Download Resonant Frequency Splitting Analysis and Optimation of

PIERS Proceedings, Moscow, Russia, August 19–23, 2012
684
Resonant Frequency Splitting Analysis and Optimation of Wireless
Power Transfer System
X. L. Huang, L. L. Tan, W. Wang, Y. L. Zhou, and H. Qiang
School of Electrical Engineering, Southeast University, Nanjing 210096, China
Abstract— Frequency optimal control for magnetic resonance coupled wireless power transfer
system to improve the system transfer efficiency, which requires the system should have one unique
stable resonant frequency. Actually, transmitted at close distance the system will appear multiple
resonant frequencies (frequency splitting) phenomenon which results in unstable and increase
control complexity. In order to study the frequency splitting mechanism, the system equivalent
load model is established on the base of bilateral capacitor parallel-compensated topology in this
paper. And then the transfer distance and load resistance threshold conditions are presented
when the resonant frequency splitting occurred. On this basis, frequency optimal control method
and solutions for system frequency splitting is proposed. Finally, simulations and experiments
verify the feasibility of the analysis.
1. INTRODUCTION
With the rapid social development and people’s pursuit of amenities, the traditional wired power
supply cannot meet the requirements. In order to get rid of the power wired problems to achieve
wireless power transmission, mountains of work have been done on this topic and lots of methods
have been proposed by scientists [1–4]. The most noteworthy is MIT’s use of magnetic resonance
coupled theory to achieve power wireless transfer in 2007 [5], which successfully avoids obstacles
to transfer power in meter-scale range, and indicates a new research direction for medium-range
(meter level) wireless power transmission. This technology quickly becomes a hot pursuit for
research institutions [6–10].
Currently, studies reveal that while at long-distance transmission, the wireless power transfer
system has only one stable resonant frequency and that the transmitting and receiving coils circuits
should be in their self-resonance respectively under this frequency to achieve efficient performance.
But, at close transfer distance, the system resonant frequency will give rise to two or three resonant frequencies which increase the power transfer instability and difficult to determine the ideal
frequency controlling points. It can be seen that the uniqueness and variation characteristics of the
system resonant frequency are core of system optimal control, and also one of the main research
contents in this paper. In order to achieve system resonance, varieties of capacitor compensated
modes can be used in transmitting and receiving circuits. In this paper, the bilateral capacitor
parallel-compensated topology is studied, and the system load equivalent model is established to
analyze the variation characteristics of the system input impedance. In conclusion, an optimization
control method is proposed to ensure the system a single stable resonant frequency and effective
work in resonant status.
2. TRANSMISSION MODEL AND PP TOPOLOGY
Theory suggests that the wireless power transfer system is composed of high-frequency power Us ,
receiving and transmitting coils L1 and L2 , capacitor C1 , and C2 , and the load RL . Only in terms
of transfer system, the specific circuit of the system is shown in Fig. 1. R1 , R2 are coils equivalent
resistance [5] respectively, M , d for mutual inductance and the distance (transfer distance) between
two coils, respectively. And the high frequency influence is ignored for simplification.
3. EQUIVALENT LOAD MODEL
Suppose that ω0 is the self-resonant frequency of receiving coil circuit, Q2 is quality factor, L1 = L2 ,
C1 = αC2 , ω = βω0 , where α for capacitance ratio, β for operating frequency ratio. Then the
relations are as follows:

√
 ω0 = 1/ L2 C2
Q2 = RL /(ω0 L2 )
(1)
 κ = M/√L L = M/L
2 1
2
Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 19–23, 2012 685
Figure 1: System PP compensated mode.
Figure 2: The normalized equivalent circuit.
Here k is coupling coefficient between two coils, and the receiving coil circuit self-impedance is
converted to the transmitting circuit (as shown in Fig. 2), the normalized receiving equivalent
impedance Zr in transmitting circuit is
£
¤
β 2 k 2 RL2 Q2 (1 + β 2 Q22 ) R2 (1 + β 2 Q22 ) + RL
£
¤
Zr =
Q2 (R2 Q2 (1 + β 2 Q22 ) + RL Q2 )2 + (βRL (1 + β 2 Q22 ) − βQ22 RL )2
−j
β 3 k 2 RL3 (1 + β 2 Q22 )(1 + β 2 Q22 − Q22 )
£
¤
Q2 (R2 Q2 (1 + β 2 Q22 ) + RL Q2 )2 + (βRL (1 + β 2 Q22 ) − βQ22 RL )2
(2)
If Zr is defined as a form of Zr = Rr − jXr , then the system input admittance Yin watched from
the transmitting side is
µ
¶
Q22 (R1 + Rr )
Q2 (βRL − Xr Q2 )
αβQ2
Yin = 2
− 2
+j
(3)
RL
Q2 (R1 + Rr )2 + (βRL − Xr Q2 )2
Q2 (R1 + Rr )2 + (βRL − Xr Q2 )2
The imaginary part of Yin should be zero while the system is working efficiently in resonant state,
and under Us , the system input power P and transfer efficiency η can be obtained by Equation (4)
P = (ReYin )Us2
(4)
4. FREQUENCY ANALYSIS AND SPLITTING PHENOMENON
For the above analysis it can be concluded that the system input impedance imaginary part ImYin is
zero when the system works in resonant, so the study of resonant frequency points can be converted
to the study of the frequency solutions of the equation ImYin = 0. Since mutual inductance’s
calculation is quite complex, we have M ≈ πµ0 r4 N2 /(d2 +r2 )1.5 [11], in which µ0 is the permeability,
N is turns of coils, r is the radius.
4.1. Resonant Frequency Splitting
The stability of the system is one of the key performances, which requires a single stable resonance
frequency. The resonant frequency is the frequency solutions which ensure zero of the system’s input
impedance imaginary part ImYin . With a fixed RL , there exists a minimum threshold distance when
frequency splitting occurs, denoted as dc . When the transfer distance is greater than dc , the system
will only have one stable resonant point. In order to solve the dc , the threshold value of coupling
coefficient kc should be given when system frequency splitting occurs. Regardless of R1 , R2 , we
propose the constraint relationship between coupling coefficient k, load resistance RL , capacitance
ratio α and the operating frequency ratio β as ImYin = F (k, RL , α, β) = 0 while frequency splitting
occurs. Substituting the relevant parameters, which can be further organized into a fourth-order
equation on k, that F (k, RL , α, β) = ak4 + bk2 + c. Here, a, b, c, are the function on the load RL ,
capacitance ratio α, operating frequency ratio β, which be expressed as

¡
¢ h
¡
¢ i
2 Q2 2 αβ 5 Q3 R2 + αβ 7 Q R2 1 + β 2 Q2 − Q2 2

a
=
1
+
β

2 L
2
2 L
2
2


¯ ¡ 3
¢
¡
¢ ¯


¯ β Q2 R2 − 2αβ 5 Q2 R2 (1 + β 2 Q2 ) 1 + β 2 Q2 − Q2 ¯

2
2
2 ¯
L
L
¯ h
i
¡
¢2
b=¯
¯
(5)
¯
¯ Q22 + β 2 1 + β 2 Q22 − Q22




h

¡
¢ i2 ¡ 3
¢

2 − βQ R2
 c = Q2 + β 2 1 + β 2 Q2 − Q2 2
αβ Q2 RL
2 L
2
2
2
PIERS Proceedings, Moscow, Russia, August 19–23, 2012
686
Then, the boundary condition of equation F (k, RL , α, β) = 0 has only one positive solution
besides zero is β = (1/α)0.5 , and the threshold value kc is
s
√
−b − b2 − 4ac
κc =
(6)
2a
Especially when α = β = 1, kc is (−b/a)0.5 = [1/(1 + Q22 )]0.5 . By Equation (3), we can see that
if k ¿ 1, ImYin is approximately proportional to β, and the frequency splitting disappears. Then
ImYin can be simplified as Q2 (β 2 α − 1)/RLβ . If α is not equal to 1, according to ImYin = 0 we
can obtain the similar relationship between β and α: β = (1/α)0.5 , which suggests that the system
resonant frequency is determined uniquely by α.
4.2. The Impact of Load Changes on Resonance Frequency
If we investigate on ImYin changed with β under different quality factor while α = 1, the results
are shown as Fig. 3. With RL changed from large to small at close distance, the system resonant
frequencies change from multiple into single. Evidently, if d is fixed, there exists a threshold load
which makes the system produce only one resonant frequency, denoted as RLc . As the distance
increases, the coupling relation between two coils will be reduced, which leads to the decreased
impact of load on system resonant frequency, as is shown in Fig. 3(b) and Fig. 3(c). By comparing
research we find that no matter how the load changes, the system has a common resonant frequency
if distance d and capacitance ratio α are fixed, which gradually approaches ω0 with the coupling
between two coils weaken. At short distance, the system resonant frequency will deviate from ω0
and the frequency splitting phenomenon may occur.
5. DESIGN AND FREQUENCY OPTIMIZATION
Based on the above analysis it can be found that at close distance, a single stable resonant frequency
of the transfer system for an easy control can be achieved as following: fixing α, and adjusting RL
to achieve only one resonant frequency at the whole distance under the context of d < dc . For the
actual system parameters simulation studying, Lp = Ls = 8.02 µH, N = 3, r = 0.3 m, C2 = 265 pF.
Simulation and experimental studies are carried out to verify the above theoretical analysis.
When α = 1, the curve of β varied with d is shown as Fig. 4. It can be easily seen if d is greater
than dc (threshold distance), the resonant frequency is almost close to ω0 , otherwise deviates from
ω0 obviously. If the power source frequency output ω can change around ω0 , this requires the
system working resonant be near ω0 with d changing, and has only one stable resonance frequency.
If RL = 1500 Ω in long-distance transmission is given, we measured dc is 0.45 m when resonance
(a) α=1, d=0.2 m
(b) α=1, d=0.3 m
(c) α=1, d=1.0 m
Figure 3: ImYin (β) at different transfer distance and load.
Figure 4: The curve of β varied with d.
Figure 5: The curve of RL varied with d.
Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 19–23, 2012 687
Figure 6: Load received power varied with d.
Figure 7: Transfer efficiency varied with d.
frequency splitting occurred. If the transfer distance is less than dc , we can adjust RL to make the
system have a unique resonant frequency, and the actual resistance value is shown in Fig. 5.
Figures 6 and 7 indicate the curves of load received power and transfer efficiency by adjusting
RL to make the system work in a single stable resonance frequency near ω0 . By comparison we
can see that through controlling the system has only one resonant frequency, which increase the
stability of the system. Due to without frequency splitting at the long-distance the system transfer
power and transfer efficiency has not been improved significantly before and after the optimization,
But at close distance, the load received power and transfer efficiency are larger.
6. CONCLUSIONS
This paper has an in-depth study of the load model of magnetic resonant coupled wireless transfer
system with PP topology. By exploring the relations between the system’s resonant frequencies,
coupling coefficient and the capacitance ratio, the variation of system resonant frequency are analyzed in detail. And then the threshold value of load and distance are investigated when system
resonant frequency splitting occurs. On this basis, an optimization method is proposed to make the
system work in a single stable resonant frequency by changing the load resistance and capacitance
ratio at different transfer distance, thus improving the system stability and controllability. Both the
simulations and experiments manifest that the system enjoys a higher transfer efficiency and power
output under this optimal control method. The research of this paper will also prove a significant
theoretical guidance and reference for the system stability control and optimized operation.
ACKNOWLEDGMENT
This work was supported by Scholarship Award for Excellent Doctoral Student granted by Ministry of Education, the Research Innovation Program for College Graduates of Jiangsu Province
(No. CXZZ11 0150) and the National Natural Science Foundation of China (No. 51177011).
REFERENCES
1. Nikola, T., “The transmission of electrical energy without wires as a means for furthering
peace,” Electrical World and Engineer, Vol. 7, 21–24, Jan. 1905.
2. Hideyuki, K., K. Tamotsu, and S. Hideo, “Power and information transmission to implanted
medical device using ultrasonic,” Japanese Journal of Applied Physics, Vol. 40, No. 5B, 3865–
3866, Feb. 2001.
3. Junji, H., K. Tae-Woong, and K. Atsuo, “Integral motor with driver and wireless transmission
of power and information for autonomous subspindle drive,” IEEE Transactions on Power
Electronics, Vol. 15, No. 1, 13–20, Jan. 2000.
4. Wang, G. X., W. T. Liu, and M. Sivaprakasam, “High efficiency wireless power transmission
with digitally configurable stimulation voltage for retimal prosthesis,” Proceedings of the 2th
International IEEE EMBS Conference on Neural Engineering, Arlington, Virginia, Mar. 16–
19, 2005.
5. Kurs, A., A. Karalis, R. Moffatt, and J. D. Joannopoulos, “Wireless power transfer via strongly
coupled magnetic resonances,” Science Express, Vol. 317, No. 5834, 83–86, Jul. 2007.
6. Zhen, N. L., A. C. Raul, and T. Ryan, “Design and test of a high-power high-efficiency loosely
coupled planar wireless power transfer system, ”IEEE Trans. on Industrial Electronics, Vol. 56,
No. 5, 1801–1812, May 2009.
7. Villa, J. L., A. Liombart, and J. F. Sanz, “Design of a high frequency inductively coupled
power transfer system for electric vehicle battery charge,” Applied Energy, Vol. 86, No. 3,
355–363, Mar. 2009.
688
PIERS Proceedings, Moscow, Russia, August 19–23, 2012
8. Segura-Quijiano, F., J. Garcia Canton, and J. Sacristan, “Wireless powering of single-chip
systems with integrated coil and external wire-loop resonator,” Applied Physics Letters, Vol. 92,
No. 7, 074102-1–074102-3, Feb. 2008.
9. Tan, L. L., X. L. Huang, and H. Huang, “Transfer efficiency optimal control of magnetic
resonance coupled system of wireless power transfer based on frequency control,” Science
China Technological Sciences, Vol. 54, No. 6, 1428–1434, Jun. 2011.
10. Sedwick, R. J., “Long rang inductive power transfer with superconducting oscillators,” Annals
of Physics, Vol. 325, No. 2, 287–299, Feb. 2010.
11. Huang, H., X. L. Huang, and L. L. Tan, “Research on the transmitter and receiver of wireless
power transmission based on magnetic resonance,” Advanced Technology of Electrical Engineering and Energy, Vol. 30, No. 1, 32–35, Jan. 2011.