fundamental research of power conversion circuit...
TRANSCRIPT
Fundamental Research of Power Conversion CircuitControl for Wireless In-Wheel Motor using
Magnetic Resonance CouplingDaisuke Gunji
The University of Tokyo / NSK Ltd.5-1-5, Kashiwanoha, Kashiwa, Chiba, 277-8561, Japan
/ 1-5-50, Kugenumashinmei, Fujisawa,Kanagawa, 251-8501, Japan
Email: [email protected]
Takehiro Imura and Hiroshi FujimotoThe University of Tokyo
5-1-5, Kashiwanoha, Kashiwa, Chiba, 277-8561, JapanPhone: +81-4-7136-3881
Email:[email protected]@k.u-tokyo.ac.jp
Abstract—The In-Wheel Motor (IWM) is the most preferabledriving mechanism of electric vehicles for vehicle motion control,energy efficiency, and vehicle design flexibility. One technicalissue of the IWM is the reliability of power and signal wires.Wireless power transfer technology is the best solution. In thispaper, a bidirectional wireless power transfer circuit using aprimary inverter and a secondary converter is proposed. Wepropose a control method of both the inverter and the converterto stabilize the secondary DC-link voltage. The proposed methodis verified by simulation and experiments using simulated testequipment.
I. INTRODUCTION
Electric vehicles (EVs) have advantages of environmentalperformance and motion control performance over internalcombustion engine vehicles due to the outstanding perfor-mance of electric motors [1]. Especially, the in-wheel motor(IWM) is the best solution because the motor is directlyconnected to a wheel.
Many IWM have been developed in previous research suchas [2]–[4]. All of them are powered using electrical wires.Reliability and safety issues of these wires are one of the short-comings of IWM. Wireless power transfer (WPT) technologyis the best solution for the above problem. We call this conceptthe “Wireless In-Wheel Motor (W-IWM)“. W-IWM is alsoapplicable to wireless power supply from power transmittingequipment which is installed under the road. Due to the motionof suspension links, the relative position will change betweenW-IWM and the vehicle chassis. Then, WPT using magneticresonance coupling [5] [6] is suitable for W-IWM. However,it is known that load voltage or current change with mutualinductance variation and load variation [7]. In order to realizeW-IWM, stabilization control of the load voltage is necessary.
Much research on power conversion circuit control of WPThave been carried out [8]–[11]. However, most focused onefficiency improvement and application for slow load variationsuch as battery charging of EVs. In the case of a W-IWM, theload is an electric motor and the load variation is very fast. Asfar as we know, there has been little research about a powerconversion circuit control method of WPT.
On-boardcommand
(wireless)
electricmotorSecondary coilPrimary coil
Inverter
Battery
In-wheel
Converter PWMinverter
Power
wheel
Transmitting coilGround (Road)
Power source
Fig. 1. Concept of W-IWM.
In this paper, we present a control method of the powerconversion circuit for W-IWM. Motor drive experiments havebeen carried out to validate the effectiveness of the proposedcontrol method, where the motor has been subjected to mutualinductance variation and rotation speed variation.
II. CONCEPT OF W-IWMA. Structure
The concept of the W-IWM is shown in Fig. 1. The W-IWMconsists of a power source (battery), primary DC-AC inverter,primary coil, secondary coil, secondary AC-DC converter,inverter for driving the electric motor, and, an electric motor.Here, primary means on-board side, and secondary means in-wheel side. The primary DC-AC inverter converts the DCsupply voltage to a high frequency AC voltage. The primarycoil is mounted on a vehicle chassis and the secondary coil ismounted on an IWM. Each coils are oppositely positioned andelectrical power is transmitted via these coils. Received poweris converted to DC by the secondary AC-DC converter. Thenthe electric motor is driven by the inverter. When the electricmotor acts regeneratively, electric power flow is inverted,meaning the secondary converter acts as a DC-AC inverterand the primary inverter acts as an AC-DC converter. Signalcommunication between the on-board side and the in-wheelside is also wireless.
B. Experimental vehicle and 1st prototype of W-IWMFig. 2(a) shows the experimental EV FPEV4-Sawyer which
was developed by our research group. The drive unit of that
(a) Experimental vehicle.
Primary
inverter
Secondary converter & electric motor
Primary coil
Secondary coil
(b) 1st prototype W-IWM unit.
Fig. 2. Experimental vehicle and 1st prototype unit.
vehicle can be easily exchanged. We are now developing the1st prototype of a W-IWM unit shown in Fig. 2(b).
C. Simulated test equipment
In this research, simulated test equipment, which is shownin Fig. 3, is used instead of the prototype unit. The structureof the equipment is same as the above W-IWM concept.
The primary coil and the secondary coil are the same shape:flat rectangular spiral coils. The primary coil is fixed, andthe secondary coil is mounted on the linear actuator. Thenposition misalignment of the two coils is controlled by thelinear actuator. This mechanism simulates the suspension linkmotion on a vehicle. Specifications of the two coils are shownin TABLE I. The resonance frequency is diffenrent forminternational standard (85 kHz) for production reasons.
A three phase geared brushless motor is used on the equip-ment. Maximum power of the motor is 50 W. The motor isconnected to a powder brake. The motor speed is controlled bythe motor driver, and the powder brake provides load torque.
III. CIRCUIT STRUCTURE AND MODELING
A. Circuit structure
The whole circuit structure of the W-IWM is shown in Fig.4. The primary coil and the secondary coil are indicated bya T-type equivalent circuit [12]. Both the primary inverterand the secondary converter are H-bridge circuits. Both theprimary resonance capacitor C1 and the secondary resonancecapacitor C2 are connectied in series to each coil. Then thecircuit structure is symmetrical and suitable for bidirectionalpower transfer.
The electric motor is driven by the voltage type three phasePWM inverter. Then the secondary DC-link voltage Vdc hasbeen controlled to fixed value. In order to realize this, someprevious studies have proposed a secondary circuit structurewhich consists of a full-wave rectifying circuit and a DC-DC converter [11]. However, in the case of the W-IWM, theavailable space of the in-wheel side is limited and downsizingof the in-wheel side equipment is required. Then the proposedcircuit structure also has an advantage of downsizing.
B. Equivalent load resistance model
We consider the condition when electrical power is trans-mitted from the primary side to the secondary side. A previousstudy [11] has suggested that if fundamental harmonics powerfactor of the secondary converter equals to 1 and electricalloss of the secondary converter can be ignored, the whole
Power conversion
circuits
Primary
coil
Secondary
coil
Linear actuatorMotor driver
Brushlessgearedmotor
Powderbrake
Fig. 3. Simulated test equipment.
TABLE ISPECIFICATIONS OF COILS.
primary coil secondary coilresistance R 0.297 Ω 0.305 Ωinductance L 71.85 µH 71.31 µHcapacitance C 35911 pF 35879 pFresonance frequency f 99.08 kHz 99.50 kHzMutual inductance Lm 12.43 µH (gap: 80 mm)
secondary converter and load are equated to an equivalent pureelectrical resistance RL.
If the load is an electrical motor which is driven by aninverter, the relation between mechanical output Pm andelectrical power is expressed as
Pm = ηmηinvVdcIdc (1)
where ηm is the motor efficiency, ηinv is the inverter efficiency,and Idc is the DC-link current. Then RL is calculated as
RL = ηmηinvVdc
2
Pm. (2)
If Vdc is controlled to a fixed value, RL depends on Pm.Therefore, the electric motor is treated as pure electricalresistance.
C. Dynamics of the power transfer circuit
The transfer function from the primary voltage to thesecondary current is expressed as
Pio(s) =b3s
3
s4 + a3s3 + a2s2 + a1s+ a0(3)
where each coefficient are defined as
a3 =L1(R2 +RL) +R1L2
L1L2 − Lm2 , (4)
a2 =R1C1C2(R2 +RL) + C1L1 + C2L2
C1C2(L1L2 − Lm2)
, (5)
a1 =R1C1 + C2(R2 +RL)
C1C2(L1L2 − Lm2)
, (6)
a0 =1
C1C2(L1L2 − Lm2), (7)
b3 =Lm
L1L2 − Lm2 . (8)
Every parameter corresponds to parameters shown in Fig. 4.The bode diagram of the transfer function is shown in Fig. 5while specifications are shown in TABLE I. At the resonance
M
S11
S12
S13
S14
S21 S23
S24S22
R1 R2C1 C2L1-Lm L2-Lm
Lm
E
Cs
Vdc
iCin
iconviinv
vinv vconv
load
PMSM
secondary converterprimary inverter
T-type equivalent circuit of coils
and resonant capacitors
RL
equivalent load resistance
Fig. 4. Circuit structure of W-IWM.
104
105
106
−80
−60
−40
−20
0
frequency [Hz]
gain
[dB
]
R
L=25 Ω
RL=50 Ω
104
105
106
−90
0
90
180
270
frequency [Hz]
phase [deg]
R
L=25 Ω
RL=50 Ω
Fig. 5. Bode diagram of T-type equivalent circuit.
frequency, the secondary current phase advances 90 deg fromthe primary voltage.
D. Dynamics of the smoothing capacitorThe transfer function from the secondary smoothing capac-
itor input current iCin to the DC-link voltage Vdc is expressedas the following equation.
PCs(s) =RL
RLCss+ 1(9)
E. Duty ratio versus DC link voltageFig. 6(a) and (b) show switching state of the primary inverter
and the secondary converter, respectively. The duty ratio of theprimary inverter dinv and the secondary converter dconv aredefined as Tp/0.5T where T is periodic time, and Tp is timeof the pulse width. Therefore, dinv=1 means a square wavewhich has ±E voltage amplitude.
According to Fig. 5, the transfer function Pio(s) has band-pass characteristic. Then, we only focus on the fundamentalharmonics of the primary inverter output voltage. The ampli-tude of the fundamental harmonics voltage Vinv1 is calculatedas
Vinv1 =4E
πsin
πdinv2
. (10)
Gate states of the secondary converter are shown in TABLEII and Fig. 7. There are three switching modes. In the caseof modes 1 and 2, the secondary current iconv passes thesecondary converter. By contrast, in the case of mode 3, thesecondary coil is shorted and the secondary current iconv doesnot flow into the load. This means load resistance is equatedto zero. Then we define the pseudo load resistance as
RLp = RL sinπdconv
2. (11)
Tp
T0.5T0.25T
0.75T
-E
+E
(a) PWM inverter.
T
0.25T
0.75T
0.5T
input current
mode 1
mode 3mode 2
(b) PWM converter.
Fig. 6. Definition of duty ratio.
TABLE IISWITCHING MODE OF THE SECONDARY CONVERTER.
mode Gate state circuit behavior1 S21, S24 = ON operate as rectifier2 S22, S23 = ON operate as rectifier3 S22, S24 = ON shorted
From eq.(3), the amplitude of iconv is
Iconv = |Piop(jωin)|Vinv1 (12)
where ωin is the driving frequency of the primary inverter, andPiop(s) is the transfer function Pio(s) with RL replaced byRLp . Assuming that the time constant of PCs(s) is sufficientlyslower than the driving frequency of the primary inverter, inputvalue of PCs(s) can be treated as an average passing currentthrough the secondary converter. The average passing currentICave is calculated as
ICave =1
π
∫ π2 +π
2 dconv
π2 −π
2 dconv
Iconv sin θdθ
=2
πIconv sin
πdconv2
. (13)
Substituting eq.(10) and (12) in (13), ICave is expressed as
ICave =8E
π2|Piop(jωin)| sin
πdinv2
sinπdconv
2. (14)
Therefore, the steady-state value of Vdc is
Vdc|t=∞ =8E
π2RL|Piop(jωin)| sin
πdinv2
sinπdconv
2. (15)
The switching timing of the secondary converter has tosynchronize with iconv in order to make the fundamentalharmonics power factor to 1. Then, we generate the PWMcarrier of the secondary converter from zero-cross timing oficonv as shown in Fig. 7.
G21 G23
G24G22
Cs
Vdc
iCinsecondary converter
iconv
zero cross saw
PWM carrier
0.25T 0.5T
iconv
vconv
Fig. 7. Voltage and current of secondary converter.
IV. DC LINK STABILIZATION CONTROL
A. Control strategy
In this section, we propose a DC-link voltage stabilizationcontrol method.
The circuit of the W-IWM has two control degrees-of-freedom: the duty ratios of the primary inverter and the sec-ondary converter. Signal communication between the primaryside and the secondary side is wireless communication. Then,there are delay and speed limitations. In order to avoid theseeffects, the primary inverter is controlled by a feedforwardcontroller and the secondary converter is controlled by a DC-link voltage feedback controller.
B. Primary inverter controller
Assuming that the torque response of the electric motor isfast enough, the equivalent load resistance R∗
L is determinedby the DC-link voltage reference value V ∗
dc, the angular speedof the electric motor, and the torque command of the electricmotor. In fact, it is useful to refer to the prepared R∗
L mapwhich describes the relation of the angular speed and thetorque of the electric motor to R∗
L. We use the nominalvalue of the mutual inductance Lm. Also, we introduce thenominal duty ratio of the secondary converter dconv in orderto determine the control margin of the secondary converter.That means, if there are no model errors or disturbance, dconvbecomes dconv by the secondary feedback controller. Then,the duty ratio command of the primary inverter d∗inv is derivedfrom eq.(15) as
d∗inv =2
πsin−1
(π2V ∗
dc
8ER∗L|Piop(jωin)| sin πdconv
2
). (16)
C. Secondary converter controller
The secondary converter controller is a two-degree-of-freedom controller, the control target plant of which is PCs(s).We use a PI controller which is designed by the pole placementmethod.
CPI(s) = Kp +Ki1
s(17)
Kp =2pR∗
LCs − 1
R∗L
(18)
Ki = p2Cs (19)
eq.(2)
or map
Pm*
Vdc*
eq.(16) PWM Piop (s)RL
*dinv
* vinv
eq.(21)CPI (s)
CFF (s)
++ ICave
*
PWMdconv
* ICavePCs (s)
+-
Vdc*
dconv
Vdc
secondary converter
primary inverter primary side (on-board)
secondary side (in-wheel)
iconv
Fig. 8. Block diagram of the proposed control method.
where −p [rad/s] is a closed loop pole (multiple root). Also,the feedforward controller is expressed as
CFF (s) =ωc
s+ ωcPCs
−1(s) (20)
where ωc [rad/s] is the cutoff frequency of the low-pass filter.The manipulated variable of the controller is the average inputcurrent to the smoothing capacitor I∗Cave. Then, the duty ratiocommand of the secondary converter d∗conv is calculated as
d∗conv =2
πsin−1
(π2I∗Cave
8E|Piop(jωin)| sinπd∗
inv
2
). (21)
The block diagram of the proposed DC-link stablization con-trollers are shown in Fig. 8.
V. SIMULATION AND EXPERIMENT
A. Duty ratio versus DC-link voltage
Experiments have been carried out to verify eq.(15). Wemeasured Vdc under the following two conditions.
• Condition 1: dinv is variable, dconv is fixed to 1.0.• Condition 2: dinv is fixed to 1.0, dconv is variable.
The DC source voltage E was set to 5.0 V, the loads were 25Ω and 50 Ω non-inductive electrical resistance, and Cs was1000 µF. Both switching signals of the primary inverter and thesecondary converter were generated from the same triangularcarrier signal. The carrier frequency was set to 100 kHz. Thephase of the primary inverter gate signal was delayed by 90deg by a delay circuit in order to synchronize the secondaryvoltage and current.
Experimental results are shown in Fig. 9(a) to (c). Themarkers on the figure are measured values, and lines arecalculated values of Vdc by eq.(15). Experimental results are inagreement with theoretical values. Fig. 9(c) shows the voltageand current waveforms of the primary inverter output and thesecondary converter input where RL = 25Ω, dinv = 0.7, anddconv = 0.5. The measured waveforms are also in agreementwith circuit simulation results.
B. Acquisition of the equivalent load resistance map
An electrical motor drive experiment was conducted in orderto verify the effectiveness of the proposed DC-link voltagestabilization control. First, the equivalent load resistance mapwas obtained by experiment where dconv = 0.5, E = 12 V,and V ∗
dc = 24 V. The experimental process is as follows:
0 0.2 0.4 0.6 0.8 10
5
10
15
20
primary invertr duty ratio [−]
DC
−lin
k v
oltage [V
]
25Ω, calc.
25Ω, exp.
50Ω, calc.
50Ω, exp.
(a) dinv vs. Vdc ( dconv = 1.0 ).
0 0.2 0.4 0.6 0.8 10
5
10
15
20
secondary converter duty ratio [−]
DC
−lin
k v
oltage [V
]
25Ω, calc.
25Ω, exp.
50Ω, calc.
50Ω, exp.
(b) dconv vs. Vdc ( dinv = 1.0 ).
−10
0
10
vin
v [V
]
−1
0
1
i inv [A
]
−10
0
10
vco
nv [V
]
0 5 10 15 20 25 30−1
0
1
time [us]
i co
nv [A
]
(c) Voltage and current waveforms.(solid: measured, dotted: simulation)
Fig. 9. Duty ratio versus DC link voltage.
motor rotation speed [rpm]
load torq
ue [N
m]
0 20 40 600
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
equiv
ale
nt lo
ad r
esis
tance [
Ω]
50
100
150
200
250
300
350
400
Fig. 10. Equivalent load resistance map.
• Perform the proposed control.• Change the motor speed every 10 rpm, and the motor
torque every 0.1 Nm.• Adjust R∗
L so that dconv equals to dconv on every drivingcondition.
• Values between measured points are calculated by linearinterpolation.
The obtained RL map is shown in Fig. 10.
C. Mutual inductance variation while motor drive
The effectiveness of the proposed control method to mu-tual inductance variation is verified by experiment. Positionmisalignment was applied to the secondary coil by the linearactuator as shown in Fig. 11(a). Then, mutual inductancevariation occurred as shown in Fig. 11(b). The rotation speedof the motor was 50 rpm, and the load torque was 1.5 Nm.In this condition, R∗
L was 40.1 Ω from the equivalent roadresistance map. Cs was assumed to be 2000 µF (it is insidethe brushless motor driver and the correct capacitance valueis unknown). Closed loop poles of the secondary convertercontroller was set to -50 rad/s. A low-pass filter with the cutofffrequency of 200 rad/s was applied to the measured Vdc valuein order to suppress measurement noise.
Simulation results are shown in Fig. 12(a) and (b). The loadresistance value is treated as fixed value in the simulation.Without the proposed control method (both duty ratios arefixed), Vdc varies with the variation of the Lm. By contrast,using the proposed method, the duty ratio of the secondaryconverter is appropriately controlled. Then, variation of theVdc is suppressed and Vdc is almost kept to the reference value.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
10
20
30
40
50
60
70
80
time [s]
positio
n m
isalig
nm
ent [m
m]
(a) misalignment.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.98
9
10
11
12
13
14
time [s]
mutu
al in
ducta
nce [uH
]
(b) mutual inductance.
Fig. 11. Misalignment and variation of Lm in experiment.
Experimental results are shown in Fig. 12(c) and (d).Without the proposed control, Vdc varied greatly with thevariation of Lm. In the case of the experiment, the equivalentload resistance also varies with Vdc. Then, the variation ofVdc was bigger than the simulation result. By contrast, theproposed method suppressed variation of Vdc and the motorstably operated.
D. Motor speed variation
The effectiveness of the proposed method on the motorspeed variation is also verified. The load torque was set to1.5 Nm. The speed command of the motor was changed from50 rpm to 20 rpm and then again to 50 rpm as shown in Fig.13(a). In this condition, R∗
L was derived from the map shownin Fig. 13(e). The closed loop poles of the secondary convertercontroller were set to -50 rad/s. The cutoff frequency of thefeedforward controller was set to 100 rad/s.
Simulation results are shown in Fig. 13(b) to (d). Withoutthe proposed control, Vdc varies greatly with variation ofthe motor speed command. By contrast, using the proposedmethod, Vdc is kept almost to the reference value. The dutyratio of the secondary converter is almost unchanged becauseof the effectiveness of the primary feedforward controller.
Experimental results are shown in Fig. 13(f) to (h). Theseresults also fit in well with the simulation results. Withoutthe proposed control, the primary inverter operation stoppeddue to the protective function of the circuit and the motoralso stopped. By contrast, the proposed method suppressedvariation of Vdc and the motor stably operated. Steady-statevalue of the secondary converter duty ratio was about 0.5.It was equal to the dconv, meaning the primary feedforwardcontroller effectively worked.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.920
22
24
26
28
30
32
34
36
time [s]
DC
−lin
k v
oltage [V
]
w/ control
w/o control
reference
(a) Vdc (simulation).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.2
0.4
0.6
0.8
1
time [s]
se
co
nd
ary
co
nve
rte
r d
uty
ra
tio
[−
]
w/ control
w/o control
(b) dconv (simulation).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.920
22
24
26
28
30
32
34
36
time [s]
DC
−lin
k v
oltage [V
]
w/ control
w/o control
reference
(c) Vdc (experiment).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.2
0.4
0.6
0.8
1
time [s]
se
co
nd
ary
co
nve
rte
r d
uty
ra
tio
[−
]
w/ control
w/o control
(d) dconv (experiment).
Fig. 12. Simulation and experimental results of variation in mutual inductance.
0 1 2 3 4 50
10
20
30
40
50
60
time [s]
rota
tion s
peed c
om
mand [rp
m]
(a) Rotation speed command.
0 1 2 3 4 520
22
24
26
28
30
32
34
36
time [s]
DC
−lin
k v
oltage [V
]
w/ control
w/o control
reference
(b) Vdc (simulation).
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
time [s]
prim
ary
in
ve
rte
r d
uty
ra
tio
[−
]
w/ control
w/o control
(c) dinv (simulation).
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
time [s]
se
co
nd
ary
co
nve
rte
r d
uty
ra
tio [
−]
w/ control
w/o control
(d) dconv (simulation).
0 1 2 3 4 530
40
50
60
70
80
time [s]
eq
uiv
ale
nt
loa
d r
esis
tan
ce
[ Ω
]
(e) R∗L.
0 1 2 3 4 520
22
24
26
28
30
32
34
36
time [s]
DC
−lin
k v
oltage [V
]
w/ control
w/o control
reference
(f) Vdc (experiment).
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
time [s]
prim
ary
in
vert
er
du
ty r
atio
[−
]
w/ control
w/o control
(g) dinv (experiment).
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
time [s]
se
con
da
ry c
on
ve
rte
r d
uty
ra
tio
[−
]
w/ control
w/o control
(h) dconv (experiment).
Fig. 13. Simulation and experimental results of variation in motor load.
VI. CONCLUSION
In this paper, a control method of a power conversion circuitis proposed as a fundamental study of the W-IWM usingmagnetic resonance coupling. We derive an equation whichexpress the relation between the secondary DC-link voltageand the duty ratio of the primary inverter and the secondaryconverter. A DC-link voltage stabilization control method wasproposed, and the effectiveness of the proposed method isverified by simulation and experimentation.
Further works are as follows:• Establish a regenerative control method for the power
conversion circuit.• Current feedback on the primary side.• Experiments using the prototype unit on the vehicle.
REFERENCES
[1] Y. Hori: “Future Vehicle Driven by Electricity and Control–Researchon Four–Wheel–Motored:“UOT Electric March II““, IEEE Trans. Ind.Electron., Vol. 51, No. 5, pp. 954-962, 2004
[2] G. Freitag, M. Klopzig, K. Schleicher, M. Wilke, and M. Schramm:“High-performance and highly efficient electric wheel hub drive in auto-motive design“, Proc. 2013 3rd International Electric Drives ProductionConference, pp.1-7, 2013, Nuremberg
[3] A. Kock, M.Groninger, and A. Mertens: “Fault Tolerant Wheel HubDrive with Integrated Converter for Electric Vehicle Applications“, Proc.2012 IEEE Vehicle Power and Propulsion Conference, pp.19-23, 2012,Seoul
[4] A.J. Rix, and M.J. Kamper: “Radial-Flux Permanent-Magnet HubDrives: A Comparison Based on Stator and Rotor Topologies“, IEEETrans. Ind. Electon., Vol.59, No.6, pp.2475-2483, 2012
[5] A.Kurs, A. Karalis, R. Moffatt, J.D. Jonnopoulos, P. Fisher, M.Soljacic,“Wireless Power Transfer via Strongly Coupled Magnetic Resonances”,Science Expression on 7 June 2007, Vol.317, No.5834, pp.83-86, 2007
[6] T. Imura, H. Okabe, T. Uchida, and Y. Hori: “Wireless Power Transferduring Displacement Using Electromagnetic Coupling in Resonance -Magnetic- versus Electric-Type Antennas-“, Trans. IEE Japan IndustrialApplication, Vol.130, No.1, pp.76-83, 2010 (in Japanese)
[7] M. Kato, T. Imura, and Y. Hori: “New Characteristics Analysis Con-sidering Transmission Distance and Load Variation in Wireless PowerTransfer via Magnetic Resonant Coupling“, Proc. 2012 IEEE 34thInternational Telecommunications Energy Conference, 2012, Scottsdale
[8] D.J. Thrimawihana, and U.K. Madawala: “A Generalized Steady-StateModel for Bidirectional IPT System“, IEEE Trans. Power Electronics,Vol.28, No.10, 2013
[9] T. Nayuki, K. Fukushima, N. Gibo, K. Nemoto, and T. Ikeya: “Pre-liminary Demonstrations of a Bi-directional Inductive Power Trans-fer System“, Electric power engineering research laboratory report,No.H10007, 2011 (in Japanese)
[10] S. Nakadachi, S. Mochizuki, S. Sakaino, Y. Kaneko, S. Abe, and T.Yasuda: “Bidirectional Contactless Power Transfer System Expandablefrom Unidirectional System“, Proc. 2013 IEEE Energy ConversionCongress and Exposition, pp.3651-3657, 2013, Denver
[11] K. Takuzaki, and N. Hoshi: “Consideration of Operating Condition ofSecondary-side Converter of Inductive Power Transfer System for Ob-taining High Resonant Circuit Efficiency“, Trans. IEE Japan IndustrialApplication, Vol.132, No.10, pp.966-975, 2012 (in Japanese)
[12] T. Imura, and Y. Hori: “Maximizing Air Gap and Efficiency of MagneticResonant Coupling for Wireless Power Transfer Using Equivalent Cir-cuit and Neumann Formula“, IEEE Trans. Ind. Electron., Vol.58, No.10,pp.4746-4752, 2011