Computational Model of Coils for Wireless Power
Transfer with Reluctance Network Analysis
Keita Furukawa

Keisuke Kusaka

Jun-ichi Itoh

Nagaoka University of Technology
Nagaoka-city, Japanarcher_FK@stn.nagaokaut.ac.jp

Nagaoka University of Technology
Nagaoka-city, Japan
kusaka@vos.nagaokaut.ac.jp

Nagaoka University of Technology
Nagaoka-city, Japan
itoh@vos.nagaokaut.ac.jp

Abstract— This paper proposes an analytical model of thecoils in wireless power transfer systems for electric vehicles
with reluctance network analysis (RNA) in order to shorten adesign process of the coils. A transmitting coil and a receiving
coil are modeled with magnetomotive force and numerous
hollow-cylindrical units of a magnetic circuit. Circular coilswith ferrite plates are modeled with RNA to demonstrate thevalidity of a RNA model. The values of the mutual inductance
between the transmitting coil and the receiving coil arecompared between the prototype and the RNA model withchanging a radius of the windings and an air gap from thetransmitting coil to the receiving coil. As a result, the error of
the values of the mutual inductance between the prototype andthe RNA model is less than 20% when the radius of thewindings is from 40 to 80 mm, and when the air gap between
the coils is from 30 to 150 mm. The RNA model is able tocalculate the mutual inductance in a short time.
Keywords—wireless power transfer, coil, magnetic circuit,reluctance network analysis

I. INTRODUCTION
Wireless power transfer systems have been actively
studied for battery charging systems of electric vehicles toavoid electric shocks for more safety [1–3]. In particular,
inductive power transfer (IPT), which transmits power withmagnetic coupling between transmitting coils and receiving
coils, is attracted to the battery charging systems in order toachieve high power and high system efficiency. Maximum
efficiency and a maximum power of IPT systems depend on
the quality factor and the coupling factor of IPT coils [3].
Thus, designing IPT coils is one of the essential processes of
the IPT-system design [1–3].
Self-inductance and mutual inductance of the transmitting
coil and the receiving coils are generally designed withsimulations or prototypes in the design process [4–5]. Boththe simulation such as the finite element method (FEM) andthe test on numerous prototypes are basically trial-and-error
methods, which are time-consuming.
Models using reluctance network analysis (RNA) areproposed for the analysis of inductors or motors to shorten
the design process [6–11]. RNA expresses the inductors or
the motors into a magnetic circuit made of numerous
reluctances and magnetomotive force [6–11]. RNA models
are useful in the design process because it is simple to design
both the shape and the size of the inductors or the motors in
the first design step in exchange for low accuracy [8–11].
However, RNA has not been used for the design processes of
the coils for IPT systems since models of the IPT coils require
accurate distribution of magnetic flux. An advantage of RNA
is expressable of the overall distribution of the magnetic flux

978-1-7281-5067-3/19/$31.00 ©2019 IEEEin a short time. However, it is impossible to assign thematerials being smaller than the unit of the magnetic circuit,which means that calculating the self-inductance is
impossible with only RNA in accuracy. Whereas, FEM isable to calculate distribution of magnetic flux in detail.
However, FEM need a long time to calculate.
This paper proposes the modeling method of the IPT coilswith RNA to shorten the design process of the mutual
inductance. The originality of this paper is the proposal of themodeling method with RNA, transformation of cores, and acircuit simulator. The time-consuming in the conventional
design processes of the IPT coils is that the mutual inductanceis calculated with FEM at numerous conditions. The selfinductance and the mutual inductance are designed withseparate processes because the variation of the selfinductance is generally less influenced by the air-gap andposition misalighnment. Thus, time of the overall design
process expects to be short when the self-inductance and themutual inductance are calculated with FEM and RNA,
respectively. A prototype of circular IPT coils is expressedwith RNA as an electric network circuit using magnetomotive
force and hollow-cylindrical units of a magnetic circuit. In
addition, the calculated mutual inductance with the RNAmodel between the transmitting coil and the receiving coil iscompared with the prototype of the circular IPT coils.
II. RELUCTANCE NETWORK ANALYSIS
A. Structure of Circular Coils for IPT System
Figure 1 shows the structure of the circular coils without
coil misalignment. Although various forms of the coils areproposed from the viewpoints of robustness for
misalignment, cost and leakage magnetic flux of the IPT
systems, the simple circular IPT coils are widely employed[1]. The ferrite plates are placed on the outer side of the pairof the windings in order to decrease reluctance of the coils
and leakage magnetic field around the IPT coils.
The IPT systems have much leakage magnetic flux due tothe wide-gap coils for in comparison with that of motors orCentral axis

Ferrite plate

Receiving coil

Receiving winding
Transmitting winding
Transmitting coilFig. 1. Sectional view of circular coils for IPT systems.

inductances because of high ratio of non-magnetic materials
to magnetic materials in the magnetic path. In addition, themagnetomotive force generated by the coils should be also
considered because the shape of the coils influences themagnetic flux distribution. Thus, a model of the coils requires
to express the three-dimensional distribution of the magnetic
flux in detail. RNA is able to express the distribution of themagnetic flux. In the next section, an outline of RNA is
explained.
B. Outline of RNA
Reluctance network analysis is one of permeance methods
using ideas that models are constructed with numberless
small units. RNA expresses the IPT coils including air aroundthe coils as a magnetic circuit with and numerous units ofmagnetic circuit. In addition, magnetomotive force generated
by windings is also set onto the units of magnetic circuit. The
reluctances are assigned into the units of magnetic circuitwith considering the three-dimensional magnetic flux. The
distribution of magnetic flux in the constructed RNA modelis calculated with circuit simulators as an electric network.
Finally, magnetic characteristics such as values of the mutual
inductance between the transmitting coil and the receiving
coil are obtained from the calculation.
The advantage of RNA is shorter computing time thanFEM [9–10]. In the previous studies, a computing time of a
proposed model based on RNA is ten or more times fasterthan computing times of FEM models [9–10]. In addition,
waveforms of magnetic flux density calculated with RNA
models almost correspond to waveforms of FEM models or
motors [8–11]. Thus, the RNA model of the IPT coils is
expected to estimate the values of the mutual inductancewhich require the accurate interlinkage magnetic flux.
C. Unit of Magnetic Circuit for Analysis of IPT Coils
Figure 2 shows the hollow-cylindrical unit of themagnetic circuit. The RNA model of IPT coils is filled withthe unit of the magnetic circuit in Fig. 2. The RNA model isconsidered under the cylindrical coordinate system because
the circular coils such as Fig. 1 is axial symmetry. The θ-axis
reluctances in the unit are unnecessary to consider because θaxis magnetic flux generated by the circular coil is zero. The
reluctances Rre, Rri, Rzt, and Rzu in the cylindrical unit of Fig.
2 are calculated from the size of the unit, and expresses asr
1
1
 2r 
Rre =  Δr
dr =
ln 

r−
Δ
Δ
πμ
πμ
2
r
d
2
d
 2r − Δr 
2............................................................................ (1),Δr
 2r − Δr 
r−
1
1
Rri =  2
dr =
ln 

r −Δr 2πμ r Δd
2πμΔd  2 ( r − Δr ) ............................................................................ (2),ΔdRzt = Rzu =.............................. (3),2πμ ( 2r − Δr ) Δr
where Δd is the height, Δr is the increment of the radius, r is
the external radius and μ is the magnetic permeability of thematerials, e.g., air and ferrite. Note that the magnetic flux
through Rre, Rri, Rzt, and Rzu are integrated value on the sidesurfaces, the upper surface or the undersurface of the unit of
the magnetic circuit.

D. Magnetomotive Force
Figure 3 shows a part of the RNA model including thewindings in which the current I flows. In general, Gauss’ law
for magnetism and Ampere’s circuital law in a magnetic
circuit can be applied similarly to Kirchhoff’s current law andvoltage law for an electrical circuit, respectively [11–12].
When an arbitrary closed-loop c is defined in the magnetic
circuit, Ampere’s circuital law is also adapted to c to estimate
a closed curve. The magnetomotive force Fc of c is shown as
Fc =  Rnφn =  H c • dr =  jc • dS ............... (4),cnSwhere Rn is one of reluctances on c, φn is the magnetic flux
through Rn, Hc is the magnetic field at a certain point on c,and jc is the current density inside the closed surface S.
When (4) is adapted to the closed-loops c1, c2, and c3 inFig. 3 (a) as examples, the magnetomotive force Fc1, Fc2 andFc3 of each closed-loop are expressed as
Fc1 = 0 ................................................................ (5),

Fc 2 = I ................................................................ (6),Fc3 = I − I = 0 .................................................... (7).
The magnetomotive force is applied onto all units of themagnetic circuit inside the coils with the value of I to fulfill(5–7). When there are plural windings in the RNA model, themagnetomotive force is set the overall closed surface of eachwinding with the value of flowing current from the principleof superposition. Moreover, the interlinkage flux is the sumof the magnetic flux through the magnetomotive force of eachwinding.
III. PROTOTYPE OF CIRCULAR IPT COILS
A. Prototype of Circular IPT Coils
Figure 4 shows the prototype of the circular IPT coils. The
prototype is constructed by a pair of facing coils: thetransmitting coil in Fig. 4 (a) and the receiving coil in

Fig. 2. Unit of magnetic circuit in RNA model.

c3
c2

c1
I

I

I
-I

Fig. 3. Distribution of magnetomotive force in RNA model.

237

237
119
79

79

119
79

79

79

70

70
105

105

210

79

210

70

70

70

70

46

46
80

80

120

120

160

160

0

0

(a) Transmitting coil.

Unit: mm(b) Receiving coil.
Receiving coil

Receiving coil

Receiving coil

210
70

70

70
150
75

80

60 55
25 20

Transmitting coil(c) Side view with 30-mm gap.

Transmitting coil(d) Side view with 75-mm gap.
Fig. 4. Prototype of circular IPT coil.

Fig. 4 (b). The transmitting coil and the receiving coil are thesame forms, material, and size for simplification.
The ferrite plates of the transmitting coil and the receiving
coil is a length of 237 mm, a width of 210 mm, and a heightof 20 mm. The ferrite plate is constructed by 18 smallersquare ferrite plates whose domain is a length of 79 mm, awidth of 70mm, and a height of 10 mm. The magnetic
material of the ferrite plate is PC40 (TDK). Moreover, theferrite plate is covered with the acrylic case whose thickness
is 5 mm.
Wires with a cross-section of 0.126 mm2 is used for eachcircular winding in Fig. 4 (a) and Fig. 4 (b). The number ofthe winding N is 10. The radiuses of the five circular windings
are 23, 40, 60, 80 and 100 mm, respectively. The five circular
windings have a role of changing the radius of the windings
rce when the prototype is compared with a proposed RNAmodel. Note that unwanted windings are open in order toignore themselves when measuring the values of the mutual
inductance because the magnetomotive force generated by
the open windings is zero.
Figures 4 (c–e) show the side view of the prototype withchanging the gap hg between the transmitting coil and theTransmitting coil

Unit: mm(e) Side view with 30-mm gap.

receiving coil. The gap is changed from 30 mm to 150 mm toevaluate the effect of the air-gap on the mutual inductance.
B. Conversion from Square Ferrite Plate to CircularFerrite Plate
Chapter II shows the RNA model of the axisymetric IPT
coils. However, the prototype is not directly translated to an
RNA model because the square ferrite plates of the prototype
are not suitable for an axisymetric RNA model. Thus the formof the ferrite plates should be changed to a cylinder which has
the same values of the mutual inductance, to express the
prototype as the RNA model. Thus, a radius rfe of the ferrite
plates whose characteristic of the mutual inductance is thesame as the prototype, is clarified with four FEM models.
The FEM models are built on JMAG (JSOL). JMAG is
one of electromagnetic-field-analysis software with the FEM.
The value of rfe is determined to be the lowest relative errorof the mutual inductance in the FEM models by comparing
the value of the mutual inductance between the FEM models
and the prototype.
Figure 5 shows the rotary surface of the FEM models andthe top view of ferrite plates on FEM models. The FEM
models are

I.

IV. RNA MODEL OF PRPTOTYPE
Figure 7 shows the RNA model employed with the IPT
coils in Fig. 4. The RNA model is used by 195 units. The
numbers of the units to the r-axis and the z-axis are 13 and 15,
respectively. The white units express air domain (the relative
permeability is one). The units colored by grey express
ferrite-plate domain (the relative permeability is 2300).
The air space around the coils including the air gap is also
considered because the magnetic flux through the air space
affects calculated values of the mutual inductance. The lengthof the r-axis is 232 mm. The length of the z-axis is from 170
mm to 290 mm depending on hg. In addition, the dimensionsof each unit Δd and Δr are determined to be lower than 30
mm. The height Δd of the units expressing the ferrite plates
Relative error to measured value M / Mm%

FEM model of the prototype whose size and form
are as same as the prototype,
II. Narrow-side model whose 105-mm rfe is equal tohalf width of prototype,
III. Average-core-length model whose 132-mm rfe isequal to the average of half width and half diagonalof prototype, andIV. Diagonal model whose 158-mm rfe is equal to half
diagonal of prototype.
Note that the depths of the circular ferrite plates and theacrylic cases in the FEM models are 20 and 5 mm as same as
the prototype in Fig. 4, respectively. In addition, the height
and width of the windings in the FEM models are 3 mm
referred to an approximate value of the prototype.
Figure 6 shows the relative error ΔM / Mm of the mutual
inductance of the four FEM models when rce and hg arechanged. The true values of the mutual inductance in Fig. 6
are measured values of the mutual inductance Mm of the
prototype under the same conditions. Figure 6 (a–c) show
ΔM / Mm with hg = 30 mm, hg = 75 mm, and hg = 150 mm,
respectively. The relative error ΔM / Mm of the the FEM
model of prototype comes from the dimension error and themeasurement error of the mutual inductance on the prototype.
Figure 6 shows that ΔM / Mm of Average-core-length
model are small. In addition, fluctuation of ΔM / Mm of theaverage-core-length model is the same as the fluctuation of
FEM model of the prototype. Whereas, ΔM / Mm of Narrowside model and Diagonal model are large when rce isincreased. It indicates that the effect of the magnetic flux near
the edge of the ferrite plates is not negligible when rce is close
to rfe. The reason why ΔM / Mm at rce = 23 mm is large in allof FEM models, is deduced from the dimension error and themeasurement error of the prototype. It is difficult to measureinductance lower than μH-order.
As a result, the RNA model should be built based on the
average-core-length model. Therefore, rfe of the circularferrite plates in the RNA model is 132 mm.

30
25
20
15
10
5
0

20

40

60

80

100

120

80

100

120

80

100

120

(a) hg = 30 mm.

30
25
20
15
10
5
0

20

40

60

(b) hg = 75 mm.

(a) Rotary surface of circular model.

50
40
30
20
10

(b) Top view of ferrite plates of each FEM model.
Fig. 5. Configuration of FEM models

0

20

40

60

(c) hg = 150 mm.
Fig. 6. Relative error of mutual inductance.

Rotational axis
r

20 mm

20 mm

20 mm

20 mm

20 mm

17 mm 15 mm

20 mm

20 mm

20 mm

17 mm 15 mm 8 mm

d
15 mm
15 mm
15 mm
ps

20 mm
5 mm

NIs

hg / 5

NIs

NIs

NIs

NIs

NIs

hg / 5

Unit of air domain(Relative permeability: 1)

hg / 5hg / 5

Unit of ferrite-plate domain(Relative permeability: 2300)

hg / 5NIp NIp

NIp

NIp

NIp

NIp

Magnetomotive force

5 mm

Central axis

keq =0.2RztRre
0.3 Rri
=

20 mm

keq
=0 0.1

Rzu

15 mm

z
15 mm

r

15 mm

Fig. 7. Representative example of RNA model (Radius rce of windings is 80 mm).

and the air domain aligned horizontally to the ferrite plates is
20 mm. The increment Δr of the units expressing the ferrite
plates and the air domain aligned vertically to the ferrite
plates are decided to meet the external diameter of the units
and windings of the prototype in Fig. 4. Note that Δr of theunits next to the central axis is 8 mm to reduce the modelingerror. A 5-mm thickness of the acrylic cases between theferrite plates and the windings is also considered with theunits of the air domain. The terminals on the boundary of the
RNA model is opened. The magnetomotive forces NIp andNIs of the transmitting coil and the receiving coil are locatedinside the units of each winding.
V. COMPARISON OF MUTUAL INDUCTANCE
In this chapter, the RNA model in Fig. 7 are comparedwith the prototype in term of the mutual inductance between
the transmitting coil and receiving coil. The RNA model ismodeled and calculated with PLECS (Plexim). The mutual
inductance between the transmitting coil and the receiving
coil is compared with changing rce and hg.
In addition, a criterion of the RNA model is shown as areference in order to confirm the validity of the RNA model;
The RNA model is available in a design process of the circular
IPT coils when the error of the mutual inductance between the
RNA model and the prototype is less than 20%. A error of
20% on the mutual inductance is decided from experiencing
things that there is a possibility to become errors of inductance
between real IPT coils and FEM models to 20%.
Figure 8 shows the mutual inductance against rce and hg
with the RNA model and the prototype. The black circles withthe full-line is the results with the proposed model of RNA,

whereas the white-painted plots with dashed line indicates theresults with the prototype. The mutual inductances arecalculated from the interlinkage magnetic flux φps of thereceiving winding and Ip when the receiving winding is open
(Is = 0). The x-axis is the ratio rce / rfe of the radius of thewindings standardized by rfe = 132 mm, whose value isdecided in Chapter III. Note that the maximum value of
rce / rfe is 0.795 because rce is limited by the width of the ferrite
plates from construction of the IPT coils.
The values of the mutual inductance of the RNA model isthe same as the value of the mutual inductance of the prototype
in the order of magnitude at same rce and hg. The result meansthat RNA is employed to the simplified design of the circular
IPT coils. In addition, the errors of the mutual inductance areless than 20% in the range of as follows: the ranges of rce / rfe
is from 0.30 to 0.76 when hg is 30 or 75 mm. The ranges ofrce / rfe is from 0.30 to 0.61 when hg is 150 mm.
The error of the mutual inductance are larger than 20% at
rce / rfe = 0.17 because of the dimension error and themeasurement error of the prototype. However, the error of themutual inductance is larger than 20% at rce / rfe = 0.76 andhg = 150 mm. Factors of the large error are neither thedimension error nor the measurement error of the prototype
because the relative error of the mutual inductance with
Average-core-length model in Fig. 6 (c) is 1%. Thus, the RNAmodel does not express the distribution of magnetic flux
completely. As supposition, the unit of the magnetic circuit inFig. 2 does not consider reluctances of the diagonal path. The
measured mutual inductance of the prototype is moreincreasing due to spreading lines of magnetic force around the
IPT coils when rce and hg are large.

Although it is impossible to calculate the self-inductancewith only the RNA model in the chapter 1 in accuracy, themaximum relative error of the self-inductance is 27.8% under
the conditions of rce / rfe = 0.30, hg / rfe = 1.14 and 3-mm
height and width of the windings. The order of magnitude of
the self-inductance is able to estimate with the RNA model.
Moreover, the computation time under the all conditions
with the RNA model is 20 second, which is shorter than 3676
second with 3-D FEM models of the prototype.
As a conclusion, the RNA model is available to the design
process of the circular IPT coils when the ratio rce / rfe are from0.30 to 0.61, and when the ratio hg / rfe are from 0.23 to 1.14.
Moreover, the RNA model contributes to shorten the design
process of the IPT coils.
VI. CONCLUSIONSThis paper examines the RNA model into the design
progress of the circular coils in IPT systems to shorten thedesign process. The RNA model would be applied at the firststep of the design progress to reduce the number of trials
conducting by the conventional time-consuming design
method such as simulations with FEM or the construction ofnumerous prototypes. The RNA model is simply constructedby numerous hollow-cylindrical units of the magnetic circuitand the magnetomotive force. In order to verify the validityof the RNA model, the mutual inductances of the transmitting
coil and the receiving coil are compared with the RNA modeland the prototype of the circular IPT coils. As a result, themutual inductance agreed with lower than 20% error in therange of the radius and the gap of the windings from 40 mm
to 80 mm and 30 mm to 150 mm, respectively. In addition,
the relation of the mutual inductance is matched in the overall
range of the radius and gap of the windings. Thus, the modelof RNA was suitable to simplify the design progress of thecoils in IPT systems.
In the future, the accuracy of the RNA model will be
improved in term of mutual inductance and self-inductance.

01

REFERENCES
[1]

05

06

07

0

0.2

0.4

0.6

0.8

1

0.8

1

0.8

1

(a) hg = 30 mm (hg / rfe = 0.23).

0.0001

1e-05

1e-06

1e-07

0

0.2

0.4

0.6

(b) hg = 75 mm (hg / rfe = 0.57).

1e-05

1e-06

1e-07

1e-08

0

0.2

0.4

0.6

(c) hg = 150 mm (hg / rfe = 1.14).
Fig. 8. Mutual inductance of prototype and RNA model.

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