Stability Analysis of Constant Power Loadand Load Voltage Control Method for

Wireless In-Wheel MotorDaisuke Gunji1, Takehiro Imura2, and Hiroshi Fujimoto2

1 The University of Tokyo / NSK Ltd., Japan2 The University of Tokyo, Japan

Abstract—In order to improve motion control performanceof electric vehicles (EVs), In-Wheel Motor (IWM) is the mostpreferred electric motor arrangement. Our research group hasproposed Wireless In-Wheel Motor (W-IWM) concept to solvesome technical problems of an IWM, which are reliability andsafety issues of power and signal wires. In this research, wedid a stability analysis of Series-Series compensated wirelesspower transfer via magnetic resonance coupling with constantpower load. Analysis result is verified by circuit simulation. Also,we propose a load voltage control method using a secondaryconverter. The effectiveness of the proposed control method hasbeen verified by simulation and experiment using small powertest equipment.

I. INTRODUCTION

Wireless power transfer (WPT) technology has receivedbroad attention in recent years especially for automotiveapplications. Typical application of WPT for automotive arewireless power charging to electric vehicles (EVs) whiledriving [1], [2] and while parking [3], [4]. Magnetic reso-nance coupling [5] is the most preferred method of WPT forautomotive applications because it can achieve high powertransfer efficiency if transmitting distance is several tens ofcentimeters.

Advantages of EVs are not only environmental performancebut also motion control [6]. In-Wheel Motor (IWM) is suitablestructure because it can achieve high control bandwidth andindependent drive of each wheel. However, there are reliabilityand safety issues because of power and signal wires. In order tosolve that problem, our research group has proposed WirelessIn-Wheel Motor (W-IWM) [7]. Concept of W-IWM is shownin Fig. 1. Electric power is bidirectionally transmitted from theprimary side (on-board side) to the secondary side (in-wheelside). Signals are also communicated by wireless protocol.Then, completely wireless structure can be achieved. W-IWMis also applicable for dynamic wireless power supply fromcoils which are set under road.

In the case of W-IWM, electrical load is permanent magnetsynchronous motor which is driven by a voltage type PWMinverter. Then, the load is considered as a constant powerload. A number of previous researches exist about constantpower load on power electronics field. In a DC-DC converterapplication, a constant power load shows unstable behaviorbecause it has negative resistance characteristics [8]. Then,stability analysis is needed in the W-IWM application.

Vehicle chassis

Wheel

Power wires

Signal wires

IWM

Ground

(a) Conventional IWM.

Wireless signalcommunication

Vehicle chassis

Wheel

IWM

Ground

powerPrimary

coil

(b) Wireless IWM.

Fig. 1. Concept of Wireless In-Wheel Motor.

In this research, we analyze stability of a constant powerload on wireless power transfer by magnetic resonance cou-pling which is Series-Series compensated circuit using sim-plified circuit model. Analysis result shows a load voltage ofa constant power load is unstable. Analysis result is verifiedby circuit simulation and experiment. We also propose a loadvoltage stabilization control method. The effectiveness of theproposed method is verified by simulation and experimentusing small power test equipment.

II. WIRELESS IN-WHEEL MOTOR (W-IWM)

A. Structure

Circuit structure of the W-IWM is shown in Fig. 2. Boththe primary and the secondary resonance capacitors are seriesconnection to coils (Series-Series compensated). The primaryside (on-board side) consists of a buck-boost converter anda full-bridge inverter. Battery voltage is converted to highfrequency which is same as the resonance frequency usingthe primary inverter. The electric power is transmitted tothe secondary side (in-wheel side) via magnetic resonancecoupling. Received electric power is converted to DC by asecondary converter which is also full-bridge circuit. Finally,a PWM inverter drives a permanent magnet synchronous motor(PMSM).

Some previous researches use a full-bridge diode rectifierand a DC-DC converter on the secondary side [9], [10].However that circuit structure is not available for bi-directionalpower transfer and downsizing. On the other hand, the abovecircuit structure is capable of bi-directional power transfer

primary inverter C1

L1

C2

L2

Lm

E

buck-boostconverter

secondaryconverter

Cs

PWM inverter

MVdc

>

CPU CPU

wirelesscommunicationtorque

command

PMSM

Fig. 2. Circuit structure of W-IWM.

(a) Experimental vehicle. (b) First trial unit.

Fig. 3. Experimental vehicle and first trial unit.

because circuit structure is symmetric between the primaryside and the secondary side.

B. First trial unit

Fig. 3(a) shows the experimental electric vehicle FPEV4-Sawyer which has developed by our research group. Drive unitof the vehicle can be easily exchanged. Then, it is possible tocompare some types of drive units on the same vehicle.

We are now developing a first trial unit of W-IWM which isshown in Fig. 3(b). Final target of electrical power is 48 kWusing four wheels, but in the first trial unit, maximum poweris 6.6 kW using two wheels. By using the first trial unit, weestablish control methods of power conversion circuit, and coildesign.

III. STABILITY ANALYSIS OF THE LOAD VOLTAGE

A. Simplified circuit model

A SS-compensated WPT circuit can be expressed using T-type equivalent circuit as shown in Fig. 4 [11]. Output voltageof the primary inverter is actually square wave. In this research,we focus on the fundamental wave components v11. In thesimilar way, we focus on the fundamental wave componentsof the secondary converter input voltage v21. In addition, weassume that the secondary current i21 is sinusoidal wave andin phase with v21.

The impedance matrix of the boxed circuit on Fig. 4 isexpressed as follows:

Z =

[R1+

1jω0C1

+jω0L1m jω0Lm

jω0Lm R2+1

jω0C2+jω0L2m

], (1)

where ω0 is angular frequency, R1,2 are respectively resistanceof coils, C1,2 are respectively resonance capacitance, L1,2 arerespectively coil inductance, Lm is mutual inductance, L1m =

TABLE ISPECIFICATIONS OF W-IWM.

Final target first trial unitNumber of IWM 4 2Max total power 48 kW 6.6 kWMax total torque 1300 Nm 475 Nm

R1 R2C1 C2L1-Lm L2-Lm

Lm

i11

v11

i21

v22

Fig. 4. T-type equivalent circuit of SS-compensated WPT circuit.

L1 − Lm, and L2m = L2 − Lm. We assume that the circuitmeets following resonance condition:

ω0 =1√L1C1

=1√L2C2

. (2)

Then, Z is expressed as follows:

Z =

[R1 jω0Lm

jω0Lm R2

]. (3)

From eq.(3), the secondary current i21 is caluculated asfollows: [

i11i21

]= Z−1

[v11

v21

], (4)

i21 = − jω0Lmv11 − v21

R1R2 + (ω0Lm)2 . (5)

Assuming that v21 is π2 phase advance to v11, RMS value of

i21 is expressed as follows:

I21 =ω0LmV11 −R1V21

R1R2 + (ω0Lm)2 , (6)

where V11 is RMS value of v11, and V21 is RMS valueof v21. If the secondary converter is operated as full-bridgerectifier, average output current of the secondary converter isis expressed as follows:

is =2√2

π

ω0LmV11 −R1V21

R1R2 + (ω0Lm)2 . (7)

Generally, ω0Lm is much bigger than R1, it can be assumedthat is is constant current source at a operating point.

If the secondary converter is operated as full-bridge rectifier,input voltage of the secondary converter is square wave. Focuson the fundamental wave component of that, RMS value ofthe secondary converter input voltage V21 is expressed by thefollowing equation.

V21 =2√2

πvL, (8)

Cs vL

iL

Constant

power

loadis

Equivalent currentsource

Fig. 5. Simplified circuit of the secondary side.

where vL is the load voltage. At a operating point, equivalentload resistance of the constant power load is expressed asfollows:

RL =vLiL

=vL

2

pL, (9)

where iL is the load current, and pL is the load power. Thesimplified circuit of the secondary side is shown in Fig. 5.

B. Stability analysis

In this section, we derive the transfer function of thesimplified circuit, and then analyze the stability of the circuit.

Circuit equation of the simplified circuit is expressed asfollows:

iL = Is − CsdvLdt

, (10)

where Cs is capacitance of the smoothing capacitor. The loadcurrent iL is expressed as follows because the load is constantpower load.

iL =pLvL

. (11)

By substituting eq.(11) into eq.(10), the following equation isderived.

dvLdt

= − pLCsvL

+isCs

. (12)

By linearizing eq.(12) around equilibrium point, the followingequation is derived.

d∆vLdt

=pL∆vL

CsVL2 +

∆isCs

, (13)

vL = VL +∆vL,

is = Is +∆is.

Then, the transfer function P∆(s) form ∆is to ∆v2 is derivedfrom Laplace transform of eq.(13):

P∆(s) =∆vL(s)

∆is(s)=

1

Cs

(s− pL

CsVL2

) . (14)

The pole of the P∆(s) is positive value in any case becausepL > 0 and VL > 0. Then, P∆(s) is unstable system. From theabove stability analysis, in the case of the constant power load,the load voltage of WPT is unstable. According to eq.(14), thepole becomes faster in any of the following cases:

1) Capacitance of the smoothing capacitor Cs is smaller.2) The load power pL is bigger.

R1 R2C1 C2L1-Lm L2-Lm

Lmv1 iL

CsvL

Constant power load

Diode bridge rectifier

pL = vL iL

Fig. 6. Circuit simulation model.

TABLE IICIRCUIT PARAMETERS.

Primary SecondaryCoil resistance R1,2 0.547 Ω 0.535 ΩCoil inductance L1,2 166 µH 167 µH

Resonance capacitor C1,2 19.9 nF 19.9 nFCoil size 200 x 200 mmCoil gap 100 mm

Mutual inductance Lm 21.8 µHcoupling coefficient k 0.132Resonance frequency 87.6 kHz

Smoothing capacitor Cs 1000 µF

3) The operating load voltage VL is smaller.Therefore, the value of Cs is designed with not only the loadvoltage ripple but also the control performance and stabilityof the load voltage.

C. Circuit simulation

Circuit simulation has been performed to verify the stabilityanalysis result using MATLAB Simulink SimPowerSystems.The simulation circuit model is shown in Fig. 6. The primaryvoltage is modeled by AC voltage source, the secondary con-verter is modeled as full-bridge diode rectifier, and the constantpower load is modeled using controlled current source. Circuitparameters are listed in TABLE II. These parameters aresame as the test equipment. Operating conditions and initialconditions are listed in TABLE III. The amplitude of theprimary voltage is set by trial and error. Simulation is stoppedwhen vL close to zero because it is impossible to simulateconstant power load.

Simulation results are shown in Fig. 7(a) and (b). Fig.7(a) shows the load voltage vL. Fig. 7(b) shows the loadcurrent iL. According to Fig. 7(a) and (b), the load voltageand the load current are diverged. Compare blue lines onfigures to red lines, time variation of the load voltage andcurrent are completely different even though difference of v1is very small. In addition, v2 decrease faster so that v2 becomessmall. That result agrees with stability analysis result. Fromabove simulation results, it is shown that SS-compensatedWPT circuit with a constant power load is unstable.

IV. LOAD VOLTAGE CONTROL METHOD

A. Operation of the secondary converter

In this chapter, we propose a load voltage feedback controlmethod. The feedback loop is closed on the secondary sideto prevent the influence of the signal communication delaybetween the primary side and the secondary side.

0 0.05 0.1 0.15 0.20

5

10

15

20

25

30

time [s]

Load v

oltage [V

]

V

1=13.9727 Vrms

V1=13.9726 Vrms

(a) Load voltage vL.

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

time [s]

Load c

urr

ent [A

]

V

1=13.9727 Vrms

V1=13.9726 Vrms

(b) Load current iL.

Fig. 7. Circuit simulation results.

TABLE IIISIMULATION CONDOTIONS.

Condition ValueLoad power 15 W

Initial load voltage 15 VInitial load current 1.0 A

Operating frequency 87.6 kHz

In order to control the transmitting power on the secondaryside, the secondary converter is operated with the followingtwo operation modes:

• Rectification mode: All switching devices are OFF. Thesecondary converter is operated as full-bridge rectifier bydiodes which are in parallel with switching devices asshown in Fig. 8 (a). Then, electric power is transmittedto the load. If the received power is bigger than the loadpower, the load voltage increases.

• Short mode: Both lower arm switching devices are ON.Electric power is not transmitted to the load because thesecondary coil circuit is shorted as shown in Fig. 8 (b).Then, the load voltage decrease because the load poweris supplied from the smoothing capacitor.

By switching above two operation modes depending on theload voltage, the load voltage can be controlled to a referencevalue. Hysteresis comparator is used to switch operationmodes in previous research [12]. In this research, we proposeanother control method.

B. Load voltage control method

Assuming that the secondary current is sinusoidal wave andfocus on the fundamental wave component, RMS value ofthe secondary current I2 is approximated by the followingequation:

I2 ≃ω0LmV11 − 2

√2

π R1VL

R1R2 + (ω0Lm)2 , (15)

where VL is the load voltage. On each operation mode, outputcurrent of the secondary converter Is are expressed as follows:

Is =

2√2

π I2 (Rectification mode)0 (Short mode)

(16)

One cycle time of above two operation mode is set to aconstant time period. Assuming that the load voltage hardly

Load

L2

C2

Cs

S1 S3

S2 S4

(a) Rectification mode.

Load

L2

C2

Cs

S1 S3

S2 S4

(b) Short mode.

Fig. 8. Operation modes of the secondary converter.

CPIVL

*

eq.(18)PL

eq.(24)

WPT

Plant

P(s) VL

eq.(19)PL

VL

*

Wireless signalcommunication

Primary side

Secondary side

Load plantFB controller

FF controller

FF controller

+ ++

-Conversion ratio

calculation

IL

*

V1cmd

αcmd IsIscmd

Fig. 9. Block diagram of the proposed control method.

changes between one period, average output current of thesecondary converter Is between one time period is expressedas follows:

Is =2√2

πI2α, (17)

where α is conversion ratio, that is time ratio of the rectifica-tion mode between one period. By manipulating α, the loadvoltage control can be achieved.

Block diagram of the proposed control method is shown inFig. 9. The primary-side controller is a feedforward controller,and the secondary-side controller is a two-degrees-of freedomcontroller.

Relation between the load power and the load currentreference I∗L is expressed as follows:

I∗L =PL

V ∗L

, (18)

where V ∗L is reference value of the load voltage. From eq.(17)

and eq.(18), the primary voltage command V1cmd is calculatedas following equation:

V1cmd =π

2√2

PL

R1R2 + (ω0Lm)

2

ω0LmV ∗Lαr

+2√2

π

R1V∗L

ω0Lm, (19)

where αr is reference value of the α. That means, if there areno model error, transmitting power becomes same as the loadpower when α = αr and V11 = V1cmd. Therefor, α indicatesmargin of the power transfer for model error, e.g. variation ofthe coupling coefficient due to coil position misalignment andgap variation.

The feedforward controller of the secondary side is same aseq.(18). The feedback controller of the secondary side is PIcontroller. The transfer function of the equivalent load plant

DC voltage

source

power conversion circuits

primary coilsecondary coil

(a) Overview.

Cs

IL

VL

Secondary converter

Primaryinverter Lm

L1 L2

C1 C2

EElectric

load

4×4×

DSP

α

(

(b) Circuit structure.

Fig. 10. Experimental equipment.

is expressed as follows:

P (s) =RL

RLCss+ 1, (20)

where RL is equivalent load resistance which is expressed asfollowing equation.

RL =V ∗L2

PL. (21)

Then, the PI controller is designed by pole placement method.

Kp =2pRLCs − 1

RL, (22)

Ki = p2Cs, (23)

where Kp is proportional gain, Ki is integration gain, and pis closed loop poles (multiple root). Therefore, the feedbackcontroller is variable with the load power.

Manipulated variable of the controller is average outputcurrent of the secondary converter Iscmd. From eq.(15) andeq.(17), the conversion ratio command of the secondary con-verter αcmd is calculated by following equation:

αcmd =π

2√2

R1R2 + (ω0Lm)2

ω0LmV1cmd − 2√2

π R1V ∗L

Iscmd. (24)

V. SIMULATION AND EXPERIMENT

In this chapter, we verify the effectiveness of the proposedcontrol method by simulation and experiment.

A. Experimental equipment

Experimental equipment is shown in Fig. 10(a). Circuitof the equipment is shown in Fig. 10(b). The equipmentconsists of a DC power supply, a primary inverter, primaryand secondary coils, a secondary converter, an electric load(PLZ1004W: KIKUSUI), and a DPS (DS1104: dSPACE).Circuit parameters are listed in TABLE II. The load voltage ismeasured on the DSP, and conversion ratio of the secondaryconverter α is controlled by the proposed method. Primaryvoltage is set by changing DC supply voltage E.

TABLE IVSIMULATION CONDITIONS.

Parameter ValueOperation mode time period 2 ms

Load power PL 2 WLoad voltage reference V ∗

L 10 VConversion ratio reference αr 0.7Closed loop pole placement 8 Hz

B. Simulation

We did a simulation using MATLAB Simulink SimPow-erSystems. Circuit model is same as Fig. 6 except for thesecondary converter. On resistance of switching devices andforward voltage of body diodes are ignored. A first order low-pass filter is applied to the feedback value of VL in order tosuppress ripple. Cutoff frequency of the filter is 100 Hz. Othersimulation conditions are listed in TABLE IV. In the case ofwithout control, conversion ratio α is set to the average valueof α which is with proposed control.

Simulation results are shown in Fig. 11(a) to (d). Fig. 11(a)shows time variation of the load voltage VL. Without theproposed method, VL decreased rapidly. On the other hand,the load voltage is successfully controlled to the referencevalue by the proposed method. Fig. 11(b) shows conversionratio of the secondary converter α. Manipulation of α by thefeedback controller is small, but that is significant to controlVL. Average value of α is bigger than αr. It is consideredthat a transient response of the secondary current is the causeof the error. Fig. 11(c) shows close-up of VL, and Fig. 11(d)shows the secondary converter output current Is. Accordingto these figures, electric power is received and VL increaseson the rectification mode, on the other hand, Is is almost zeroand VL decreases on the short mode. The average value of thesecondary converter output current is controlled to I∗L by theproposed control method. From these simulation results, theeffectiveness of the proposed method is verified.

C. Experiment

Experiment has been carried out using the experimentalequipment. Experimental condition is same as the simulationcondition as listed in TABLE IV. Cutoff frequency of thelow-pass filter to suppress the measurement value of VL isset to 20 Hz. On the experiment, feedforward value of the

0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

time [s]

Load v

oltage [V

]

w/

w/o

(a) Load voltage VL.

0 0.2 0.4 0.6 0.80.7249

0.725

0.7251

0.7252

0.7253

0.7254

time [s]

Convers

ion r

atio α

[−

]

w/

w/o

(b) Conversion ratio α.

0 0.002 0.004 0.006 0.008 0.019.7

9.8

9.9

10

10.1

10.2

10.3

time [s]

Load v

oltage [V

]

w/

(c) Load voltage (zoom).

0 0.002 0.004 0.006 0.008 0.01−0.2

0

0.2

0.4

0.6

0.8

1

time [s]

Convert

er

outp

ut curr

ent [A

]

w/

w/ (ave.)

(d) Converter output current Is.

Fig. 11. Load voltage control simulation results.

0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

time [s]

Load v

oltage [V

]

w/

w/o

(a) Load voltage VL.

0 0.2 0.4 0.6 0.80.694

0.696

0.698

0.7

0.702

0.704

0.706

time [s]

Convers

ion r

atio α

[−

]

w/

w/o

(b) Conversion ratio α.

0 0.002 0.004 0.006 0.008 0.01−0.3

−0.2

−0.1

0

0.1

0.2

0.3

time [s]

Rip

ple

of th

e load v

oltage [V

]

w/

(c) Load voltage ripple (zoom).

0 0.002 0.004 0.006 0.008 0.01−0.2

0

0.2

0.4

0.6

0.8

1

time [s]

Convert

er

outp

ut curr

ent [A

]

w/

w/ (ave.)

(d) Converter output current Is.

Fig. 12. Load voltage control experimental results.

primary voltage by eq.(19) is insufficient due to dead-time ofthe primary inverter, forward voltage of body diodes on thesecondary converter, and modeling error. Therefore, we adjustthe DC voltage E to be α = αr when with the proposedcontrol.

Experimental results are shown in Fig. 12(a) to (d). Theseresults are in good agreement with simulation results. The loadvoltage is successfully controlled to the reference value by theproposed method. According to Fig. 12(b), manipulation of αis little bigger than the simulation result. Fig. 12(c) showsripple of the load voltage. Ripple width and time variationfit in well with the simulation result. Fig. 12(d) shows outputcurrent of the secondary converter. The current is successfullycontrolled by two operation modes.

From the above simulation and experimental results, theeffectiveness of the proposed load voltage control method isverified.

VI. CONCLUSION

In this research, we did a stability analysis of SS-compensated WPT circuit with constant power load. Accord-ing to the analysis result, that system is unstable. In orderto stabilize the load voltage of the constant power load, wepropose a load voltage feedback control method using thesecondary converter. The effectiveness of the proposed methodis verified by simulation and experiment. Future work is im-provement of rectification method of the secondary converter.

ACKNOWLEDGMENT

This research was partly supported by the Ministry ofEducation, Culture, Sports, Science and Technology grant(number 26249061).

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