Sensitivity Analysis and Parameter Optimization of Inductive Power Transfer

There are several resonance frequencies points in the high-order resonant network to realize constant current/constant voltage output, but the disturbance of the resonant network parameters will affect the output characteristics of the system. Therefore, selecting the appropriate resonant frequency is necessary to achieve stable output characteristics of wireless power transmission. This paper analyzes the influence of sensitivity of the double-sided LCC compensation network parameters on resonance point and output characteristics. Moreover, this paper proposes a design method to realize load-independent constant current and constant voltage output under two ZPA frequencies, and the resonance parameters are optimized for the realization of ZVS. Under constant current and constant voltage output resonant points, the transmission efficiency of the double-sided LCC IPT system reaches 88.9% and 88.5%, respectively.


I.INTRODUCTION
Lithium-ion batteries are considered the most competitive battery for electric vehicles because of their high-power density, long cycle life, safety, and environmental protection. [1][2][3]. Lithium-ion batteries can be safely, efficiently, and quickly charged with inductive power transfer (IPT) or capacitive power transfer (CPT). Wireless power transfer is widely used in battery charging applications ranging from low-power battery charging for mobile phones to high-power charging for electric vehicles [4][5][6]. Therefore, magnetic field or electric field coupling is the development trend to realize the safe and efficient charging of lithium-ion batteries for electric vehicles.
Constant current/constant voltage (CC/CV) charging prolongs lithium-ion batteries cycle life and service time. During lithium-ion battery charging, the equivalent load of the battery changes from a few ohms to a few hundred ohms as the charging state changes [7]. Therefore, it is necessary to realize load independent constant current and constant voltage output in an extensive load range [8][9]. In IPT systems, DC/DC converters are usually added at the front or back end to realize CC/CV output. Furthermore, it is necessary to track the zero-phase angle between the input voltage and current of the resonant network to realize loadindependent CC/CV output [10][11]. Due to frequency bifurcation, frequency conversion control may be unstable, resulting in electromagnetic compatibility problems [12][13]. Constant frequency duty cycle control can effectively avoid frequency bifurcation, but it is not easy to achieve zero voltage switching (ZVS) under light load. Therefore, it is an effective solution to realize the CC/CV output under zero phase switch (ZPA) in an IPT system with specific resonance parameters. The four basic compensation topologies in wireless power transmission can achieve different loadindependent output characteristics: P-S topology can achieve CV output, P-P topology can achieve CC output, S-S topology can achieve CC output, and S-P topology can achieve both CC and CV output. However, S-S topology loses ZPA conditions in CV mode, and S-P topology loses ZPA conditions in CC mode [14][15][16][17]. In order to realize ZPA, wang proposed a hybrid topology of S-S and P-S. The hybrid topology requires additional switches and driver circuits to realize topology conversion, increasing the number and loss of components and reducing transmission efficiency [18][19][20]. Therefore, for an ideal IPT charger, there are at least two load independent ZPA operating frequencies. The resonant network can realize the load-independent CC output and the load-independent CV output to meet the battery charging characteristics. Esteban B proposes a double-sided LCC compensation network, achieving system output characteristics independent of load and coupling coefficient. The load-independent CC and CV outputs are achieved at two different operating frequencies, but the ZPA at two different frequencies is not achieved [21]. V.-B. Vu and X.Qu have realized load-independent constant current and constant voltage outputs of double-sided LCC compensation networks under ZPA conditions [22][23]. Furthermore, J. Lu has found that load-independent CC and CV outputs can be achieved with multiple resonant frequency points in a double-sided LCC compensation network under ZPA conditions. [24][25][26]. However, selecting the appropriate operating frequency in multiple resonant points of a high-order resonant network has attracted little attention. J. C derived the input impedance angle model of double-sided LCC compensation network using mutual inductance model and analyzed the influence of series compensation capacitor on the system input impedance angle [27]. Nonetheless, the problem of how to select the appropriate resonant frequency has not been solved. The sensitivity analysis of high-order resonant network parameters to constant current/constant voltage output is not enough and selecting the appropriate resonant frequency operating point still lacks theoretical analysis.
In this paper, the double-sided LCC topology is modeled by T-type circuits and π -type circuits, and the resonance conditions of load-independent CC/CV output under ZPA conditions are obtained. The sensitivity of topology parameters to the output of the resonant network is analyzed. The simulation model reveals the relationship between the output characteristics of the IPT system and topology parameters. Finally, the paper selects the appropriate working frequency from the multi-resonant frequency and establishes the experimental prototype to verify the accuracy of the theoretical analysis.

II. LOAD-INDEPENDENT CC AND CV MODES WITH ZPA CONDITIONS OF DOUBLE-SIDED LCC COMPENSATION TOPOLOGIES
A. T-and πnetwork T or π circuit is formed by adding reactance into Lcircuit. Figure.1 From (2), inT Z is purely resistive when 23 0 TT XX += . It means that load independent ZPA conditions can be implemented, and the purely resistive input impedance inT R is given by The T circuit is equivalent to two L-shaped circuits, as shown in Figure 1 The CV output characteristics can be achieved when The output voltage and the output impedance of the T-circuit are XX + , inT Z cannot be a pure resistance in CV output mode, the T-circuit cannot realize ZPA in the CV output mode. The T network can achieve both CC and CV output but load independent ZPA can only be implemented in CC mode. The π -circuit is equivalent to the connection of an L-circuit and reverses L-circuit, and the analysis process is similar to T-circuit, which is shown in Figure 2. The πcircuit can also realize CC and CV output, but the ZPA condition can only be realized in CV mode. For the T-circuit and π -circuit, CC and CV outputs can be realized. However, the ZPA operating condition is only achieved for the Tcircuit in CC mode or the π-circuit in CV mode.

B. Load-Independent CC Outputs with ZPA Conditions
The resonant compensation network works in the ZPA state, reducing the reactive power and making the inverter realize ZVS easily. The double-sided LCC compensation network in the IPT system is shown in Figure 3. is the load impedance. The coupling coefficient of the coil is:

Fig.3. Double-sided LCC resonant compensation network
The equivalent circuit of the double-sided LCC compensation network is shown in . Fig .4. Leakage inductance equivalent circuit referred to the primary side of the double-sided LCC compensation network.
The double-sided LCC compensation network is driven by the modulation voltage of the full-bridge inverter, and the driving voltage is approximately the first harmonic of the inverter's output voltage . The double-sided LCC equivalent model in CC mode is obtained as shown in figure 5, consisting of an LC-circuit and two π -circuits in series. The variables that can be expressed are: is the resonant angular frequency of the double-sided LCC resonant circuit to realize loadindependent CC output and 2 It can be obtained by analyzing T-circuit and π -circuit that the resonance conditions and output current of the double-sided LCC resonant network to achieve CC output can be calculated as: This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
I are the output current of equivalent π -circuit and LC-circuit in figure 5. The output current of IPT system as The resonant network's input impedance and input current in CC output mode are equation (14) and equation (15), respectively.
I are the output voltage and current of the IPT system in CC output mode. It can be seen from formula (14) that ZPA can be realized when the double-sided LCC IPT system is in the CC mode.

C. Load-Independent CV Outputs with ZPA Conditions
In order to achieve load-independent CV output of double-sided LCC compensation network, the leakage inductance model in fig. 4 is equivalent to the series connection of three T-circuit in Fig. 6. The conversion of the equivalent variable in Figure 6 is shown in Formula 16. Based on the above analysis of CV output characteristic of T-circuits, the resonance condition of the loadindependent constant voltage output of double-sided LCC resonant compensation network can be obtained: The impedance and output voltage of the double-sided LCC compensation network in the CV output mode can be obtained: ' ' 1 2 ZPA in the CV output mode of the double-sided LCC compensation network can be realized when the equation (21)

D. Sensitivity analysis of parametric uncertainty in IPT systems
The double-sided LCC compensation network can realize load-independent CC and CV output in ZPA mode. However, in practical application, it is difficult to ensure that the actual parameters are consistent with the design parameters, so it is not easy to realize the complete resonance of the resonant network. It is necessary to consider the inconsistency between design parameters and actual parameters. Furthermore, the double-sided LCC compensation network is a high-order resonant network, and multi-parameter perturbation analysis is complex. It is difficult to change the compensation inductor in the double-sided LCC compensation network after the design is completed, and the integrity of the LC resonance network is essential to maintain the constant current of the primary coil and the stability of the system. The primary/secondary side series compensation capacitor is an integral part of the coil resonance compensation network in the double-sided LCC compensation network. Therefore, the series compensation capacitor parameters in the compensation network are selected for the disturbance analysis. The impedance characteristic of the system is affected by the variation of the series compensation capacitance in the resonant network, and the realization of the soft switch is related to the input impedance characteristic of the whole system. This article analyzes the disturbance of the series compensation capacitor parameters C 1 and C 2 , which will change the impedance characteristics of the circuit and the power factor of the system. Therefore, it is necessary to analyze the disturbance of the series compensation capacitor C 1 and C 2 obtain the condition of realizing ZVS in the wireless power transfer system. The output voltage of the inverter circuit in the wireless power transmission is a square wave, but there are large numbers of odd-numbered highorder harmonics in the circuit, which makes the inductive degree of the system difficult to quantify. The turn-off current is an important parameter that affects the realization of ZVS. Therefore, the circuit's inductance can be quantified by the turn-off current, and the turn-off current can be designed in a reasonable range so that the inverter can easily achieve ZVS. However, the turn-off current contains the fundamental component of the output current and the highorder harmonics. Therefore, the realization of the ZVS inverter can achieve a suitable power factor and a slight cutoff loss [29]. To achieve the ZVS of the inverter circuit, the switching current of MOSFET needs to charge and discharge the junction capacitors in a dead time. Equation 22 can be used to represent the conditions under which MOSFET implements ZVS.  can be calculated: The fundamental wave component of the turn-off current can be obtained when the secondary side series capacitor 2 C changes.
The optimized value of the primary-side compensation capacitor is ' 1 C , and , the primary-side current ab I can be obtained as: The magnitude of the off current off I is related to the current on the primary side. The fundamental wave off I is obtained by 2 the imaginary part in equation (36). Moreover, the high-frequency component off According to formulas (30), (35), (37), changing the turn-off current of the primary side compensation capacitor 1 C and the secondary side compensation capacitor 2 C can be expressed as: The input impedance is inductive, the full-bridge circuit can realize ZVS, and the charge-discharge time of the shunt capacitor at both ends of the switch tube is less than the dead time. The turn off current off I required to achieve soft switching is as follows: In this part, the parameter disturbance of series compensation capacitor in double-sided LCC compensation network is studied, and the cut-off current is used as the intermediate to realize ZVS.

A. Parameter Design
In order to verify the above analysis, MATLAB simulation and experimental prototype of IPT system for double-sided LCC compensation network are established. The design method for the parameters of the double-sided LCC compensation network is shown in Figure 8. The design parameters of the IPT system based on a double-sided LCC compensation network are shown in Table 1.

B. Sensitivity analysis of the double-sided resonant network
According to the parameters in Table 1, the relationship between the current gain, voltage gain, and frequency of the double-sided LCC compensation network under different loads can be calculated. There are three resonant frequency points (51.715,81.198,97.786kHz) in the double-sided LCC compensation network for CC output and four resonant frequency points (50.534,53.427,90.315,100.644kHz) in the double-sided LCC compensation network for CV output. Take the resonant network parameters in Table 1 as an example. The resonance frequency offset of the IPT system under constant current and voltage output is analyzed by perturbation of capacitor parameters in series with a doublesided LCC compensation network. Figures 10(a) and (b) show the effect of changing the primary/secondary side series compensation capacitor on the constant current resonance frequency. The constant current resonance frequency of 2 f decreases with the increase of the series compensation capacitance of the secondary side, but the change of the series compensation capacitance of the primary side has little effect on the constant current frequency of 2 f .The constant current output frequency is affected equally by the series compensation capacitance of the primary and secondary sides. The variation of the secondary series compensation capacitance will not affect the constant current output frequency of 1 f .It can be seen from figure 10 that the resonant frequency of constant voltage output is changed when the original side series compensation capacitance is less than 32.3nF. The constantvoltage output resonant frequency has high stability in the process of compensation capacitance change when the original side series compensation capacitance is more than 32.3nF. Due to the symmetrical structure of the double-sided LCC compensation network, the series compensation capacitor has the same effect on the constant voltage frequency point.

Fig. 10 Constant voltage frequency point versus compensation capacitor
The change of the primary and secondary side series compensation capacitor has a greater impact on the current gain of 1 f under constant current output. At the constant current output, the output current gains of 2 f and 3 f are less sensitive to the primary side series compensation capacitance. The voltage gain of 1 f varies greatly when the primary compensation capacitance is larger than 47nF, and the voltage gains of 2 f , 3 f and 4 f have higher stability. The voltage gain of each resonance point varies greatly under constant voltage output when the secondary series compensation capacitance is less than 47nF. 1 f , 3 f , 4 f are less sensitive to the secondary side compensation capacitance when the secondary series compensation capacitance is larger than 47nF.    Figure 16 (a) is the experimental waveforms of the output voltage and current of the double-sided LCC resonator in the constant current mode (81.198 kHz). The load is switched from 15ohms to 35ohms, and the output current of the wireless power transfer system remains constant. The system achieves load-independent CC output. Fig.16(b) is the experimental waveform of the output voltage and current of wireless power transfer system at a constant voltage (90.315 kHz). The load is switched from 35ohms to 55ohms, and the output voltage of the system remains constant. The system realizes the load-independent CV output characteristic. In figures 17 and 18, Oscilloscope Channel 1 is the output voltage of the inverter, channel 2 is the output current of the inverter, and channel 3 is the output voltage of the compensation network at the receiving end, channel 4 is the output current of the receiving loop compensation network.

Ⅳ. Conclusion
In this paper, T/π circuit is used to model double-sided LCC compensation network, and the load-independent CC and CV outputs of double-sided LCC compensation network in ZPA mode with two different frequencies are analyzed. The series compensation capacitor of a double-sided LCC resonant network is optimized, which is helpful to realize ZVS. The sensitivity of the series compensation capacitor parameter disturbance to the output of the IPT system is analyzed. On this basis, an optimization method of series compensation capacitor for double-sided LCC compensation network is proposed. A 3.3 kW prototype was built, and it was proved that the IPT system could realize CC and CV outputs in ZPA mode. The relevant design and analysis method in this paper is helpful to select the appropriate working frequency among the multiple resonant frequencies of the high-frequency resonant network, thus achieving stable output, reducing power consumption, and improving efficiency.