Load and mutual inductance estimation of wireless

power transfer system for electric vehicles

Xiulan Liu Yanxia Chen Shufan Li*
State Grid Beijing Power Research State Grid Beijing Power Research Key Laboratory of Power Electronics
Institute Institute and Electric Drives, IEECAS
Beijing, China Beijing, China Beijing, China
xiulan83@163.com chenbepc@163.com lishufan@mail.iee.ac.cn

Qian Zhang Zhimeng Liu Yuan Jin

State Grid Beijing Power Key Laboratory of Power State Grid Beijing Power
Research Institute Electronics and Electric Drives, Research Institute
Beijing, China IEECAS Beijing, China
zq15120075877@163.com Beijing, China 18600105060@163.com
liuzhimeng@mail.iee.ac.cn

Abstract—Wireless power transfer (WPT) system has resonant circuit, therefore the estimation errors may increase
gathered attention for years with its safety, convenience and when the system are put into practical use. Wang et al. [6]
flexibility. Control of the whole system has always been a key sampled the peak value of the primary resonant current by
issue for WPT technology. A lot of methods of parameter controlling the switches of the IGBTs in a voltage-fed IPT
estimation have been presented in relevant literature, providing system, and estimated the equivalent resistance load by
accordance for controlling. In this paper, an online parameter analyzing the energy transfer procedure. Both simulation and
estimation method for WPT system is proposed based on the experiments verified the theory that the heavier the load, the
measurement of input voltage and current of the inverter. A DC
faster the drop rates of the primary resonant current. This
input side model for the inverter was built, and the equivalent
estimation method is limited to an IPT system for kitchen
load of the inverter can be determined by the input voltage and
current. Then, by studying the characteristics of the tangent of
appliances, and only suitable for the voltage-fed system.
the reflected angle, a curve fitting method was proposed to Experiments have shown that the accuracy of this load
estimate the system load. The mutual inductance of coils was also detection method is more than 85%. Yin et al. [7] built a
estimated to provide information of the system. The experiments mathematical model of an n-Coil wireless power transfer
results showed that the proposed method can reduce the system, and estimated the load impedance, output voltage,
estimating error of the load to 7%. output current, output power and the loop currents by
measuring the input voltage and current. Such method was
Keywords—Online parameter estimation, DC input side verified with a wireless power domino-resonator system
measurement, Load and Mutual inductance estimation, Wireless consisting eight coil resonators. Despite of the high accuracy
power transfer system. of this load estimation method, it is only based on the ideal
condition of the WPT system.
I. INTRODUCTION Another important parameter for the control of WPT
Over the past decade, wireless power transfer (WPT) system is the mutual inductance of coils. The mutual
technology has developed rapidly because of its safety, inductance may change due to the coils misalignment, the
convenience and flexibility [1]-[4]. vertical distance change of two coils, or the change of coils
parameters. Once the mutual inductance changes, not only the
To keep a WPT system operating steadily and efficiently, power transfer efficiency would be affected, but the accuracy
key system parameters are often needed to know. However, of parameter estimation would be damaged. Recently, some
some parameters are difficult to measure directly. For example, researches have been done to study the change of mutual
an EV’s battery can be taken as an equivalent load of a WPT inductance of coils. In [8], the high-misalignment condition is
system, and its value varies during the charging process. Also, taken into consideration when designing the compensation
the lateral misalignment of coupling coils can reduce the topology. Takehiro Imura et al. calculated the mutual
mutual inductance of coils, which is also hard to monitor in inductance with Neumann formula and demonstrated it the
the charging process. same with electromagnetic field analysis results [9]. In [10],
To solve the above problems, some researches have been the coupling coefficient of the WPT system is identified by
focused on the estimation of the load parameters of WPT measuring the DC variable including the information of the
systems [5-7]. In [5], the output voltage of the IPT (Inductive load. However, the information of the load is usually
Power Transfer) system is estimated by calculating the track transferred via wireless communication, which can be
voltage of the primary side. Also, a phase error that can be interfered and also increase the cost of the system. Therefore,
used in controlling the track current is estimated by comparing a method estimating both the load and the mutual inductance
the estimated output voltage and the reference (Vref), which is in need.
represents the required output voltage. However, this
estimation method were based on the assumption of an ideal

This work was supported by the Science and Technology Project of State
Grid Corporation of China under Grant Nos. 52020118000L.

978-1-7281-3153-5/19/$31.00 ©2019 IEEE

 This paper presented an online parameter estimation U ab = U d (1)
method for WPT system based on the measurement of input
voltage and current of the inverter. A DC input side model for Id = I p (2)
the inverter was built, and the equivalent load of the inverter
can be determined by the input voltage and current. Then, by (2) Sa = 0, Sb = 1
studying the characteristics of the tangent of the reflected
angle, a curve fitting method was proposed to estimate the U ab = −U d (3)
system load. The mutual inductance of coils was also
estimated to provide information of the system. Id = −I p (4)

II. DC INPUT SIDE ANALYSIS Then, the input current Id2 and output voltage Uab can be
described as:
A. Modeling of the inverter
U ab = (Sa - Sb )U d (5)
In a wireless power transfer system shown in Fig.1, a high
frequency inverter is used to transform the DC power. To
deeply analyze the relation between the DC input current and I d2 = (Sa - Sb )I p (6)
the inverter output current, an inverter model is built, as
The Fourier transform of Sk is:
shown in Fig.2. The input voltage is labeled as Ud, and a
capacitor Cd is used to stabilize the input voltage of the 1 2  1
inverter. An R-L series(Req and Leq) is used to simulate the Sa = +  sin(n  t) (7)
2  n =1,3,5 n
load of the inverter. The input current of the inverter is Id2 ,
while the current through the voltage source is Id1 . The output 1 2  1
Sb = −  sin(n  t) (8)
current of the inverter is Ip, and the voltage of the two ports of 2  n=1,3,5 n
the inverter are Ua and Ub respectively.
where ω is the angular frequency of the switches.
Id L11 IP
CP CS
L21 IL
The output current of the inverter Ip(fundamental
component) is
Ud
DC Cd Inverter Zeq C11 ZSP ZS C21
4U d sin(wt -  )
CL RL

Ld RLP
LP LS
RLS Ip = (9)
MPS  R 2 +(wL)2

Fig.1. WPT system where  = arctan( wLeq Req ) is the phase angle of Ip compared
In a wireless power transfer system, a high frequency to the fundamental output voltage.
inverter is used to transform the DC power. To deeply analyze
The input current Id2 can be derived with(6)-(9)
the relation between the DC input current and the inverter
output current, an inverter model is built, as shown in Fig.2. 8U d 
1
The input voltage is labeled as Ud , and a capacitor Cd is used
Id 2 =
 2
R + ( L)
2 2
 n cos (n − 1) t +   − cos (n + 1) t −  
n=1,3,5
(10)
to stabilize the input voltage of the inverter. An R-L series(Req
and Leq) is used to simulate the load of the inverter. The input If we take into consideration only the dc component and
current of the inverter is Id2 , while the current through the two-time-frequency component in Id2 , the input current Id2 can
voltage source is Id1 . The output current of the inverter is Ip, be shown as
and the voltage of the two ports of the inverter are Ua and Ub
8U d  1 
respectively. Id 2 = cos  - cos(2t -  ) + 3 cos(2t +  )  (11)
 2
R + ( L)
2 2

Id1 Id2
Leq Req Further, we can divide Id2 into two parts, the dc component
Ip
Ua Id1 and two-time-frequency component Idac, which can be
Ud Ub
Cd described as
8U d cos 
Id 1 = (12)
Fig.2. Inverter model of the WPT system  2
R 2 + ( L)2

With the use of state space averaging method, we analyze 8U d  1 

the working principle of the inverter in a cycle. A switching
Idac = − cos(2 t −  ) + 3 cos(2 t +  )  (13)
 2
R + ( L)
2 2

function Sk(k=a,b) is used to describe the on/off state of the

four switches of the inverter. The values of Id2 and Uab change A DC input side model of the inverter can be built based
with the change of Sk. In fact, there are two conditions: on the above analysis. As Fig.3 shows, the current through R1
is the dc component Id1, and the current through R2 is the two-
(1) Sa = 1, Sb = 0 time-frequency component Idac.
 In II, we found the relation between the equivalent load of
Idac the inverter and the input voltage and current. However, it is
Id1 R2 obvious that the real part and imaginary part of the load cannot
Ud R1 be derived by the parameter A. In Fig.4, the phase angle of the
Ue inverter is taken into consideration when analyzing the
relation between A and the equivalent load of the inverter, in
which A is determined by
Fig.3. DC input side model
8U d cos 2 
A= (23)
According to (12),  2 I d1

 2 R 2 + ( L)2 100
R1 = (14)
8cos  80

8U d R2  1 
Ue = Ud + cos(2 t −  ) − cos(2 t +  )  60

R (Ω)
 (15)
 2 R 2 + ( L)2  3  θ=0.72°

eq
40 θ=15.12°
θ=29.52°
Equation (15) can be simplified as 20
θ=43.92°
θ=58.32°
θ=72.72°
8U d R2 K θ=87.12°
Ue = Ud + cos(2 t −  ) (16)
0
20 40 60 80 100 120
 2
R 2 + ( L)2 A

(a)
where
60
4 16
K= cos 2  + sin 2  (17) 50
9 9 40

30
 = arctan( 2tan ) (18)

X (Ω)
20 θ=0.72°

eq
θ=15.12°
10 θ=29.52°
B. Inverter equivalent load analysis 0
θ=43.92°
θ=58.32°

The real part and imaginary part of the inverter equivalent -10 θ=72.72°
θ=87.12°
load are labeled as Req and Xeq respectively. The cosine value -20
20 40 60 80 100 120
of the phase angle of the inverter load is A

(b)
Req
cos  = (19) Fig.4. Relation between “A” and equivalent load of the inverter.(a)real
Req 2 + X eq 2
part;(b)imaginary part.

Therefore, (12) can be rewritten as With the knowledge of A and the phase angle θ, we can
derive the real part and imaginary part of the equivalent load
8U d Req of the inverter by two dimensional surface fitting method. As
Id 1 = (20)
 ( Req 2 + X eq 2 )
2
is shown in Fig.5 and Fig.6, parameter A and phase angle θ
are input values, and the real part(Req) and imaginary part(Xeq)
8U d of the equivalent load of the inverter are output values.
We assume a parameter A =  2 I , and according to (20),
d1

A = ( Req 2 + X eq 2 ) / Req (21)

i.e.

A = Req + X eq 2 / Req (22)

Equation (22) reveals a possibility that the equivalent load

of the inverter can be determined by the input voltage and
current if we neglect the higher harmonic in them.
Fig.5. Fitting surface of equivalent load of the inverter(real part)

III. PROPOSED ONLINE PARAMETER ESTIMATION

A. Inverter equivalent load estimation

 We use variable λ to represent the tangent value of the
reflected impedance angle θ, then
 = − X S RS (31)
which is unrelated to the mutual inductance MPS, and just
related to the secondary loop impedance. Fig.7 verifies the
above conclusion. When the mutual inductance of the coils is
17μH, 21μH and 27μH, the reflected impedance angle θ
Fig.6. Fitting surface of equivalent load of the inverter(imaginary part) declines in a same trend with the system load RL increasing,
and the tangent value of θ also has the same trend with the
The surface fitting functions are as follows, mutual inductance varying.
FR ( x, y ) = p00 + p10 x + p01 y + p20 x 2 + p11 xy + p02 y 2 + p30 x 3 + p21 x 2 y 5
M=17μH
+ p12 xy 2 + p03 y 3 + p31 x 3 y + p22 x 2 y 2 + p13 xy 3 + p04 y 4 (24) 0
M=21μH

reflected angle(°)
-5 M=27μH
where FR(x,y)=Req, x=A, y=θ, p00=36.15, p10=-0.7022 ,p01 =- -10

1.069, p20=0.02497, p11=0.04326, p02=0.0001467, p30=- -15

0.0001176, p21=-0.0005051, p12=-8.694e-05, p03=0.0001243, -20

p31=1.543e-06, p22=1.274e-06, p13=-1.509e-06, p04=-6.193e-07. -25

-30

FX ( x, y ) = p00 + p10 x + p01 y + p20 x 2 + p11 xy + p02 y 2 + p30 x 3 + p21 x 2 y -35

30 40 50 60 70 80 90 100

RL(Ω)
+ p12 xy + p03 y + p31 x y + p22 x y + p13 xy + p04 y
2 3 3 2 2 3 4
(25)
(a)
where FX(x,y)=Xeq, x=A, y=θ, p00=-2.969, p10=-0.1563, 0.1

p01=2.174, p20=0.003369, p11=-0.05846, p02=-0.05177, p30=- 0

M=17μH
M=21μH
3.505e-05, p21=0.0003772, p12=0.001024, p03=0.0004257, -0.1 M=27μH
p31=1.108e-08, p22=-3.659e-06, p13=-3.89e-06, p04=-1.229e- -0.2

06 . -0.3

λ
-0.4

B. Reflected impedance angle analysis -0.5

-0.6

In a wireless power transfer system shown in Fig.1, LCCL -0.7

topology is used both in the primary and secondary side.

30 40 50 60 70 80 90 100

RL(Ω)

The transfer model for wireless power transfer system can (b)
be described by Fig.7. Relation between load and reflected impedance angle.(a) reflected
impedance angle;(b)tangent value of reflected impedance angle
U p   j LP + RLP + 1/ ( jCP ) j M PS   I P 
 0 =  (26) Also, based on the above analysis, it can be found that the
   j M PS Z S   I S  relation between tangent value of the reflected impedance
angle λ and the system load RL is linear monotonically.
Where ZS is the equivalent impedance of secondary loop
Further, simulation results show a linear relation between Z S
circuit, and
and RL, as is shown in Fig.8.
1 12

Z S = RLS + j ( LS − ) + Z 21 (27) M=17μH

CS
11
M=21μH
10
M=27μH

We assume that Rs is the real part of ZS, and Xs is the 9

R (Ω)

imaginary part of ZS. Then the reflected impedance ZSP, which

8
s

7
is the impedance reflected to the primary side from the 6

secondary side, can be described by (28): 5

( M ) ( M )
4
2 2 30 40 50 60 70 80 90 100

ps ps RL(Ω)
Z SP = RS −j XS (28)
RS2 + X S2 RS2 + X S2 (a)

Then Rsp (the real part of ZSP) and Xsp (the imaginary part
of ZSP) are

( M )
2

RSP =
ps
RS (29)
RS2 + X S2

( M )
2

X SP = −
ps
XS (30)
RS2 + X S2
 3.5 In a wireless power transfer system, the high sampling rate
3 is needed to improve the precision for equivalent impedance
2.5 calculating in primary side. We use the FPGA processor
2
EP2C8Q208I8 to realize the load detection method, which can
X (Ω) 1.5
work at 50MHz system frequency. The real-time signals of
s
1
uP(t) and iP(t) are measured by sensors, and then translated into
0.5 M=17μH digital data by AD unit. The voltage sensor is Yubo CHV-25P
M=21μH
0
M=27μH with the response time of 10μs and measurement bandwidth of
-0.5
30 40 50 60 70 80 90 100 0-100kHz. The current sensor is LA55-P/SP50 produced by
RL(Ω)
LEM Company, with the response time of 1μs and
(b) measurement bandwidth of 0-200kHz. ADC9218 is used as
Fig.8. Relation between load and secondary circuit impedance.(a)real AD unit, with 10 independent A/D converters and the
part;(b)imaginary part. converting speed of 105MSPS. The load detection unit obtains
the digital data from the AD unit using parallel data
C. Parameter estimation method communication, and calculates parameters RL, MPS, η, IL and
According to section B, the value of RL can be determined UL. The system information is displayed by LED digital in
with the knowledge of the reflected impedance angle, and the signal indication unit. Fig.10 is the proposed online multi-
secondary loop equivalent impedance ZS can also be parameter load detection circuit.
determined thereafter. With the use of curve fitting method in CP
L11
Matlab, RL, RS and XS can be derived,
Secondary
RL=7.0370λ2-83.6746λ+39.7622 (32) Ud Cd Inverter C11
Side
LP LS
XS=-0.4255λ5-3.1564λ4-8.934λ3-12.4788λ2-10.0051λ- R11 RLP
MPS
0.1027 (33) Current Voltage
Sensor Sensor Driver
Circuit
RS=0.8378λ5+6.2732λ4+17.8900λ3+24.879λ2+19.067λ+10. ADC9218

41 (34) EP2C8Q208 Estimation

FPGA Results
Then the coils mutual inductance can be determined with
Fig.10. Parameter estimation diagram.
M = RSP ( R + X ) /  RS
2
S
2
S
2
(35)
TABLE I.
SYSTEM PARAMETERS OF THE ESTABLISHED WIRELESS POWER
TRANSFER SYSTEM

IV. EXPERIMENT RESULTS

Parameter Value Parameter Value
A. System configuration Distance 20cm Frequency 50kHz
In order to further verify the proposed parameters LP 164.3 μH LS 56.57μH
RP 0.112Ω RS 0.122Ω
estimation method, we design a WPT experiment system. The L11 89.4μH C11 175.51nF
system elements values are listed in Table. I. Fig. 9 are CP 101.28nF CS 115.89nF
photographs of the resonant coils of the established system. C21 161.16nF L21 85.85μH
Magnetic disks made of ferrites are added to improve the CL 3μF Input voltage 300V
power transfer efficiency, and coils are made of Litz wire. M 27.77μH

B. Experiment results
The results of parameter estimation experiments are shown
in Fig.11-12. A range of 40-300Ω for the system load was
estimated in Fig.12. Under different conditions (in the pictures,
“dis” means the lateral misalignment distance of the coils), the
estimation error of the estimation was less than 7%.
Fig. 9. Three resonant coils of the wireless power transfer system.
 350 V. CONCLUSION
actual value
300 dis=0 This paper presented an online parameter
dis=10cm
250 estimation method for WPT system based on the
200 measurement of input voltage and current of the inverter.
R (Ω)

A DC input side model for the inverter was built, and the
L

150

100
equivalent load of the inverter can be determined by the
50
input voltage and current. Then, by studying the
characteristics of the tangent of the reflected angle, a
0
-3 -2.5 -2 -1.5 -1 -0.5 0 curve fitting method was proposed to estimate the
λ
(a)
system load. The mutual inductance of coils was also
estimated to provide information of the system.
0.06
dis=0 Experiments results showed that the proposed method
0.04 dis=10cm
can reduce the estimating error of the load to 7%.
0.02
Error(100%)

-0.02 REFERENCES
-0.04

-0.06 [1] S. Y. Hui, “Planar wireless charging technology for portable

-0.08
electronic products and Qi,” in Proc. IEEE, vol. 101, no. 6, pp. 1290–
0 50 100 150 200 250 300 1301, Jun. 2013.
RL(Ω)
[2] Qifan Li and Yung C. Liang, “An Inductive Power Transfer System
(b) With a High-Q Resonant Tank for Mobile Device Charging,” IEEE
Fig.11. Estimation results of load. (a)estimation curves. (b) estimation error Trans. Power Electron., vol. 30, no. 11, pp. 6203–6212, 2015.
curves [3] V. Cirimele, M. Diana, F. Freschi, and M. Mitolo, "Inductive Power
Fig.12 shows the estimation results of the coils mutual Transfer for Automotive Applications: State-of-the-Art and Future
Trends," IEEE Transactions on Industry Applications, vol. 54, no. 5,
inductance when the system load was set to 65Ω, 100Ω and pp. 4069-4079, 2018.
235Ω respectively. Within the lateral misalignment distance [4] Z. Zhang, H. Pang, A. Georgiadis, and C. Cecati, "Wireless Power
of 15cm, the estimation error of the coils mutual inductance Transfer—An Overview," IEEE Transactions on Industrial
was less than 5%. Electronics, vol. 66, no. 2, pp. 1044-1058, 2019.
[5] U. K. Madawala and D. J. Thrimawithana, “A single controller for
30
actual value
inductive power transfer systems ,” in Proc. IEEE Ind. Electron. Conf.,
28 R =65Ω
L
pp. 109–113, 2009.
26
R =100Ω
L [6] Wang, Z. H., et al. “Load Detection Model of Voltage-Fed Inductive
R =235Ω
L Power Transfer System, ” IEEE Trans. Power Electron., vol.28, no.11,
M(μH)

24
pp. 5233-5243, Nov. 2013.
22 [7] Jian Yin, Deyan Lin, Chi-Kwan Lee, et al., “A Systematic Approach
20 for Load Monitoring and Power Control in Wireless Power Transfer
Systems without Any Direct Output Measurement,” IEEE Trans.
18
Power Electron., vol.30, no.3, pp. 1657-1667,. 2015.
16 [8] J. L. Villa, J. F. Osorio and A. Llombart,” High-misalignment tolerant
0 5 10 15
dis(cm) compensation topology for ICPT systems,” IEEE Trans. Ind.
(a) Electron., vol. 59, no. 2, pp. 945-951, 2012.
0.15 [9] Takehiro Imura, Yoichi Hori. “Maximizing Air Gap and Efficiency of
R =65Ω
L
Magnetic Resonant Coupling for Wireless Power Transfer Using
0.1 R =100Ω
L
Equivalent Circuit and Neumann Formula,” IEEE Trans. Ind.
R =235Ω Electron., vol. 58, no. 10, pp. 4746-4752, 2011.
Error(100%)

L
0.05 [10] X. Dai, X. Li, Y. Li, P. Deng, and C. Tang, "A Maximum Power
Transfer Tracking Method for WPT Systems with Coupling
0 Coefficient Identification Considering Two-Value Problem," Energies,
vol. 10, no. 10, p. 1665, 2017.
-0.05

-0.1
0 5 10 15
dis(cm)
(b)
Fig.12. Estimation results of mutual inductance of coils.(a)estimation
curves(b)estimation error curves