Identification of magnetostriction vibrations in power transformers

ZESZYTY NAUKOWE POLITECHNIKI ŁÓDZKIEJ
Nr xxx
ELEKTRYKA, z. 1xx
2006
BEATA WAWRZYNIAK
Technical University of Łódź
Institute of Mechatronics and Information Systems
THE APLICATION OF THE FAMILY OF
ORTHOGONAL FUNCTIONS FOR MODELLING
THE GENERATORS WITH PERMANENT
MAGNETS
Reviewer: dr hab. Paweł Witczak prof. PŁ
Manuscript received 2007.03.20
Abstract - The paper presents the computing method making possible
the replace the inductance in the description of the electric machine,
with three dimentional distribution of magnetic flux density in the airgap fot chosen states of work: no-load, armature reaction directed in
d and q axes. The input data comes from magnetostatic solution of
FE model. The method converts the distributions of the flux density
into the space of periodic functions, what allows the similar
procedure as in the classic dq method.The presented approach allow
to solve the radial and axial flux topology of the machine.The effects
of the slot skew is consider using the low-pass filter in spectral
domain. Experimental measurements of the existing machine were
performed and they were compared with calculations.
1. INTRODUCTION
The most popular representation of an AC machine is based on lumped
circuit consisting of resistances and inductances converted to dq plane. This
approach gives very quick solution, but it is limited to sinusoidal quantities. On
the other hand the numerical models in 2D and 3D space are also used. It allows to
consider the machine with flux density distribution distorted from sinusoidal
shape. Unfortunately, the calculations using for example finite element are time
consuming, especially for 3D objects.
The aim of the work is to present the combine method, where the inductances
will be replaced by flux density distribution in the air-gap obtained from finite
element analysis at different load conditions: no-load, armature reaction directed
in d and q axes.
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Beata Wawrzyniak
2. ANALYZED MACHINES
Two kind of machines were analyzed – radial flux machine with surface
mounted magnets (machine A), radial flux with embedded magnets (machine B)
and axial flux machine with surface mounted magnets (machine C). Topologies of
the machines are shown in fig.1.
machine A
machine B
machine C
Fig.1. Topology of exemplary machines.
The machine A is existing one with the fractional, three-phase winding with
skewed stator slots, the rest of machines are virtual. Numerical experiment of
machine B showed that the contribution of end-winding into flux density
distribution is comparable to calculation accuracy and can be neglected. The radial
flux machines were analyzed by means of 2D models with the vector potential
approximation 3rd order. In case of axial flux machine the 3D model was necessary
– the scalar potential approximation 2nd order. The reduction of the models size
obtained using adequate boundary and periodicity conditions. The meshes are
presented below.
Fig.2. Finite element meshes of reduced models of PM generators.
2. PRINCIPLE OF THE METHOD
The input flux density distributions were obtained for no-load state, armature
reactions of the phase current with given RMS value IN in direct and quadrature
axes. From 2D models of radial machines obtained one-dimensional vectors, but
The application of the family of orthogonal functions to modelling the generators
with permanent magnets
3
for axial machine data was two-dimensional, therefore, to use the same postprocessing algorithm, it had to be averaged in radial direction following equation
(1), for the phase belt consisting of k coils (N turns each one) and positioned at
angles { 1k, 2k}.
α1k Ω t L
Ψ(t)
B(α, r) r dr dα
N
k
(1)
α 2k Ω t 0
where L is the core size in transverse direction to the field movement and is the
mechanical angular velocity of the rotor.
Afterwards the data was filtered using DFT/iDFT technique to remove these
harmonics, which do not fulfil non-dispersive wave condition (2)
(2)
μ Cν
where , are spatial and time harmonic orders, and C is integer multiplier.
Fig. 3a presents the exemplary 2D one-sided spectrum comes from timestepping of the radial flux machine at rated load. The selection of slotting and subharmonics components can also be done having only the 1D DFT – fig 3b,
because the slotting space harmonics are of order kNs p, where k=1,2,… and Ns is
the number of stator slots. This simplification can be applied, when the number of
slots per pole and phase is greater than one. In other cases other filtering technique
should be used.
fundamental, 0.71 T
a.
b.
Fig.3. Spectra of normal flux density component in q axis (a) 2D one-sided DFT and (b) 1D
one-sided DFT.
Considering the field variation along radius equation (1) can be replaced by (3):
α1k Ω t
NS
(3)
Ψ(t)
B( ) dα
2
k α 2k Ω t
where S denote active surface of the air-gap. If the flux wave is non-dispersive,
only the speed component of the electromotive force exists.
NSδ
(3)
e(t)
[B(α1k Ω t) B(α 2k Ω t)]
t
2
k
Using classic Gram-Schmidt algorithm it is possible to convert (3) into set of
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Beata Wawrzyniak
orthogonal basis functions m( t) – details of procedure can be find in [3].
The resultant e.m.f. formula using the basis functions [4] for any load conditions is
presented in (4)
e(ψ, I, t)
NSδ
{[b 0
2
(b d0
[b d1CC1
b 0 ) i r CC0
(b q0
b 0 ) i r CS0 ]
b q1CS1 ] i r 1 (Ω t) b q2CS2 i r
2
0
(Ω t)
(4)
(Ω t)}
where
b0
bm n
e0
em ,
n
CC m
m
( ),
CSm
m ( ),
m
(α ψ)
(5)
m (α π/2 ψ)
and < , > means dot product of basis functions. The load angle =0, when exist
the demagnetizing armature reaction only. CC and CS are termed as autocorrelation factors. Coefficient ir is the relative value of the phase current I/IN.
a.
b.
Fig.4. Exemplary set of normalized basis functions for machine A without skew (a) and
with skew (b)
Knowing the components of electromotive force at any load it is possible to
calculate the distribution of phase current following (6)
i(ψ, i r , t)
ir IN
i e (ψ, i r , Ω t)
ie
(6)
To calculate the value of electromagnetic torque TE for basis functions the
equation (7) can be used
3 2
(7)
TE
E k (ψ, i r )I k (ψ, i r )
Ωk0
where Ek and Ik are the orthogonal components respectively of the e.m.f. and
current. The results of calculations compared with the measurement for the
machine A are presented in fig.5
The application of the family of orthogonal functions to modelling the generators
with permanent magnets
5
Fig.5. Torque vs. load angle for machine A (OF – orthogonal functions, FE – finite
element)
3. COMMUTATION PHENOMENA
Using the rectifier bridge as the load of generator impel to consider the
commutation phenomena. The describe of the voltage in selected winding can be
done using (8)
uV
di V
dt
e0V (t) L
(8)
Ri V
with the sinusoidal quantities assumption it is possible following [1] to calculate
(9) the commutation wyprzedzenia angle 0, related with the voltage on resistance:
ωt 0
arcsin
RI
2 3E 0
RI
6E 0
ν0
(9)
where R – sum of the winding resistance and the diode, I – average value of the
rectified current. Considering the moment, when phase W took the current from V,
it is possible to assume the equalities (10):
di V
di W
(10)
i V (t) i W (t) I and
dt
dt
Balancing the voltages for the commutation moment and including (10) obtains:
di V
(L V
dt
L W ) 2Ri V
e 0V (t ) e 0W (t ) RI
(11)
It is suggested to apply the elliptical approximation of inductance as follows:
L( )
Ld cos 2α Lqsin 2α
(12)
6
Beata Wawrzyniak
upper envelope
W
lower envelope
U
d axis
V
W
0
t
Fig.6. Load angle at immediate commutation, U,V,W denote the phase carrying the current.
Based on fig. 6 for t=0 possible to note the equations describing the angle
for phase which return current (13) and take over (14)
(13)
2
2
t
0
3
2
2
t
0
3
(14)
Putting (12), (13) and (14) into (11) we receive the value of current iV(t),
which, after insert into formula (8) allows to find the searching voltage curves
during commutation. It is worth to mention, that for permanent magnet
synchronous generators, there is dependence inductances of the armature reaction
from the rotation angle, coming from different magnetic associations in d and q
axes. Including this fact it is possible to formulae inducing voltage in one of the
phases, during commutation between the others:
d
(Φ VU
dt
UV
3ωi U cos(2(ωt
Ld
Φ VW )
ν0
Lq
2
3 di U
sin
2 dt
ψ)) ωIsin 2 ωt
(15)
π
3
ν0
ψ
where VU, VW denote the flux induced in phase V by the current flowing
respectively in phase U and W.
The influence of rotational currents on field dumping depends on properties
(resistivity) and position of the permanent magnets. It is possible to consider its
influence replacing the inductance Ld by the Ld” [2]. Finally to calculate the value
of inductance the equation (16) was used:
Ld
(bd 0 b0 ) 2 bd21
kd
2
pIN
and Lq
bq 2
2
pI N
kq
(15)
where b0, bd0, bd1, bq2 means r.m.s. value of the adequate components of e.m.f. For
further calculations assumed the value of the coefficient kd”=0,71 for machines
with surface mounted magnets and kd”=1 for machine with embedded magnets.
The proposed algorithm was used to set the curves of the voltage and current in
The application of the family of orthogonal functions to modelling the generators
with permanent magnets
7
the analyzed machines.
4. EXPERIMENTAL VERIFICATION OF THE THEORY
The experimental verification of the method was done using the permanent
magnet machine with rated data TN=38 Nm, nN=2000 rpm, p=4. The machine
has winding with fractional number of slots per pole and phase. The air-gap field
was measured by the search coils - number 1, 2, 3 in fig.7– follow the scheme of
phase winding, coil s was placed on single tooth.
Fig.7. Outlook of investigated permanent magnet machine.
The measurement of electromotive force for machine loaded by the r.m.s.
current I=13,9 A was done and compared with the results of calculations – fig.8
search coil EMF, [V]
1.5
measured
1
calculated
0.5
0
1
-0.5
-1
-1.5
time
Fig.8. Measured and calculated electromotive force in search coil under rated load.
Table 1. Comparison of main parameters at generator duty, n=1500 rpm
(symmertical load)
item
measurement
calculation
maximum torque [Nm]
rated torque [Nm]
rated r.m.s. voltage [V]
50
38
109
49.8
36.2
102.7
One of analyzed states of work was the machine loaded by rectifier bridge.
The results of experiment in this case are presented below.
8
Beata Wawrzyniak
U [V] 200
I [A] 12
calculations
measurements
calculations
measurements
8
100
4
T
0
T
0
-4
-100
-8
a.
-12
b.
Fig.9. Phase voltage (a) and phase current (b)for machine loaded by 6D rectifier bridge.
-200
Table 2. Comparison of measurements and calculation for machine loaded by
rectifierr bridge.
current
[A]
14.2
6.4
item
measurement
calculation
P [W]
U [V]
P [W]
U [V]
4402
111.3
2246
124.4
4706
113.0
2291
124.9
6. CONCLUSIONS
The method presented in this paper enables the computation of PMS machines
by means of the family of orthogonal functions. The main advantage of proposed
algorithm is the independence on the shape of the field in the air-gap. This
approach allow to include the influence coming from different in inductances in d
and q axes. It is worth to mention that also diverse topologies of machine can be
analyzed using the same procedure.
The possibility of expand the family of basis functions according to necessaries
(include the magnetic saturation effects), make this method very useful to more
precise analyses. The advantage against time stepping follow the availability of
interim results showing the influence of many geometric and construction factors.
REFERENCES
[1] Jabłoński M.: Przekształtniki, PŁ, Łódź 1978.
[2] Paszek W: Dynamika maszyn elektrycznych prądu przemiennego, HELION,
Warszawa 1986
[3] Wawrzyniak B.: Zastosowanie rodziny funkcji ortogonalnych do modelowania
The application of the family of orthogonal functions to modelling the generators
with permanent magnets
9
prądnic z magnesami trwałymi współpracujących z turbinami wiatrowymi, rozprawa
doktorska, Politechnika Łódzka, Łódź 2007
[4] Witczak P., Wawrzyniak B.: Determination of exploitation parameters of permanent
magnet synchronous machine following the finite element solutions, CEM, pp. 177178, Aachen 2006 .
ZASTOSOWANIE RODZINY FUNKCJI
ORTOGONALNYCH DO MODELOWANIA PRĄDNIC
Z MAGNESAMI TRWAŁYMI
Streszczenie
Praca dotyczy metody obliczeniowej, pozwalającej na zastąpienie
indukcyjności w opisie maszyny elektrycznej przestrzennymi rozkładami indukcji
magnetycznej w szczelinie wyznaczonymi w wybranych stanach pracy – stan
jałowy, reakcja twornika w osi q i d. Proponowane podejście pozwala na
uwzględnienie rozmieszczenia cewek uzwojeń fazowych, skosu żłóbków stojana
oraz dopuszcza analizę maszyn o strumieniu radialnym i osiowym za pomocą tego
samego algorytmu. Dodatkową zaletą jest możliwość rozszerzenia rodziny funkcji
bazowych o kolejne składniki odwzorowujące np. wpływ prądów wirowych.
W pracy zaproponowano również połączenie metody funkcji ortogonalnych
z obwodowymi równaniami opisującymi zjawiska zachodzące w mostku
prostowniczym, przez co możliwe jest również analizowanie wpływu
niesymetrycznego obciążenia na parametry maszyny. Dokonano porównania
wyników obliczeń z pomiarami rzeczywistego obiektu.
Promotor dr hab. inż. Paweł Witczak, prof. PŁ
Recenzenci pracy doktorskiej:
1. prof. dr hab. inż. Marian Łukaniszyn – Politechnika Opolska
2. dr hab. inż. Adam Pelikant, prof. PŁ – Politechnika Łódzka