Model and Frequency Control for Three-Phase Wireless Power Transfer System

In order to the eliminate the “dead spot” in the traditional three-phase wireless power transfer (WPT) system, a three-phase WPT systemwith an asymmetricmagnetic circuit is presented in this paper. Additionally,mathematicalmodel of the system is established and the system parameters are optimized. Based on the fact that the resonant frequency and efficiency are greatly varied with the load, amethod based on impedance conversion is further proposed to improve the frequency stability and system efficiency. Finally, simulation and experimental results show that the proposed method is reliable and feasible to eliminate the “dead spot.”


Introduction
Wireless power transfer (WPT) can transmit electric energy from the power sources to the loads without any wire between the two sides, which offers advantages in terms of flexibility, reliability, and safety [1][2][3][4].Therefore, it has been widely used in some special fields, including wireless power desktop supply platform, two-way power supply for tram [5], and mining application [6].
Most of the previous research is limited in single-phase WPT system.With rapid development of electrical equipment, the three-phase WPT system has been further developed [7][8][9].A new three-phase and bidirection WPT system that is suitable for rapid charging of EVs is presented [10].A new three-phase bipolar inductive power transfer system can provide power across the entire width of a roadway [11].However, a three-phase WPT system usually suffers the "dead spot" problem in pick-up side.Some researchists present a solution about the pick-up "dead spot" in a three-phase WPT system [12][13][14].It is suitable for desk wireless power supply system.There will be a tiny pick-up voltage when the load is exactly in the pick-up "dead spot."Besides, the system will automatically switch one-phase to another to avoid "dead spot" problem.However, it suffers low efficiency, complicated operation, and extra detection circuit.Furthermore, the impacts of the load variation on system stability and efficiency are not considered in this system.Some papers did some research on the efficiency of the three-phase WPT system from magnetic structure and system modeling [15].Therefore, further studies on the three-phase WPT system, especially the dead spot elimination, frequency stabilization, and efficiency improvement are strongly required.A lot of papers study the method of impedance conversion [16,17].However, this method is rarely studied to control frequency stability in three-phase WPT system.
In this paper, to eliminate the "dead spot" in the traditional three-phase WPT system with symmetric circuit, a new asymmetric magnetic circuit is studied.Mathematical model of the system is established to optimize the system parameters.In addition, an impedance conversion based method is proposed to improve the frequency stability and system efficiency as both the resonant frequency and efficiency are heavily affected by the load variation.An experimental prototype is established to verify the validity of the proposed method.Simulation and experimental results indicate that the "dead spot" can be always eliminated regardless of the load position.

Overview of Three-Phase Wireless Power Transfer System.
A typical three-phase WPT system with a symmetrical 2 Mathematical Problems in Engineering magnetic circuit is shown in Figure 1.This system consists of DC power supply (  ), high frequency inverter ( 1 ∼ 6 ), primary side (  ,   ,   and   ,   ,   ), and pick-up side (  and   ) as well as load ( eq ).Meanwhile, * is dotted terminal of the three-phase primary coils.The mutual inductances between primary side and pick-up side are denoted by  1 ,  2 , and  3 .Primary-side power converter generates a threephase symmetrical high frequency sinusoidal current in the primary coils.Then the pick-up side, which is magnetically coupled to primary side, obtains power from the primary side to supply load through power conversion.
It has been confirmed that such a system in Figure 1 suffers from "dead spot" problem in the pick-up side.And a new three-phase WPT system with asymmetrical magnetic circuit which can effectively eliminate the "dead spot" is presented [18].However, system parameters optimization and the influence of load variation on system stability were both ignored in this paper.

Characteristic Analysis of "Dead Spot"
. By the analysis of three-phase full-bridge inverter, currents in the three primary coils, with the same value of amplitude  0 , can be written as The pick-up voltage of secondary coil is where   ,   , and   are the pick-up voltages of the secondary coil induced from three primary coils.
According to (2), the vector graph of pick-up voltage   is shown in Figure 2.
From (2) and Figure 2, it can be assured that pick-up   equals zero when three mutual inductances  1 - 3 have the same value.That is, "dead spot" exists when secondary coil is located in the center of three primary coils.

New Type of Three-Phase WPT System with Asymmetric Magnetic Circuit
3.1.Analysis of Eliminating "Dead Spot".In order to solve the problem of "dead spot" in three-phase WPT system, a three-phase WPT system with an asymmetric magnetic circuit is proposed in Figure 3.The dotted terminal of  phase (the third phase of three phase coils) is reversed so that the magnetic circuit becomes asymmetric.
Figure 4 shows the equivalent model of the pick-up side.In this case, the pick-up voltage   of the pick-up side can be written as where   is the current amplitude and  is the work frequency of the system.It can be seen from ( 3) that, regardless of the value of three mutual inductances  1 ,  2 , and  3 , "dead spot" never exists compared to the traditional symmetrical magnetic circuit, ensuring that the load on any position can achieve efficient power transmission.
Figure 3: Three-phase WPT system with asymmetric magnetic circuit.

Design and Optimization of Parameters.
As shown in Figure 3, the instantaneous value of current flowing in the three primary coils can be represented by where   ,   , and   are all the current instantaneous value.
In Figure 4,   is self-inductance of secondary windings.  is the pick-up resonant capacitor.  can be expressed as The pick-up current   is According to Ohm's law, a phase of primary coils can be represented by where   is the reflected impedance.By ( 6) and (7),   can be expressed as  So,   and   can be expressed as Based on mutual inductance coupling principle, the pickup side can be reflected into primary side.The equivalent model is shown in Figure 5.According to the actual demand of the system, the real part of   ,   , and   should be more than zero to ensure the system resonance; then The sums between imaginary parts of   ,   ,   , and  should be more than zero.
In summary, the mutual inductance and self-inductance of three-phase WPT system with an asymmetric magnetic circuit should meet the following equation: When the primary side is at resonance, (15) should meet So the resonant capacitor of the three-phase primary winding can be written as Obviously, actual operation frequency is affected by load  eq .That is, there is a one-to-one correspondence between corresponding direct parameters (inductances and capacitances) and a load value.However, the value of load is usually variational due to heat and temperature variation.So, common platform is more meaningful for three-phase WPT system.16), the values of the primary resonance capacitor are associated with ,  eq , and .Fixed and variable frequency control are two main control strategies.It is obvious that fixed frequency control is more suitable for this system.If the load is placed in the center of three-phase primary coils, the mutual inductance  is fixed.Therefore, the values of the primary resonance capacitor are only associated with  eq .In practical threephase WPT system, power transfer and efficiency are greatly affected by the change in resonant frequency due to the load variance.To solve the problem mentioned above, a fixed frequency control strategy based on impedance conversion was presented in next section.

Basic Principle of Impedance Conversion.
In the pickup side, for the sake of achieving the frequency stability of system, a rectifier-filter circuit and DC/DC converter are introduced to achieve the fixed frequency of system, which can maintain the equivalent impedance of load through regulating the duty cycle of DC chopper.The schematic circuit of impedance conversion based on the Buck-Boost converter is shown in Figure 6, where  1 is the equivalent impedance after rectifier-filter circuit regulation and  eq is the equivalent impedance across the pick-up power compensation part.
As shown in Figure 6, pick-up coil generates a direct voltage  1 after resonance compensation and rectifier-filter circuit.Then, this voltage ( 1 ) can provide power for load (  ) across the Buck-Boost converter, so that the output voltage of load becomes   .

Theoretical Derivation.
Assume that Buck-Boost circuits work in the soft-switching mode and the switching loss is ignored.To simplify the system, the load   is a resistant load; then When the Buck-Boost circuit works under the continuous current mode, the relationship between  0 and  1 can be expressed as where  is the duty cycle of switch.From ( 17) and ( 18), the relationship between  1 and   can be given by The equivalent ac resistance  eq for the rectifier with capacitive filter [19] is given by The relation between the duty ratio of switch  and the load   can be calculated from ( 19) and (20).
where  is in the range 0∼1.The relationship between duty cycle  and variable load   is shown in Figure 7 when equivalent load  eq is held constant.From Figure 7, for a given   , there will be  to hold  eq constant.

Simulation and Experimental Results
The viability of the proposed system is verified by simulation and experiments.The prototype system is shown in Figure 8.For the following confirmatory experiments, 10 Ω load will be powered by a 500 V DC voltage source.The system resonant frequency is 20 kHz.The three primary coils inductors   ,   , and   are 91.78mH.The pick-up coil   is 67 mH.The mutual inductances of  1 ,  2 , and  3 are 30 mH.According to (5), the capacitances of secondary   side can be calculated as 0.94 F.Furthermore, from (16), the capacitances of primary side   ,   , and   are 0.565 F, 0.906 F, and 0.686 F, respectively.
In Figures 9-14, (a) is the waveform of simulation and (b) is the waveform of experiment.
5.1.Elimination of "Dead Spot".Pick-up voltage with a symmetric magnetic circuit is illustrated in Figure 9. From Figure 9, it is evident that the primary-to-pick-up mutual inductances are approximately equal; therefore, the induced voltage,   , will become zero if the pick-up coils are placed in the center of three-phase primary coils.Figure 10 shows the pick-up voltage    with an asymmetric magnetic circuit.Even if the load is placed in the center of three-phase primary coils, the induced voltage    is not zero.Comparing Figure 9 with Figure 10, the proposed three-phase WPT system with an asymmetric magnetic circuit can effectively eliminate the pick-up voltage "dead spots."

Parameter Verification.
Figure 11 shows the voltage and current waveforms of capacitor   in asymmetric system (Figure 3).As evident from the illustration, the waveform of the voltage and current is not distorted.And its frequency and amplitude are very stable so that the parameters of the system are designed properly.topology of primary side remains the same.In addition,   is 100 Ω,  1 is 220 F,   is 3.9 mH, and   is 10 F.
Figure 13 shows the voltage waveform  1 which is before impedance conversion circuit.From Figure 13, the load changes from 100 Ω to 50 Ω at 0.05 s.Meanwhile, according to (21), duty cycle  of switch  7 is changed from 0.741 to 0.649 for maintaining constant equivalent load  eq in Figure 6.The DC voltage  1 remains stable after dynamic change in a period of time, which means reflected impedances from secondary side to primary side maintain being constant. 1 rises to 54 V after load change.However, the error is about 1.48%, which indicates that the pick-up voltage holds stable basically after adding the impedance conversion circuit.
In conclusion, the input voltage  1 and impedance  1 of impedance conversion circuit in Figure 6 can be unchanged by adjusting duty cycle  at the time of load change.So, the output power of load   will be stable according to (17) whether   is changed or not.
Figure 14 shows the voltage waveform of load   .Comparing Figure 14(a) with Figure 14(b), the output voltage lowers in actual system because of the existence of parasitic resistances which are ignored in calculation and simulation.
From Figure 14(b), the output power is 201.64 W while it reduces to 200 W after load change.However, the error is only 0.8%, which indicates that the output power of system can maintain being constant.
According to the above analysis, the proposed system improves frequency stability and efficiency by applying impedance conversion circuit into three-phase WPT system.

Conclusion
Aimed at the problem of "dead spot" in symmetric threephase WPT system, a three-phase WPT system with an asymmetric magnetic circuit has been presented.The parameters of the system are designed and optimized.However, there is a one-to-one correspondence between corresponding direct parameters (inductances and capacitances) and a load value.It means the system will be in nonresonant state if the load is changed, which causes low efficiency and high total harmonic distortion.But, in fact, the value of load is usually variational due to heat and temperature variation.So, common platform is more meaningful for three-phase WPT system.As transfer power and efficiency are greatly affected

Figure 2 :
Figure 2: Vector graph of pick-up voltage   in symmetric threephase WPT system.

Figure 4 :
Figure 4: Equivalent model of the secondary circuit.

Figure 5 :
Figure 5: Primary equivalent inductance coupling model of asymmetric circuit.

Figure 7 :
Figure 7: One-to-one correspondence between duty cycle  and variable load   for constant equivalent load  eq .