Finite Element Analysis of a Contactless
Transformer
Jianyu Lan, Houjun Tang, and Xin Gen Key Laboratory of Control of Power Transmission and Transformation, Ministry of Education, Shanghai Jiao Tong
University, Shanghai, China
Email: [email protected]
Abstract—Inductively coupling power transfer is an
emerging technique, which enables power transfer to loads
through air. The contactless transformer is the key
component of it, and the design of a transformer is a time-
consuming work with a large number of tests. In this paper,
a design method of contactless transformer with finite
element analysis is presented. First the contactless
transformer model is deduced from Maxwell Equations, and
the self inductance and mutual inductance computational
equations are given as well. Then the magnetic field
distributions of contactless transformer with different air
gaps are presented by simulation of MAXWELL ANSOFT.
Furthermore, the skin and proximity effects are analyzed as
well. At last, the results are compared with the experimental
results with the same dimension and material. The analyses
show that there has a good agreement with each other. So by
this method, the design period of a contactless transformer
will be shorter than before.
Index Terms—Inductively Coupling Power Transfer,
Contactless Transformer, Finite Element Analysis
I. INTRODUCTION
Some applications require a power transfer without
direct contacts with source supplies. The wireless power
transfer (WPT) system allows transferring energy to
electronic appliances through an air gap [1]-[2]. As an emerging technique, the WPT has been a hot topic for
researchers. Currently, there are three types of WPT
techniques have been reported: electromagnetic radiation,
inductive coupling and magnetic resonance coupling [3].
Among these, the magnetic inductive coupling technique
can transfer higher power than others with higher
efficiency. The power transfer system using inductive
coupling technique is called Inductive coupled power transfer (ICPT) [4]. Because it has many advantages,
ICPT is often applied to transfer power in special
occasions. First, high voltage equipments applied ICPT
technique may increase their security level because there
is no need for user to handle the plugs and cables [1].
Second, in under-water, under-mine and corrosive
environments, the lifetime of this system will be longer
because of no exposition of coppers. Then, the ICPT systems allow removing brushes of mobile loads or
rotating systems, which often cause blasts in the case of
oil fields. In addition, in implanted medical applications,
it would be better to avoid any physical link between
inner body of the patient and the environment, as in [4].
So far efforts have been made to improve the efficiency
and enlarge transfer range of ICPT systems. Gyu
proposed an energy transmission system for an artificial
heart using leakage inductance compensation of transformer [5], and a contactless electrical energy
transmission system for portable telephone is discussed in
[1]. Besides, a design method of wireless power transfer
system based inductively coupled is presented by Shinya
[2]. Furthermore, other researchers focus on the
mathematic model and controller design about
bidirectional wireless power transfer system [6]. On the
other hand, Chen proposed a capacitive coupled contactless power transfer system [7], and an
optimization model of a wireless power transfer system
form medical implanted devices is presented in [8].
Nevertheless above mentions are mainly based on circuit
topologies, and not focused on the contactless power
transformer, which is a key element of an ICPT system.
In ICPT systems, the energy transfer through a
contactless transformer to loads by magnetic induction coupling. It is very important to have an accurate model
in order to design the transformer avoiding a large
number of tests on prototyping stages. Different methods
to model the transformer are proposed in the literature
[9]-[10]. But these models are based on analytical
equations, which do not have enough accuracy with
different separations between the primary and secondary
sides. Because those works did not take into account the frequency effects, such as skin and proximity effects. On
the other hand, models based on Finite Element Analysis
(FEA) are presented in [11]-[15], which have more
accurate results. A homogenization method and three
dimensional FEA have been used to estimate the losses of
a two-winding transformer with large air gap [16]-[17]. In
[18] Jesus presented an analysis of the mutual inductance
between two planar circular windings. Besides, a rectangular contactless transformer is proposed by use of
FEA [19]. In addition, Pascal employed FEA to analysis
the skin and proximity effects of a coreless transformer.
Based on which the losses of transformer are deduced
[20]-[22]. However, up to date, little work presents the
analysis of comparison of FEA simulation and
experimental results. In this paper, a transformer design
method by FEA is discussed, and the simulation results are compared with the experimental. The analyses show
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that using FEA method can acquire an accurate
transformer model, and thus the design cost and time are
reduced.
This paper is organized as follows. In section II the
operation principle of ICPT system is reviewed. In
section III, the model of contactless transformer is
deduced by Maxwell Equations. In section IV, a design method of contactless transformer by FEA is presented.
First, the EE type core contactless transformer is
compared to the CC type core. Then, the skin and
proximity effects with different frequency are analyzed.
Furthermore, the experimental results with a contactless
transformer are compared to the results form FEA by
MAXWELL ANSOFT. At the end, the conclusions are
given in section V.
II. REVIEW OF ICPT SYSTEM
A typical ICPT system consists of the high frequency
plus-current generator, a resonant converter, a rectifier
and loads, shown in figure 1. The contactless power
transformer and the compensated net form a resonant
converter.
LoadResonant
Converter
Figure 1. Typical structure of ICPT System.
Because of a large air gap existed between the primary
coil and the secondary coil, the mutual coupling
inductance within ICPT systems is generally weak. So a resonant tank circuits should be added into both the
primary and secondary parts. Normally, the compensated
net includes the PP, PS, SP and SS or other high level
compensator [5]. Figure 2 shows the schematic of an
ICPT system studied in this paper, in which a half bridge
inverter is chose and the PP compensated net is applied.
sCM
R
pC1Q
2Q
dU
pL sL
Figure 2. Schematic of proposed half bridge ICPT system.
As seen from figure 2, a square wave voltage is
produced at the mid-node of the half bridge inverter by driving switches Q1 and Q2 alternately with 50% duty
cycle. The resonant tank consists of the contactless
transformer and capacitors connected in series both on
the primary coil and the secondary coil. In figure 3, Lp
and Ls denote the primary coil self-inductances and the
secondary coil self-inductances, respectively, M is the
mutual inductance between the primary coil and
secondary coil and R is the equivalent AC resistor of loads. By means of mutual inductance theory, the
equivalent circuit of this system is deduced and shown in
Figure 3.
MpCsC
sLpL
rZ cU
R
Figure 3. Mutual inductance equivalent circuit of the ICPT system.
In figure 3, Zr is the reflected impedance from the secondary side and Uc is the induced voltage of
secondary side, and Zr can be expressed by (1).
2 2 /r sZ M Z (1)
In which Zs is the lumped impedance of secondary side,
which is expressed in (2).
1/ ( )r s sZ j L j C R (2)
Substituting (2) into (1) the reflected resistance and
reactance from the secondary coil to the primary is,
respectively:
4 2
2 2 2 2 2Re
( 1)
s L
s s s L
C RZr
C L C R
(3)
And
3 2 2
2 2 2 2 2
( 1)Im
( 1)
s s s
s s s L
C M C LZr
C L C R
(4)
Then, the equivalent impedance looking from the input
side of half bridge inverter is:
1/ ( )eq p p p rZ j L j C r Z (5)
Therefore, the output power is deduced as equation (6)
[8]:
2
2 2
8Re r dc
out
eq
Z UP
Z (6)
And the output voltage is as following:
p L
out
s
j MI RU
Z
(7)
Define Mv as the gain of output voltage, which is
expressed as
out
v
in
UM
U (8)
Substituting (1) to (7) into(8) the voltage gain is:
(.)
(.)v
NM
M (9)
In which, N(.) and M(.) is expressed in (10) and (11).
2
2
2(.)
1 2 ( )(2 )
n
n
n
j f QknN
j f Q nf
(10)
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2 2 2 2
22
2
(.)
(2 )12 1
(2 )1 2 ( )
(2 )
n
n
nn
n
M
f k Q nj f Q
fj f Q n
f
(11)
where fn=f/fr, and /s pn L L . The couple coefficient is
defined as k, and is the capacitor rate defined Cp/Cs.
The quality factor Q is defined as:
2 r pf L
QR
(12)
According to (9), the frequency response of the system
can be illustrated in figure 4.
vM
(Hz)nf
0.9 10.8 1.1 1.2
2
4
3
0
1
1.3 1.4
=1Q
=4Q
=2Q
=10Q
Figure 4. Voltage gain curves at different Q.
Figure 4 shows the curves of voltage gain vs. operation
frequency at different load, which mean that at resonant frequency the voltage gain is at maximum point, and with
the increasing of load the gain curve become sharper,
which mean that the voltage gain is sensitive with the
change of load.
III. MODEL OF CONTACTLESS POWER TRANSFORMER
Figure 5 shows the equivalent circuit of a contactless
power transformer, in which Lp and Ls are the self
inductances of primary and secondary winding, respectively, while M is the mutual inductance; Rp and Rs
are the equivalent series resistance of the primary and
secondary winding , respectively, and Rm is the mutual
resistance between windings.
The energy stored in the transformer is associated with
magnetic flux and magnetic field created by windings,
which is show in (1) [23].
Re2
p s p sB B H HE dv
(13)
On the other hand, the energy can be indicated by self
inductances and mutual inductances which are expressed
in (8).
2 21 1
2 2p p s s m p sE L I L I L I I (14)
where Ip is the rms current in the primary winding, Is is
the rms current in the secondary winding. Applying the
superposition theorem to solve the equations (13) and
(14), the inductance values can be calculated by
2
1p p p
p
L B H dvI
(15)
2
1s s s
s
L B H dvI
(16)
1
2m p s s p
p s
L B H B H dvI I
(17)
And, then the coefficient of mutual inductance is given
by
p s
Mk
L L (18)
To calculate the resistance, the equation to calculate
the power losses is about the function of the in current density associated with transformer, and it is expressed as
follows.
Rep s p sJ J J J
P dv
(19)
where σ is the electric conductivity, the power losses in
the model are calculated by
2 2 2p p s s m p sP R I R I R I I (20)
Comparing (19) and (20), and then apply the
superposition theorem; the resistances are calculated by
2
1 p p
p
p
J JR dv
I
(21)
2
1 s s
s
s
J JR dv
I
(22)
1
2
p s s p
m
p s
J J J JR dv
I I
(23)
IV. FEA ANALYSIS OF CONTACTLESS POWER
TRANSFORMER
A. Analysis of EE Core and CC Core
In this section, the EE core and CC core of contactless
power transformer will be analyzed to employ FEA. By
MAXWELL ANSOFT, the magnetic field distribution of
EE core with the air gap of 2mm is shown in figure 6,
while figure 7 shows the magnetic field distribution of a
CC core with the same dimension and air gap.
Comparing figure 5 and figure 6, it is easy to found
that the magnetic field distribution of EE type core is well distributes, while CC type core magnetic lines
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distribution is asymmetrical. Also, it can be seen that the
magnetic flux density of EE core is more than that of CC
core.
pR sR
pLsL
mR
M
Figure 5. Model of contactless power transformer.
Figure 6. Flux distribution of EE type core.
Figure 7. Flux distribution of CC type core.
Figure 8. Flux distribution of EE type core.
In the following part, the excitation position of coil
current in the EE core will be discussed. Figure 8 and figure 9 are the magnetic line distribution of EE type core
with the excitation on different position. Comparing
figure 6, figure 8 and figure 9, it is clearly seen that, the
magnetic flux density in figure 6 is more than others. On
the other hand, from the computational results of
MAXWELL ANSOFT, the mutual inductance of figure 6
is largest. So, when the excitation coil is in the position
like figure 6, the coefficient of contactless power
transformer will be largest, thus the efficiency of ICPT
system is highest than in other positions.
Figure 9. Flux distribution of EE type core.
B. Analysis of Skin and Proximity Effects
High frequency currents (10 kHz – 5MHz) are injected
into the primary coil in order to generate a coupling
magnetic field, and a voltage is induced by varying
magnetic field on secondary coils. Both skin and
proximity effect are caused by the high frequency
currents. This section presents an analytical model of the
current distribution in a conductor of the cross-section as a function of the frequency. Based on this model, a
numerical integral resolution can be acquired by
computer. Thus, it is possible to determine the resistance
and internal inductance of the conductor.
This model of the current distribution for a single
conductor is generated from Maxwell equations. If one
assume that the conductor is infinite in length and that the
current flows through the z axis direction. Then, the cross-section is just situated in the x, y plane. The current
density J(x, y) is defined by the following integral
equation [24]-[27]:
0
2 2
0
( , ) ( , )2
ln ( ) ( ) ( , )
S
jJ x y J x y
x x y y dxdy J x y
(24)
Where ω is the frequency of the current, μ0 is the
permeability of free space, σ is the conductivity of the
conductive material, here σ = 58.82e6 S/m for copper, S is
the surface of the cross-section, Jω=0(x, y) =-σV is the
current density generated by the voltage supply and refers
to as the current density at low frequencies. Resolving this equation gives the current density distribution over
the cross section.
Knowing the distribution of the current over the cross
section, it is possible to calculate the per-unit-length
resistance and internal self inductance of the conductor
through energy relations as:
2
1
2
1
( , )
( , )
N
n n n
n
N
n n n
n
J x y S
r
J x y S
(25)
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2
1
2
0
1
1
( , )
N
n n
n
iN
n n n
n
B S
l
J x y S
(26)
By FEA method, the arrayed copper coils with the
outside excitation current of 10 ampere in the frequency
2MHz are investigated. The current density distributions
are shown in figure 10.
Figure 10. Skin and proximity effects of copper coils.
Observed form figure 10, it can be known that the
strongest current density are displayed at the edge of
cooper coils, which demonstrates the skin and proximity effects well.
01
2
3
0.5 1 1.5 2 2.5
Frequency (MHz)
P-u
-l r
esis
tance
(/
)O
hm
m
4
5
6
Figure 11. Computed p-u-l resistance of copper coil.
To achieve the features of coil-resistance to the
frequency of excitation current, a group of data is computed by the software MAXWELL ANSOFT. The
results are plotted in figure 11, which shows that with the
increasing of excitation currents frequency, the per unit
length resistor will increases. This agrees with the above
theoretic analysis.
C. Experimental Result
In this section a contactless transformer with EE type
magnetic core is presented by FEA. The distributions of
magnetic field are shown by ANSOFT MAXWELL, and
different gaps between coils are analysis. Also, the
coefficients with the change of gaps are computed by
ANSOFT. At last the results are compared with the experimental results, which show good agreement with
each other. Figure 12 is the core studied in this paper, and
the material used is ferrite with relative permeability of
1000. The dimension values are shown in table 1.
a
b
c
d
d
Figure 12. EE type ferrite core.
(a) Flux distribution
(b) B vector distribution
(c) B amplitude distribution
Figure 13. Magnetic field distribution of contactless transformer with
gap at 2 mm.
Figure 14 and figure 13 show the magnetic field distribution with the excitation current of 10 ampere at
primary side. The figure 13 (a) is the flux distribution, (b)
is the B vector distribution and (c) is the magnetic density
distribution.
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(a) Flux distribution
(b) B vector distribution
(c) B amplitude distribution
Figure 14. Magnetic field distribution of contactless transformer with
gap at 10 mm.
TABLE I. DIMENSION OF EE CORE
Parameters values
a 20 mm
b 8 mm
c 16 mm
d 11 mm
Primary winds 10 turns
Secondary winds 10 turns
The figure 13 shows the field distribution with the air
gap of 2mm, while figure14 shows the distribution with
the air gap of 10mm. Comparing with figure 13 and
figure 14, it is clearly shown that the field density at 2mm
is strong than 10mm. Furthermore, the mutual inductance and coefficient are
computed by ANSOFT with the gap from 2mm to 10 mm,
and the result is shown in table 2. At the same time, tests
on a prototype shown in figure 15 with the same core
dimension analyzed in simulation have been performed.
These tested results are compared with the simulation
results in table 2.
Figure 15. Prototype of contactless power transformer.
TABLE II. SIMULATION AND EXPERIMENTAL RESULTS
D (mm)
Simulation results Experimental results
M (μH) k M (μH) k
2 31.7 0.92 33.1 0.93
4 28.5 0.88 27.8 0.87
6 18.2 0.65 19.1 0.67
8 11.2 0.5 12.5 0.53
10 8.7 0.45 8.2 0.41
To indicate magnetic field features, figure 16 shows
the mutual inductance of the contactless transformer with
the increasing of the gap distance. It is clearly that with
the gap become bigger the mutual inductance of
contactless transformer will be small, and there is a good
agreement of the simulation results and experimental
results with each other.
0
10
20
30
2 4 6 8 10distance (mm)
Mutu
al i
nduct
ance
(H
)
Experiment
Simulation
Figure 16. Mutual inductance with different distances.
0
0.2
0.4
1
2 4 6 8 10distance (mm)
Coupli
ng c
oef
feci
ent
Experiment
Simulation
0.6
0.8
Figure 17. Coupling coefficient with different distances.
Similarly, the coupling coefficients of EE core
transformer are computed and tested by MAXWELL
ANSOFT and experimental setups in different distances.
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The curves are shown in figure 17. From this plot, we can
see that the coupling coefficient drops with the increasing
of gap, which agrees with the theoretical analysis.
Figure 18 shows the skin and proximity effects of
primary coil. The tested results of resistors of primary
coil are compared with the computational results from
MAXWELL ANSOFT. The plot shows that with the increasing of excitation frequency, the resistor will
increases at the same time because of the skin and
proximity effects. And the experimental results are
according with the computational results well.
0
0.2
0.4
1
1 2 3 4 5frequency (MHz)
Res
ista
nce
Experiment
Simulation
0.6
0.8
1.2
Figure 18. Primary coil resistors with excitation frequency.
V. CONCLUSTIONS
Most problems related to contactless power transfer are
usually employing equivalent electric circuits as a tool for their analysis. Furthermore, its values usually acquired
from measurement results performed in prototypes, which
is an expensive and time consuming task. In this paper,
however, a FEA approach based on the Maxwell
Equations is presented, and the magnetic field
distributions are displayed by ANSOFT MAXWELL.
Moreover, experimental results from a prototype are
compared with the simulation results, which show good agreement with each other.
ACKNOWLEDGMENT
This work is supported by the National Natural
Science Fund of China (51277120).
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Jianyu Lan was born in Fujian province, China, in 1980. He received the B.S. degree in electrical engineering from Zhengzhou University, Zhengzhou, China, in 2002, and M.S. degree in electrical engineering from Shanghai Maritime University, Shanghai, in 2009. He is currently working toward his Ph. D.
degree in electrical engineering at Shanghai Jiao Tong University. His
research interests include power electronics emerging at power transfer. Houjun Tang was born in Shandong province, China, in 1957. He received B.S. and M.S. degrees in 1982 and 1988 respectively. He received Ph.D. degree in Yamagata University, Japan, 1997. He is professor of the Department of Electrical Engineering, Shanghai Jiao Tong University. His research interests include power electronics emerging at power transfer.
Xin Gen was born in Heilongjiang province, China, in 1980. He received the B.S. and M.S. degree in electrical engineering from Shanghai Jiao Tong University, Shanghai, China, in 2002, 2008. He is currently working toward his Ph. D. degree in electrical engineering at Shanghai Jiao Tong University. His research interests include power electronics emerging at power transfer.
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