d-q Equivalent Circuit Representation of
Three-Phase Flux Reversal Machine with Full Pitch
Winding
D . S . More, Hari Kalluru and B . G . Fernandes
Department of Electrical Engineering, Indian Institute of Technology Bombay,
Powai, Mumbai - 400 076, INDIA.
Email: dsmore@ee.iitb.ac.in khari@ee.iitb.ac.in bgf@ee.iitb.ac.in
Abstract— In this paper a d-q equivalent circuit for flux reversal machine (FRM) is proposed. In order to improve the power
density, full pitch winding is proposed. FRM with this winding
winding (FPFRM) is compared with conventional concentrated
stator pole winding FRM (CSPFRM). The output power of
FPFRM is twice that of CSPFRM for the same machine dimensions, electrical and magnetic loadings. The results obtained
using proposed d-q circuits are compared with those obtained
from FEM analysis. Steady state and dynamic performance of
FPFRM and CSPFRM is evaluated with proposed d-q circuits.
I. I NTRODUCTION
Single phase flux reversal machine (FRM) was first introduced in 1997 by R. P. Deodhar and et al for automobile
application to replace the standard claw pole alternator [1]. It
has numerious advantages such as simple construction, low
inertia, high power density and is suitable for high speed
application due to stationary permanent magnets and stator
winding. This single phase configuration is fully explored as
a high speed automotive generator. Three phase FRM was
introduced by C. Wang and et al in 1999 [2]. The design of the
machine was optimized to ensure (i) high PM flux linkage in
the winding, (ii) low cogging torque and PM weight. The basic
machine configuration is 8 salient pole rotor and 6 pole stator
with concentrated windings. Permanent Magnets are fixed to
stator pole. Fig. 1 shows this machine configuration.
FRM for low-speed servo drive application was introduced
by Ion Boldea and et al in 2002 [3]. This low speed machine
has 28 rotor poles and 12 stator poles with two permanent
magnet pairs on each stator pole. This machine is designed for
128 rpm at 60 Hz. Using vector control high torque density
with less than 3% torque pulsation was achieved. In order to
reduce the cogging torque, rotor teeth pairing method has been
proposed [4]. Attempts were made to reduce the leakage flux
by providing flux barrier on the rotor poles at its edges [5].
Power density comparison of doubly salient permanent magnet
electrical machines has been made. It is concluded that FRM
has higher power density in comparison with other machines
in the same class [6]. Full pitch winding flux reversal machine
(FPFRM) was proposed to improve the power density of the
machine as compared to conventional concentrated stator pole
winding flux reversal machine (CSPFRM) [7].
978-1-4244-1668-4/08/$25.00 ©2008 IEEE
Fig. 1.
Cross-section of 6/8 pole concentrated stator pole winding FRM.
In this paper, concept of fictitious ‘Electrical Gear’ is
proposed based on the flux pattern of the machine. The dq equivalent circuits for FRM based on this gear is proposed.
In order to validate d-q equivalent circuits, two dimensional
FEM analysis [8] is carried out on CSPFRM and FPFRM.
Optimized machine dimensions are obtained from C. Wang
and et al [2]. The important dimensions of FRM are given in
Table I for ready reference.
Section II describes the flux linking to the stator winding
of the machine and there from the concept of full pitch stator
winding arrangement is discussed. Section III proposes the
fictitious ‘Electrical Gear’ concept applicable to FRM. Section
IV proposes the d-q equivalent circuit for FRM based on this
fictitious gear. Section V describes the FEM simulation results
to validate the d-q equivalent circuit. Section VI compares
the power density of FPFRM with CSPFRM and finally
conclusions are drawn.
II. F ULL P ITCH S TATOR W INDING FOR FRM
Geometry of 6/8 pole three-phase FRM (as per Table 1)
and the flux distribution in this machine at no load is shown
in Fig. 2. FRM machine has 6 stator poles and 8 pole variable
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Normal component of flux density in air gap
4th degree
0.06
2
Flux density ( Wb/m )
0.04
0.02
0
−0.02
−0.04
0
50
100
150
200
250
Distance along the air gap (mm)
Fig. 2.
Flux distribution in 6/8 pole FRM at no load
Fig. 4.
reluctance rotor. The normal component of flux density at the
middle of stator pole along the periphery of the machine is
shown in Fig. 3. The observation of this normal component
of flux density plot reveals that the machine has two pole flux
pattern.
Phase flux linkage in FRM is sinusoidal in nature and hence
the induced voltage [2]. Considering a linear load the phase
current is also sinusoidal. Normal component of armature
reaction along the air gap at one instant of time is shown in
Fig. 4. This flux pattern also reveals that machine has effective
two pole flux pattern. In other words machine has two effective
poles.
FRM has 6 slots and two pole flux pattern, hence electrical
angle per slot is 60◦ . CSPFRM stator winding has a coil span
of 60◦ . Fundamental pitch factor of the stator winding is 0.5.
As electrical angle between the slots is 60◦ , full pitch winding
is possible. The arrangement of this full pitch winding is
shown in Fig. 5 and fundamental pitch factor of stator winding
is unity. Hence, voltage induced in FPFRM is twice that of
CSPFRM for the same number of turns.
0.8
0.6
Normal component of armature flux density along the air gap with
magnets are de-energized
Normal component of flux density
4th degree
TABLE I
D IMENSIONS OF FRM
Sr. No.
1
2
3
4
5
6
7
8
9
10
11
12
Description
Air gap (mm)
Magnet thickness (mm)
Rotor pole span angle
Stator pole span angle
Stator pole span (mm)
Rotor pole span (mm)
Stator pole height (mm)
Rotor pole height (mm)
Outer dia. of rotor (mm)
Outer dia. of stator (mm)
Number of turns /phase
Stack length (mm)
Symbol
g
hpm
βr
βs
τps
τpr
hps
hpr
Di
Do
Nph
lsk
Value
1
3
16.2◦
42.6◦
27.8
10.3
15
18
72
129
52
86
Flux density ( Wb/m2)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
50
100
150
200
250
300
350
Distance along the periphery of the machine at the middle of the stator pole ( mm)
Fig. 3.
Normal component of flux density along the periphery of machine
at middle of the stator pole
Fig. 5.
Full pitch winding arrangement in FPFRM
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TABLE II
G EAR R ATIO FOR VARIOUS FRM C ONFIGURATIONS
III. F ICTITIOUS E LECTRICAL G EAR
The frequency and speed relationship for FRM is given by
[3]
60 × f
(1)
n=
nr
where, n = rotor speed in rpm.
nr = number of rotor teeth (poles).
f = frequency in Hz.
The three-phase 6/8 pole FRM has two effective poles, and
hence flux pattern speed for supply frequency f Hz is given
by
nf = 60 × f
(2)
where, nf = flux pattern speed in rpm.
Equations (1) and (2) reveal that rotor speed and flux pattern
speed is different. The shaft speed is nr times less than flux
pattern speed. In conventional machines, flux pattern speed
and rotor speed is same. Pictorial representation of 6/8 pole
FRM motor is shown in Fig. 6 while a pictorial representation
of 2 pole PMSM is shown in Fig. 7. The difference in
speed between rotor and flux pattern speed is represented by
a fictitious step-down gear and is called ‘Electrical Gear’.
Electrical gear ratio (K) is defined as ratio of flux pattern
speed to the shaft speed.
Supply Frequency = f Hz
Stator
Flux pattern speed
= 60 x f
Equivalent
2 pole PM Rotor
Shaft speed
n rpm
Stator
Sr. No.
Machine
type
No.of
magnets
Gear
ratio
1
2
3
4
5
6/8 pole
12/16 pole
6/14 pole
12/28 pole
12/40 pole
12
24
24
48
60
8
8
14
14
20
The generalised equation for electrical gear ratio is given as
nr
K=
(3)
Peq /2
where, Peq = no. of flux pattern poles.
Hence 6/8 pole FRM can be analysed as 2 pole PMSM with
a gear ratio of 8. Gear ratios for various FRM configurations
are given in Table II. It can be observed that no. of flux pattern
poles in FRM are 2 for 6 stator poles and 4 for 12 stator poles.
IV. d − q E QUIVALENT C IRCUITS FOR FRM
Permanent magnet synchronous machine (PMSM) is analyzied with d-q equivalent circuits [9]. Transient and steady
state behaviour of the PMSM is obtained with these equivalent
circuits. Fig. 8 shows the d-q equivalent circuit for PMSM.
Back EMF in FRM is sinusoidal in nature. Self inductance
and mutual inductance is almost constant with rotor position. It
requires sinusoidal stator current to produce a constant torque.
Hence d-q equivalent circuit can be used to analysis the steady
state and transient behaviour of FRM.
d-q equivalent circuits of FRM are derived from PMSM.
The major difference between PMSM and FRM is the relationship between speed and frequency. Three phase 6/8 pole
FRM can be considered as 2 pole PMSM with gear ratio (K)
of 8 as shown in Fig. 6.
Mathematical model of FRM is similar to PMSM except
n = (60 x f)/n r
id
Fictitious electrical Gear
Fig. 6.
No. of Flux
pattern
poles
2
4
2
4
4
Representation of 6/8 pole FRM
R
−
e λq
+
L ld
+
Lmd
Vd
Supply Frequency = f Hz
If
Stator
Flux pattern speed
= 60 x f
iq
2 pole PM Rotor
Fig. 7.
+
e λd
−
L lq
+
n = 60
L mq
Vq
Shaft speed
n rpm
Stator
R
x f
Representation of 2 pole PMSM
Fig. 8.
d-q equivalent circuit of PMSM
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id
e λq
+
K
R
−
L ld
FPFRM
CSPFRM
0.05
+
0.04
Lmd
iq
K
R
+
e λd
−
If
Phase flux linkages (Wb)
Vd
L lq
+
0.03
0.02
0.01
0
−0.01
−0.02
−0.03
L mq
Vq
−0.04
−0.05
0
5
10
15
20
25
30
35
40
45
Rotor position (mech. degrees)
Fig. 9.
d-q equivalent circuits of FRM
Fig. 10.
the gear ratio (K). This gear ratio (K) is considered in the
modeling of FRM. The following assumptions are made while
deriving these equivalent circuits [9].
• Saturation in the machine is neglected.
• The induced EMF is sinusoidal.
• Eddy currents and hysteresis losses are negligible.
• There are no field current dynamics.
• There is no cage on the rotor.
With these assumptions, The d-q equations in synchronously
rotating reference frame of FRM are
vd = Rid + pλd − Kωe λq
(4)
vq = Riq + pλq + Kωe λd
(5)
λq = Lq iq
(6)
λd = Ld id + λaf
(7)
The electrical torque Te is given by
Te =
3 Peq
×
× K(λaf iq + (Ld − Lq )id iq )
2
2
(8)
where,
R = stator resistance (Ohm).
id , iq = d and q axes stator currents (A).
Ld , Lq = d and q axes inductances (H).
p = derivative operator.
λd , λq = d and q axes flux linkages (Wb).
λaf = mutual flux linkages due to PM (Wb).
ωe = rotor speed (rad/sec).
FRM machine considered for simulation has 6/8 pole structure and has equal d and q axes inductance [2]. Therefore
torque equation reduces to
Te =
3 Peq
×
× Kλaf iq
2
2
where,
Peq = no. of flux pattern poles of the machine= 2
(9)
Phase flux linkage of FPFRM and CSPFRM
K =8
FRM is controlled with constant flux upto base speed by
maintaining id equals to zero. Under this condition and at
steady state, (4) to (7) are reduce to
vd = −Kωe λq
(10)
vq = Riq + Kωe λd
(11)
λq = Lq iq
(12)
λd = λaf
(13)
A. Steady State Torque Calculation
FRM machine design data is obtained from [2] and is
shown in Table I. Physical dimensions of the machine and
number of turns/phase are kept same in FPFRM and CSPFRM,
only the winding arrangement is changed. FEM analysis is
carried out to determine the variation of phase flux linkage
of both machines with rotor position. This variation for both
the machines without skewed rotor is shown in Fig.10. Flux
linkage variation is shown for one rotor pole pitch (i.e. 45◦
mech.). Figure clearly shows that FPFRM stator winding flux
linkage is approximately twice that to the CSPFRM.
Values of λaf obtained from Fig. 10 for CSPFRM and
FPFRM. They are 0.021 Weber and 0.041 Weber respectively.
The steady state torque of both machines for Iph = 15 A
is obtained from (9). The calculated values of steady state
torques of CSPFRM and FPFRM are 5.34 Nm and 10.43 Nm
respectively.
V. FEM S IMULATION TO VALIDATE THE d − q
E QUIVALENT C IRCUITS
FEM motor simulations of CSPFRM and FPFRM for constant torque operation are carried out. The linking between
the FEM winding regions to coil components of the circuit for
FRM is shown in Fig.11. B1 to B6 are winding regions; where
as b1 to b6 are corresponding coil components in the circuit.
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Phase current
Phase voltage
80
Voltage (V) and current (A)
60
40
20
0
−20
−40
Fig. 11.
−60
Coupling between FE regions and electrical circuit of FPFRM.
−80
TABLE III
F ULL L OAD T ORQUE (N M ) OF CSPFRM
Parameter
Calculated using d-q circuits
Obtained from FEM simulation
AND
CSPFRM
5.34
5.18
0
0.5
1
1.5
2
2.5
Time (seconds)
3
3.5
4
−3
x 10
FPFRM
Fig. 13.
FPFRM
10.43
10.14
Phase voltage and phase current supplied to CSPFRM.
TABLE IV
PARAMETERS OF FRM
L1 ,L2 and L3 are the end turn leakage inductance/phase.
R1 ,R2 and R3 is stator winding resistance/phase. Motor is
supplied from three phase sinusoidal current source I1 , I2 and
I3 . Vector control is obtained with id equal to zero and iq is
maintained in phase with back EMF of the machine.
Simulated steady state torque of both machines using
Flux 2D software is shown in Fig.12. The average value of
steady state torque of CSPFRM and FPFRM is 5.18 Nm and
10.14 Nm respectively. The average torque obtained using d-q
equivalent circuits and that obtained from FEM simulation at
full load is shown in Table III and it can be seen that there is
a good agreement between these results.
Steady state waveforms of voltage and rated current supplied to the CSPFRM and FPFRM at 1995 rpm obtained
Sr. No.
1
2
3
4
5
6
Parameter
K
ωe (rad/sec.)
R (ohm)
Ld = Lq (mH)
λaf (Wb)
iq (A)
FPFRM
8
208.9
0.107
3.67
0.041
21.21
CSPFRM
8
208.9
0.050
0.94
0.021
21.21
using FEM simulation are shown in Fig. 13 and Fig. 14
respectively. Peak value of fundamental component of supply
voltage obtained from FEM simulation study for CSPFRM and
FPFRM is 48.47 V and 165.88 V respectively. Peak value of
fundamental component of supply voltage is calculated from
(10) to (13). The data required for these equations is given
in Table IV. The relationship between peak value of supply
14
FPFRM
CSPFRM
Phase voltage
Phase current
200
12
Voltage (V) and Current (A)
150
Torque (Nm)
10
8
6
4
100
50
0
−50
−100
2
0
−150
0
Fig. 12.
20
40
60
80
100
Rotor angle (mech. degrees)
120
−200
140
FEM simulation of full load torque of CSPFRM and FPFRM
0
Fig. 14.
0.5
1
1.5
2
2.5
Time (seconds)
3
3.5
4
−3
x 10
Phase voltage and phase current supplied to FPFRM.
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Parameter
Calculated using d-q circuits
Obtained from FEM simulation
OF
CSPFRM AND FPFRM
CSPFRM
49.17
48.47
60
FEM Simulation
d−q equivalent circuit
FPFRM
148.24
165.88
50
voltage Vs , vd and vq is given by
Vs = vd2 + vq2
(14)
Calculated peak value of supply voltage and value obtained
from FEM simulation is shown in Table V and they are in
good agreement.
Terminal voltage ( V )
TABLE V
P EAK VALUE OF S UPPLY VOLTAGE (V)
vqg = Kωe λaf − Riq − pλq iq − Kωe Ld id
(16)
λq = Lq iq
(17)
λd = λaf − Ld id
(18)
The data required to calculate the values of vd and vq are
given in the Table IV. Terminal voltage regulation obtained
from FEM simulation and d-q equivalent circuit for CSPFRM
is shown in Fig. 15, while these plots for FPFRM are shown
in Fig. 16.
20
0
FEM based simulation is carried on CSPFRM and FPFRM
generator at 2000 rpm to determine the voltage regulation. The
proposed d-q equivalent cicuit is used to caculate the terminal
voltage of the machine. The d-q equations for FRM generator
are given below.
(15)
30
10
A. Voltage Regulation of CSPFRM and FPFRM from d − q
Equivalent Circuits
vdg = Kωe λq − Rid − pλd id
40
0
1
2
Fig. 16.
3
4
5
Load current ( A )
6
7
8
Voltage regulation of FPFRM
VI. P OWER D ENSITY C OMPERISION OF CSPFRM AND
FPFRM
Physical dimensions and number of turns/phase are same
in both machines, only winding arrangement is changed. Both
machines have same rated current. FEM analysis at rated
current is performed on both machines to determine the rated
torque of both machines. Output torque of CSPFRM and
FPFRM is 5.18 Nm and 10.14 Nm respectively. The torque
constant of FRM is given by
Kt =
3 Peq
× Kλaf
×
2
2
(19)
28
d−q equivalent circuit
FEM simulation
26
24
10
22
Torque (Nm)
Terminal voltage (V)
CSPFRM
FPFRM
12
20
18
16
8
6
4
14
2
12
10
0
2
Fig. 15.
4
6
Load current (A)
8
Voltage regulation of CSPFRM
10
0
12
0
1000
2000
3000
Speed (rpm)
4000
5000
6000
Fig. 17. Steady state speed torque capability curve of CSPFRM and FPFRM.
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Torque constant mainly depends upon λaf . The values of
λaf for both machines are shown in Table IV. λaf for FPFRM
is twice as that of CSPFRM and hence torque production
capability of FPFRM is twice as that of CSPFRM in constant
torque zone.
Steady state speed-torque capability curve at same rated
current for both machines is deduced and is shown in Fig. 17.
Speed higher than base speed is obtained with flux weakening.
In the flux weakening region supply voltage and input current
are maintained at rated values. id is increased with speed
to reduce the flux in the machine. As id increases iq has
to decrease resulting in reduction in torque capability of the
machine. Speed range in constant power region is higher for
CSPFRM as compared to FPFRM.
VII. C ONCLUSION
Full pitch winding concept for FRM is introduced which
increases the output power of FRM approximately twice that
of FRM with concentrated stator pole winding. Concept of
fictitious electrical gear is introduced. The d-q equivalent
circuits for FRM are proposed and same are validated with
steady state FEM analysis.
R EFERENCES
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[2] C. Wang, S. A. Nasar, I. Boldea “Three phase flux reversal machine
(FRM),” IEE Trans. Electrical power application ., vol. 146, No. 2,
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[3] Ion Boldea, Jichum Zhang, S. A. Nasar, ”Theoretical characterization
of flux reversal machine in low speed servo drives-The pole PM
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[4] Tae Heoung Kim, Sung Hong Won, Ki Bong and Ju Lee, “Reduction
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[8] CEDRAT, France “Flux 2-D FEM Software, ”.
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