Estimation of coupling coefficient for wireless power transfer
KAZUYA YAMAGUCHI
Miyazaki university
Engineering
Miyazaki City
JAPAN
tc11033@student.miyazaki-u.ac.jp
ICHIJO HODAKA
Miyazaki university
Engineering
Miyazaki City
JAPAN
hijhodaka@cc.miyazaki-u.ac.jp
YUTA YAMAMOTO
Miyazaki university
Engineering
Miyazaki City
JAPAN
nc12006@student.miyazaki-u.ac.jp
Abstract: In order to implement a highly efficient wireless power transfer, it is necessary to cause electromagnetic
resonance by adjusting the frequency of AC power supply as a transmission side. The position relationship between
two coils in transmitter and receiver influences on the coupling coefficient between the coils. This paper proposes
to estimate the coupling coefficient instead of the position relationship. Estimation is performed by system identification approach. This paper discusses how to decide the hold time and sampling period for system identification.
This will provide one of methods online applicable to the variation of the position relationships of two coils.
Key–Words: wireless power transfer，system identification，coupling coefficient
1
Introduction
Recently, the concept of wireless power transfer has
attracted the attention. It aims wireless power transfer over the distance (0.1 to 1 meter) longer than the
conventional transfer. This special situation is known
to be realized by causing a resonance phenomenon.
Therefore in order to implement effective wireless
power transfer, we should detect at least whether or
not the resonance phenomenon appears. We focus
on a coupling coefficient between transmitting and
receiving coils as a parameter for detecting the phenomenon.
As for an ordinary transformer, which is also one
of device for wireless power transfer, the coupling coefficient between two coils is fixed, of course. However our wireless power transfer supposes that two
coils are independent. Consequently, the coupling coefficient is given by each task of power transfer, and
it is not known beforehand if we suppose various applications. Thus we should adjust the frequency of
power supply for each task so as to cause resonance
phenomena. At last we should measure or estimate
the coupling coefficient for each task.
We propose to estimate the coupling coefficient
of a circuit for wireless power transfer by using so
called system identification method since it is difficult
to measure the coefficient directly. This identification
method estimates a black box with a given input data
and the resultant output data. However, the estimation
requires an appropriate sampling period and hold time
in order to convert a discrete time signal into a continuous time signal, and vice versa. This paper discusses
how to decide the sampling period and hold time in order to estimate the coupling coefficient as accurately
as possible. This leads to implement a highly efficient
wireless power transfer system.
2 Estimation steps of the coupling
coefficient
This section considers a typical circuit for wireless
power transfer. We will try to decide a hold time
by inspecting step response. With the hold time we
will perform a simulation of the behavior of the circuit in continuous time. Then we will obtain discrete
time signals of voltages over transmitting and receiving coils with a temporary sampling period at least
shorter than the hold time. Finally we will have an estimated coupling coefficient after performing system
identification algorithm with the discrete time signals.
A circuit used in this simulation is shown in Fig.1.
+
+
-
-
Figure 1: circuit of simulation
We set that R1 = R2 = 1kΩ, L1 = L2 =
10mH, C1 = C2 = 200nF. The transformer in Fig.1
is noticed, and equations of voitage y1 , y2 of transmit-
ting and receiving coils are written as below.
√
di1
di2
+ k L1 L2
dt
dt
√
di1
di2
= k L1 L2
+ L2
.
dt
dt
y1 = L1
y2
(1)
k(0 < k < 1) is the coupling coefficient.
From Fig.1 and the equation (1), the following
circuit equation can be obtained.
From Fig.2, it is seen that the response converges
approximately 3 msec. The input signal by M series
with a long period repeats on and off at least three
times in one cycle, we can assess that Th = 1msec
so that y1 and y2 are both converge. The continuoustime input signal is generated by holding discrete-time
M series whose period is N = 63 with Th = 1msec.
The generated input signal is shown in Fig.3.


√
i̇
L
k
L
L
0
1
1
1
2


 √
L2
0   i̇2 
 k L1 L2
0
0
C2
ẏ2


6
4

y1


y2
=
.
y2
− R2 − i2
Voltage [V]
2
(2)
0
-2
From the equation (2), the trunsfer function G(s) from
y1 to y2 is obtained in the following.
k
L1 L2 C2 (1−k2 )
.
1
1
R2 C2 s + L2 C2 (1−k2 )
√
G(s) =
s2
+
-6
(3)
1.14
1.16
1.18
1.2
1.22
1.24
time [sec]
Figure.3 M series
Since all the coefficients in the denominator are positive, G(s) is a stable transfer function. Set the co1
efficients as f1 = R21C2 , f2 = L2 C2 (1−k
2 ) , f3 =
k
√
.
L1 L2 C2 (1−k2 )
If these coefficients are estimated, the
coupling coefficient k to be found will be easily recovered.
3
-4
With the input signal y1 and the resultant output
signal y2 , an estimation algorithm are performed to
estimate the transfer function G(s). The output signal
is discretized with a sampling period Ts = 10µsec.
The calculation is done by MATLAB.
We have an estimated
Problem Solution
A step response experiment is performed in order to
decide Th with a unit step input u = 5 in Fig.1. The
result of the simulation with the coupling coefficient
k = 0.8 is shown in Fig.2.
G(s) =
1.112 × 108
,
s2 + 5.002 × 103 s + 1.389 × 108
(4)
for the true
6
5
G(s) =
Voltage [V]
4
1.1 × 108
.
s2 + 5.0 × 103 s + 1.4 × 108
(5)
3
2
1
0
-1
-2
0
0.5m
1m
1.5m
2m
time [sec]
Figure.2 step responce
2.5m
3m
Notice here that the target value to be found is the
coupling coefficient k in the coefficients. There are
five unknown numbers of R2 , L1 , L2 , C2 , k whereas
three estimated equations concerned with f1 , f2 , f3 .
Hence we assume that we know R2 = 103 , L1 = 0.1
and the value of k is found by comparing coefficients
of the equations (4) and (3). Finally k = 0.8002 is
obtained. The result for different Ts , Th for each value
of k is shown below.
sampling period [sec]
-5
-4 -6
5× 10
=0.8
=0.6
=0.4
=0.2
-1
10
-2
error
-5
10
1010
5×5×
10
References:
-4
-4
10
5×10
5×
-3
= 10
-3
10
-4
10
Figure.4 various sampling period
hold time [sec]
=0.8
=0.6
=0.4
=0.2
-3
10
error
-5
= 10
5×-4
10
-4
10
-4
-4
5× 10
10
-3
10
-3-3
5×10
10
5×
-2
10
Figure.5 various hold time
4
Conclusion
According to Fig. 4 and 5, we can see that the errors
are small for shorter sampling periods Ts and longer
hold times Th . This observation is quite natural, but
shorter Ts or longer Th generally requires longer period for each calculation for identification. Thus we
suggest that the value of Ts should be taken as approximately one-hundredth of Th . We summarize the
best condition as below.
Th
Te
Ts
Table.1 best conditions
1msec
125msec（2 periods of M series）
10µsec
If we can estimate the coupling coefficient in a
given wireless power transfer system by the proposed
method, we expect that we can decide a frequency to
cause resonance with knowledge of the coupling coefficient. This will lead to more efficient wireless power
transfer in future.
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Strongly Coupled Magnetic Resonances, Science 317, 83, 2007.
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identification approach to diadnostics for switching circuits, Proceedings of the 2012 International Conference at Singapole, 2012.
[3] T. Mita, digital control theory, shokodo, 1984.
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Use with MATLAB, The MathWorks, Inc.,
2006.
[5] S. Adachi, The foundation of System Identification, Tokyo electrical machinery univercity,
2009.
[6] K. Yanagisawa, The foundation of circuits theory, electric society, 1986.