Download Flux Displacement in Rectangular Iron Sheets and Geometry

Flux Displacement in Rectangular Iron Sheets
and Geometry-Dependent Hysteresis Loss
F. Bachheibl, D. Gerling

Abstract – This paper suggests a possible explanation to the
geometry-dependence of core loss by solving Maxwell’s
equations for the flux distribution inside a magnetic conductor
carrying eddy currents. The solution reveals a displacement of
flux towards the edges of the magnetic sheet. Core loss with and
without the displacement effect are compared for the tooth of a
highly excited and high-frequency electric machine. A
significant difference from standard calculation methods is
obtained.
Index Terms -- Analytical Calculation, Core Loss, Excess
Loss,
Flux
Displacement,
Magnetic
Skin
Effect,
Rectangular Cross Section
derivation of the flux distribution is obtained by solving the
dual problem to the distribution of current in rectangular
conductors, as investigated in [1].
y
x
h
w
E
I.
INTRODUCTION
LECTRIC MACHINES are subject to various sources
of loss, the most prominent being conduction loss and
iron loss. Conduction loss is reliably quantifiable,
whereas iron loss is still subject to great uncertainty when
designing an electrical machine. As conduction loss is the
major loss source in electrical machines, up to now, iron loss
has been of minor importance, resulting in an inaccuracy in
calculation of as much as up to 2000% in extreme cases [8].
The striving for efficiency and compactness of electrical
machines now puts an increasing focus on iron losses, that is
to say eddy current loss and hysteresis loss. Minimization of
eddy current loss in electrical machines is achieved by
lamination of the flux-conducting parts. This measure
reduces the impact of two loss-mechanisms: First of all,
eddy currents produce conduction losses in the magnetic
core. Secondly, the field is displaced out of the center of the
magnetic body towards the edges of the material, resulting in
a locally higher flux density and therefore a higher overall
hysteresis loss. This paper investigates the second effect and
tries to establish that it might be one of the causes for the
aforementioned inaccuracies which should be taken into
consideration for future design of electric machines and for
the estimation of losses.
II.
Fig. 1. Sheet of iron under investigation
In Maxwell’s equation (1) for the electric field, the
permeability µ is considered a constant in order to facilitate
the derivation:
rot ( E ) 



E  E x (x, y, t)  ex  E y (x, y, t)  ey
(2)
whereas the magnetic flux and field only have a nonzero
component in z-direction:


H(x, y, t)  H z (x, y, t)  e z
(3)
Additionally, it is assumed that field quantities are
independent of z, which is consistent to the assumption of
infinite length along that axis. From (1), (2) and (3), the
following conclusions can be drawn:
E y
H x
0
dt
z
H y
E x
 
0
dt
z



E y Ex
Hz

 
dt
x
y
A.
Derivation of the field distribution
The sheet of transformer steel under investigation is
depicted in Fig. 1. Its geometric properties are the width w
in direction of the x-axis and the height h following the yaxis. It is assumed to be of infinite length in z-direction. The
M.Sc. Florian Bachheibl is with University of Federal Defence,
Chair for electrical Drives and Actuators, D-85577 Neubiberg, Germany
(E-mail: [email protected]) ©2014 IEEE
978-1-4799-4775-1/14/$31.00
(1)
It is assumed that the electric Field E, and therefore the
current density, only exist in the x-y-plane,
FLUX DISPLACEMENT IN RECTANGULAR IRON CORES
This work was supported by SIEMENS AG, Industry Sector, Large Drives
H
 B
 
t
t
 
( 4.1 )
( 4.2 )
( 4.3 )
From Maxwell’s Equation (5) for the magnetic flux,
disregarding the displacement current:
1187
Prof. Dr.-Ing. Dieter Gerling is with University of Federal Defence,
Chair for electrical Drives and Actuators, D-85577 Neubiberg, Germany
(E-mail: [email protected])


  dD
rot(H)  j 
 E
dt
(5)
and (2) and (3), we obtain the following conditions:
H z
  Ex
y
( 6.1 )
H z
  E y
x
( 6.2 )
Inserting equations (6.1) and (6.2) into (4.3), we get:
2 H z 2 H z
H z

  
2
2
x
y
dt
(x≥0, y≥0) is regarded in the following to calculate k and l .
In order to obtain a complete enclosure to the Volume V, the
covers in the drawing plane would have to be considered,
but can be neglected due to the assumption of infinite length.
Additionally, Volume V is enclosed by the boundaries of the
region under investigation. These boundaries towards the
environment may be disregarded for the solution of (12) as
they cannot be penetrated by current and therefore (14) must
apply:
 
 
E  A   j  A  0
(7)
y
Which is identical to the result obtained in [1] for the
electric field intensity in rectangular conductors.
B.
Determination of constants
As derived in [1] for the electric field, the magnetic field
intensity on the sheet is described by (8):
H z (x, y, t)  C  cosh  kx   cosh  ly   e jt
( 14 )
A1 z
h/2
A2
x
(8)
w/2
where
Fig. 2: Current flow in upper right quarter of the domain
2
2
k  l  j
(9)
Inserting (6.1) and (6.2) into (13) yields the following result:
 h
 w
cosh  l    cosh  k  
 2
 2
C can be obtained by assuming that the flux passing through
the area be independent of the frequency. Therefore, it must
be guaranteed that:

 f (t) 
w /2 h /2
 

 H z (x, y, t)dydx   f 0 (t)
And therefore:
k l w h
 w
 h
sinh  k  sinh  l 
 2
 2
( 11 )
 is to be chosen identical to the DCThereby, the value for B
case with a homogenous distribution of flux. In order to
obtain k and l , Maxwell’s equation for the electric field is
applied to the problem:
 
E   0

 
 
 
E

dA

E

dA

E



  dA
A1
C.
Characteristic Distribution of Fields
The highest induction in electrical machines appears in
the teeth and therefore one section of electrical steel located
in the tooth of an electrical machine is chosen to
demonstrate the effect of flux displacement due to eddy
currents. Table 1 contains all relevant information on an
electric turbocharger motor developed at the Institute for
Electrical Drives and Actuators, discussed in [6], which is
selected for its high supply frequency.
Table 1: Machine Data
( 12 )
Here, ρ is the density of volume charge in the region.
Equation (12) in its integral form reads:
V
( 16 )
Notably, for square geometries with h=w, we obtain that
k=l, as also found by [1]. Equations (9) and (16) easily allow
the calculation of l and k .
Solving (10) for C yields:
B̂
4
l  h  k  w
( 10 )
 w /2  h /2
C
( 15 )
A2
( 13 )
Rated power
Rated torque
Tooth width
Tooth induction
Maximum speed
Maximum supply Frequency
Sheet material
Sheet conductivity (150°C)
Sheet relative permeability (1.2 T)
230 W
0.12 Nm
8.5 mm
1.2 T
18000 min-1
3200 Hz
30 EX 1200
1.2195·106 Sm-1
5970
The areas A1 and A2 represent the areas in Fig. 2 colored in1188Material data for 30 EX 1200 was obtained from a steel
black and blue, respectively, which cut the region of Fig. 1 datasheet [10].The slice of material considered below has
into four identical pieces. One fourth of the iron sheet the tooth width set as width w and the material thickness set
as height h. As Fig. 3 indicates, the operating point of flux
density at 1.2 T is situated just at the saturation point of the
material. The formulae developed above lose their validity
beyond this point of saturation, as constant permeability had
been assumed.
The effect of flux displacement causes local field intensity
and flux density to increase to values well above the
homogenous case. Figures 6 to 11 therefore only convey
qualitative information on the effect flux-displacement has
on the distribution of magnetic field.
Fig. 5: Magnetic field intensity for f=50 Hz at t=0.55/ω
Fig. 3: B-H-Curve of 30 EX 1200 (Manufacturer Data)
Figures 4 to 7 show the distribution of magnetic field
intensity, assuming a flux corresponding to a flux density of
1.2 T. The distribution of field and flux density has been
evaluated at different time steps for 50 and 3200 Hz,
respectively. There is an obvious phase shift in the field
intensity between the edges and the center of the conductor,
resulting from a behavior similar to a wave travelling from
the edges towards the inside of the conductor. Figures 8 and
9 illustrate the electric field intensity for t=1/ω and for
t=0.4 ω for a frequency of 3200 Hz.
Fig. 6: Magnetic field intensity for f=3200 Hz at t=1/ω
Fig. 7: Magnetic field intensity for f=3200 Hz at t=0.55/ω
Fig. 4: Magnetic field intensity for f=50 Hz at t=1/ω
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Fig. 8: Electric field intensity for f=3200 Hz at t=1/ω
As already mentioned, field intensities as obtained for
3200 Hz excite the material well into saturation. In order to
calculate the flux density, the B-H-curve shown in Fig. 3 is
fitted by a Langevin-Function, which is very suitable for
modelling saturation and also used in the Jiles-Atherton
Model [3]:
1 

B  a, b, H   a   coth  b  H  

bH 

( 17 )
The parameter fit is performed by a nonlinear least square
algorithm and obtains the following values for a and b:
Table 2: Coefficients for Langevin-Function
Fig. 9: Electric field intensity for f=3200 Hz at t=0.4 ω
Figures 10 and 11 display the maximum values of field
intensity over time, as attained by each point in the domain.
Fig. 10: Distribution of peak magnetic field intensity for f=50 Hz
a
b
1.379
0.06431
Using (17), the flux densities corresponding to the field
intensities shown in Fig. 6 and Fig. 11 can be calculated.
The resulting flux density distribution and flux density at
t=1/ω are shown in Figures 13 to 15. The flux densities
shown do result from eddy currents and their respective
fields as displayed in Figures 4 to 11. However, it needs to
be said that eddy currents are driven by a change in flux
density and not in field intensity. This change, being subject
to saturation effects, is too small for an effect at that scale.
The chain of effects shown in Fig. 12 elucidates these
dependencies which take effect simultaneously, according to
Maxwell’s equations in general and their application in this
paper in particular. An externally excited magnetic field
causes flux inside the material, which is subject to
saturation. Changes in flux density lead to an electric field
and therefore to eddy currents. Those cause an internal
magnetic field that adds to the external, homogenous
magnetic field as fields superimpose by simple addition. The
nonlinear dependency of flux and field intensity determines
the flux excited by external and internal magnetic field, thus
closing the loop of Fig. 12. In Section E, the implications of
saturation as well as a possible remedy are discussed in
order to enlarge the scope of applicability of the formulae
developed.
Fig. 11: Distribution of peak magnetic field intensity for f=3200 Hz
Fig. 12: Chain of effects
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Fig. 15: Total core loss versus flux density
ANSYS Maxwell’s material library provides a convenient
tool to calculate coefficients for loss separation using core
loss data, conductivity and thickness of the material. The
following coefficients can thus be obtained for the steel in
use:
Fig. 13: Distribution of peak magnetic flux density for f=3200 Hz
Table 3: Loss coefficients for 30 EX 1200, scaled for W/kg
Hysteresis loss coefficient kH
Eddy-current loss coefficient kE
0.0157
22.9*10-6
Using these coefficients, it is possible to calculate the total
core loss by ( 20 ):
 
2  k f B

Pv  k H f B
E
Fig. 14: Magnetic Flux Density for f=3200 Hz at t=1/ω
D.
Increase in Hysteresis Loss
As discussed in Section C, the maximum flux density in
the iron sheet is higher than in the homogenous case, which
leads to an increase in hysteresis loss. Material loss data for
electrical steel is generally provided to fit the Steinmetz loss
equation presented in [7]:


p v  Cm f  B
( 18 )
Where Cm, α and β are experimentally obtained constants.
Figure Fig. 15 contains manufacturer data for core loss at
different peak polarizations [10]for the material 30 EX 1200
at 1 kHz.
2
( 20 )
Hysteresis loss for one loop is independent of the speed of
magnetization and therefore, it is assumed to be only linearly
dependent of the frequency, whereas eddy-current loss is
squarely proportional to frequency. The results presented in
section C, however suggest a different behavior. Flux is
displaced from the center of the iron sheet to its corners,
resulting in a significantly higher flux density and therefore
also in a higher hysteresis loss overall. As indicated in
Section A, µ was considered to be constant for the
derivation of formulae. Therefore it is crucial for the
following investigation of hysteresis loss not to exceed the
range of validity of this assumption. Referring to the B-HCurve in Fig. 3, it can be concluded that saturation occurs at
1.2 T and that the material behaves linearly beneath that
point. That being said, every conclusion drawn from a setup
of which no point experiences a peak flux density higher
than 1.2 T is valid on the grounds of satisfying the
assumptions made. For frequencies up to 3200 Hz,
homogenous flux density is therefore limited to 0.01 T to
avoid saturation in the edges. The results shown in Fig. 16
have been calculated for flux densities from 0.001 T to 0.01
T. It has been found that the flux density has no effect on the
relative increase in hysteresis loss as long as no saturation
occurs. The relative increase ΔP has been calculated
according to:
Usually, loss curves are used to obtain factors for hysteresis, eddy current and excess loss for calculation purposes, an
approach introduced by Jordan [9] and Bertotti [2]. Despite
many efforts in the domain of loss modelling, according to
[8], “Many papers report that calculation errors between
200%-2000% can occur” when calculating Iron Losses in
electrical machines. In the following section, the influence
of flux-crowding on hysteresis loss is investigated and it is
shown that it can have a significant effect which may
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account for some of the prediction errors.
P 
PDisplacement  PHomogenous
PHomogenous
( 19 )
The same holds true for all effects induced by this change,
notably eddy currents and electric field changes in the
electrical steel sheet and its isolation. In summary,
displacement currents may be neglected for the investigation
of flux displacement.
III. CONCLUSION
Fig. 16: Relative increase in hysteresis loss vs. frequency
E.
Scope of validity and future work
The assumptions limiting the scope of applicability of
equations (8), (9), (11) and (16) are the following:
i.)µ is considered to be constant
ii.)Displacement currents inside and outside of the sheet
are neglected
As shown in Section D, disregarding saturation
(assumption i) restricts the range of validity, as local flux
densities increase so drastically that even flux densities of
0.01 T for the homogenous case lead close to saturation in
the edges. In future work, an attempt will be made at
substituting the constant interrelationship between H and B
by a Langevin-type dependency, which may create the need
to solve the system numerically.
Neglecting displacement currents (assumption ii) is
necessary to obtain an integral formulation for H. The effect
of displacement currents is dependent on the permittivity of
the insulation material and on the rate of change of the
electric field. The permittivity ε0εr is of order 10-11 for resin
insulation [4]. As also the rate of change of the field plays a
role in the magnitude of displacement currents, it has to be
investigated in the light of inverter-feed. Voltage rise times
of IGBT-based inverters are typically close to 50 ns as in
[5], resulting in dU/dt-Values of 12V/ns for a 600V switch.
This rate of change of order 1010 elevates displacement
current to a non-negligible quantity in conductors. Yet, these
rates of change have only minimal effect on the field:
 t 2   U t
Switch    U
 DC
( 21 )
 being the rate of voltage rise and the integration
With U
interval being equal to the switch time of 50ns, the change of
flux linkage caused by a switching event is of order 10-9, as
the right side of (21) simplifies to:
 Switch   U DC  t Switch
( 22 )
The flux distribution in a conductive magnetic sheet subject
to sinusoidal external magnetic fields has been derived from
Maxwell’s formulae. Hysteresis loss has been calculated for
the obtained distorted and the uniform distribution of flux
density. It was shown that a significant increase of hysteresis
loss occurs for high frequencies, caused by Flux
Displacement towards the Edges.
The assumption of constant permeability, however, demands
that the maximum flux density in the iron sheet not exceed
the linear range, thus limiting the scope of applicability of
this work. Therefore, a numerical model including
nonlinearities is under development and will soon be
presented.
IV. REFERENCES
D. Gerling. (2009) "Approximate analytical calculation of the skin
effect in rectangular wires", Electrical Machines and Systems, 2009.
ICEMS 2009. IEEE International Conference on, 15-18 Nov. 2009
[2] G. Bertotti. (1988) “General properties of power losses in soft
ferromagnetic materials” Magnetics, IEEE Transactions on, 24(1),
pp. 621-630
[3] D. C. Jiles and D. L. Atherton. (1983) “Ferromagnetic hysteresis”,
Magnetics, IEEE Transactions on, 19(5), pp. 2183-2185
[4] A. Helgeson. (2000) “Analysis of dielectric response measurement
methods and dielectric properties of resin-rich insulation during
processing”, Ph.D. dissertation, Department for Electric Power
Systems, KTH Stockholm, 2000
[5] Datasheet
Infineon. (2011), "SIGC223T120R2C", Infineon
Technologies
[6] D. Gerling and G. Dajaku. (2004) “Comparison of Different
Calculation Methods for the Induction Motor with Multilayer Rotor
Structure”, Electrical Machines, 2004. ICEM 2004. International
Conference on, 5-8 Sept. 2004
[7] C. Steinmetz. (1984) "On the law of hysteresis." Proceedings of the
IEEE. 72(2) pp. 197-221
[8] J. Reinert, A. Brockmeyer and R. De Doncker. (2001) “Calculation of
losses in ferro- and ferrimagnetic materials based on the modified
Steinmetz equation”, Industry Applications, IEEE Transactions on
[9] H. Jordan. (1924) “Die ferromagnetischen Konstanten für schwache
Wechselfelder”, Elektr. Nach. Techn., Berlin
[10] Datasheet NKK. (1998) “NK SuperE-Core Magnetic Property
Curves”, NKK Corp.
[1]
V.
M.Sc. Florian Bachheibl graduated with a Master of Science in
Mathematical Engineering from the University of Federal Defense Munich
in 2011. He is a research assistant at the Institute of Electrical Drives and
Actuators at the University of Federal Defence in Munich, led by Prof. Dr.Ing. D. Gerling.
Prof. Dr.-Ing. Dieter Gerling received his diploma and Ph.D. degrees in
Electrical Engineering from the Technical University of Aachen, Germany
in 1986 and 1992, respectively. From 1986 to 1999 he was with Philips
Research Laboratories in Aachen, Germany as a Research Scientist and later
as Senior Scientist. In 1999 Dr. Gerling joined Robert Bosch GmbH in
Bühl, Germany as Director. Since 2001, he is full Professor and Head of the
Institute of Electrical Drives at the University of Federal Defense Munich,
Germany.
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