A Novel Compensation Circuit for Capacitive Power Transfer System to Realize Desired Constant Current and Constant Voltage Output

Abstract

Capacitive power transfer (CPT) technique possesses the advantages of safety, isolation, low cost, and insensitivity to conductive barriers. To charge lithium-ion batteries, CPT should possess the output profile consisting of first constant current (CC) output and later constant voltage (CV) output. To fulfill the output profile, many power switches or compensation components are added in the CPT circuit, which is not expected due to the bulky size and additional losses. To reduce the redundancy of the CPT system, an L_x-PS CPT circuit with only five compensation components is proposed in this paper. After a systematic analysis and a parameter design procedure, the proposed CPT circuit can realize input ZPA at both CC and CV modes. In addition, the output current at CC mode and the output voltage at CV mode are all adjustable based on the charging demands of different loads. Finally, simulations are done to prove the analysis in this paper. Compared to previous research, the CPT circuit proposed in this paper can not only achieve the charging demands of lithium-ion batteries, but also reduce the redundancy of the whole system.

1. Introduction

Compared to the traditional plug-in charger, the wireless power transfer (WPT) technique possesses the advantages of convenience, safety, and isolation [1,2]. The inductive power transfer (IPT) technique [3,4,5] and capacitive power transfer (CPT) technique [6,7,8] are two main solutions of WPT. The coupler of IPT is two physically separated coils of a loosely coupled transformer. High-frequency magnetic fields are generated by the loosely coupled transformer to realize power transfer. As an alternative solution to IPT, metal plates are required in a CPT system to form a capacitive coupler that utilizes high-frequency electric fields to transfer power [9].

Since the equivalent impedance of a load may change during the charging process, a WPT charger should realize the load-required constant current (CC) or constant voltage (CV) output. In addition, a high-performance WPT charger is also expected to have a near resistive input impedance for decreasing component power ratings in the case of load variations. For example, energy storage applications are universally equipped with lithium-ion batteries. In much research, IPT chargers are designed to realize the input zero phase angle (ZPA) and load-required CC and CV outputs [10,11,12,13] to fulfill the charging demands of lithium-ion batteries. The authors of [10] propose hybrid compensation circuits to meet the requirement of the transition from CC output to CV output. The output current or voltage of hybrid compensated CPT circuits are load-independent. However, these IPT systems need additional switches, and the output current or voltage highly relies on the coupler. In [11], CC and CV outputs are achieved while the variations of the coupler and loads are taken into consideration. However, two additional intermediate coils with resonant capacitors are adopted, which are bulky. A cascade buck-boost DC/DC converter is utilized at the secondary side of an IPT circuit in [12] to realize the desirable outputs. An additional DC/DC converter would increase the system volume, losses, and cost. To save the back-end DC/DC converter, two operating frequencies for load-independent CC and CV outputs are derived in [13], and the transition of CC and CV mode is realized by simply switching the frequencies. The aforementioned IPT systems can realize the input ZPA for the improvement of the system efficiency and power transfer capability. Moreover, the input impedance is modulated to be slightly inductive for zero-voltage switching (ZVS) of MOSFETs to reduce power losses and switching noises.

The CPT technique has raised significant concern in recent years due to its advantages of low cost, low weight, insensitivity to conductive barriers and misalignment of the coupler [14,15,16]. Research about the capacitive coupler and the comparisons between IPT and CPT was reported in [17]. As concluded in [17], CPT is more suitable for short-distance applications that have relatively larger equivalent capacitances of the capacitive couplers. Reference [18] proposes a compensation design method for achieving the maximum power of CPT systems under coupling voltage constrains. With predesigned output current and voltage stresses on the capacitive coupler, a parameter design method was proposed in [19] to achieve input ZPA considering the inductance detuning. In [20], 144 compensation circuits were proposed. However, most CPT chargers [19,20,21,22,23,24] can only realize CC output or CV output. The authors of [25,26] propose a design method to realize CC and CV outputs with input ZPA for both IPT and CPT systems. However, only IPT systems are systematically analyzed. Moreover, a high-performance CPT system should satisfy the following constraints:

(1)

A high operating frequency is needed: Equivalent capacitances of the capacitive coupler are always in pF range in most applications due to the small dielectric constant of air [16,21,22,23]. To eliminate reactive power, the small equivalent capacitances must resonate with a relatively large inductor, which is not expected due to bulky size and corresponding losses, so that CPT converters always work at a high operating frequency.

(2)

The output characteristics of the CPT circuit should be independent of the capacitive coupler parameters: The structure of a capacitive coupler is always constrained by the volume of a CPT system. Thus, the parameters of a capacitive coupler are determined and they cannot be designed to satisfy the charging demands of loads [27].

(3)

External capacitors are needed to elevate the equivalent capacitances of the capacitive couplers in some situations: With the pF-range equivalent capacitances of the capacitive coupler, it is very difficult to compensate for the reactive power induced by the capacitive coupler. Thus, external capacitors are always added to parallel with the capacitive coupler to elevate the small equivalent capacitances [16,28].

As stated in [22,29], the double-sided LC compensated CPT circuit with four compensation components cannot realize both CC and CV outputs while the input ZPA is guaranteed. As stated in [30], the LCLC-LC compensated CPT circuit with six compensation components has two CC frequencies and one CV frequency to realize input ZPA and load-independent CC and CV outputs. In addition, the CC and CV output gains of the LCLC-LC compensated CPT circuit are adjustable. However, one CC frequency is enough to realize CC output and multiple components in the compensation circuit would increase the system cost and power losses. To satisfy the charging demands of batteries, six constraints should be realized: CC output, CV output, input ZPA at CC mode, input ZPA at CV mode, designable output current, and designable output voltage. In theory, taking CC or CV frequency as a design variable, a CPT circuit with five compensation components may realize desired CC and CV outputs with input ZPA.

In this paper, a CPT circuit with five compensation components is designed to realize both CC and CV outputs. After systematical analyzation, a CC frequency and a CV frequency with input ZPA are derived to realize the predesigned CC and CV outputs. The effect of the change in the length of air gap between the plates on CC/CV condition is studied. To permit the ZVS of MOSFETs in both CC/CV modes, the sensitivities of input impedance and output current or voltage to variations of compensation parameters are analyzed. Finally, a CPT simulation prototype is built to verify the analysis in this paper.

2. Characteristics of the Proposed CPT Circuit

As shown in Figure 1a, the typical capacitive coupler is composed of four metal plates and it can be equivalent to a six-capacitor model where capacitors C₁₂, C₁₃, C₁₄, C₂₃, C₂₄, C₃₄ are formed by plates P₁ and P₂, P₁ and P₃, P₁ and P₄, P₂ and P₃, P₂ and P₄, P₃ and P₄ respectively. Furthermore, the six-capacitor model can be converted to a Π-type model where C_P is the primary capacitor, C_S is the secondary capacitor, and C_M is the mutual capacitor, as shown in Figure 1b.

The relationship between the six-capacitor model and the Π-type model satisfies

$$\left\{\begin{array}{c}{C}_P{=C}_{12}+\frac{\left(C_{13}{+C}_{14}\right)\left(C_{23}{+C}_{24}\right)}{C_{13}{+C}_{14}{+C}_{23}{+C}_{24}}\\ C_S{=C}_{34}+\frac{\left(C_{13}{+C}_{23}\right)\left(C_{14}{+C}_{24}\right)}{C_{13}{+C}_{14 }{+C}_{23}{+C}_{24}}\\ C_M=\frac{C_{24}{C}_{13 }-{ C}_{14}{C}_{23}}{C_{13}{+C}_{14}{+C}_{23}{+C}_{24}}\end{array}\right.$$

(1)

Since the dielectric constant of air is 8.85 × 10⁻¹² F/m, the capacitors in the six-capacitor model and the Π-type model are always in the pF range. To elevate the equivalent capacitances, external capacitors C_ex1 and C_ex2 are added to parallel with the capacitive coupler as shown in Figure 2a, defining C₁ = C_P + C_ex1 and C₂ = C_S + C_ex2. To simplify the analysis, the Π-type model together with C_ex1 and C_ex2 can be further converted to a T-type model as shown in Figure 2b where capacitors C_A, C_B, and C_C satisfy the equation:

$$C_A=\frac{C_1{C}_2-{ C}_M^2}{C_{1 }-{ C}_M},C_B=\frac{C_1{C}_2-{ C}_M^2}{{ C}_M},C_C=\frac{C_1{C}_{2 }-{ C}_M^2}{C_{2 }-{ C}_M}$$

(2)

IPT has four basic compensation circuits, i.e., series-series (SS), series-parallel (SP), parallel-series (PS), and parallel-parallel (PP) as shown in Figure 3, where M is the mutual inductance, L_P is the primary self-inductance, L_S is the secondary self-inductance, C_pr is the primary compensation capacitor, and C_se is the secondary compensation capacitor. The SS and SP IPT circuits are powered by a high-frequency sinusoidal voltage source and the PS and PP IPT circuits are powered by a high-frequency sinusoidal current source. Taking the capacitive coupler and the external capacitors as a whole, CPT also has four basic compensation circuits as shown in Figure 4 where L₁ and L₂ are compensation inductors and the capacitive coupler is equivalent to a T-type model.

However, the CPT circuit with four compensation components cannot realize the input ZPA and CC/CV outputs to satisfy the charging demands of batteries. Thus, to prove a CPT circuit with five compensation components can achieve input ZPA and desired output, an additional inductor L_x is added into the PS and PP CPT circuits and these CPT circuits are powered by a high-frequency voltage source, as shown in Figure 5.

2.1. The Analysis of the L_x-PS CPT Circuit

With the Π-type model of the capacitive coupler, if inductors L_x and L₁ resonate with capacitor C₁ − C_M, the CPT circuit can be converted to Figure 6b and the power source is equivalent to a current source i₁ as shown in Figure 6b. To realize load-independent CV output, inductor L₁ should resonate with capacitor C₂ − C_M as shown in Figure 6c,d. According to the aforementioned analysis, if the CV angular frequency is defined as ω_v, it should satisfy the constraint:

$${\omega }_v=\sqrt{\frac{L_1{+L}_x}{L_1{L}_x{(C}_1-{ C}_M)}}=\frac{1}{\sqrt{L_2{(C}_2-{ C}_M)}}$$

(3)

According to Figure 6a, the input impedance Z_v at ω_v can be simplified as:

$$Z_v=\frac{{\omega }_v^4{{(C}_2-{ C}_M)}^2{L}_x^2{C}_M{R}_E}{C_M-{ j\omega }_v{(C}_2-{ C}_M{\left)\right[C}_{2 }{+\omega }_v^2{C}_M{L}_x{(C}_2-{ C}_M{\left)\right]R}_E}$$

(4)

To achieve input ZPA, the input impedance Z_v should be resistive. Thus, the CV angular frequency ω_v should satisfy the constraint:

$$L_x=\frac{C_2}{{\omega }_v^2{C}_M{(C}_{M }-{ C}_2)}$$

(5)

Due to C₂ = C_ex2 + C_P and C_P > C_M, capacitor C_M − C₂ is smaller than zero. With L_x > 0, the Equation (5) is not valid. Thus, the input ZPA cannot be achieved.

Based on the T-type model of the capacitive coupler and external capacitors, the L_x-PS CPT circuit can be drawn as Figure 7a. If capacitor C_A resonates with inductors L_x and L₁ and capacitor C_C resonates with L₂, the CPT circuit in Figure 7b can realize the load-independent CV output. With the aforementioned analysis, if the CV angular frequency is defined as ω_v1, it should satisfy the constraint:

$${\omega }_{v1}=\sqrt{\frac{L_1{+L}_x}{L_1{L}_x{C}_A}}=\frac{1}{\sqrt{L_2{C}_C}}$$

(6)

According to Figure 7a, the input impedance Z_v1 at ω_v1 can be simplified as

$$Z_{v1}=\frac{{\omega }_{v1}^4{C}_A^2{L}_x^2{R}_E}{{1+j\omega }_{v1}{(C}_A{+C}_{B }-{ \omega }_{v1}^2{C}_A^2{L}_x{)R}_E}$$

(7)

To obtain a resistive input impedance, the CV angular frequency ω_v1 should satisfy:

$${\omega }_{v1}=\sqrt{\frac{C_A{+C}_B}{L_x{C}_A^2}}$$

(8)

Then, Equation (7) is simplified as

$$Z_{v1}{=\omega }_{v1}^4{C}_A^2{L}_x^2{R}_E$$

(9)

Until now, the input ZPA and CV output of the CPT circuit are all achieved at ω_v1. To work at a CC mode, a CC frequency for the L_x-PS CPT circuit with input ZPA and CC output should be derived.

The CPT circuit can be converted to Figure 8. With (C_A + L_x‖L₁) ‖ C_B = ∞, a load-independent CC output is realized. With the aforementioned analysis, if the CC angular frequency is defined as ω_c, it should satisfy the constraint:

$${\omega }_c=\sqrt{\left(\frac{1}{C_A}+\frac{1}{C_B}\right)\left(\frac{1}{L_1}+\frac{1}{L_x}\right)}$$

(10)

With Figure 8, the input impedance Z_c at ω_c can be calculated as

$$Z_c=\frac{{j\omega }_c^3{C}_M^2{L}_x^2}{C_2{+\omega }_c^2{C}_M^2{L}_{x }-{ \omega }_c^2{C}_2^2{L}_2{+j\omega }_c{C}_2^2{R}_E}$$

(11)

To obtain a resistive input impedance, the CC angular frequency should satisfy:

$${\omega }_c=\sqrt{\frac{C_2}{C_{2 }^2{L}_2{+C}_{M }^2{L}_x}}$$

(12)

Then, the input impedance Z_c can be simplified as

$$Z_c=\frac{{\omega }_c^3{C}_M^2{L}_x^2}{C_2^2{R}_E}$$

(13)

The output current i_rps can be simplified as

$$i_{rps}=\frac{{j\omega }_c{C}_M^2\left(C_1{+C}_2-{ 2C}_M\right)}{C_2\left(C_1-{ C}_M\right)\left(1 -{ \omega }_c^2{C}_2{L}_2\right)}{v}_{in}$$

(14)

According to Equations (13) and (14), the input ZPA is realized in the CC mode. With the above analysis, the L_x-PS CPT circuit can realize the input ZPA in both CC and CV modes.

However, according to Equations (6) and (8), there are three constraints for the realization of input ZPA and CV output. As stated in the Introduction, six variables with one frequency and five compensation components are adopted to satisfy six design constraints. Thus, a CV frequency with two constraints to realize the input ZPA at CV mode is needed.

The CPT circuit can be converted to Figure 9. If the impendence of the purple box is zero, the CPT circuit can directly realize load-independent CV output. And the CV angular frequency defining as ω_v2 should satisfy

$$\left(C_1{C}_2-C_M^2\right)L_1{L}_2{L}_x{\omega }_{v2}^4-\left(L_1{L}_2{C}_2{+L}_1{L}_x{C}_1{+C}_2{L}_2{L}_x\right){\omega }_{v2 }^2{+L}_{1 }{+L}_x=0$$

(15)

The input impendence of the L_x-PS CPT circuit at ω_v2 is

$$Z_{v2}=\frac{{\omega }_{v2}^4{C}_M^2{L}_1^2{R}_E}{\left[{\omega }_{v2}^4{L}_1{L}_2{(C}_M^2{-C}_1{C}_2{)+\omega }_{v2}^2\left(C_1{L}_1{+C}_2{L}_2\right)-1\right]\left[{\omega }_{v2}^4{L}_1{L}_2{(C}_M^2{-C}_1{C}_2{)+\omega }_{v2}^2\left(C_1{L}_1{+C}_2{L}_2\right){-1+j\omega R}_E{(\omega }_{v2}^2{C}_1{C}_2{L}_1{-C}_2{-\omega }_{v2}^2{C}_M^2{L}_1)\right]}$$

(16)

To make Z_v2 resistively, ω_v2 should satisfy

$${\omega }_{v2}=\sqrt{\frac{C_2}{L_1\left(C_1{C}_2-C_M^2\right)}}$$

(17)

Then, Equation (16) is simplified as

$$Z_{v2}=\frac{C_2^2{R}_E}{C_M^2}$$

(18)

At ω_v2, the load-independent output voltage v_r2 is

$$v_{r2}=\frac{C_M}{C_2}{v}_{in}$$

(19)

According to Equations (15) and (17), the input ZPA is realized in the CV mode. With the above analysis, the L_x-PS CPT circuit can realize the input ZPA in both CC and CV modes with four constraints. According to Equations (14) and (19), the output current at CC mode and the output voltage at CV mode are all adjustable which means the L_x-PS CPT circuit can satisfy the charging demands of different kinds of loads.

2.2. The Analysis of the L_x-PP CPT Circuit

Similar with the L_x-PS CPT circuit, an additional inductor L_x is inserted between the voltage source and the primary side of a PP compensation circuit to form the L_x-PP CPT circuit as shown in Figure 10a. With Figure 10b, if C_M resonates with (L_x‖L₁‖C₁ − C_M), the input voltage v_in can be converted to a load-independent voltage source v_pp1. In this way, the CV angular frequency ω_cv can be calculated as

$${\omega }_{cv}=\sqrt{\frac{1}{C_1}\left(\frac{1}{L_1}+\frac{1}{L_x}\right)}$$

(20)

With Figure 10, the input impedance Z_cv at ω_cv is simplified as

$$Z_{cv}=\frac{{j\omega }_{cv}^5{C}_M^2{L}_2{L}_x^2{R}_E}{{j\omega }_{cv}{L}_2{+R}_E\left(1 -{\omega }_{cv}^2{C}_2{L}_{2 }{+\omega }_{cv}^2{C}_M^2{L}_2{L}_x\right)}$$

(21)

To eliminate the imaginary part of Z_cv, the CV angular frequency ω_cv should satisfy

$$1+\left({\omega }_{cv}^4{C}_M^2{L}_x-{\omega }_{cv}^2{C}_2\right)L_2=0$$

(22)

Similarly, with the constrain of Equation (22), the input ZPA at CV mode is achieved. And the simplified input impedance Z_cv is simplified as Equation (23).

$$Z_{cv}{=\omega }_{cv}^4{C}_M^2{L}_x^2{R}_E$$

(23)

Until now, the input ZPA and CV output are realized. According to (24), the output voltage v_ppr at ω_cv is adjustable.

$$v_{ppr}=\frac{v_{in}}{{\omega }_{cv}^2{C}_M{L}_x}$$

(24)

If L_x resonates with L₁ and C₁ − C_M and L₂ resonates with C₂ − C_M, the L_x-PP CPT circuit is converted to a load-independent current source as shown in Figure 11.

If the CC angular frequency is defined as ω_cc, it should satisfy two constraints:

$${\omega }_{cc}=\sqrt{\frac{1}{C_1-C_M}\left(\frac{1}{L_1}+\frac{1}{L_x}\right)}=\frac{1}{\sqrt{L_2\left(C_2-C_M\right)}}$$

(25)

The simplified input impedance Z_cc at ω_cc is expressed as

$$Z_{cc}=\frac{{j\omega }_{cc}^3{C}_M{L}_x^2}{{1+\omega }_{cc}^2{C}_M{L}_x{+j\omega }_{cc}{C}_M{R}_E}$$

(26)

The imaginary part of Z_cc cannot be eliminated. Thus, the input ZPA cannot be guaranteed.

To derive a CC frequency with a resistive input impedance, the L_x-PS CPT circuit using the T-type model is shown in Figure 5b. As shown in Figure 12a, if C_A resonates with L_x paralleling with L₁, v_in is transformed to a voltage source v_pp2. If L₂ resonates with C_C, the CPT circuit is converted to a load-independent current source as shown in Figure 12b. The CC angular frequency defined as ω_cc1 should satisfy

$${\omega }_{cc1}=\sqrt{\frac{1}{C_A }\left(\frac{1}{L_1}+\frac{1}{L_x}\right)}=\frac{1}{\sqrt{L_2{C}_C}}$$

(27)

The input impendence Z_cc1 at ω_cc1 is simplified as

$$Z_{cc1}=\frac{{j\omega }_{cc1}^3{C}_A^2{L}_x^2}{{j\omega }_{cc1}{C}_C^2{R}_{E }{+\omega }_{cc1}^2{C}_A^2{L}_x-{ C}_{A }-{ C}_B-{ C}_C}$$

(28)

If the CC angular frequency ω_cc1 satisfy

$${\omega }_{cc1}=\sqrt{\frac{C_{2 }-C_M}{L_x{C}_M\left(C_1-C_M\right) }}$$

(29)

the input impendence is resistive. Then, Z_cc1 is simplified as

$$Z_{cc1}=\frac{1}{{ \omega }_{cc1}^2{C}_M^2{R}_E }$$

(30)

If the impendence of the green box in Figure 13 is zero, the L_x-PS CPT circuit is converted to a voltage source v_pp3. The operating angular frequency ω_cv1 is

$${\omega }_{cv1}=\sqrt{\left(\frac{1}{C_A}+\frac{1}{C_B{+C}_C}\right)\left(\frac{1}{L_1}+\frac{1}{L_x}\right)}$$

(31)

Since Equation (31) can be simplified as Equation (20), the CV angular frequency ω_cv1 is equal to ω_cv.

Similar to the angular frequency ω_v of the L_x-PS circuit, there are three constraints for the L_x-PP circuit at ω_cc1 to realize input ZPA and CC output according to Equations (27) and (29). Thus, a CC frequency with two constraints to realize the input ZPA at CC mode is needed. The CPT circuit can be converted to Figure 14.

If the impendence of the blue box in Figure 14 is infinite, the L_x-PS CPT circuit is converted to a current source i_pp3. And the operating angular frequency ω_cc2 should satisfy

$${\omega }_{cc2}^4{L}_1{L}_2{L}_x\left(C_1{C}_2-C_M^2\right)-{\omega }_{cc2}^2\left(L_1{L}_2{C}_2{+L}_1{C}_1{L}_x{+C}_2{L}_2{L}_x\right){+L}_1{+L}_x=0$$

(32)

The simplified impendence of the L_x-PP circuit at ω_cc2 is calculated as

$$Z_{cc2}=\frac{{j\omega }_{cc2}{\left(L_x{+L}_1-{ \omega }_{cc2}^2{C}_1{L}_1{L}_x\right)}^2}{L_1-{ \omega }_{cc2}^2{C}_1{L}_1^2{+L}_{x }-{2\omega }_{cc2}^2{C}_1{L}_1{L}_x{+\omega }_{cc2}^4{C}_1^2{L}_1^2{L}_x{+j\omega }_{cc2}^3{C}_M^2{L}_1^2{R}_E}$$

(33)

To realize input ZPA, the L_x-PP circuit at ω_cc2 should satisfy

$${\omega }_{cc2}=\sqrt{\frac{1}{C_1{L}_1}}$$

(34)

Then Equation (33) can be calculated as

$$Z_{cc2}=\frac{1}{{\omega }_{cc2}^2{C}_M^2{R}_E}$$

(35)

And the expression of output current i_cc2 is

$$i_{cc2}{=j\omega }_{cc2}{C}_M{v}_{in}$$

(36)

As mentioned above, the mutual capacitance C_M of a regular capacitive coupler is always in pF-range. Thus, according to (36), even if the CV frequency is MHz-level, the output current i_cc2 is relatively small, which cannot satisfy the charging demands of most loads. With the aforementioned analysis, the L_x-PP CPT circuit cannot realize desired outputs at both CC and CV modes.

3. Parameter Design Procedure of the L_x-PS CPT Circuit

As analyzed in Section 2, the L_x-PS CPT circuit can satisfy the charging demands of batteries. The schematic of a L_x-PS CPT circuit is shown as Figure 15. The whole circuit is powered by a direct voltage V_DC. The inverter uses a full-bridge inverter. The rectification uses a full-bridge rectifier and a filter capacitor. The voltage V_DC is converted to a high-frequency square-wave voltage v_AB by the inverter. The fundamental component of v_AB is v_in which is calculated as

$$v_{in}\left(t\right)=\frac{{4V}_{DC}}{\pi }sin\frac{\pi D}{2}sin\omega t$$

(37)

And D is the duty cycle of v_AB. The load of the whole circuit is R_L. The relationship between R_L and R_E is

$$R_L=\frac{{\pi }^2}{8}{R}_E$$

(38)

Since the maximum value of i_r is I_r, the relationship between I_O and I_r is

$$I_O=\frac{2}{\pi }{I}_r$$

(39)

Since the maximum value of v_r is V_r, the relationship between V_O and V_r is

$$V_O=\frac{\pi }{4}{V}_r$$

(40)

As analyzed in Section 2, the CC angular frequency ω_c and the CV angular frequency ω_v2 are chosen as the operating frequencies of the L_x-PS CPT circuit. Given the load-required output current I_O and output voltage V_O, the compensation parameters L_x, L₁, L₂, C₁, and C₂ of the L_x-PS compensation circuit can be calculated as Equations (41)–(45) by the constraints of Equations (10)–(19).

$$L_x=\frac{C_2\left({\omega }_c^2-{ \omega }_{v2}^2\right)}{{2C}_M^2{\omega }_c^2{\omega }_{v2}^2}$$

(41)

$$L_1=\frac{C_2{\left({\omega }_c^2-{ \omega }_{v2}^2\right)}^2}{{2C}_M^2{\omega }_c^2{\omega }_{v2}^4}$$

(42)

$$L_2=\frac{{\omega }_c^2{+\omega }_{v2}^2}{{2C}_2{\omega }_c^2{\omega }_{v2}^2}$$

(43)

$$C_1=\frac{4\left({\omega }_c^4{+\omega }_{v2}^4\right)C_M{V}_O}{\pi {\left({\omega }_c^2-{ \omega }_{v2}^2\right)}^2{V}_{DC}}$$

(44)

$$C_2=\frac{{\pi C}_M{V}_{DC}}{{4V}_O}$$

(45)

The relationship between ω_c and ω_v2 can be simplified as

$${\omega }_{v2}{=\pi \omega }_c\sqrt{\frac{I_O}{{\pi }^2{I}_O{+16\omega }_c{C}_M{V}_{DC}}}$$

(46)

Until now, with predesigned CC angular frequency ω_c, output current I_O, and the output voltage V_O, the CV angular frequency ω_v2 and compensation parameters L_x, L₁, L₂, C₁, and C₂ are all determined. The output current i_O and the output voltage v_O can be simplified as

$$i_O=\frac{16}{{\pi }^2}\frac{{\omega }_c{\omega }_{v2}^2{C}_M{V}_{DC}}{j\left({\omega }_c^2-{ \omega }_{v2}^2\right)}$$

(47)

$$v_O=\frac{8}{{\pi }^2}\frac{C_2^2{R}_L}{C_M^2}$$

(48)

With predesigned CC angular frequency ω_c and the expression of Equation (47), the output current i_O at CC mode is adjustable by changing the CV angular frequency ω_v2. Similarly, the output voltage v_O at CV mode is also adjustable by changing the compensation parameter C₂. The output current i_O at CC mode is in direct proportion to ω_v2 and the output voltage v_O at CV mode is in direct proportion to C₂. Thus, the maximum power of the L_x-PS CPT circuit is limited by the insulating properties of the whole system and the actual output power is determined by the charging demands of loads.

Thus, the L_x-PS CPT circuit can be designed to charge lithium batteries.

4. Certification of the Proposed L_x-PS CPT Circuit

To verify the above analysis, a L_x-PS CPT circuit is designed. The constant output current is 2 A and the constant output voltage is 50 V. The CC operating frequency f_c is designed as 500 kHz and the input voltage V_DC is chosen as 100 V. The structure of the capacitive coupler is shown as Figure 1a. The power transfer distance is designed as 1 mm. The plates of the capacitive coupler are aluminum and the length of a plate is 380 mm. The distance between plate P₁ and plate P₂ or plate P₃ and plate P₄ is 10 mm. The power transfer distance is 18 mm. The mutual capacitance C_M is 352.2 pF, the primary and secondary capacitances C_P and C_S are 352.6 pF and 352.7 pF. With the predesigned output current, output voltage, the CC operating frequency, and the capacitive coupler, the CV operating frequency and compensation parameters can be calculated by Equations (37)–(48), which have been summarized in

Table 1. The CV operating frequency and compensation parameters.

Parameter	CV Frequency	Lx	L1	L2	C1	C2
Value	479 kHz	25.8 μH	2.31 μH	150.5 μH	47.9 nF	704 pF

.

According to Figure 16, the output current keeps at 2 A when R_L is equal to 12 Ω or 25 Ω, which means the CC output is successfully achieved. The input voltage v_AB is in phase with the input current i_AB, which means the input ZPA is achieved and most reactive power is eliminated.

According to Figure 17, when the operating mode of the L_x-PS CPT circuit is transferred to CV mode, the output voltages at 25 Ω and 50 Ω are all 50 V. Similarly, the load-independent CV output is achieved and the input ZPA is guaranteed.

However, the power transfer distance may change during the power transmission process. Since the metal plates P₁, P₂, P₃, and P₄ are the same size, capacitors C₁₃ = C₂₄, C₁₂ = C₃₄, and C₁₄ = C₂₃. Compared to C₁₃ and C₂₄, C₁₄ and C₂₄ are negligible. Then, Equation (1) can be approximately simplified as

$$\left\{\begin{array}{c}{C}_P{=C}_{12}+\frac{C_{13}}{2}\\ C_S{=C}_{12}+\frac{C_{13}}{2}\\ C_M=\frac{C_{13}}{2}\end{array}\right.$$

(49)

Based on the CV operating frequency and compensation parameters shown in Table 1, the effect of the change in the length of air gap between the plates on CC/CV conditions is shown in Figure 18.

With the increasing of the air gap, the capacitance C₁₃ would decrease. As shown in Figure 18, the output current increases at CC mode and the output voltage decreases at CV mode.

Figure 19 gives the curves of input phase angle, output current and voltage varying with the operating frequency, which can also verify the theoretical design.

If the plate misalignment happens, the secondary plates of P₃ and P₄ may move in X-axis or Y-axis direction as shown in Figure 1a. Thus, capacitances C₁₃, C₂₄, C₁₄, and C₂₃ would change. Taking P₃ and P₄ moving in X-axis as an example, the cross-coupling capacitors C₁₄ and C₂₃ are still negligible. Similar with the aforementioned analysis, C₁₃ and C₂₄ decrease when the plate misalignment happens. Based on the CV operating frequency and compensation parameters shown in Table 1, the output current at CC mode would increase and the output voltage at CV mode would decrease.

To permit ZVS of MOSFETs Q_1,2,3,4, the input impedance should be slightly inductive in both CC and CV modes. Figure 20 shows the output current or voltage versus the different normalized parameters and load conditions for the two modes of operation. From Figure 20a,b in CC mode, the output current is not sensitive to the variation of L₂ and L_x. From Figure 20c,d in CV mode, the output voltage is not sensitive to the variation of L_x. Thus, L_x is chosen to realize the ZVS of MOSFETs in the two modes of operation. Figure 21 shows the input phase angle versus normalized parameters at two charging modes and two loading conditions. A slight variation of L_x can permit a slight inductive phase angle for ZVS operation.

5. Discussion

A high-performance CPT circuit should be designed to realize input ZPA and load-independent outputs. Since lithium-ion batteries have been universally used, many CPT circuits have been proposed in different research to fulfill the charging demands of batteries.

The proposed L_x-PS CPT circuit can realize input ZPA and CC/CV outputs. Moreover, the output current and output voltage of the proposed circuit are all adjustable, which means the L_x-PS CPT circuit can satisfy the charging demands of different loads. Above all, to reduce the redundancy of the CPT system, the L_x-PS CPT circuit only has five compensation components. Compared to previous research, the number of compensation components is minimum and the whole CPT charger is designed considering all constraints of CPT technology.

6. Conclusions

An L_x-PS CPT circuit with only five compensation components is proposed in this paper to fulfill the charging profile of lithium-ion batteries. A CC frequency and a CV frequency with input ZPA and desired output gain are all derived. An L_x-PS CPT circuit with output current of 2 A and output voltage of 50 V was built and simulation was done to prove the analysis in this paper. According to simulation waveforms, CC output and CV output were all achieved. The input ZPA was realized in both CC and CV modes, which means the input reactive power was zero. Finally, the charging demands of batteries were fulfilled and the redundancy of the CPT circuit was successfully reduced.