電気学会研究会資料（半導体電力変換 / 家電・民生 / 自動車合同研究会）
The papers of Joint Technical Meeting on “Semiconductor Power Converter”,“Home and Consumer Appliances” and “Vehicle Technology”, IEE Japan

SPC-23-259
HCA-23-049
VT-23-042

Basic Study on Ground Coil Current Control for WPT with theConsideration of Time-Varying Mutual-Inductance
Binh-Minh Nguyen＊, Ryo Matsumoto, Masahiro Mae, Toshiyuki Fujita, Hiroshi Fujimoto（The University of Tokyo）

Utilizing the circle criterion, this paper formulates the design condition for the ground coil current control in WPT. The condition
properly addresses the time-varying parameter of mutual inductance. The effectiveness of the proposed condition has been verified
by theoretical analysis, graphical tests, simulation tests, and experimental tests.
Keywords: absolute stability, current control, wireless power transfer, time-varying system.

1.

Introduction

To tackle the main problem of short mileage per charge forelectric vehicles (EVs), wireless power transfer (WPT) using
magnetic resonance coupling has been considered as one ofpromising technologies [1]. Especially, dynamic wireless powertransfer (DWPT) has attracted much interests recently [2]. As

Fig. 1. Example of mutual inductance map of DWPT coils.

shown in Fig. 1, the mutual inductance 𝑀 changes in accordance

inductance. The study [3] experimentally verified that the proposedwith the coil positions. Thus, 𝑀 is a time-varying parameter in

PI controller is robust to a sharp change of 50% in coupling

the DWPT system. Consequently, the coil currents might

coefficient. However, this performance was not supported by any

exaggeratedly exceed the rated value at the charging starting

theoretical result. Similarly, the study [8] focuses on the problem

moment, and they might fluctuate during the charging period. Suchof maximizing the energy transfer instead of stability analysis.

phenomena increase the device and maintenance costs. Therefore,

With respect to the aforementioned issues, this paper focuses on

many previous studies have focused on power transmission by

the primary side current control for DWPT with the S-S topology.

dealing with the primary side control [3] – [6]. A method toThis paper shows that the system can be modelled as a feedback

improve transient response of the start-up current control using theconnection of a time-invariant part and a time-varying part withsecondary side was presented [7]. A method, which simultaneously

sector-bounded characteristics. Based on the circle criterion, thisincorporates power and current control, was proposed in [8].

paper formulates the design condition that guarantees the absolute

Investigating the literature, we have recognized the following

stability of the system as the mutual inductance varies between a

two issues. First, from a practical point of view, it is meaningful tolower-bound and an upper-bound. The approach allows us ainvestigate a simple but effective controller that tracks the actual

graphical test of system stability, which is convenient for practicalcurrent with the reference value. Almost the previous studies

application. Utilizing the graphical test, this paper designs a PI

utilized the proportional-integral (PI) controller and neglects thecontroller which allows large position misalignment.

discussion on the suitability and effectiveness of such controller.
Second, from a theoretical point of view, it is essential to develop

2.

Modelinga systematic approach to design and analyze current control with

〈2･1〉

Derivation of the envelop modelFig. 2 shows

respect to the time-varying parameter 𝑀 . The study [6] onlyan S-S equivalent circuit model for the study of WPT. The self-

analyzed the worst case by assuming the coupling coefficientinductance and resonant capacitor of each coil are denoted as 𝐿

equals to zero. In [4], the one cycle control-proportionaland 𝐶 , respectively. The currents are denoted as 𝑖 . The

differential control was merely designed at a given mutual

subscripts 𝑖 = 1, 2 mean the primary and secondary sides,

SPC-23-259
HCA-23-049
VT-23-042
C1

i1

C2

i2

M



v1

L1

From (7) and (8), the envelop model is derived as follows:
RLV1  2L2

L2

RL

dV1  t dt

 4L1L2d 2 I1  t dt2

 2L1RL

dI1  t dt

 o M  I1  t  .. (9)2Applying the Laplace transformation to (9), the transfer function
from 𝑉 to 𝐼 can be written as:

1
RL
s
2 L1
4 L1L2

Fig. 2. S-S equivalent circuit model for DWPT.

respectively. The mutual inductance is denoted as 𝑀 . The load
resistance is denoted as 𝑅 . The capacitors are designed to satisfyGs 

s2 

the following resonance condition:〈2･2〉

 M 
RLs
2 L2
4 L1L221s   0.................... (10)s 2  1s   0

Block diagram representation

This study

o2 LiCi  1 ...................................................................... (1)examined the DWPT system with the parameters summarized inwhere 𝜔 is the resonant frequency. The primary voltage is the

Table 1. 𝑀

output of the inverter, and it is originally square waves. Thanks tostarting power transmission. Fig. 3 shows the Bode diagram ofthe bandpass characteristics of the WPT circuit, the currents are

𝐺(𝑠) with different values of M between 𝑀

sine waves at the resonant frequency. Let 𝑉 (𝑡), 𝐼 (𝑡) and 𝐼 (𝑡)To rigorously treat the time-variation of 𝑀 , our idea is tobe the fundamental amplitudes of 𝑣 (𝑡) , 𝑖 (𝑡) , and 𝑖 (𝑡) ,equivalently represent 𝐺(𝑠) as the feedback connection system inrespectively, the waveforms can be approximately expressed as

Fig. 4 where: s  0

1
P1  s   21
, P2  s   2, ................. (11)s  1ss  1s
4 L1 L2

v1  t   V1 sin ot .................................................... (2) I1 2   t   I1 2  sin o t 

Appling the Kirchhoff’s voltage law to Fig. 2, we havev1  t  

di1  t 1
i1  t  dt   L1  M 
C1 dt

 di  t  di2  t  M 1 ... (3)
dt 
 dt

 di  t  di  t  di  t  1
M  1  2    L2  M  2   i2  t  dt  RLi2  t  . (4)
dt
dt
dt
C2



is the minimum allowable mutual inductance for
and 𝑀

.

  t    2 M 2 ............................................................... (12)Table 1. Parameters of DWPT system used in this study.
Parameter
𝐿
𝐿
𝑅
𝜔
𝑀
𝑀
𝑉,
𝑇

MeaningPrimary coil inductance
Secondary coil inductanceLoad resistanceResonant frequency
Minimum mutual inductance
Maximum mutual inductance
Maximum voltage of inverter
Delay of sensor measurementValue
246 𝜇𝐻106 𝜇𝐻
5.7 W
2𝜋 × 85000 𝑟𝑎𝑑/𝑠
5 𝜇𝐻
20 𝜇𝐻
30 𝑉
20 𝜇𝑠

Since 𝜔 is commonly very big, the following approximations
can be obtained by neglecting the small terms of high orders [7].

11i1  t  dt   I1  t  sin ot  dt
C1 
C1



dI1  t 1 
cos  o t  dt  .................... (5) I1  t  cos ot   
oC1 
dt1
1 dI1  t 
I1  t  cos ot   2
sin ot 
oC1
o C1 dt

 dI  t  M  2 cos  ot   o I 2  t  sin ot  dt ............ (6) o MI 2  t  sin ot M

di2  t 
dt

Fig. 3. Bode diagram of 𝐺(𝑠) with 𝑀 ∈ [𝑀V1  t   2 L1

dI1  t dto MI1  t   2 L2 o MI 2  t  .......................................... (7)dI 2  t 
dt

 I1

C s


V1

P1  s 

P2  s P2  s  RL I 2  t  ...................................... (8)].

Controller

Substituting (2) into (3) and (4) with respect to (1), (5) and (6),the following equations are derived, respectively.

,𝑀

Gs

I1

 t 

Fig. 4. Block diagram of the primary current control system.

SPC-23-259
HCA-23-049
VT-23-042
3.

Stability condition of current control system
Problem setting〈3･1〉

The study of thispaper is based on the current control system shown in Fig. 4. How
to detect the starting moment of WPT [7] and how to calculate thereference current 𝐼 ∗ [8] is not the main goal of this paper. 𝐶(𝑠)is the controller to be designed. It outputs the voltage signal 𝑉
which can be realized by adjusting the width of switching pulsesof the primary side inverter. From (12), 𝜑(𝑡) varies between alower-bound 𝜑 = 𝜔 𝑀
and an upper-bound 𝜑 = 𝜔 𝑀.

Fig. 6. Definition of disk D.

Stability condition

The stability of theas shown in Fig. 7. This means the actual system does not satisfy

system in Fig. 4 can be discussed via the feedback connection of

the absolute stability condition. To validate this conclusion,

𝐻(𝑠) and 𝜑(𝑡) in Fig. 5, where:

simulation tests were conducted as follows with the reference〈3･2〉

P2  s H  s 

current of 1.5 A.

1  C  s  P1  s ................................................... (13)Case A: The positions of two coils are fixed, such that there isno misalignment between them. In this case, the mutual inductance

Applying the circle criterion with Theorem 7.2 in [9], thealways equals to the nominal value (see Fig. 8a). A good current
tracking performance can be realized as shown in Fig. 8b.

stability condition can be stated as follows:
Stability condition: The current control system is absolutelyCase B: The power transmission starts with a large misalignment

stable if the Nyquist plot of 𝐻(𝑗𝜔) does not enter disk D definedby 𝜑 and 𝜑 in Fig.6.

between two coils, such that the mutual inductance varies from the4.

System analysis with simulation demonstration〈4･1〉

Conventional PI control design

minimum value (see Fig. 8a). Consequently, the current vibrates
seriously (see Fig. 8b).

We

firstly examine a conventional design approach which neglects thevariation of the mutual induction. To this end, we consider the
WPT system with the parameters shown in Table 1. The nominaltransfer function from 𝑉 to 𝐼 is expressed as:
Gn  s   M n 
1s   0
,  0n ........................... (14)2s  1 s   0 n
4 L1 L2

For instance, 𝑀

2

is set as the value when there are no

misalignments between two coils ( 𝑀 = 19.5𝜇𝐻 ). By a fine-Fig. 7. Graphical test: conventional PI controller design.

tuning process using the nominal plant, we select the PI controller
𝐶(𝑠) = 𝐾 + 𝐾 /𝑠 with 𝐾 = 0.677 , 𝐾 = 2.041 × 10 . With
this selection, the nominal close-loop transfer function 𝑃
𝐶(𝑠)𝐺 (𝑠)/[1 + 𝐶(𝑠)𝐺 (𝑠)]hasstable

poles

(𝑠) =

−20.16 ±548.70𝑗 and −3.70 × 10 . However, 𝐻(𝑗𝜔) enters the disk D,(a) Mutual inductance.

(b) Envelope current.
Fig. 5. Equivalent diagram of the current control system.

Fig. 8. Simulation test: conventional PI controller design.

SPC-23-259
HCA-23-049
VT-23-042
Proposed PI control design〈4･2〉

SubstituteFor 𝑘 = 1 ∶ 𝑁(11) into (13) with the notice that 𝛽 = 𝛽 𝛼 , 𝐻(𝑠) is written asCalculate the PI gains using (19) with 𝜌 = (𝜌/𝑁)𝑘.

follows with the PI controllerGraphical test: Verify if the Nyquist plot of 𝐻(𝑗𝜔) doesH  s s2
 s  1   s  1K p s  1K i  .............................. (15)

not enter disk D or not.


required performance of settling time and overshoot, andIt can be seen that the poles of 𝐻(𝑠) are also the poles of𝑃 (𝑠) from 𝐼∗

to 𝐼 in the worst case such that the mutual

inductance is zero. The real and imaginary parts of 𝐻(𝑗𝜔) are:Re H 

 2  1 K i  11   2 

 11 K i  1  1K p   
2
1

2 2

1

i

2
1

The upper-bound is found to be 𝜌 = 3.6 × 10 . The fine-tuning
was performed with 𝑁 = 100 . This paper demonstrates the2p𝑘 = {2, 10, 28, 56, 100}for cleardiscussion. The graphical test and simulation test are summarized
in Figs. 9 and 10, respectively. At all steps, the Nyquist plot of𝐻(𝑗𝜔) does not enter disk D. Furthermore, it can be shown that:      K      K  2

the saturation of the inverter voltage does not occur.

results of some steps.................... (16) 2  12   1Ki     12 K p2 2 
2 2

2Im H 

Simulation test: Verify if the controller satisfies the2.................... (17)

is very poor. By increasing 𝜌 (i.e., 𝑘 = 10 ), theSolving Re = 0, it can be found that 𝐻(𝑗𝜔) intersects with
the imaginary axis at two points symmetrical about the real axis.

If 𝜌 is too small (i.e., 𝑘 = 2), the tracking performancetracking performance can be improved.



If 𝜌 is too big (i.e., 𝑘 = 56), the steady state error andSolving Im = 0, 𝐻(𝑗𝜔) is shown to intersect with the real

the settling time can be further enhanced. However, theaxis at two separate points. One point is the origin, and the otherovershoot is increased. For the safety of the DWPT

point is placed at the right-hand side of the imaginary axis. The
distance 𝐿 between two points is 𝑅𝑒 (𝜔 ), where:im 

system, this overshoot is not allowable.


If 𝜌 reaches the upper-bound at 𝑘 = 100 , thefluctuation happens due to the voltage saturation.

11 K i........................................................ (18)
1  1 K p

is the non-zero solution of the equation Im = 0.
From (15), 𝐻(𝑠) has a stable pole which equals to −𝛼 .
Placing other two poles at the stable value −𝜌, we have K  2 11 
2
s 2  1K p s  1K i   s      p
1 2 ................. (19) K i  1 
The maximum value of the inverter voltage should beconsidered when designing the feedback controller. Thus, 𝜌should be limited by a threshold 𝜌. If 𝜌 is chosen bigger than thisthreshold, the system might suffer fluctuation due to the saturationof the inverter voltage. Recently, we proposed in IEEE-IECON
Fig. 9. Graphical test: proposed PI controller design.

2023 a method to rigorously design the feedback controller withrespect to the saturation of the control signal [10]. This paper aims
at a practical design for current control in DWPT. Therefore, we
considered the following strategy for tuning 𝜌 . The selected 𝜌a simulation test by performing a simulator that imitates the real

I1 [A]

must satisfy the graphical test given by the stability condition, and
DWPT system shown in Table 1. In this study, Matlab/Simulink
R2021b is used to establish the simulator. The simulation test is thesame as Case B presented in the previous sub-section.
Step 1 (Rough tuning): By increasing 𝜌 , we find the upperbound 𝜌 such that the current start to fluctuate.
Step 2 (Fine tuning): The range from 0 to 𝜌 divided by N steps.

Fig. 10. Simulation test: proposed PI controller design.

SPC-23-259
HCA-23-049
VT-23-042

Fig. 11. Relationship between 𝐿 and desired poles.

To compromise the aforementioned trade-off, it is possible toFig. 13. Testbench system.

select the desired pole at 𝑘 = 28 (𝜌 = 3.6 × 10 ). With thisselection, the system only suffers a small overshoot of 13% even if
the power transmission starts at 𝑀.

control period of the system is 5.9 𝜇𝑠.
〈5･2〉

The relationship between the distance L and the desired poles

Experimental results

Several tests weperformed, and there results are summarized in Fig. 14.

are shown in Fig. 11. The increasing of 𝜌 results in the decreasing

Test 1 (Without current control): The longitudinal and lateralof 𝐿. When 𝜌 = 𝜌 or 𝑘 = 𝑁 = 100, the distance L is reduced

misalignments are set to be zero. The duty cycle of the inverter isto the minimum value of 0.0033. With the selected poles at 𝑘 =fixed at 0.5. As can be seen in Fig. 14a, the current reaches a peak

28, the distance is 0.0164. With 𝑘 = 2, the distance is increased

value which is more than 2 times of the reference value. The

to 0.2475. From this discussion, the distance 𝐿 can be used as an

current overshoot would be more serious if the coil misalignments

index to select the desired poles to compromise the trade-off

are non-zero.

between steady state error and transient overshoot.

Test 2 (PI controller designed by conventional approach): This

Varying the coil inductances and the load resistance in the range

test is to evaluate the PI controller obtained in sub-section 4.1. Theof [+20% , −20%] around the nominal values, the Nyquist plots

power transmission starts at the longitudinal and lateralof 𝐻(𝑗𝜔) with 𝑘 = 28 are shown in Fig. 12. It can bemisalignments of 450 mm and 60 mm, respectively. Hence, therecognized that, the proposed PI controller guarantees a certain

mutual inductance is only 5.5 𝜇𝐻 . As shown in Fig. 14b, thelevel of robustness under model uncertainty.

current fluctuates considerably as the power transmission starts.

5.

This result, which is a failure example of the conventional

Experimental validation
Experimental setting

approach, matches with the theoretical analysis in previous section.
Fig. 13 describes

Test 3 (PI controller designed by proposed approach): The

the test-bench with the nominal parameters shown in Table 1. Thispower transmission starts with large longitudinal and lateral

testbench allows us to adjust the relative position between themisalignments, which are similar to that of Test 2. As can be seen

primary coil and secondary coil. A FPGA board is used to calculate

in Fig. 14c, the envelop current quickly follows the reference

the envelop of the current. The PE-Expert4 computer serves as thevalue. The overshoot is reduced to approximately 75% in

main controller. C programing is used to perform the feedback

comparison with that of Test 1.

〈5･1〉

current control, display and store the experimental data. The

6.

Conclusions

Using the Circle criterion, this paper formulates the stability
condition for the ground coil current control in DWPT. The
condition properly addresses the time-varying parameter of mutual
inductance. Thanks to the graphical test, the condition allows us apractical approach to analyze and tune the current controller.
Testbench experiments show that stable current control can beachieved with less overshoot, even if the system suffers a large
misalignment between two coils.
However, this paper only performed power transfer in a shortperiod of time. Also, the compensation of current delay was not
Fig. 12. Robustness of the proposed PI controller with 𝑘 = 28.

considered in this paper. In future, we will develop the controller

SPC-23-259
HCA-23-049
VT-23-042(3)(a) Test 1: without current control.

(b) Test 2: PI controller designed by conventional approach.

(c) Test 3: PI controller designed by proposed approach.
Fig. 14. Experimental results of envelope current.

to compensate the delay of sensor measurement. Experiments withhigher power using moving coil will be conducted to further
evaluate the proposal of this paper. Besides, we are interested in
the power control in the outer loop to properly generate thereference current. Furthermore, the proposed method will be
extended to current control in both primary and secondary sides.
Acknowledgements
This research is based on results obtained from a project,
JPNP21005, subsidized by the New Energy and Industrial
Technology Development Organization (NEDO).

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