Zeszyty Problemowe Maszyny Elektryczne Nr 4/2014 (104) 117

Krzysztof Ludwinek
Kielce University of Technology

REPRESENTATION OF THE STATOR TO ROTOR SELF- AND

MUTUAL INDUCTANCES IN A SALIENT POLE SYNCHRONOUS
GENERATOR IN THE NO-LOAD STATE

Abstract: This paper presents a representation of the stator to rotor self and mutual inductance and derivative
distributions in the 5.5 kVA salient pole synchronous generator with damping circuits, both with and without a
rotor skew in the no-load steady state. In the circuital linear and nonlinear model, the stator to rotor mutual
inductances are determined in the FEM program. Presented distributions allow to express the waveforms of
induced phase stator voltages in the no-load steady state using a circuital model in the natural reference frame
of the stator and rotor.

1. Introduction The damping circuits allow for shortening

A 3-phase synchronous generator is used as a many transients stages, e.g. the hunting [6],
constantly increasing reserved power source. reducing higher harmonics in the field winding
A synchronous generator as a source of power current or increasing of higher harmonics in
has good quality if the contents of the higher armature currents [7]. As will be shown in the
harmonic induced in armature windings are no-load steady state, the discrete distribution of
very low [1, 2]. The content of higher harmonic the damping bars due to higher harmonic
voltages induced in the armature windings is current has an influence on higher harmonic
particularly evident in low power up to several contents in induced voltages.
kVA salient pole synchronous generators In the no-load of the steady state of a salient
especially [3]. The largest content of harmonics pole synchronous generator the distribution of
in the induced phase voltage (in low power magnetic flux density in the air gap is distorted
generators) occurs with the synchronous due to: saturation of the magnetic circuit
generators with the single-layer stator winding (mainly low order odd harmonics) [8, 9], the
without the rotor (or stator) skew. In 3-phase influence of the stator slot opening or rotor
low power synchronous generators (less than 10 damping cage slot opening [10], magnetic rotor
kVA) dominate single layer windings. Apart asymmetry [5, 9, 11, 12] and static and dynamic
from that, many low-cost synchronous stator and rotor eccentricity [9, 11, 12].
generators in the Polish market have the stator In circuital modelling an analytical way of
and rotor structures without a skew. No skew describing a distribution of the self- and mutual
causes a significant increase in higher harmonic inductances with taking into account the
order in stator and rotor inductance electrical angle of the rotor position and current
distributions [3, 4] in the induced voltages [1, 2, are described in [9, 10, 11, 13]. However, this
4] and in currents during powering of various method requires correction of coefficients
types of electronic equipment (AV, etc.), describing the self- and mutual inductances [14,
computers, notebooks, UPS, compact 15]. These coefficients the most frequently are
fluorescent lamps, etc. The presence of higher expressed in the form of Fourier series or based
harmonics in the phase voltage and current on the co-energy are determined as a function
waveforms of synchronous generators has of the electrical angle of the rotor position and
negative economic consequences. Because of current using analytical or FEM methods [12 -
additional power losses these generator sets 16]. The correction of the analytical form of
consume more fuel. coefficients the most often is carried out on the
In dynamic states the electromagnetic basis of experimental investigation or using the
properties of a synchronous machine depend on FEM methods (if the detailed geometry and
the presence of the damping circuits [5, 6]. construction-material data are known) [14].
There are many methods to obtain the This article examines the influence of
parameters of the damping circuits from representation of the stator to the rotor winding
experimental, analytical or FEM methods [5, 6]. mutual inductances on the induced voltages in
118 Zeszyty Problemowe Maszyny Elektryczne Nr 4/2014 (104)

the 3-phase armature winding in a 5.5 kVA, Figure 1 shows the equivalent circuit
salient pole synchronous generator both with parameters of a synchronous generator in the
and without rotor skew under no-load steady stator and the rotor natural reference frame in
state conditions. In part 2 of this article, the the no-load state (ia, ib and ic are equal to zero).
higher harmonic contents in induced stator The equivalent circuit represents the stator
phase voltages are calculated when the DC windings, the field winding and shorted
voltage or DC current is powering the field equivalent 5-damping bars per pole.
winding. The symmetry of the stator windings For such arranged equivalent circuit (Fig. 1),
and field winding with 10 equivalent shorted the voltage ua, ub, uc of a salient pole
damping bars, with the linearity and synchronous generator induced in the three-
nonlinearity of the magnetic circuit have been phase armature windings, taking into account
assumed. The self- and mutual inductances are the field winding, shorted damping bars and the
determined using Flux 2D Skew and FEMM electrical angle of the rotor position can be
programs [3, 4]. The end effects (leakage derived from the equations, in stator
inductance and resistance) are calculated using coordinates (for the stator windings) and in
the analytical technique. rotor coordinates (for the rotor windings).
2. Model of a salient pole synchronous
d a d b d c
generator in the stator and rotor natural = ua , = ub , = uc (1)
dt dt dt
reference frame
d f
The Parks transformation of the higher + Rf i f = u f (2)
harmonics of the stator self- and mutual dt
inductance distributions to the dq0-axes [9, 14, d r ( k ) ir ( k )
+ Rr ( k ) ir ( k ) + Ler =0 (3)
17] does nothing and only introduces additional dt dt
unnecessary calculations. In this case the self- d
and mutual inductance distributions in the dq0- = and ir (1) + ir ( 2 ) + ... + ir (10 ) = 0 (4)
dt
axes are dependent on the angle position of the
rotor. Hence, the induced phase stator voltages Where: a, b and c indexes of stator windings,
ua, ub and uc in simulations is easier to carry out f field winding index, r(k) index of kth-
with a circuital model of a salient pole damping bar, k = {1, 2, 10},a, b, c
synchronous generator in the stator and rotor stator linkage fluxes, ua, ub, uc stator phase
natural reference frame. voltages, electrical angle of the rotor
a) b) position, = mpb, m mechanical angle of the

rotor position, = (1/pb)d/dt electrical
angular velocity, Rf resistance of the field
winding, if field current, ir(1), ..., ir(10) current
in equivalent 10 damping bars and ring
elements, Rr(1), ..., Rr(10) resistance of the

equivalent 10-damping bars and ring elements,
respectively, whereas, Rr(k) resistance of
c) equivalent damping bar and ring elements Rr(k)=
Rpr + Rer/{2sin((k)/Qr)}, Rpr, Rer resistance of
Ler Ler Ler Ler damping bar and ring elements, respectively,
(k) is the angle between the equivalent k
Rr(1) Rr(5) Rr(6) Rr(10)
rotor damping bar (with ring elements) and the
ir(1) ir(5) ir(6) ir(10)
d r(1) d r(5) d r(6) d r(10) rotor reference axis, Qr number of rotor bars,
dt dt dt dt Ler inductance of end ring elements.

Fig. 1. Equivalent circuit parameters of a The equation for differential linkage fluxes (1) -
salient pole synchronous generator in the stator (3) taking into account the influence of currents
and rotor natural reference frame in the no- and electrical angle of the rotor position can
load state a) field winding, b) stator windings, be derived from the equations [9,13, 18]:
c) damping circuits
 Zeszyty Problemowe Maszyny Elektryczne Nr 4/2014 (104) 119

LLLL
iiii

iiii
iiii
LLLL
iiii
d d ( , )
= + ( , ) d (5) 3. Determination of the self- and mutual
dt dt dt inductances
The self- and mutual inductances in the
Based on the expressions (1) - (3) and (5) and
expressions (6) - (7) are calculated on the basis
the mutual inductances of the stator to rotor
of the real construction data (non-uniform air
windings and the self inductance of the field
gap at the periphery of the stator and rotor)
winding, the induced phase stator voltages ua,
received from the manufacturer of synchronous
ub, uc can be expressed as: LLLL generators that are used frequently in generator
sets in Poland. While calculating the inductance
LLLL
sr di r
us = ir + sr (6) distributions (as a function of the electrical
dt
rotor position angle , with 10 damping bars
and nonlinearity) in the Flux 2D Skew and
However, the equation describing the excitation
FEMM software [3, 4], the factory single-layer
of the field winding and damping bars can be
winding placed in the stator slots is taken into
expressed as
account. The self- and mutual inductances are
LLLL

carried out for a salient pole synchronous

RRRR

LLLL

fr di fr
(7)
u fr = + fr i fr +
fr generator (with and without the rotor skew)
dt
rated: SN = 5.5 kVA, UN = 400 V (Y), nN = 3000
rpm, IN = 7.9 A, cosN = 0.8, Qs = 24 (number
Where: us = [ua, ub, uc]T matrix of induced
of stator slots), q = 15 (factory rotor skew
Laf Lar (1) ... Lar (10)
equal to stator slot pitch), pb = 1 (number of
stator phase voltages, L sr = Lbf Lbr (1) ... Lbr (10)
pole pairs).
Lcf Lcr (1) ... Lcr (10)
Figure 2 presents magnetic flux distribution
matrix of mutual inductances of stator-to- lines of the examined 5.5 KVA nonlinear
rotor damping bars and ring elements, Laf, Lbf, salient pole synchronous generator in the no
Lcf mutual inductance of stator-to-field load steady state with shorted equivalent
windings, Lar(1), Lbr(1), Lcr(1), Lar(10), Lbr(10), damping 5 bars per pole with initial rotor
Lcr(10) mutual inductance of stator windings to position 0 = 0. A method of determining the
10 damping bars (and ring elements), self- and mutual inductance distributions in
FEMM program is detailed presented in [18].
L f + Lef L fr (1) ... L fr (10)

L fr = L fr (1) Lr (1) + Ler ... Lr (1, 10) matrix
... ... ...
L Lr (10,1) ... Lr (10) + Ler
fr (10)

of self- and mutual inductances of field-to-

damping bars and ring elements, Lf self
inductance of the field winding, Lfr(1), Lfr(10)
mutual inductance of field winding to 10
damping bars and ring elements, Lr(1), Lr(10)
self inductance of 10 damping bars and ring
elements, ir = [ir(1), ... , ir(10)]T matrix currents
in damping bars, ifr = [if, ir(1), ... , ir(10)]T matrix Fig. 2. Magnetic flux distribution lines in the
of field winding current and currents in 5.5-kVA salient pole synchronous generator
damping bars, ufr = [uf, 0, ... , 0]T matrix of Figure 3 shows the comparison of the
voltages of field winding and in shorted 10 distributed stator to field winding mutual
damping bars and ring elements, respectively, inductances and their derivatives for linear and
Rfr = [Rf, Rr(1), ... , Rr(10)] diagonal matrix of nonlinear 5.5 kVA salient pole synchronous
resistance of the field winding and 10 damping generator with and without the rotor skew. The
bars and ring elements, Lr(m,n) mutual mutual inductances for nonlinear model are
inductance of damping bars and ring elements very similar to the linear ones. The differences
(m n) and m, n = {1, , 10} [18]. are only visible after calculation of the product
of Lafn/ and due to Fourier analysis.
120 Zeszyty Problemowe Maszyny Elektryczne Nr 4/2014 (104)

a) Figure 3 indicates: s index with the rotor

1
Laf skew, n index for nonlinear model. Detailed
0.5 L analysis of the self- and mutual inductance
- [H]

afn
0 distributions for the salient pole synchronous
generator is described in [3, 4].
af
L

-0.5
Figure 4 shows the contents of harmonic
-1
0 60 120 180 240 300 360
magnitudes due to Fourier analysis in
- [deg] inductances Laf and derivatives Laf/ for
b) linear and nonlinear model (Figs. 3a - 3c) with
400
d L /d - [ H/s ]

and without the rotor skew.

200
The contents of harmonic in inductances (Fig.
0
dLaf/d 4) from 43rd to the 75th harmonic are less than
af

-200 dLafs/d 0.05%. Figure 4 shows that the greatest

-400 reduction of the harmonic content is achieved
0 60 120 180 240 300 360
- [deg] by using the rotor skew (or stator skew).
c)
400
1
d L /d - [ H/s ]

200 0.75

k - [-]
0.5
0
q
afn

dLafn/d 0.25
-200 0
dLafsn/d
-400 -0.25
0 12 24 36 48 60 72 84
0 60 120 180 240 300 360 harmonics
- [deg]

Fig. 3. Comparison of the stator to field Fig. 5. Skew factor kq for the q = 15 and pb = 1
winding mutual distributions of a) inductances
a)
and b), c) the product of Laf/ for linear 4 x 10
-4

Lar(1)
and nonlinear model with and without the rotor
- [H]

2 Lar(2)
skew, respectively Lar(3)
0
ar(k)

Lar(4)
L

-2
a) Lar(5)
8 -4
Laf 0 60 120 180 240 300 360
- [deg]
6 Lafs
- [%]

b)
Lafn
/d- [H/ s]

0.2
4
afv

Lafsn 0.1
L

2
0
ar(3)

0 dLar(3) /d
3 5 7 9 11 13 15 17 19 21 23 25 27 29
dL

-0.1
harmonics dLar(3)s /d
b) -0.2
0 60 120 180 240 300 360
8 - [deg]
dLaf/d c)
dL /d - [%]

6 0.2
/d- [H/ s]

dLafs/d
4 dLafn/d 0.1
afv

dLafsn/d
2 0
ar(3)

dLar(3)n /d
-0.1
dL

0 dLar(3)sn /d
3 5 7 9 11 13 15 17 19 21 23 25 27 29
harmonics -0.2
0 60 120 180 240 300 360
c) - [deg]
8
dL /d - [%]

dLaf/d Fig. 6. Comparison of the stator to damping bar

6
dLafs/d mutual distributions of a) inductances for damping
4 dLafn/d
bars Lar(1) - Lar(5) (Lar(6) - Lar(10)), b), c) the product of
afv

dLafsn/d
2 Lar(3)/ for linear and Lar(3)n/ for nonlinear
0
43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75
model with and without the rotor skew, respectively
harmonics

Fig. 4. Comparison of harmonic contents for Figure 5 shows the values of the skewing factor
linear and nonlinear model of the stator (phase kq(v) for the -th harmonic of the examined
a) to field winding mutual distributions of a) synchronous generator (q = 15 - equal to the
inductance from 3rd to the 29th ones, b), c) stator slot pitch and pb = 1). Figures 4 and 5
derivative from 3rd to the 29th and 43rd to the show that the skew of the rotor (or stator)
75th ones, respectively
 Zeszyty Problemowe Maszyny Elektryczne Nr 4/2014 (104) 121

causes a significant reduction in higher From expressions (6) and (7) results that ua, ub
harmonic order. Lower-order harmonics caused and uc depend on the field current and currents
by e.g. the saturation of the magnetic circuit in damping bars. The field current (7) depends
due to the rotor skew are only slightly reduced. on the method of powering the field winding
Figure 6 shows the comparison of the stator to [3] and on the self- and mutual inductances LfD.
damping bars Lar(1) - Lar(5) (Lar(6) - Lar(10))
distribution of mutual inductances and the a)
5
product of Lar(3)/ for the linear- and

L - [H]
Lf
Lar(3)n/ for the nonlinear salient pole 4.9 Lfs

f
synchronous generator. Lfn
4.8
The distributed stator to damping bar mutual Lfsn

inductances for nonlinear model are very 4.7

0 60 120 180 240 300 360
similar to the linear ones and therefore not - [deg]

shown in Figure 6. The differences are only b)

300

dL/d - [ H/s ]
visible after the calculation of the derivatives, 200
100
the comparison of which, due to Fourier 0
dL /d
f
dL /d
analysis (in relation to the fundamental fs

f
-100
component of Lar(3)), is shown in Figure 7. -200
-300
0 60 120 180 240 300 360
- [deg]
a) c)
100
300
- [ %]

Lar(1)
80
dL /d- [ H/s ]

200
Lar(1)s
60 100
Lar(1)n
ar(1)

dLfn/d
40 Lar(1)sn 0
L

dLfsn/d
fn

20 -100
-200
1 3 5 7 9 11 13
harmonics -300
0 60 120 180 240 300 360
b) - [deg]
100
Lar(3) d)
- [ %]

80 Lar(3)s 70
60
d L /d - [ % ]

60 Lar(3)n dLf/d
50
ar(3)

40 dLfs/d
Lar(3)sn 40
L

30 dLfn/d
20
f

20 dLfsn/d
0 10
1 3 5 7 9 11 13
harmonics 0
24 48 72 harmonics
c)
100
Fig. 8. Comparison of the a) field winding self
/d - [ %]

dLar(1) /d
80
60
dLar(1)s /d inductance distributions for linear and
dLar(1)n /d
40 nonlinear model with and without the rotor
ar(1)

dLar(1)sn /d
20 skew, b), c) products of (Lfs/), (Lf/)
dL

0
1 3 5 7 9 11 13 15 17 19 21 23 25 27
harmonics
and (Lfsn/), (Lfn/), d) detailed
d) harmonic contents of self inductance
100
dLar(3) /d
derivatives for 24th, 48th and 72nd order
/d - [ %]

80
dLar(3)s /d
60
dLar(3)n /d Figure 8 shows the distributions of the field
40
winding self inductance, the product of
ar(3)

dLar(3)sn /d
20
dL

0 (Lf/) and their harmonic contents in

1 3 5 7 9 11 13 15 17 19 21 23 25 27
harmonics relation to the constant component of Lf.
Fig. 7. Harmonic contents of the stator to
damping bars of a), b) mutual inductances Lar(1)
and Lar(3), c), d) derivatives Lar(1)/ and
Lar(3)/ for linear and nonlinear model with
and without the rotor skew
 122 Zeszyty Problemowe Maszyny Elektryczne Nr 4/2014 (104)

a)
-3
(Lafn/)ifn, (Lbfn/)ifn, (Lcfn/)ifn
2 x 10 (product of and derivatives of mutual
Lfr(1)
- [H]

1 Lfr(2) inductances of stator to field winding and

0 Lfr(3) field winding current),
fr(k)

Lfr(4)
(Lfn/)ifn, (product of and derivatives
L

-1
Lfr(5)
-2 of self inductances of field winding and field
0 60 120 180 240 300 360
- [deg] winding current).
b) Magnetic saturation has a very small influence
0.6
/d - [ H/s ]

dLfr(1)n /d on changing harmonic contents in derivatives of

0.4
dL
fr(1)sn
/d mutual inductance of field winding to damping
0.2
bars Lfr(1)n/ - Lfr(5)n/ (Lfr(6)n/ -
fr(1)n

0 Lfr(10)n/).
d L

-0.2
0 60 120 180 240 300 360
The greatest reduction of the harmonic content
- [deg] in the self- and mutual inductance distributions
c) is achieved by using the skew of the rotor.
4
4 x 10 x 10
/d - [%]

dLfr(1) /d
3 5. Bibliography
dLfr(1)s /d
2
dLfr(1)n/d [1]. S. Keller, M. Tu Xuan, J.-J. Simond:
fr(1)

1 dLfr(1)ns/d Computation of the no-load voltage waveform of

dL

0
24 48 72 73
laminated salient-pole synchronous generators.
harmonics IEEE Transactions on Industry Applications, Vol.
Fig. 9. Distributions of a) the field winding to 42, no. 306/2006, pp. 681-687
damping bars mutual inductance, b) the [2]. H. May, R. Palka, P. Paplicki, S. Szkolny, W.
products (Lfr(1)sn/), (Lfr(1)n/), c) R. Canders: Modified concept of permanent magnet
excited synchronous machines with improved high-
harmonic contents of the mutual inductance
speed features. Archives of Electrical Engineering,
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[3]. K. Ludwinek: Influence of DC voltage and
Figure 9 shows the comparison of the mutual current of field winding on induced stator voltages
inductance distributions of field winding to of a salient pole synchronous generator.
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The differences between the linear and 2013, pp. 103-111
[5]. L. Vicol, A. Banyai, I.-A. Viorel, J.-J. Simond:
nonlinear mutual inductance distributions are
On the damper cage bars currents calculation for
very small and therefore are only shown for salient pole large synchronous machines. Advances
nonlinear model in Figure 9. Moreover, the in Electrical and Electronic Engineering. vol. 7,
other components of the products (Lfr(k)s/), no. 1, 2, 2008, pp. 165-170
(Lfr(k)/) and (Lfr(k)sn/), (Lfr(k)n/) for [6]. R. Nadolski, J. Staszak, L. Harbaoui:
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243
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Nonsinusoidal and asymmetrical influence of
4. Conclusion electric power system on field voltages and currents
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496 k.ludwinek@tu.kielce.pl
[14]. T. J. Sobczyk: Extreme possibilities of circuital Kielce University of Technology,
models of electric machines. Electrical Power
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approaches to modelling of electrical machines with