D
Journal of Energy and Power Engineering 7 (2013) 110-117
DAVID
PUBLISHING
Brushless Doubly-Fed Induction Machine as a Variable
Frequency Transformer
Ayman Abdel-Khalik1, Ahmed Elserougi1, Ahmed Massoud2 and Shehab Ahmed3
1. Electrical Engineering Department, Alexandria University, Alexandria 21544, Egypt
2. Electrical Engineering Department, Qatar University, Doha 2713, Qatar
3. Electrical and Computer Engineering Department, Texas A & M University at Qatar, Doha 23874, Qatar
Received: April 01, 2012 / Accepted: June 06, 2012 / Published: January 31, 2013.
Abstract: VFT (variable frequency transformer) has been recently used as an alternative to HVDC (high voltage direct current) to
control power flow between asynchronous networks. VFT consumes less reactive power than a back-to-back HVDC system, provides
faster initial transient recovery, and has better natural damping capability. VFT is simply a DFIM (doubly-fed induction machine)
where the machine torque controls the power flow from stator to rotor and vice versa. The main disadvantage of this VFT is the slip
rings and brushes required for the rotor circuit, especially in bulk power transmission. The BDFM (brushless doubly-fed machine) with
nested cage rotor machine is proved to be a comparable alternative to conventional DFIM in many applications with the advantage that
all windings being in the stator frame with fixed output terminals. In this paper, the BDFM is used as a BVFT (brushless variable
frequency transformer). A prototype machine is designed and simulated to verify the system validity.
Key words: Variable frequency transformer, brushless doubly-fed machines, AC machines, induction machine.
Nomenclature
V
I
λ
p
R
L
M
ωm
θm
Voltage
Current
Flux linkage
Number of pole-pairs
Resistance
Self-inductance
Mutual inductance
Mechanical angular speed
Rotor angular position
Stator peripheral angle
1, 2
s, r
d,q
Stator windings
Stator and rotor
d-axis and q-axis

Subscripts
1. Introduction
Interconnections between power networks can be
classified into synchronous and asynchronous ties [1].
Corresponding author: Ahmed Massoud, Ph.D., research
fields: power electronics, energy conversion, power quality,
and renewable energy. E-mail: ahmed.massoud@qu.edu.qa.
Synchronous ties are the most common type in
practical power systems. However, synchronization
between involved networks is necessary to ensure
effective power transfer between them, which is a
challenging task especially with weak networks. On the
other hand, asynchronous ties can be used to directly
connect two power networks experiencing a slight
difference between their frequencies. HVDC (high
voltage direct current) transmission is widely
employed as being advantageous for long-distance
(more than 800 km), bulk power delivery, and
asynchronous interconnections [1].
Recently, VFTs (variable frequency transformers)
are recognized as a promising candidate for
bi-directional asynchronous power transfer offering
many advantageous over HVDC transmission [2]. The
VFT technology was introduced in the early 2000s [3].
The world’s first VFT for bulk system power transfer
between large asynchronous power networks has been
successfully installed and tested at Hydro-Quebec’s
Brushless Doubly-Fed Induction Machine as a Variable Frequency Transformer
Langlois substation in Canada [3]. VFT provides
isolation between networks, improves grid stability,
meets power transfer needs during normal conditions
and supports during fault conditions [4]. A comparison
between a back-to-back HVDC system with series
compensation, and a variable frequency transformer,
when used to control the power flow between weak AC
networks, is given in Ref. [4]. The steady state and
dynamic simulations show that both technologies are
able to control power flow accurately. The merits of
variable frequency transformer can be summarized in
the following points:
 less reactive power;
 less harmonics;
 faster initial transient recovery;
 better natural damping capability.
VFT is a simply doubly-fed rotating machine which
is connected to the involved asynchronous networks
(one on the stator side and the other on rotor side) to
achieve its function as asynchronous tie between the
two networks. The rotor speed is controlled to operate
at the slip frequency between the two networks [5].
When two networks with the same frequency are
connected, the torque on the shaft is controlled at zero
speed [4]. The angle of the rotor is positioned to
achieve a scheduled power flow by means of DC drives
[5]. The basic concepts and electromagnetic design of
VFT are presented in Ref. [6] while its mathematical
model is given in Refs. [7, 8] and simulated under
different operational conditions. Being an effective
mean to control power flow in weak interconnections,
the VFT is also used to reduce the power fluctuations
of a large-scale grid-connected offshore wind farm [9].
The BDFM (brushless doubly-fed machine) with
nested cage rotor machine topology provides
advantages that may warrant its application despite an
apparent power density penalty [10-12]. It was
proposed as an attractive alternative to conventional
DFIM (doubly-fed induction machine) in many
applications, especially in variable speed drive for
large induction machines [10], wind applications [11],
111
and contactless power delivery systems [12]. The main
advantage of BVFT (brushless variable frequency
transformer) is that all machine windings in the stator
with fixed output terminals.
In this paper, a BDFM with nested cage rotor is used
to functionally replace the conventional VFT to
interconnect two asynchronous networks. BVFT is
presented to overcome the brushes problem. The theory
of operation of the proposed system is firstly described.
A prototype machine is then designed [13, 14] and its
inductances are calculated using modified winding
function [15]. Based on suitable transformations, the
dq-model [16-19] for this machine is then presented
and used to simulate the machine in different
operational modes.
2. Proposed Brushless Variable Frequency
Transformer
The proposed BVFT is simply a BDFM with a
nested cage rotor. The stator comprises two three-phase
windings with p1 and p2 pole pairs. The rotor is a nested
cage rotor, with number of nests equals to the sum of
the stator pole pairs (p1 + p2) [16]. The nested cage
rotor has the same construction as conventional
squirrel cage rotor, however, the main difference
between them is that, in conventional squirrel cage
rotor all bars are connected from both sides with two
end rings. However, in nested cage rotor the bars are
divided into (p1 + p2) groups. The selection of the
suitable number of poles is a design factor and depends
on many factors as magnetic saturation and radial
forces. In the proposed design, the selected number of
poles is 6/2 stator winding pole number combination.
The selection gives an approximate quasi square flux
pattern and shows great promise in many applications
such as wind-power generators and pump drives [20].
The general layout of the proposed system is shown in
Fig. 1 and the machine construction is shown in Fig. 2.
The capacitor banks shown in Fig. 1 are designed to
supply the reactive power required for machine
magnetization. The operation of the proposed BVFT is
112
Brushless Doubly-Fed Induction Machine as a Variable Frequency Transformer
cage induction machine is used to build a prototype
BVFT. The stator is rewound to accommodate two
stator windings with 6/2 pole combination. The number
of turns per phase for both windings and their cross
sectional areas are selected to ensure same acceptable
air gap flux and power. A nested cage rotor is designed
to fit into the machine stator. Since the prototype
machine is still under construction, this paper
introduces the machine design, modeling, and
simulation to prove the idea. Experimental verification
will be considered in future.
Fig. 1 Proposed BVFT.
3. Machine Design
Fig. 2 Construction of the proposed brushless variable
frequency transformer.
similar to the conventional VFT using DFIM [6],
where the BVFT is used to connect two networks
experiencing a slight difference between their
frequencies. The two networks are connected to the
two stator windings. The power can be transferred from
one side to the other side by controlling the torque
applied to the machine. Generally, DC motor is
mechanically coupled on the BDFM shaft for
controlling the shaft torque and hence the amount of
transferred power between the two involved
asynchronous networks [5]. For example, if the torque
is positive, the power is transferred from first stator
winding to the second stator winding and vice versa.
The rotor of the VFT spins at a speed that is
proportional to the difference between the two
networks frequencies f1 and f2 [16].
m  2
f1  f 2
p1  p2
(1)
When two networks with the same nominal
frequency are interconnected, the torque on the shaft of
the VFT must be controlled at stall or near stall
conditions. The stator frame of an existing 3 hp squirrel
The prototype machine will be constructed using an
existing 380 V, 3 hp squirrel cage rotor induction
machine. The design is restricted with the stator
dimensions given in Table 1. The stator is rewound to
house two stator windings with dissimilar pole
numbers (p1, p2). The suggested number of poles are (6,
2), respectively. However, a nested cage will be built
with same diameter as old rotor.
3.1 Stator Design
For simplicity, the stator is wound with two single
layers windings. The lower layer comprises the 6-pole
winding while the upper layer houses the 2-pole
winding. Since an existing stator is used, the only
design criterion is the number of turns for each winding.
The stator coils are fully pitched because each winding
is wound with only one layer. For total number of slots
36, the number of slots per phase per pole for the 6-pole
and the 2-pole windings are 2 and 6, respectively. Since
it is required to replace the single three-phase winding
of the existing induction machine by two separate
three-phase windings, hence, the rated voltage for both
stator windings is selected to be half that of the existing
machine of 110 V. The current rating for both winding
will be selected the same as that of the induction
machine. Conventional design steps as in Refs. [13, 14]
are followed to select the required number of turns for
both windings. The design should fulfil the following
requirements:
113
Brushless Doubly-Fed Induction Machine as a Variable Frequency Transformer
Table 1 Existing 3 hp induction machine ratings and
dimensions.
220 V
5A
1,420 rpm
117.6 mm
90.7 mm
0.5 mm
14.4 mm
21.1 mm
36
44
(1) The magnitude of the MMF (magneto motive
foce) produced by the 6-pole winding is 1/2 that of the
2-pole winding to ensure that the peak value of the total
MMF equals that of the 2-pole winding when fed
separately with rated current, as shown in Fig. 3;
(2) The back iron flux with both windings carrying
full-load current will not cause core saturation.
The suitable numbers of turns per phase for the
6-pole and the 2-pole windings are found to be 300 turn
and 198 turn, respectively.
3.2 Rotor Design
In nested cage rotor, the number of rotor nests is
chosen equal to the sum of the pair-poles of the two
stator windings (p1 + p2). Hence, for the selected number
of pole-pairs, the required number of nests will be four.
The number of loops per nest, n, is chosen to minimize
rotor MMF harmonic content. As the number of loops
per nest increases, the harmonic content decreases, but
with corresponding reduction in the inter-loop space
through which the flux passes. Typically, 3-6 loops per
nest are used [20]. The selected number of bars is 40,
where each nest has five loops as shown in Fig. 4. Same
bar dimensions as the existing rotor will be assumed for
the designed rotor and is used to calculate the
resistance of different nested loops.
4. Machine Modelling
The machine is modeled using conventional dq
model [16-20]. The machine modeling commences
with inductance calculation using modified winding
6-pole MMF
2-pole MMF
Total MMF
1
0.5
MMF, pu
Rated phase voltage
Rated current
Rated speed
Stator inner diameter (D)
Rotor stack length (L)
Air gap length (g)
Stator back iron
Slot height
Number of stator slots
Number of rotor bars
1.5
0
-0.5
-1
-1.5
0
60
120
180
240
Peripheral angle  , deg
300
360
Fig. 3 MMF distribution with stator peripheral angle.
Fig. 4 Nested cage rotor.
function [15] and based on machine dimensions then
the corresponding dq-model is presented.
4.1 Inductance Calculation
The first step for machine modeling is to calculate
the machine inductance matrix. The modified winding
function method [15] is used to calculate the machine
inductance matrix. The general expression for mutual
inductance between any two circuits i and j in any
electrical machine is given in Eq. (2).
Lij   o L
D
2
2
 N  ,    N  ,    g  ,    d
1
i
m
mj
m
m
(2)
0
where:
N i  ,  m  is the turn function of winding i;
N mj  , m  is the modified turn function of winding j;
g 1  ,  m  is the inverse air gap function. For
cylindrical rotor, it is constant and equals the reciprocal
of the air gap length, g.
For the prototype machine type, there are six turn
functions for the stator windings and 20 turn functions
for the rotor nested loops. Firstly, all turn functions are
calculated as a function of the peripheral angle. Then,
substituting in Eq. (2), the inductance matrix elements
as a function of rotor position are calculated [15]. In this
114
Brushless Doubly-Fed Induction Machine as a Variable Frequency Transformer
case, the inductance matrix dimension will be 26 × 26.
To simplify the machine model, to reduce the number of
equation, and to eliminate the inductance dependency
on rotor position, dq-model is obtained from the phase
model through suitable transformations [16, 18]. The
final dimension of the resulting inductance matrix will
be 14 × 14 but with constant coefficients. The complete
calculation of the final inductance matrix is shown in
Refs. [17, 19] and is given in the appendix.
4.2 dq-Model
In dq-model, each stator winding is represented by
two dq-coils. Each group of loops, having same colour
in Fig. 4, of the nested cage rotor is represented by two
dq-coils for more exact representation [17, 20]. Hence,
the rotor will be represented by n coils in the d-axis and
n coils in the q-axis. For the prototype machine, the
total number of coils will be 14, as shown in Fig. 5.
Hence, the dq-model for the BDFM, in rotor reference
frame, is given in Eq. (3) [17].
vd 1  R1id 1  pd 1  p1 m q1
vq1  R1iq1  pq1  p1m d 1
vd 2  R2id 2  pd 2  p2 m q 2
(3)
vq 2  R2iq 2  pq 2  p2m d 2
v drk  Rrk idrk  pdrk
v qrk  Rrk iqrk  pqrk
where k = 1, 2, ..., n.
The flux linkage vector is given in Eq. (4).
 dq1 21   Ls1 22
M s1r 210   idq1 21 
0
 

Ls1 22 M s1r 210   idq 2 21  (4)
 dq 2 21    0
dqr   M rs1 102 M rs 2 102 Lr 1010  idqr  
101 
 101 

The inductance matrix is calculated as mentioned in
the previous section and the machine parameters are
given in the appendix.
Finally, the machine torque is given in Eq. (5).
Te  3 2 p1 d 1iq1  q1id 1  3 2 p2 d 2iq 2  q 2id 2 (5)




5. Simulation Results
The system is simulated with the same controller of
the conventional VFT presented in Refs. [7, 8]. This
controller ensures the following requirements:
Fig. 5 dq representation of BVFT.
The rotor speed is regulated at a synchronous speed
that depends on the difference between the two
networks frequencies and is given in Eq. (1).
Shaft torque is controlled by means of the DC
machine to achieve a scheduled power flow between
the two networks.
The proposed system is simulated with a 110 V, 50
Hz network, denoted as network (A), is connected to
the 6-pole winding, while the 2-pole is connected to
another 110 V network, denoted as network (B), that
experiences frequency variation. The frequency of the
second network is assumed 49 Hz. Hence, the
corresponding synchronous speeds for the two
frequencies will be 15 rpm. Firstly, the two networks
are synchronized as shown in Ref. [8]. The DC
machine is controlled with reference torque shown in
Fig. 6a. The two stator winding powers are shown in
Fig. 6b. The machine speed is shown in Fig. 6c, where
the speed remains synchronized at 15 rpm irrespective
to load change. It is clear that the power transfer
between the two networks follows the torque reference.
With a positive torque applied to the shaft, power is
transferred form network A to network B and vice
versa. The relatively large difference between the two
stator winding powers is due to the relatively high
machine losses corresponding to such low power rating.
However, for systems with high power rating, this
difference should be negligible. The rms currents for
both stator windings are shown in Figs. 6d and 6e,
while for nested loops are shown in Fig. 6f. It is evident
that the outer loops of the nests carry much higher
current than the inner loops. This problem has been
115
Brushless Doubly-Fed Induction Machine as a Variable Frequency Transformer
20
3000
Ps1
2000
Ps2
Power, W
Torque, Nm
10
0
1000
0
-1000
-10
-2000
-20
0
10
20
30
40
-3000
50
0
10
20
Time, s
50
6
6
0
-50
Is2,A
8
4
2
0
10
20
30
Time, s
40
50
0
0
10
(c)
20
30
Time, s
4
40
0
50
0
10
(d)
20
30
Time, s
40
50
(e)
1000
Ir1
500
Ir1
Ir2
400
Ir3
300
Ir4
Rotor loop currents, A
Rotor loop currents, A
50
2
600
Ir5
200
100
0
40
(b)
8
Is1, A
Speed, rpm
(a)
100
-100
30
Time, s
0
10
20
30
40
50
Time, s
(f)
Ir2
500
Ir3
Ir4
0
Ir5
-500
-1000
0
0.005
0.01
0.015
0.02
Time, s
0.025
0.03
0.035
0.04
(g)
Fig. 6 System performance with f1 = 50 Hz and f2 = 49 Hz: (a) torque; (b) winding powers; (c) speed; (d) RMS Is1; (e) RMS Is2;
(f) RMS rotor loop currents for a given nest; and (g) instantaneous rotor loop current for a given nest.
investigated in Ref. [18] which cause unequal current
distribution between different rotor bars yielding
unequal rotor heat distribution. Equal current
distribution can be tackled by using rotor bars with
different cross sectional area, larger cross sections for
outer loops and lower cross sections for inner loops. An
important reason that justifies why multiple rotor-loop
model is used in this paper rather than the much simple
single loop representation [17] is the non-negligible
phase shift angles between currents in different loops in
the same nest, as shown in Fig. 6g. This angle changes
as the machine is loaded [18]. This gives a significant
error if the whole nest is represented by a single coil.
Comparing the simulation results with those are given
in Refs. [7, 8], the proposed BVFT is shown to be an
attractive alternative to conventional VFT while
retaining the advantage of the absence of brushes and
the presence of all windings in the stator. The BVFT
characteristic curves in per unit (pu), representing the
machine winding powers, current, and efficiency, are
shown in Fig. 7. The base values are defined as the
values correspond to rated winding current.
The efficiency is defined as the ratio between the
transmitted power and received power between the two
networks. It is notable that the output rated transmitted
power at rated current is approximately 60% that of the
existing 3 hp squirrel cage machine. This is due to the
non-optimal design of the machine since an existing
machine is used to build the prototype one. However,
machine optimal design is expected to increase the
machine power density. Moreover, this machine type is
generally disadvantageous due to its relatively lower
power density when compared with squirrel cage
machine. However, when it is compared with DFIM
116
Brushless Doubly-Fed Induction Machine as a Variable Frequency Transformer
1
1
80
P s1
0.8
Power transfer
from network
B to A
Power transfer
from network
A to B
-0.5
70
Is2
60
Efficiency, %
0
P s2
Current, pu
Power, pu
0.5
Is1
0.6
0.4
50
40
30
0.2
20
-1
-1
-0.5
0
Torque, pu
0.5
1
0
-1
-0.5
0
Torque, pu
0.5
1
10
-1
-0.5
0
Torque, pu
0.5
(a)
(b)
(c)
Fig. 7 Characteristic curves in pu for torque versus: (a) winding powers; (b) winding currents; and (c) efficiency (Pbased =
1,850 W, Ibase = 5 A, and Tbase = 13.6 Nm).
with same power, the power density penalty will be less.
This may warrant its application with the fixed output
terminal advantage obtained.
6. Conclusions
In this paper, a brushless variable frequency
transformer is proposed, based on conventional
brushless doubly-fed machine, to control power flow in
weak interconnections. Its two stator windings are used
to connect two networks experiencing a slight
difference between their frequencies. By controlling
the shaft torque using a DC machine, the power flow
between the two networks can be controlled much
similar to conventional VFT. A prototype model which
is still under construction is designed and simulated to
verify the validity of the proposed system. The winding
function method is used to calculate the machine
inductance matrix which is then used to develop the
machine dq-model. Despite an apparent power density
penalty, simulation results show that the proposed
BVFT may be an attractive alternative to conventional
VFT while retaining the advantage of the absence of
brushes and the presence of all windings in the stator
with fixed output terminals, which is much suitable for
bulk power transmission.
Acknowledgments
This publication was made possible by NPRP
(National Priorities Research Program) grant NPRP
08-504-2-197 from the Qatar National Research Fund
(a member of Qatar Foundation). The statements made
herein are solely the responsibility of the authors.
References
[1]
M.P. Bahrman, Overview of HVDC transmission, in:
2006 IEEE PES Power Systems Conference and
Exposition, Atlanta, 2006, pp. 18-23.
[2] R.
Piwko,
E.
Larsen,
Variable
frequency
transformer—FACTS technology for asynchronous power
transfer, in: 2005/2006 IEEE PES Transmission and
Distribution Conference and Exhibition, Dallas, 2006, pp.
1426-1428.
[3] D. Nadeau, A 100-MW variable frequency transformer
(VFT) on the Hydro-Québec TransÉnergie network—the
behavior during disturbance, in: IEEE Power
Engineering Society General Meeting, Tampa, 2007, pp.
1-5.
[4] B. Bagen, D. Jacobson, G. Lane, H. Turanli, Evaluation of
the performance of back-to-back HVDC converter and
variable frequency transformer for power flow control in a
weak interconnection, in: IEEE Power Engineering
Society General Meeting, Tampa, 2007, pp. 1-6.
[5] P. Truman, N. Stranges, A direct current torque motor for
application on a variable frequency transformer, in: IEEE
Power Engineering Society General Meeting, Tampa,
2007, pp. 1-5.
[6] A. Merkhouf, P. Doyon, S. Upadhyay, Variable frequency
transformer—Concept and electromagnetic design
evaluation, IEEE Energy Conv. 4 (2008) 989-996.
[7] R. Yuan, Y. Chen, G. Chen, Y. Sheng, Simulation model
and characteristics of variable frequency transformers
used for grid interconnection, in: IEEE Power & Energy
Society General Meeting, Calgary, 2009, pp. 1-5.
[8] E.T. Raslan, A.S. Abdel-Khalik, M.A. Abdulla, M.Z.
Mustafa, Perfromance of VFT when connecting two
power grids operating under different frequencies, in:
Conf. PEMD IET, Manchester, UK, 2010, pp. 1-6.
[9] L. Wang, L. Chen, Reduction of power fluctuations of a
large-scale grid-connected offshore wind farm using a
variable frequency transformer, IEEE Sustainable Energy,
3 (2011) 226-234.
[10] S. Shiyi, E. Abdi, R. McMahon, Low-cost variable speed
1
Brushless Doubly-Fed Induction Machine as a Variable Frequency Transformer
[11]
[12]
[13]
[14]
[15]
drive based on a brushless doubly-fed motor and a
fractional unidirectional converter, IEEE Ind. Electron. 1
(2012) 317-325.
Q. Wang, X.Chen, Y. Ji, Control for maximal wind energy
tracing in brushless doubly-fed wind power generation
system based on double synchronous coordinates, in:
International Conference on Power System Technology,
Chongqing, 2006, pp. 1-6.
A. Abdel-Khalik, M. Masoud, B. Williams, A.
Mohamadein, M. Ahmed, Steady-state performance and
stability analysis of mixed pole machines with
electromechanical torque and rotor electric power to a
shaft-mounted electrical load, IEEE Tran. Ind. Electron. 1
(2010) 22-34.
X. Wang, R. McMahon, P. Tavner, Design of the brushless
doubly-fed (induction) machine, in: IEEE International
Electric Machines & Drives Conference, Antalya, 2007,
pp. 1508-1513.
I. Boldea, S. Nasar, The Induction Machine Handbook,
CRC Press LLC, United States, 2002.
A. Dehkordi, P. Neti, A. Gole, T. Maguire, Development
[16]
[17]
[18]
[19]
[20]
117
and validation of a comprehensive synchronous machine
model for a real-time environment, IEEE Energy Conv. 1
(2011) 34-48.
F. Barati, S. Shao, E. Abdi, H. Oraee, R. McMahon,
Generalized vector model for the brushless doubly-fed
machine with a nested-loop rotor, IEEE Trans. Ind.
Electron. 6 (2011) 2313-2321.
J. Poza, E. Oyarbide, D. Roye, M. Rodriguez, Unified
reference frame dq model of the brushless doubly fed
machine, IEE Electr. Power Appl. 5 (2006) 726-734.
A. Kemp, M. Boger, E. Wiedenbrug, A. Wallace,
Investigation of rotor-current distributions in brushless
doubly-fed machines, in: Conf. IEEE IAS Annu. Meeting,
San Diego, 1996, pp. 638-643.
A. Wallace, R. Spee, H. Lauw, Dynamic modeling of
brushless doubly-fed machines, in: Conf. Rec. IEEE IAS
Annu. Meeting, 1989, pp. 329-334.
S. Boger, A. Wallace, R. Spee, Investigation of
appropriate pole number combinations for brushless
doubly-fed machines applied to pump drives, IEEE Ind.
Appl. 1 (1996) 189-194.
Appendix
The machine calculated parameters are as follows:
R1 = 1.5 Ω, R2 = 1 Ω, L1 = 244 mH, L2 = 917 mH,
Rr1 = 0.18 mΩ, Rr2 = 0.16 mΩ, Rr3 = 0.14 mΩ, Rr4 = 0.12 mΩ, Rr5 = 0.1 mΩ
0
15
0
11.4
0
7.7 0
19.2
 0
19.2
0
15
0
11.4 0 7.7

0
15.3
0
11.4
0. 7.7 0
 15

0
15
0
15.3
0
11.4
0 7.7

11.4
0
11.4
0
11.6
0
7.7 0
 Lr   
11.4
0
11.4
0
11.6 0 7.7
 0
 7.7
0
7.7
0
7.7
0
7.8 0

7.7
0
7.7
0
7.7
0 7.8
 0
 3.9
0
3.9
0
3.9
0
3.9 0

3.9
0
3.9
0
3.9
0 3.9
 0
3.9 0 
0 3.9 

3.9 0 

0 3.9 
3.9 0 
 H ,
0 3.9 
3.9 0 

0 3.9 
4
0 

0
4 
0.8 0 0.93 0 0.9 0 0.72 0 0.41 0 
 M s1r   
 mH ,
 0 0.8 0 0.93 0 0.9 0 0.72 0 0.41
0
3 0
2.3 0 1.6
0
0.82
0 
3.6
 M s2r   
 mH.
0
3.6
0
2.9
0
2.3
0
1.6
0
0.82






