The analysis and comparison of leakage inductance in different winding arrangements for planar transformer

Abstract

The coupling of the windings can be easily increased by using multiply stacked planar windings connection. Interleaving is a well-known technique used to reduce leakage inductance and minimize high-frequency winding losses. The paper aims to analyze leakage inductance based on magneto motive force (MMF) and energy distribution in planar transformer and correct the formula of leakage inductance proposed by previous publications. The investigation of different winding arrangements shows significant advantages of interleaving structure. In this work, a novel half turn structure is proposed to reduce leakage inductance further. Some important issues are presented to acquire desired leakage inductance. The design and modeling of 1 kW planar transformer is presented. In order to verify the analytical method for leakage inductance in this paper, finite element analysis (FEA) and measurement with impedance analyzer are presented. Good matching between calculation, FEA 2D simulation and measurement results is achieved.

Full text

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The Analysis and Comparison of Leakage Inductance in Different Winding
Arrangements for Planar Transformer

Ziwei Ouyang, Ole C. Thomsen, Michael A. E. Andersen
Department of Electrical Engineering,
Technical University of Denmark
Kgs. Lyngby, 2800, Denmark
zo@elektro.dtu.dk

Abstract -- The coupling of the windings can be easily
increased by using multiply stacked planar windings connection.
Interleaving is a well-known technique used to reduce leakage
inductance and minimize high-frequency winding losses. The
paper aims to analyze leakage inductance based on magneto
motive force (MMF) and energy distribution in planar
transformer and correct the formula of leakage inductance
proposed by previous publications. The investigation of different
winding arrangements shows significant advantages of
interleaving structure. In this work, a novel half turn structure
is proposed to reduce leakage inductance further. Some
important issues are presented to acquire desired leakage
inductance. The design and modeling of 1 kW planar
transformer is presented. In order to verify the analytical
method for leakage inductance in this paper, finite element
analysis (FEA) and measurement with impedance analyzer are
presented. Good matching between calculation, FEA 2D
simulation and measurement results is achieved.

Index Terms-- leakage inductance, magneto motive force
(MMF), finite element analysis (FEA), interleaving, half turn,
planar transformer
I. INTRODUCTION
In recent years, planar transformers have become
increasingly popular in high frequency power converter
design because of the advantages they achieved in terms of
increased reliability, reproducibility, and increased power
density. In terms of circuit performance one of the advantages
of planar transformer is low profile and repeatable leakage
inductance [1]. The leakage inductance causes the main
switch current at the device input to vary at a low slope
between zero and rated value and reduces the rate of
commutation between output diodes. In addition, the stored
energy in the leakage inductance leads to the generation of
voltage spikes on the main switch which, besides creating
EMI problems, increases the switching losses and lowers the
efficiency [2]. Therefore, most designers expect the leakage
inductance to be as small as possible. However, in some
applications such as a phase-shift-modulated soft switching
DC/DC converter, the magnitude of the leakage inductance
determines the achievable load range under zero-voltage
operation, and a relatively large leakage inductance is
desirable. This paper aims to calculate the leakage inductance
stored in planar transformer by analyzing magneto motive
force (MMF) and energy distribution. Section ІІ defines
leakage inductance using the perspective of energy. The
energy associated with leakage inductance should be equal to
the sum of energy stored in each element layer inside the core
window. The section also analyzes the magnetic field
strength in each layer and finite element analysis 2D model is
simulated to demonstrate the correctness of the analytical
method. As presented in previous publications [3-5], the
formula (see eqn.6) is generally used to calculate the leakage
inductance. However, it must be noted that the formula
doesn’t provide precise results. It assumes that the magnetic
field strength along the height of insulator layer between non-
interleaved sections varies linearly but actually it should keep
constant during the whole area of insulator layer. In order to
correct the previous formula, a new formula suited for
symmetrical winding arrangement is proposed in this paper.
The error analysis on the two calculations is also presented.
Section ІІІ proposes a novel half turn structure to reduce
leakage inductance further. The MMF distribution curve for
half turn arrangement is analyzed and leakage inductance is
computed. Section ІV describes some important issues to
require desired leakage inductance including copper
thickness, the thickness of insulator layer and the number of
turns. Section V evaluates the good matching between
calculation, FEA 2D simulation and measurement with
impedance analyzer (PSM1735+ Impedance Analysis
Interface and Kelvin Fixture) which indicates the correctness
of the analytical method and the proposed calculation.
Section V provides the conclusion.
II. BASIC DEFINITION AND CALCULATION FOR LEAKAGE
INDUCTANCE
A. Basic Definition of Leakage Inductance
Not all the magnetic flux generated by AC current
excitation on the primary side follows the magnetic circuit
and link with the other windings. The flux linkage between
two windings or parts of the same winding is never complete.
Some flux leaks from the core and returns to the air, winding
layers and insulator layers, thus these flux causes imperfect
coupling. If the secondary is short-circuited, the main flux
which links both windings will be negligible because the
primary and secondary ampere turns almost cancel. So the
leakage flux parts don’t lose their individual identities. It is
seen from Fig.1 that within the winding area the mutual
repulsion causes the leakage flux to lie approximately parallel
to the winding interface. The leakage inductance referred to
the primary can be accessed by the energy stored in a
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magnetic field,
2
p
2
1
2
1
I
lk
L
total
dVHB
energy
E
⋅⋅=
∫
⋅⋅=
(1)

11IN
×
٠
P
S
22IN
w
b
x
d
symmetry axis
MMF Distribution

Fig. 1 The leakage flux paths and magneto motive force variation (MMF)
B. Leakage Inductance Calculation
For simplification to analyze MMF created by the
windings, turns ratio 1:1 and total number of turns 8 are used
as example. The MMF varies linearly in winding layers (see
Fig.2) can be assumed when operation frequency is not very
high. When the frequency is increased, MMF distribution will
concentrate on the surface of rectangle conductor because of
eddy current effect. In practice, as frequency grows, the
leakage inductances slightly decrease. Indeed, relative
variation of leakage inductance as the frequency changes is
quite small [6]. The leakage inductance for non-interleaving
structure can be calculated as follows:

Fig.2 Analytical scheme of MMF distribution for non-interleaving structure

The differential volume of each turn is
the total energy is sum of the energy stored in each
elementary layer which can be expressed by
∑∫
0
2
where
w
turn, h represents the thickness of each winding layer. Fig.1
shows the thickness
x
inner surface of the secondary winding. The field strength
along the flux path which includes this layer depends on the
number of ampere turns linked by the path. Since the flux
disperses rapidly on leaving the winding, the associated
energy is much reduced and the reluctance of the path within
the magnetic core can be ignored compared with that of the
path in the winding, therefore the flux path can be expressed
dx
w
b
w
l
⋅⋅
, therefore
⋅⋅⋅=
h
wwenergy
dxblHE
2
0
μ
(2)
l is the length of each turn,
w
b is the width of each
d , situated at a distance x from the
by the width
be taken as the field strength in the winding layer which is
assumed to be constant along the plane of layer, thus, for first
primary layer,
x
b
w
according to the eqn.2, the energy in the total winding space
can be deduced then
⎡
∫
+⋅⋅⋅
w
b rather than the full closed flux path. H may
1
1
h
I
H
⋅=
(3)
2
p
44
3
)
21
(46
2
0
2
)
1
4
b
(
)2
21
(
2
)
1
3
b
()2
21
(
2
)
1
2
b
(
1
0
2
0
)2
21
(
2
)
1
w
(
2
)
2
1
w
(4
2
)
1
1
w
(4
2
0
Ih
hh
w
b
w
l
h
w
I
hhh
w
I
hhh
w
I
hh
hhh
b
I
dx
h
x
b
I
dx
h
x
b
I
w
b
w
l
energy
E
⋅ ⎥
⎦
⎤
⎢
⎣
⎡
Δ
+
+
⋅=
⎥
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
Δ
⋅+
Δ
++⋅+
Δ
++⋅+
∫
Δ
++⋅+⋅⋅⋅
⋅⋅=
μ
μ
(4)
where
2
respectively,
Δ
and Fig.5 show energy distribution (a) and magnetic field
strength distribution (b) in non-interleaving structure, P-P-S-
S-P-P-S-S structure and
respectively. Fig.4 and Fig.5 represent partial interleaving
and complete interleaving
respectively. It is obvious to see that interleaving structure
provides significant advantage
inductance. The analytical MMF distribution (c) can be
verified by magnetic field strength distribution based on FEA
simulation results (b). The good matching between (b) and (c)
illuminates the correctness of analytical method. Based on the
above calculation, a new general formula suited for
symmetrical winding arrangement (symmetrical MMF
distribution) is proposed which can be expressed by,
⎡
+
Δ
3
w
b
w
b
(5)
where
MM
N ,
2
respectively, M is the number of section interfaces.
1 h and
h are the thickness of primary and secondary
h is the height of insulator layers. Fig.3, Fig.4
P-S-P-S-P-S-P-S structure
winding arrangements
in reducing leakage
⎥
⎦
⎥
⎤
⎢
⎣
⎢
∑
2
−
=
Δ
+⋅+
∑
1
−
=
Δ
+⋅⋅+
⋅+⋅
⋅=
1
2
0
)
2
(
2
)
2
w
(
1
1
0
)
1
(
2
)
1
w
(
2
)
1
(
2211
0
K
K
hh
b
K
K
K
hh
b
K
h
K
M
hKhK
w
l
leakage
L
μ
N
K
N
K
2
2
;
1
1
==
,
1
N are the number of turns on the primary and secondary
C. The error analysis
The formula (see eqn.6) was published in the previous
reference [3-5] to compute the leakage inductance for the
symmetrical interleaving structure, which has been generally
used to compute leakage inductance for most of designers.
2
0
⎥⎦
⎤
⎢⎣
⎡
∑Δ
+
∑
⋅⋅=
x
x
M
N
w
b
w
l
leakage
L
3
2
μ
(6)
where N is the number of turns on the winding which the
leakage inductance is to be referred; M is the number of
section interfaces; ∑x is the sum of all section dimensions
perpendicular to the section interfaces and ∑Δ
all inter-section layer thickness. It must be noted that the P-P-
x is the sum of
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S-S-S-S-P-P and P-S-S-P-P-S-S-P structures mentioned in
publication [3] have same leakage energy with the P-P-S-S-P-
P-S-S and P-S-P-S-P-S-P-S structures respectively.
However, it assumes that the magnetic field strength along
the height of insulator layer between non-interleaved sections
varies linearly which is shown by the blue line in the Fig.6.
Actually there is no extra flux path link with the insulator
layer, the MMF curve should, therefore, keep constant in the
area of insulator layer (see black line in Fig.6). The
correctness of the latter analytical MMF distribution can be
proved by FEA 2D simulation.

Fig.6 Comparison of MMF distribution in two different analytical methods

As an example, a planar transformer has been built with EI
64/5/50 core, the length of each turn is 202mm, copper width
is 20mm and the thickness of primary and secondary are both

Fig.3 Non-interleaving structure (a) Energy distribution in FEA 2D simulation (b) Magnetic field strength distribution in FEA simulation (c)
Analytical MMF distribution

Fig.4 P-P-S-S-P-P-S-S structure (a) Energy distribution in FEA 2D simulation (b) Magnetic field strength distribution in FEA simulation (c)
Analytical MMF distribution

Fig.5 P-S-P-S-P-S-P-S structure (a) Energy distribution in FEA 2D simulation (b) Magnetic field strength distribution in FEA simulation (c)
Analytical MMF distribution
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0.2mm, the thickness of insulator layer is 0.3mm. The
winding arrangements P-P-P-P-S-S-S-S, P-P-S-S-P-P-S-S
and P-S-P-S-P-S-P-S will be seen as case 1, case 2 and case 3
respectively. Table I describes the error between the proposed
eqn.5 and the previous eqn.6.

TABLE I . THE ERROR ANALYSIS TABLE

Calculation (eqn.5)
Previous
Calculation (eqn.6)
291 nH
82.9 nH
25.8 nH
The error
Case 1
Case 2
Case 3
245 nH
62.6 nH
22 nH
19%
32%
17%
III. NOVEL HALF TURN STRUCTURE
The interleaving, partial interleaving and non-interleaving
structures cause a significant difference in leakage inductance
because of MMF distribution. From the MMF distribution
curve, maximal magnetizing force in each layer determines
the value of leakage inductance. Therefore half turn structure
could be proposed to optimize leakage inductance further.
One solution is to physically form half turn in top layer and
bottom layer respectively. The other solution is to parallel the
top layer with bottom layer so as to sustain half current to
flow, the other layers are still in series, only one turn in each
layer.
As can be seen from Fig.7, the MMF distribution has been
shifted to be a symmetrical curve on the X axis. The maximal
magnetizing force is reduced to half of primary current.
Taken together eqn.1-3, the energy in the total winding space
can be found as follows,
I
h
⎢
⋅⋅=
2
p
2
48
2
4
1
11
⎢
⎣
2
0
2
)
2
1
b
(8
00
)
2
(6
2
)
2
2
1
b
(8
2
)
1
2
1
b
(2
2
0
12
1
0
2
1
1
b
2 /
2 /
Ih
hh
w
b
w
l
h
w
⎤
I
⋅
dx
h
x
w
I
⋅
dx
h
x
w
I
⋅
dx
h
x
w
w
b
w
l
energy
E
h
h
⋅
⎥
⎦
⎥
⎢
⎡
Δ
⋅+
⋅+⋅
⋅=
⎥
⎦
⎥
⎥
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎢
⎡
Δ
⋅⋅+
∫∫
⋅⋅⋅+⋅⋅⋅+⋅⋅
⋅
⋅
∫
μ
μ
(7)

Fig.7 Analytical scheme of MMF distribution for half turn structure
Obviously, the energy enclosed in the winding space gets
a significant deduction, the leakage inductance therefore can
be computed by
nH h
hh
w
b
w
l
leakage
L4 . 8 2
48
411
0
21
=
⎥⎦
⎤
⎢⎣
⎡
+
+
⋅⋅=
Δ
μ

This structure not only reduces the leakage inductance, but
also benefits the winding loss caused by skin effect and
proximity effect. Referring to Dowell equation [7], the
quantity m represents the ratio of the MMF
ampere-turnsNI. The value of m can directly affect proximity
loss of winding [7-10]. Interleaving windings can
significantly reduce the proximity loss when the primary and
secondary currents are in phase. Regarding the interleaving
structure, the value of m is equal to 1 for each layer. Further,
the value of m also can form to 0.5 by the half turn structure
which will decrease proximity loss a lot. Of course, the
maximal magnetizing force can be reduced further by
paralleling more layers. The MMF curve can almost be
distributed into a line which overlaps with X axis if there are
sufficient layers to be parallel. However, I have to mention
that it doesn’t make sense because of sacrificing winding
space.
)(hF
to the layer

Fig.8 Magnetic field strength distribution and flux vector for half turn
structure
IV. IMPORTANT ISSUES FOR LEAKAGE INDUCTANCE
A. The thickness of copper foil
As we can see from eqn.5, leakage inductance can be
influenced by the thickness of copper foil. It should be as
small as possible if leakage inductance is to be reduced. Fig.
9 shows that the thicker the copper, the higher the leakage
inductance will be achieved. However, the winding loss
might be sacrificed if the thin copper foil is used. It is
necessary to note that the ratio of ac-resistance and dc-
resistance will be reduced because of the lower skin effect,
although dc-resistance is increased. Therefore, there is an
optimal value on the thickness of copper foil which can
balance leakage inductance and winding loss.
B. The thickness of insulator layer
Leakage inductance can be influenced further by the
thickness of insulator layer which also can be observed from
the equ.5. From Fig.9, the leakage inductance decreases when
the thickness of insulator layer is reduced. Considering
capacitor effect between intra-windings and insulator
strength, the thickness of insulator won’t be too low. The
designer should find a balance between leakage inductance
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and self-capacitor. Reducing the insulation layer thickness
below a certain level will result in a considerable increase of
the total losses [11].
C. The number of turns
Comparing the charts (a), (b) and (c) in Fig.9, it can be
seen that the number of turns provide a significant difference
in leakage inductance. The more number of turns, the higher
the leakage inductance will be. However, if the number of
turns is increased, winding loss will be increased which is not
desirable. In reverse, core loss will be reduced because of the
variation of flux density is decreased. Therefore trade-off
becomes an essential design property.
D. The others
As known from eqn.5, the permeability of copper foil and
insulator, the length and width of conductor are also related
to the leakage inductance. The relative permeability can be
controlled by different materials. Therefore, a leakage layer
which consists of ferrite film could be used to realize higher
leakage inductance without sacrificing winding loss. This
leakage layer can be used in half –bridge resonant converter
and many phase-shift applications to realize ZVS.
V. EXPERIMENTAL VERIFICATION
The design and modeling of 1 kW planar transformer is
built (see Fig.10). The fold technique on the planar copper
winding is adopted to avoid some undesirable problem
caused by the terminal connection. Furthermore, the different
winding arrangements are quite flexible to be realized if the
fold technique is used.
In order to verify the analytical method for leakage
inductance in this paper, the results based on measurement
with impedance analyzer (PSM1735+ Impedance Analysis
Interface and Kelvin Fixture) is presented. Fig.11 shows good
matching between calculation, FEA 2D simulation and
measurement results is achieved. The proposed novel half
turn structure has been seen as case 4. Obviously half turn
arrangement has best result in leakage inductance. There is no
doubt that small error exists between measurement and
calculation because of complex magnetic flux in actual
model. The tolerance of insulator thickness and short-loop in
secondary side also might cause a slight error between
measurement and calculation. In addition, extra connection
also leads inaccuracy result.

Fig. 10 The prototype of 1 kW planar transformer using in DC-DC converter


Fig.11 Comparison between calculation, FEA simulation and measurement
VI. CONCLUSION
The purpose for this paper is to find a solution to acquire a
desired leakage inductance. An analytical computation of
leakage inductance has been introduced. Several different
winding arrangements have been investigated. Computed
results are in good agreement with those obtained by FEA 2D
simulation. The interleaving structure provides significant
advantage in reducing leakage inductance. The previous
formula has been corrected. In order to optimize leakage
inductance and winding loss further, a novel half turn was
proposed in this paper. Computed results shows a half turn
structure benefit low leakage inductance extremely. Some
important issues including copper thickness, insulator
thickness and number of turns were concluded to guide
designer to obtain desired value. The analytical method has
been experimentally validated based on a planar core
transformer. Good matching is achieved between calculation,
FEA simulation and measurement.

Fig.9 FEA simulation results for interleaving structure with different issues
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