International Journal of Latest Research in Science and Technology
Volume 4, Issue 4: Page No.101-105, July-August 2015
http://www.mnkjournals.com/ijlrst.htm
ISSN (Online):2278-5299
MODELLING OF AXIAL FLUX PERMANENT
MAGNET SYNCHRONOUS MACHINE AND ITS
STABILITY ANALYSIS
1
N V SIVAKUMARI & 2 Mr V RAMESH BABU
1
PG Student, VNR VJIET, Hyderabad, India
2
Associate professor, VNR VJIET, Hyderabad, India
1
Email:sivak232@gmail.com 2Email:rameshbabu0506@vnrvjiet.in
Abstract- Nowadays Axial-flux permanent-magnet machines are becoming significant machinery in different applications. Using
mathematical modeling, single rotor single stator configuration of axial flux permanent magnet synchronous machine was simulated in
MATLAB environment. And also linear mathematical model of axial flux permanent magnet synchronous generator was presented. For
large systems the state space model has been used more frequently in connection with system described by linear differential equations.
The system is whether stable or unstable is determined by Eigen-values of the system coefficient matrix A. These Eigen-values are
determined with the parameters of the machine.
Keywords - Axial Flux permanent magnet synchronous machine, linearization, mathematical modeling, stability
I. INTRODUCTION
Depends on the flux flow in the air gap, the
electromechanical energy conversion machines are
categorized as Radial and Axial Flux Machines. The
operating principle concerned with both the machines is alike
but varies in its structure .In general radial flux machines the
conductors are positioned in axial and the direction of air gap
flux is perpendicular to the shaft axis . In the axial flux
machines air gap flux is axial to the shaft axis and the
conductors are positioned in perpendicular to the shaft. With
the price competitiveness of high power permanent magnets,
PM machines are ever more becoming leading machines.
These machines recommend many distinctive features. They
are becoming more proficient because of the fact that field
excitation losses are eliminated resulting in significant rotor
loss reduction. Thus, the motor efficiency is significantly
enhanced and higher power density is achieved. Moreover,
magnetic thickness is also small which results in less
magnetic dimensions. They can be designed to have a higher
power-to-weight ratio resulting in less core material.
Moreover, they have planar and simply variable air gaps. The
noise and vibration levels are less than the radial machines.
Also many discrete topologies can be obtained by varying the
direction of main air gap flux.
In axial flux machines the structures of stator and rotor
are ring and disc shaped respectively. For the production of
the torque ,the radial length from the stator internal radius to
the external radius is the active part . Axial length is
dependent on the exact yoke design of the stator and the rotor
i.e., the flux density in the stator and rotor yokes. Even the
number of poles increases the active radial part of the
machine remains unaffected and the axial length depends on
the flux density in stator and rotor yokes. When the ratio of
motor overall length to motor outer diameter > 1then the
radial type is chosen and when it is < 0.3 axial flux type
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machine is chosen. Thus because of higher pole number
machine can be an substitute for the low speed applications.
II. STRUCTURE OF AXIAL FLUX MACHINES
The most attractive aspects of axial flux machines is the
extensive range of topologies. The simplest axial flux
structure is the single air gap structure as it is seen in Figure
1. The stator consists of a ring type winding fixed in epoxy
like material and an iron disc which is contrived from a plain
tape wound iron core. The rotor is shaped from a solid steel
disc on which the magnets are fixed. The main difficulty to
conquer in axial flux designs with the single-stage structure is
the great axial force exerted on the stator by the rotor
magnets.
Fig 1. Single-rotor-single-stator axial flux PM machine
structure
TORUS machine (TORUS-NS) is a typical dual-rotorsingle-stator, axial flux, PM, disc type structure. An idealized
version of the machine structure is shown in Figure 2. The
machine has a single stator inserted between two PM rotor
discs. The stator of the machine can be formed by tape
wound core with poly phase AC air gap windings which are
enfolded around the stator core with an end-to-end
connection. The rotor configuration is produced by arch101
International Journal of Latest Research in Science and Technology.
shaped surface mounted Nd-Fe-B magnets, rotor core and
shaft. The two disc shaped rotors carry the axially
magnetized Nd-Fe-B magnets mounted axially on the internal
surfaces of the both rotor discs.
The machine should have N stators and N+1 rotors (N ≥
1) for internal stator and external rotor (TORUS) types and
N+1 stators and N rotors (N ≥ 1) for external stator and
internal rotor (AFIR) types. If N is chosen as 1 for external
rotor and internal stator structure, the minimum TORUS
machine structure can be derived. Based on the slotting
machines can be divided as slotted and non slotted.
Vas = Rs Ia + p øma
Vbs = Rs Ib + p ømb
Rs Ic + p ømc
Vcs
=
(1)
Where, øas, øbs, and øcs are the instantaneous flux linkages
induced by the three-phase AC currents and the PMs. The
flux linkage equations are
øas = Laa Ia + Lab Ib + Lac Ic + øma
øbs = LbaIa + LbbIb + Lbc Ic + ømb
øcs = Lac Ia + Lbc Ib + Lcc Ic + ømc,
(2)
Meanwhile, flux-linkages at the stator windings due to the
permanent magnet are
øma = øm cos 
ømb =
øm cos ( - 120)
ømc = øm cos ( + 120)
(3)
Fig 2. Axial flux TORUS type non-slotted surface mounted
PM motor configuration (TORUS-NS)
In order to create the appropriate flux path, the magnets
facing each other on each rotor should be N and N poles or S
and S poles in TORUS NN type and, N and S or S and N
poles in TORUS NS type. Therefore, the direction of
armature current must be changed appropriately so as to
create torque. Since the windings in the perpendicular
direction are used for torque production, the end windings of
the NN type machine are much shorter than that of NS type
machine. In other words, the winding structure used in the
NS type machine results in a longer end winding which
implies a bigger outer diameter, high copper loss, reduced
efficiency and power density compared to its NN counterpart.
Where, øm, is the rotor flux linkage caused by the permanent
magnet.
Consider the stator winding 3ö voltage equations in matrix
notation.
V A 
 V    V A B C   p A B C  R A B C i A B C
(4)
s
s
s

 B  s
 V C 
Applying the transformation matrix i.e. represented by
Tdq0(è) to equation (4)
V d
V
 q
 V 0

  V d q 0    T
  A B C 
  s   d q 0     V s

(5)
Where,
 T d q 0    

 co s 

2 
  sin 
3 
 1

2

2 

cos   

3 

2 

 sin   

3 

1
2
4  

co s   

3  


4  

 sin   

3 


1

2

Then the d-q voltage equations are as follows
vq = Rsiq + ùr (LdId+ ëm) + p LqIq
Fig 3. Structures and 2D flux directions of the (a) NN type
TORUS (b) and NS type TORUS
III. MATHEMATICAL MODELLING
All rotating electrical machines are made equivalent to a
primitive machine model in the process of obtaining there
mathematical models. But many poly phase AC machines are
constructed in different manner than primitive machine.
These poly phase AC machines have many distinct phase
windings on stator and rotor which produce the rotating
magnetic field(RMF) when they are excited by poly phase
supply. Hence the machine inductances change especially in
non-uniform air gap machines with respect rotor position
which is a function of time. It is therefore necessary to apply
certain transformation in series to convert the equations into
linear equations with constant coefficients.
The electrical dynamic equation in terms of phase variables
can be written as
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vd = Rsid - ùr LqIq + p( LdId + ëm)
and the developed is,
Te =(3/2)( ) ( ëdiqs -(Ld -Lq) idsiqs)
(6)
(6)
(7)
The above differential equation is solved by using
SIMULINK in MATLAB to obtain the d-q axis currents of
both stator and rotor. The simulation results are shown
below.
Fig 4: d-q axes currents of axial flux permanent magnet
machine
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International Journal of Latest Research in Science and Technology.
=
-
(9)
Fig 5: steady state torque of an axial flux permanent magnet
machine
Here a mathematical model of the synchronous machine for
the stability analysis is developed. State space formulation of
the machine equations is used. The synchronous machine
under consideration is assumed to have three stator windings,
one permanent magnet and two damper windings. These
windings are magnetically coupled. The magnetic coupling
between the windings is a function of rotor position. The
instantaneous terminal voltage of any winding is in the form
of,
v = ±Ri ± ∑
=
The voltages of preceding section are not in convenient form.
One difficulty is numerically awkward values with stator
voltages in Kilo voltage range and remaining voltages are
low(zero). This problem can be solved by normalizing the
equations to a convenient base value and expressing all
voltages in pu(in percent) of base.
Referring to the flux linkage equations, let id = IB and iD = IDB
be applied one by one with other currents set to zero. If we
denote the magnetizing inductances(ȴ - leakage inductances)
as
Lmq. Lq - ȴq H
Lmd Ld - ȴd H
LmQ LQ - ȴQ H
LmD LD - ȴD H
and equate the mutual flux linkages in each winding,
= kMDIBIDB
Lmd = LmD
Lmq = LmQ
= kMQIBIQB
and this is the fundamental constraint among base currents.
and the requirement for equal SB we compute
VDB/VB = IB/IDB = (LmD/ Lmd)1/2 = kMD/ Lmd = LmD/ kMD kD
VQB/VB = IB/IQB = (LmQ/ Lmq)1/2 = kMQ/ Lmq = LmQ/ kMQ kQ
-
These basic constraints permit us to compute
RQB = RB Ù
RDB = RB Ù
LDB =
Fig 6. Schematic diagram of Permanent
magnet
synchronous machine
As damper windings are short circuited, the voltage induced
in the damper bars is zero.
0 = RDiD +
RQiQ +
0
LB H
LDB =
LB H
and since the base mutual must be the geometric mean of the
self-inductances.
Normalized voltage equation is
=
-
=
(8)
(10)
Flux linkage from the permanent magnets is constant. so
,
= 0.
Normalized swing equation is
=
+
(11)
and the relation between ù and ä can be written as
=ù-1
(12)
By incorporating above equations , we obtain
=
Fig 7. dq-axes equivalent circuit
By writing d-q voltage equations of both stator windings and
damper bars in state space form we get,
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+
(13)
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International Journal of Latest Research in Science and Technology.
This matrix equation is in the desired state-space form =
f(x,u,t). It is clear that above equation is non linear , as inputs
as v and Tm.
When the system is subjected to a small load change, it
tends to acquire a new operating state. During the transition
between the initial state and the new state the system
behaviour is oscillatory. If the two states are such that all the
state variables change only slightly (i.e, the variable xi
changes from xio to xio + xiÄ where xiÄ is a small change in xi
), the system is operating near the initial state. The initial
state may be considered as a quiescent operating for the
system. To examine the behavior of the system it is perturbed
such that the new and old equilibrium states are nearly equal,
the system equations are linearized about the quiescent
operating conditions, so that first order approximations are
made for the system equations. The new linear equations thus
derived are assumed to be valid in a region near the quiescent
condition.
By substituting the parameters of pu values A matrix can be
written as
A=
The Eigen values of the A matrix are negative,
which shows that the system is stable.
V.RESULTS
MATLAB/SIMULINK program was used in the stability
analysis of the synchronous machine infinite bus system. The
M-file and SIMULINK model can be combined by the
commands [num, den]=ss2tf(A, B,C, D, 1);c=step(num, den,
t); plot(t, c); title; x label and y label. The following figures
show the step response of the rotor angle and frequency with
step input and D-Q axes currents after linearization, when
machine is conducted to infinite bus.
D e l ta , d e g re e s
26
=
24
22
20
18
16
(14)
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
t, sec
When synchronous machine connected to infinite bus
vd = -Ksin(ä - á) + Reid + Le + ùLeiq
vq = Kcos(ä - á) + Reiq + Le + ùLeid (15)
final linearized machine equation is as follows
=
F re q u e n c y , H z
60.08
60.06
60.04
60.02
60
59.98
59.96
0
1
2
3
4
5
t, sec
Fig 8:
Step response of the rotor angle and
frequency with
state space Modeling with step function
(16)
These matrix M and K are included the transmission line
constants, synchronous machine parameters and the infinite
bus voltage. Therefore, state space model of the system is as
= - M-1 Kx - M-1v = Ax +Bu
(17)
where A = - M-1 K.
Stability characteristics
may be
determined
by
examining(17). x is an n vector denoting the states of the
system and A is a coefficient matrix.
For the following parameters of the permanent magnet
machine , stability study is analyzed.
TABLE I: machine Parameters
S=5.6 MVA
Rs = 1.28ohm
V= 400v
RD=3.59ohm
Vph = 230v
RQ=4.23 ohm
I= 8A
Ld =23mH
Cos ϕ=0.93
Lq=24mH
kMD= 5.785mH
kMQ = 2.779mH
Re = 1.2 ohm
Le = 5.43H
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Fig 9: d,q axes currents when synchronous connected to
infinite bus
VI. CONCLUSION
A thorough literature survey is conducted on Axial flux
permanent magnet machines and their credentialsl are
highlighted. A single airgap Axial flux permanent magnet
synchronous machine is modelled in MATLAB
environment.To examine the behavior of the system it is
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International Journal of Latest Research in Science and Technology.
perturbed such that the new and old equilibrium states are
nearly equal, the system equations are linearized about the
quiescent operating conditions, a physical model is obtained.
The Eigen values of this physical model which were obtained
using SIMULINK were negative, which shows that the
system is stable for given parameters. This will demonstrate
the advantages of using MATLAB for analyzing steady state
power system stability and its capabilities for simulating
transients in power systems.
ACKNOWLEDGMENT
We like to express our sincere appreciation and deepest
gratitude to our colleagues, family and friends.
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