Electr Eng (2013) 95:33–42
DOI 10.1007/s00202-012-0237-7
ORIGINAL PAPER
Core lamination selection for distribution transformers
based on sensitivity analysis
Juan C. Olivares-Galvan · Pavlos S. Georgilakis ·
Eduardo Campero-Littlewood · Rafael Escarela-Perez
Received: 27 April 2010 / Accepted: 11 February 2012 / Published online: 29 February 2012
© Springer-Verlag 2012
Abstract In this paper, the sensitivity analysis is used to
select the core lamination thickness of single-phase distri-
bution transformers rated from 5 to 50 kVA. Three different
magnetic materials (M2, M3 and M4) with thicknesses of
0.18, 0.23 and 0.27 mm are considered. Transformer designs
are compared based on the total owning cost as well as on
the transformer bid price. The impact of the different lami-
nations on total owning cost and bid price is calculated for
a total of 144 transformers (72 for each criterion). All trans-
formers fulfill all the operating and construction constraints.
The paper considers the impact on core losses of the space
factor (core-assembling pressure) and of the building factor
and also describes howcorelossesareaffected by core design
parameters such as the number of laminations per step, air
gap and overlap. It is concluded that for the analyzed power
range, M3 lamination is the best choice since all of the stud-
ied cases have smaller bid price and 79% of the studied cases
have lower total owning cost. This paper gives guidelines
to select the appropriate thickness and can help transformer
manufacturers to select the optimal thickness for distribution
transformers.
Keywords Costing · Design engineering · Magnetic cores ·
Magnetic materials · Losses · Transformer · Transformer
cores
J. C. Olivares-Galvan (
B
) · E. Campero-Littlewood ·
R. Escarela-Perez
Universidad Autonoma Metropolitana-Azcapotzalco,
Mexico, Mexico
e-mail: jolivare_1999@yahoo.com
P. S. Georgilakis
National Technical University of Athens (NTUA), Athens, Greece
e-mail: pgeorg@power.ece.ntua.gr
1 Introduction
Due to the importance of improved electrical core perfor-
mance, researchers are very active in the development of
better transformer cores and core modeling techniques [1–7].
Better manufacturing techniques have been developed as a
consequence of a better understanding of the factors that
influence magnetic properties. Nowadays,thequalityofelec-
trical steel has been substantially improved. Factors that
impact core loss of electrical steel are reported in [8,9]: (a)
quality of sheet insulation, (b) percentage of silicon in the
alloy, (c) chemical impurities, (d) grain size, (e) crystal ori-
entation control and (f) core lamination thickness.
A useful model in literature is presented in [1]; the model
covers steady-state unbalanced conditions of three-phase
transformers including three-legged, five-legged and triplex
core designs. In this model, Córcoles et al. [1] used phase
and sequence nodal equations for all winding connections.
Guerra et al. [2] used a nonlinear electric circuit to describe
the behavior of magnetic cores in low-frequency conditions.
In this electric circuit, the hysteresis modeling takes into
account minor loops and remanent magnetic flux. Classic
eddy current losses and anomalous losses are represented by
a linear resistor and a nonlinear resistor, respectively.
Zirka et al. [3] compared the two-component method
where total loss is subdivided into hysteresis (static) and eddy
current (dynamic) components with the three-component
method where the total loss is subdivided into hysteresis,
classical and excess components. Authors of [3] found that
total losses obtained with the three-component method are
more accurate than that calculated using the two-component
method. In [4], the authors modeled the dynamic loops and
losses in grain-oriented electrical steels under arbitrary mag-
netization regimes using the concept of magnetic viscosity;
they found that the steel could be modeled, for frequencies
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34 Electr Eng (2013) 95:33–42
up to 200 Hz, using the thin sheet model (the instantaneous
value of the applied field is subdivided into hysteresis field,
classical eddy current field and excess field) and at higher
frequencies using a finite-difference solver.
Schultz et al. [5] built and tested oil-cooled, amorphous-
core (Metglas TCA) distribution transformers prototypes;
they installed four 100 kVA and one 50 kVA transformers
on the Hydro-Québec power system and tested them over
a period of 1 year. They recommend using amorphous-core
transformers in areas where the cost of the no-load losses
exceeds USA $5.20/W. In [6] Thompson describes the main
advancements in grain-oriented silicon iron for transform-
ers (reduction of magnetostriction, improvements of mag-
netic properties, reduction of core loss) and suggests possible
lines of research (basic properties improvements, variation
of thickness within a lamination, domain behavior). In [7]
Kefalas et al. propose an iron loss minimization of wound
core transformers using a combination of different grade
steels; they used permeability tensor finite element model
and simulated annealing.
Overall, improvements in core materials and manufactur-
ing processes have a significant impact on the total cost of the
transformer. When the performance of the magnetic material
improves, the size of the core can be reduced. Use of better
material with improved core joints allows higher operating
flux density. Nevertheless, theproductioncost per unit weight
of electrical steels increases rapidly as lamination thickness
is reduced. While the thinnest materials may be necessary
for certain applications, the use of laminations thinner than
necessary is wasteful.
Transformer prices are normally compared in terms of the
total owning cost (TOC) and the purchasing price usually
called bid price. TOC of a transformer is the sum of the pur-
chasing price plus the cost of transformer losses throughout
the transformer life [10]. Electric utilities usually purchase
transformers based on the TOC, i.e., they select the offer that
minimizes TOC. On the other hand, industrial users usually
purchase transformers based on transformer bid price, i.e.,
they select the offer with minimum purchasing price. Conse-
quently, transformer manufacturers have to minimize either
the TOC or the bid price depending on customer and trans-
former specifications.
This paper gives clear guidelines to select the appropri-
ate thickness for core lamination in distribution transform-
ers based on the minimization of either TOC or bid price.
Three different grades of magnetic materials are considered:
M2, M3 and M4. The importance of this research is based
on the fact that the cost of magnetic steel in single-phase
shell-type distribution transformers (Fig. 1) represents 27–
38% of the total cost of materials as shown in Table 1 [11].
This table was obtained taking into account shell-type trans-
formers designed with M3 lamination. Figure 2 shows the
core manufacturing parameters and the values used in the
Core
L
V
H
V
L
V
H2
H1
Fig. 1 Single-phase shell type transformer (LV low voltage, HV high
voltage, H1 and H2 high voltage terminals)
Table 1 Percent of material cost in the manufacturing of distribution
transformers
Transformer material Cost (%)
Magnetic steel 32.5± 5.5
Windings (copper or aluminium) 22± 6
Insulation 14.1± 5.5
Carbon steel 16.4± 8.5
Fabricated parts 15± 9
Total 100
Fig. 2 Core with step-lap joint (g = air gap, d =lamination thickness.
T
i
= insulation thickness, L
o
= overlap, n
s
= number of laminations
per step), where 1 mm < g < 2mm, L
o
= 1cm, T
i
/2 = 0.0001 cm
per surface
wound-core distribution-transformer ratings of this research.
Section 2 includes the model used in the analysis and gives
an insight on how core loss components are affected by lam-
ination thickness. Section 3 describes in detail the manufac-
turing factors that can produce a core loss increment and
how they are included in the analysis. Section 4 describes
the software program and objective functions used to obtain
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Electr Eng (2013) 95:33–42 35
the transformer designs with the three different magnetic
materials and includes the obtained results. Appendix A
describes how core losses are affected by core design param-
eters such as the number of laminations per step, air gap and
overlap.
The lamination thickness for 60 Hz transformers is usu-
ally in the range of 0.17–0.27 mm, depending on the relative
importance of core losses in the total losses of the trans-
former and on price criteria. There is no general agreement
among transformer manufacturers about the optimal thick-
ness of laminations.
2 Core lamination modelling
The term “electrical steels” has been universally accepted as
the designation for flat rolled magnetic materials in which
silicon is an important alloying element [12]. The American
Iron and Steel Institute (AISI) designation for electrical steel
grades consists of the letter M (magnetic material) followed
by a number (e.g., M2) to specify the type of lamination.
At the time the AISI system was adopted, the type num-
ber assigned to each grade was approximately ten times the
core loss expressed in watts per pound for a given thickness.
Today, type numbers do not have this specific association
with core loss. The thicknesses of electrical steels studied in
this paper are presented in Table 2.
In the analysis performed in this paper, the model used to
calculate core losses for the three magnetic materials (M2,
M3 and M4) is that of [13], where the core loss is in watts
per kilogram for the considered laminations at 60 Hz as a
function of the peak magnetic flux density B
p
(T):
w
M2
kg
=−21.11312203 + 8.583546123 · B
p
+ 1.390035903 · B
2
p
+0.113207533 · B
3
p
− 0.004609366 · B
4
p
+ 7.54374 · 10
−5
· B
5
p
(1)
w
M3
kg
=−45.94322511 + 17.94316167 · B
p
− 2.787213965 · B
2
p
+0.21646225 · B
3
p
− 0.008382569 · B
4
p
+ 0.000129908 · B
5
p
(2)
w
M4
kg
=−0.08058632 + 0.07744565 · B
p
− 0.01948912 · B
2
p
+0.00350717 · B
3
p
− 0.0002352 · B
4
p
+ 5.9045 · 10
−6
· B
5
p
(3)
Graphs for equations (1)–(3)areshowninFig.3, where
it can be observed that M2 and M3 materials have a very
similar behavior and cores manufactured with thicker mate-
rials have less loss per unit weight, although transform-
Table 2 Electrical steel thicknesses
Grade AISI designation (ASTM
designation)
Thickness in inches (mm)
M2 (Type 18G041) 0.007 (0.18)
M3 (Types 23G045 and 23H070) 0.009 (0.23)
M4 (Types 27G051 and 27H074) 0.011 (0.27)
Note: M2 is approximately equivalent to ASTM Type 18G041
Fig. 3 Core loss as a function of magnetic flux density [13]
Fig. 4 Number of core laminations versus transformer rating
ers built with thinner laminations need less core material.
The distribution transformers analyzed here use a differ-
ent number of laminations depending on the selection of
the magnetic material. The number of laminations needed
to form the transformer cores (Fig. 1) for the ratings of
the analysis (5–50 kVA) are given in Fig. 4. The aver-
age increase in the number of laminations is 19.1% when
using M2 instead of M3 and is 22.0% when using M3
instead of M4. Thus, the impact of lamination thickness in
losses and costs can only be evaluated in terms of the core
weight and hence on the design of the transformer. Inci-
dentally, it is more difficult and time consuming to han-
dle and process thinner laminations. Equations (1)–(3)are
included in the computer program used to design the trans-
formers.
Although the model used for the sensitivity analysis is the
one described in (1)–(3), it is interesting to use other mod-
els that represent core loss components to briefly illustrate
the difficulty in finding the lamination thickness to have the
minimum core losses. The core loss is summarized using the
conventional technique [14]:
P = P
cl
+ P
h
+ P
ex
(4)
where P
cl
are the classical eddy current losses, P
h
are the
hysteresis losses and P
ex
are the excess losses.
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36 Electr Eng (2013) 95:33–42
Classical eddy current losses per unit volume at power
frequency excitation can be expressed as [15]:
P
cl
=
t · π · B
p
· f
2
6 · ρ
(5)
where ρ is the electrical resistivity of the material, t is the
lamination thickness, B
p
is the peak sinusoidal magnetic flux
density and f (Hz) is the frequency. It is evident from (5) that
lamination thickness reduction means a squared reduction of
eddy current losses.
Hysteresis loss per unit volume at power frequencies is
[16]:
P
h
=
2 fSB
2
p
μ
(6)
where μ (H/m) is the permeability of the material, and S is
the shape factor.
An expression that describes excess loss in terms of clas-
sical core loss has been derived by Pry and Bean [17]:
P
ex
=
1.628
2L
t
− 1
P
cl
, when 2L/t >> 1(7a)
P
ex
<< P
cl
, when 2L/t << 1 (7b)
Equation (7) shows that a fundamental parameter to char-
acterize excess losses is the ratio 2L/t between the domain
width (2L) and the lamination thickness (t).
The excess eddy current loss is a direct consequence of the
domain structure of the material and arise from the currents
localized at the moving domain walls [18]. The excess loss
P
ex
can be minimized further by refining domain structure
or by metallurgically pinning domain walls.
At power frequencies, the difference between measured
and calculated losses, the so-called anomalous or excess loss,
may be not significant for non-oriented steel used in motors
and generators. The percentages of hysteresis, classical eddy
current and excess losses for 0.27 mm thick grain oriented
steel are 42, 21 and 37%, respectively [19]. Losses were mea-
sured for other materials in [20] and they are listed in Table 3.
The decrease in the lamination thickness leads to a qua-
dratic decrease of the classical eddy current loss as can be
seen in (5). There is experimental evidence that hystere-
sis loss increases as lamination thickness decreases below
0.20 mm [21]. Excess loss is impacted by the lamination
thickness as can be seen in (7). It is important to mention
that the domain width (2L) increases as lamination thickness
Fig. 5 Lamination factor versus test pressure for the most widely used
forms of grain-oriented silicon steel produced by AK Steel Corporation
decreases [21]. Lamination thickness has a different impact
on core losses components, thus the total core loss as a func-
tion of thickness has a minimum [21,22]. The choice of lami-
nation thickness is a compromise between loss reduction and
transformer cost.
3 Space factor and building factor
Space factor or lamination factor is the measure of com-
pactness of an electrical steel core. This is also referred to
as stacking factor. Space factor is the ratio of the equivalent
“solid” volume, calculated from the weight and density of the
steel, to the actual volume of the compressed pack. Figure 5
illustrates howthe spacefactor variesas a function of pressure
for the laminations compared in this paper. Pressure com-
pressing the sheets should not exceed a maximum limit of
1.0 MPa to avoid excessive reduction of resistivity of the
lamination sheet insulation [15].
Bandages as the ones shown in Fig. 6 are used in distri-
bution transformer cores for a uniform pressure distribution.
If a typical core assembling pressure of 20 psi (0.14 MPa),
as the one obtained with manual strapping machines, is con-
sidered, the space factors for M2, M3 and M4 result in 96.8,
Table 3 Losses (W/kg)
measured with the
three-component method for
various materials
Copyright © 1988 IEEE [20],
reprinted with permission
Material Thickness (10
−4
m) P
h
P
cl
P
ex
Grain-oriented 3% SiFe longitudinally cut 2.9 0.34 0.29 0.53
Grain-oriented 3% SiFe transversally cut 3.4 1.7 0.31 1.8
Nonoriented 3% SiFe 3.5 1.8 0.29 0.33
Amorphous METGLAS 2605 CO 0.3 0.29 8.5 × 10
−4
0.094
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Electr Eng (2013) 95:33–42 37
Fig. 6 Use of bandages in transformer cores for a uniform distribution
of pressure
97.6 and 97.8%, respectively (Fig. 5)[12]. When core-assem-
bling pressure is high, the effective superficial resistance is
reduced [23]. Normally, cores are assembled with manual
strapping machines that represent a pressure of 0.14 MPa.
To increase the core-assembling pressure, pneumatic strap-
ping machines are used, as a result a 0.63 MPa pressure is
obtained [24]. Two cores of 15 kVA single-phase transform-
ers were used for experimentation, one core assembled with
manual straps and the other one with pneumatic straps. Mea-
surements were performed and an increment of 4% in core
loss was observed when coil core was assembled with pneu-
matic machines. The difference in core losses can be seen in
Table 4.
There are other factors that are considered as possible
causes of increase in core losses: (a) improper handling of the
core steel during transformer manufacturing; (b) poor insu-
lation coating within lamination layers (Fig. 7); (c) improper
arrangements of core joints; (d) burrs forming at slit edges or
at the cut joints (if burrs are present in the lamination, inter-
lamination short circuits can occur); (e) incomplete stress
relief annealing. The additional losses due to all these fac-
tors were taken into account by including a building factor.
The building factor is the ratio of the test measured core loss
per weight (W/kg) for a fully assembled core, to the speci-
fied manufacturer loss (W/kg) for the considered magnetic
material.
Table 4 Core loss measurements of 15 kVA transformers with different
core assembling pressure
Sample Core loss (W) for 0.63 MPa Core loss (W) for 0.14 MPa
assembling pressure assembling pressure
1 44.5 43
2 45.5 44
Fig. 7 The insulating coating of the laminations can be damaged when
there is improper handling of the core steel during steel manufacturing
or core assembling. The canals 1–1
and 2–2
(highlighted in white
on the edges) were done intentionally to show the appearance of the
lamination when insulation is damaged
In all designs of all the ratings of the single-phase trans-
formers considered in this research, the space factors for the
three lamination thicknesses are 96.8% for M2, 97.6% for
M3 and 97.8% for M4, and a building factor of 1.06.
4 Results and discussion
The software program for the optimal design of single-phase
shell-type distribution transformers uses equations (1)–(3)
in Sect. 2 and includes the space factor and building factor
given in Sect. 3 in order to obtain the dimensions of the core,
core weight and the no-load losses, based on the algorithm
described in Table 5. More details on the optimization meth-
odology and the software can be found in [25]. This com-
puter program was validated with the design, construction
and laboratory tests of a 25 kVA transformer. The intention
is to obtain the design of distribution transformers from 5
to 50 kVA considering the three different magnetic materi-
als and to optimize the design using two different objective
functions [26]:
(1) Minimizing the transformer bid price (usually the objec-
tive when transformers are for industrial and commer-
cial users).
(2) Minimizing TOC (usually the objective for transform-
ers that are purchased by electric utilities).
The transformer bid price, BP ($), is computed as follows
[26]:
BP =
MC + LC
1 − SM
or BP =
TMC
1 − SM
(8)
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