagram for the flux-linkage components
shows that the total flux-linkage isn’t always in phase with
current but is shifted away for the angle γ depending on
the load angle δ .
Referring to the above phasor diagram, the following
relation for the flux-linkage of phase u (peak value) can be
obtained:
2
(8)
The remaining v, w phase inductances may be calculated
using the same procedure as above. Since the windings of
the electrical machines are identical and symmetrically
distributed, the phase inductances are equal. The uvw
phase inductances now may be expressed as:
(5)
2
LB ⋅ sin ( 2δ )


 LA + LB ⋅ cos ( 2δ ) 
γ = atan 
The flux-linkage of the phase u can be written as sum of
two components,
ψˆ u =  LA′ + LB′ ⋅ cos ( 2δ ) +  LB′ ⋅ sin ( 2δ )  ⋅ iˆ
(6)
where
LA′ =
(7)
And, the angle between flux-linkage and phase current can
be derived as,
ψ u = LA ⋅ iˆ ⋅ cos (ω t + δ ) + LB ⋅ iˆ ⋅ cos (ω t − δ )
ψ u = ψ uA + ψ uB
2
2
3
LA + LB ⋅ cos ( 2δ )  +  LB ⋅ sin ( 2δ ) 
2
3
3
LA , LB′ = LB
2
2
Fig. 5. Total phase inductance versus load angle
Using the relation L = ψˆ / iˆ , the correct expression for the
total inductance of the phase u is:
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δ
.
winding inductances of a PM machine. The machine type
chosen for the analysis is a synchronous motor with
interior magnets design in the rotor (V-magnets rotor
topology). Figure 8 shows the flux distribution under load
condition of the studied PM machine.
For an easy illustration of the variation of the phase
inductance with the load angle, the xy-plane is used in the
following, see figure 6. The phase inductance in the
xy-plane can be explained using a vector L ( δ ) , which
depending on the load angle moves around the periphery
of a circle. The length of the L ( δ ) vector represents the
actual value of the phase inductance depending on the
load angle δ (being conform with the fact that the
inductance is a number and not a vector). The radius of
the circle is L′B and its origin is shifted for L′A away from
the origin of the xy-plane.
Fig. 8. Flux distribution inside the studied PM machine.
In the following figures, the phase inductance L ( δ ) and
the shifting angle γ of the analysed PM machine obtained
from the FEM and the analytical expressions (equ. (9) and
(8)) are presented and compared. Firstly the L ( δ ) and γ
parameters versus load angle are derived using FEM for
50 A stator current (linear operation case). Then, by
knowing the phase inductance values at two different load
angle conditions, the parameters LA and LB can be
obtained using the equ. (10). From the FE results, the
parameters LA and LB at 50A load condition are:
Fig. 6. Variation of the total phase inductance with the load angle.
Furthermore, from the equ. (8) for δ = 0° and δ = 90°
we get:
γ ( δ = 0° ) = γ (δ = 90° ) = 0
(11)
that means that for these operation conditions the total
flux-linkage is in phase with the stator current. Otherwise,
the angle γ versus load angle δ (between q- and
d-position) changes as is shown in the figure 7. The
following simulations are made for the same LA and LB
parameters as in section 2.
Fig. 7. The angle
γ
versus load angle
δ
LA = 2,487 ⋅10−4 H ,
LB = 8,1284⋅10−5 H . Using these
parameters the variation of the phase inductance and the
shifting angle γ versus load angle δ are simulated using
the expressions (8) and (9).
.
Fig. 9. Total phase inductance versus load angle
III.2 COMPARISON BETWEEN ANALYTICAL AND FE
RESULTS
In the following, the derived analytical expression for the
phase inductance is validated using the finite element
method. The ANSYS software is used to predict the
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δ
.
that means the flux linkage and the phase current are
always in phase for this type of electrical machines (linear
case).
Equation (14) shows that the stator phase inductance of
the electrical machines with non-salient rotor geometry is
equal to 3/2 times self-inductance.
L phase =
Fig. 10. The angle
γ
versus load angle
δ
.
V. CONCLUSION
In this paper, the analytical expressions for the self-,
mutual-, and phase inductances are presented. For the
electrical machines with salient pole rotor topology, it is
shown that due to the rotor saliency the self- and mutual
inductance vary sinusoidally with the rotor position.
Deriving a correct formula for the total phase inductance
for this type of electric machines, it is shown that this
parameter is constant with rotor position, and doesn’t vary
as is assumed in many literatures. The analytical
expression for the phase inductance is verified and
validated by comparing with FE methods. Thanks the new
derived expression for the phase inductance, in [4] a new
and simple mathematical model for the PM machine is
developed.
δ = 0° to Ld at δ = 90° as is shown in the figure 9. Also,
from the figure 10 it is shown that the angle γ is zero (the
total flux-linkage is in phase with current) for δ = 0° and
δ = 90° , and different from zero (the total flux-linkage is
not in phase with the current) for 0° < δ < 90° .
IV. WINDING INDUCTANCES OF THE NON-SALIENT POLE
MACHINES
The winding inductances of the electrical machines with
non-salient rotor geometry (surface mounted PM machine,
induction machine) may be expressed from the inductance
relationhips given for the salient-pole synchronous
machine. In the case of such machines the air-gap is
uniform. Thus, the 2θ variations in the self- and mutual
inductances do not occur. This variation may be
eliminated by setting LB = 0 in the inductance
relationships given for the salient pole-synchronous
machines. All stator self-inductances are equal,
Luu = Lvv = Lww = LA
REFERENCES
[1] Krause, P. C: “Analysis of electric machinery”,
McGraw-Hill Book Company, 1986, ISBN: 0070354367.
[2] Kwak, S. Y.; Kim, J. K.; Jung, H. K.: “Inductance
and torque characteristics analysis of multi-layer buried
magnet synchronous machine”, KIEE International
Transactions on EMECS, Vol. 4-B, No. 4, 2004.
[3] Kulkarni, A.; Ehsani, M.: “A novel position sensor
elimination technique for the interior permanent magnet
synchronous motor drive”, IEEE Transactions on Industry
Applications, Vol. 28, No. 1, 1992.
[4] Dajaku, G.: “Electromagnetic and thermal modeling
of highly utilized PM machines”. Ph. D. Thesis, 2006,
University of Federal Defence Munich, Institute for
Electrical Drives and Actuators.
(12)
Likewise all stator-to-stator mutual inductances are the
same,
1
Luv = Lvu = Luw = Lwu = Lvw = Lwv = − LA
(13)
2
The stator phase inductances are also the same,
3
LA
2
(14)
And, the angle between flux-linkage and phase current is,
γ =0
(15)
With this it is shown that concerning the phase inductance
the non-salient pole machine can be regarded as a special
case of the salient pole machine.
It is shown that the results obtained from the analytical
expressions are in good agreement with the FEM results,
that means that the derived analytical expressions for the
phase inductance L ( δ ) and the shifting angle γ
accurately describe these parameters. The phase
inductance changes with the load angle from Lq at
LU = LV = LW =
3
Lself
2
(15)
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