A Double-Side Feedback Pulse Train Control for the Output Voltage Regulation of Two-Stage Wireless Power Transfer System

Abstract

Pulse Train (PT) control is a nonlinear voltage regulation method with favorable characteristics of quick response time and simple structure. In this paper, a PT-based feedback control strategy is proposed for stabilizing the output voltage of two-stage wireless power transfer (WPT) systems. Low-frequency voltage oscillations can be observed in PT controlled front-stage power converters, which significantly degrades the output power quality of the WPT system. To solve this problem, by using feedback variables sampled from both output sides of a power converter and WPT, a capacitor current and output voltage feedback PT (CC&OV-PT) controlled two-stage WPT system is further devised in this paper. A discrete time model of the two-stage WPT system is established, and the performance of low-frequency voltage oscillations suppression for the double-side feedback CC&OV-PT controlled WPT system is analyzed. Simulation and experimental verification have been conducted, both of which show that the CC&OV-PT controlled WPT system achieves fast response with effective low-frequency oscillation suppression.

1. Introduction

Wireless power transfer systems can deliver electric power from the power supply side to the load side without requiring any physical connections [1]. Possessing the advantages of contactless operation, safety, low maintenance and environment friendliness [2], WPT systems have been widely used in many applications, such as transportation, medical electronics, consumer electronics, and aerospace [3,4,5,6].

With the high nonlinearity [7], WPT systems are highly sensitive to many disturbances, such as input voltage fluctuation, coil misalignment and load variations [8,9]. Therefore, many control strategies have been proposed to improve the output voltage quality. The PWM control strategy is one of the most commonly used control strategies in WPT systems. In [10], a control scheme without the secondary side measurement is proposed. It only uses the current and voltage measurements of the primary side to estimate mutual inductance and load in real time. The PWM driving pulse is adjusted according to the changes in mutual inductance and load to stabilize the output voltage. Authors in [11] proposed a multi-transmitter and multi-receiver (MTMR) WPT PWM-based control strategy that can balance current distribution and achieve high efficiency and stable output voltage. In [12], a novel control strategy combining one cycle control and proportional differential control (OCC-PD) is proposed to ensure the swiftness of the transient response of the WPT system. The approximated boundary of the start-up transient time of the WPT stage is solved analytically by dividing the resonant circuit into three equivalent parts and analyzing their transfer functions, respectively. Authors in [13] proposed a new duty cycle control (DCC) strategy for a dual-side LCC resonant converter with a semi-bridgeless active rectifier (SAR). With such a control strategy, the gate signals of the primary- and secondary-side circuits do not need to be synchronized, which could reduce the complexity and the cost of the WPT system. A novel system topology for WPT of fuel cell (FC) energy is introduced in [14]. The pre-regulating WPT system is developed with a quadratic boost converter (QBC) located on the transmitting side is applied between the FC unit and the inverter to increase the output voltage of the FC to a higher voltage level, which is sufficient for WPT transmission. However, the transient response speed of many WPT control strategies needs to be improved in order to meet the operational requirements in applications, such as mining and medical power supply [15,16]. Concerning the well-established PWM-based methods, the advantage of the PT control method is its rapid-transient response speed and simple implementation [17]. The major characteristic of the PT control is that the error amplifiers and corresponding compensation circuits are not required, ensuring a relatively high response speed when the load varies [18].

Due to simple implementation, PT control methods have been applied in DC/DC power converters [19,20,21,22]. However, when the PT controlled DC/DC converters are operated in continuous conduction mode (CCM) [23,24], low-frequency voltage oscillations at the output side can be observed due to the phase difference between the control and the response. The tradeoff between improving response speed and increasing system stability is a challenge for PT control. Hence, many research efforts have been made on suppressing the low-frequency oscillations of PT controlled power converters. Authors in [25] proposed a PT control strategy with inductor current ripple injection feedback (ICRIF) to avoid such oscillations. The valley current-based PT (VCM-PT) control is proposed in [26] to ensure that the variation of inductor current in each switching cycle is equal to zero, and low-frequency oscillations can be eliminated. However, due to the use of preset valley current, the allowable output power range of the converter is limited. It has been proven in [27] that low-frequency oscillations can be suppressed by the PCC-PT control technique, which uses the peak capacitor current as feedback. Moreover, the PCC-PT controlled buck converter features a simple design, fast transient response, small output voltage ripple, and a wide load range. Different from single stage DC/DC converters, when the traditional PT control method is introduced into the two-stage WPT system, the low-frequency oscillations at the output voltage can be further amplified due to the asynchronization of oscillations between the input voltage and the output voltage, leading to an unstable output voltage.

In this paper, a double-side feedback control strategy for regulating the output voltage of two-stage WPT systems based on capacitor current and output voltage feedback is proposed. Compared to the conventional PT control strategy, the two feedback state variables in the proposed method are sampled from the output sides of the front-stage power converter and back-stage WPT. Hence, different from all reported PT control strategies, the proposed CC&OV-PT control strategy can compensate for the phase difference between control and response and eliminate most low-frequency oscillations by combining the filter capacitor current of the front-stage buck converter together with the output voltage of the WPT system as feedback variables for regulating the output voltage. This control strategy is then tested, which displays higher robustness and faster response compared with existing methods; this can provide a new reference for designing voltage regulators for WPT systems. It is also more suitable for some scenarios requiring high stability and response speed, such as the wireless charging of implanted medical devices, underwater and mining wireless power supply.

The rest of the paper proceeds as follows. The structure diagram of the two-stage WPT system and the analysis of low-frequency oscillations are presented in Section 2. Section 3 illustrates the control principle of the proposed double-side CC&OV-PT control. The dynamic model of the two-stage WPT system is established in Section 4. Section 5 simulates the PWM control, the traditional PT control and the proposed CC&OV-PT control methods, respectively. The output load variation and input voltage variation are studied, and the circuit parameter design is presented. In Section 6, experimental results are presented to verify that the proposed double-side CC&OV-PT method has the best control performance.

2. CC&OV-PT Controlled Two-Stage WPT System

2.1. Equivalent Model of the WPT System

Regarding WPT systems, the Serial-serial (SS) topology is widely used since it can achieve constant current output. The SS topology of a magnetically coupled resonance WPT system is shown in Figure 1.

DC voltage V₁ is converted into a high-frequency AC voltage v_P with frequency f₁ through a full-bridge inverter, and electric energy is wirelessly transferred to the secondary side through a resonance device composed of a capacitor and an inductor. AC voltage v_s is induced with frequency f₁ on the secondary side, which is then converted into a DC voltage by the full-bridge rectifier to energize load R_L.

Generally, high-order harmonic components of square wave voltage v_P can be attenuated to zero after passing through the resonant network. The fundamental voltage of v_P here is defined as $v_{\mathrm{p}\mathrm{x}}$, which can be expressed by:

$$v_{\mathrm{Px}}=\frac{4}{\pi }{V}_1\sin {\omega }_{1}t$$

(1)

By defining R_Lx as the equivalent load resistance and referring to [28], R_Lx can be equivalently written by:

$$R_{\mathrm{Lx}}=\frac{8}{{\pi }^2}{R}_{\mathrm{L}}$$

(2)

For calculation convenience, the equivalent circuit model can be constructed by using controlled sources for the WPT system, as shown in Figure 2.

Define R₁ as the resistance of the primary coil and R₂ as the resistance of the secondary coil. With the equivalent circuit shown in Figure 2, the currents of primary and secondary sides can be obtained by:

$$\left\{\begin{array}{l}{\dot{I}}_{\mathrm{P}}=\frac{4{\dot{V}}_1(R_2 + R_{\mathrm{Lx}})}{\pi \left[{\omega }^2{M}^2 + R_1\left(R_2 + R_{\mathrm{Lx}}\right)\right]}\\ {\dot{I}}_{\mathrm{S}}=\frac{4j\omega M{\dot{V}}_1}{\pi \left[{\omega }^2{M}^2+R_1\left(R_2+R_{\mathrm{Lx}}\right)\right]}\end{array}\right.$$

(3)

According to energy conservation, the input power of the rectifier circuit is equal to the output power, namely,

$${\left(\frac{I_S}{\sqrt{2}}\right)}^2{R}_{\mathrm{Lx}}=\frac{V_{\mathrm{o}}^2}{R_{\mathrm{L}}}$$

(4)

Thus, the output voltage V_o can be calculated as

$$V_{\mathrm{o}}=\frac{8{\omega }_{1}M{R}_{\mathrm{L}}{V}_1}{{\pi }^2\left[{\omega }_1^2{M}^2+R_1\left(R_2+R_{\mathrm{Lx}}\right)\right]}$$

(5)

Equation (5) reveals that the output voltage is in proportion to the input voltage when the WPT system operates in a steady state. Hence, the output voltage V_o can be timely regulated by adjusting the input voltage V₁. A DC/DC converter is usually employed as the first power stage to regulate the output voltage of the WPT system.

2.2. The Proposed Double-Side Feedback PT Control Strategy

2.2.1. WPT System Description

The overall topology of the proposed voltage-regulating WPT system is shown in Figure 3. The power circuit is composed of a Buck converter and a magnetic coupling resonant WPT circuit. The output voltage can be tuned by regulating the front-stage Buck converter connected to the H-bridge inverter. When the PT controller is applied to the Buck converter (S_e2 is off), the controller generates a high and low power pulse P with duty cycle D_v to drive the switch S₁ of the Buck converter. We have:

$$D_v=\left\{\begin{array}{l}{D}_{\mathrm{H}}\hspace{1em}P=P_{\mathrm{H}}\hspace{1em}\\ D_{\mathrm{L}}\hspace{1em}P=P_{\mathrm{L}}\end{array}\right.$$

(6)

The output voltage V_o is sampled and sent to the primary side through the wireless communication module, which is then compared with reference voltage V_ref. When V_o is higher than V_ref, a low power pulse is generated to drive the Buck converter to reduce the output voltage; when V_o is lower than V_ref, a high power pulse is produced to increase the output voltage. We have:

$$P=\left\{\begin{array}{l}{P}_{\mathrm{H}}\hspace{1em}{V}_{\mathrm{o}}<V_{\mathrm{ref}}\hspace{1em}\\ P_{\mathrm{L}}\hspace{1em}{V}_{\mathrm{o}}>V_{\mathrm{ref}}\end{array}\right.$$

(7)

When the WPT system operates in a steady state, the control pulse train will be generated with constant cycles of high and low pulses.

2.2.2. Cause and Analysis of Low-Frequency Oscillations

With regards to the Buck converter, the variation of the output voltage can be presented in terms of the inductor current,

$$\mathrm{\Delta }{v}_1=\frac{1}{C_1}{\displaystyle {\int }_0^{T_0}{i}_{\mathrm{C}}dt=\frac{1}{C_1}{\displaystyle {\int }_0^{T_0}(i_{\mathrm{L}}-i_1)dt}}$$

(8)

where i_C and i_L are the output capacitor current and inductor current of the Buck converter, respectively. Term i₁ is the input current of the full-bridge inverter, and Δv₁ is the voltage variation of capacitor C₁ inside a switching cycle.

Assuming that the switching frequency of the Buck converter is f₀, and the switching frequency of the inverter is f₁, for the design convenience, the following relation is applied in this work,

$$f_1=\frac{m{f}_0}{2}\hspace{1em}m=1,2,3\cdot \cdot \cdot N$$

(9)

Therefore, their corresponding angular frequencies and periods can be described by:

$$\left\{\begin{array}{l}{\omega }_1=2\pi f_1=\frac{m}{2}{\omega }_0\\ T_1=\frac{1}{f_1}=\frac{2}{m}{T}_0\end{array}\right.$$

(10)

where ω₀ and ω₁ are the equivalent angular speed of the Buck converter and the inverter, m is an integer, and T₀ and T₁ are their switching periods, respectively.

Regarding the high power pulse and low power pulse, the waveforms of inductor current i_L, capacitor current i_C as well as the input current i₁ of the inverter are shown in Figure 4.

The variation slope k_v of the inductor current is expressed by:

$$k_v=\left\{\begin{array}{l}{k}_r=\frac{V_{\mathrm{i}}-V_1}{L}\hspace{1em}n{T}_0<t\le (n+D_v)T_0\\ k_f=-\frac{V_1}{L}\hspace{1em}(n+D_v)T_0<t\le (n+1)T_0\end{array}\right.$$

(11)

Inside a switching cycle, the variation of inductor current can be calculated by:

$$\mathrm{\Delta }{i}_{\mathrm{L}}=i_{\mathrm{L}}(n{T}_0+T_0)-i_{\mathrm{L}}(n{T}_0)=\frac{\left(V_{\mathrm{i}}{D}_v-V_1\right)T_0}{L}$$

(12)

According to the operation principle of the inverter circuit, we have:

$$i_1=\frac{4\left(R_2+R_{\mathrm{Lx}}\right)V_1\left|\sin {\omega }_{1}t\right|}{\pi \left[R_1\left(R_2+R_{\mathrm{Lx}}\right)+{\omega }_1^2{M}^2\right]}$$

(13)

According to Figure 4, since the calculation of the shadow area corresponding to the high power pulse and the low power pulse is the same, only the high power pulse is taken as an example. The area can be calculated by:

$$S_{\mathrm{H}1}={\displaystyle {\int }_{n{T}_0}^{(n+1)T_0}{i}_{\mathrm{L}}(t)dt}=\frac{T_0^2}{2L}(-V_{\mathrm{i}}{D}_{\mathrm{H}}^2+2V_{\mathrm{i}}{D}_{\mathrm{H}}-V_1)+T_0{i}_{\mathrm{L}}(n{T}_0)$$

(14)

$$S_{\mathrm{H}2}={\displaystyle {\int }_{\mathrm{n}{T}_0}^{(n+1)T_0}{i}_1(t)}dt=\frac{8m\left(R_2+R_{\mathrm{Lx}}\right)V_1}{\pi {\omega }_1\left[R_1\left(R_2+R_{\mathrm{Lx}}\right)+{\omega }_1^2{M}^2\right]}$$

(15)

$$\mathrm{\Delta }{v}_1=\frac{1}{C_1}{S}_{\mathrm{H}3}\hspace{1em}=X+Y+Z$$

(16)

where X, Y and Z are expressed by,

$$\left\{\begin{array}{l}X=\frac{T_0^2}{2L{C}_1}\left(-V_{\mathrm{i}}{D}_{\mathrm{H}}^2+2V_{\mathrm{i}}{D}_{\mathrm{H}}-V_1\right)\\ Y=\frac{T_0}{C_1}{i}_{\mathrm{L}}\left(nT\right)\\ Z=-\frac{8m\left(R_2 + R_{\mathrm{Lx}}\right)V_1}{{\omega }_1{C}_1\left[R_1(R_2 + R_{\mathrm{Lx}}) + {\omega }_1^2{M}^2\right]}\end{array}\right.$$

(17)

It can be seen that Δv₁ is decided by three terms, X, Y and Z. The values of X and Z are in fact constant when the system parameters are determined, and the value of Y is decided by i_L(nT) and hence Δv₁ is decided by i_L(nT).

According to (16), with regards to high and low power pulses, we have

$$\mathrm{\Delta }{v}_1=\left\{\begin{array}{l}{X}^{\mathrm{H}}+Y+Z\hspace{1em}P=P_{\mathrm{H}}\\ X^{\mathrm{L}}+Y+Z\hspace{1em}P=P_{\mathrm{L}}\end{array}\right.$$

(18)

where X^H and X^L are the values of X corresponding to high power pulse and low power pulse states, respectively.

Assuming Δv_C = 0, i_L(nT) can be obtained by:

$$i_{\mathrm{L}}(nT)=\left\{\begin{array}{l}-\frac{C}{T}(X^{\mathrm{H}}+Z)=i_{\mathrm{b}1}\\ -\frac{C}{T}(X^{\mathrm{L}}+Z)=i_{\mathrm{b}2}\end{array}\right.$$

(19)

Define i_b1 and i_b2 as the boundary currents of either high or low power pulses for driving the power switch, respectively. We have:

$$i_{\mathrm{b}1}<i_{\mathrm{b}2}$$

(20)

The inductor current curve i_L(nT) can be divided into three regions, namely Regions I, II and III, as shown in Figure 5.

In a switching period, the variation of the output voltage of the front-stage converter is in fact dependent on the region the inductor current i_L(nT) sits in. As shown in

Table 1. Influence of i_L(nT) on input voltage of inverter.

Pulse P	Inductor Current Variation	iL(nT) Region	Capacitor Voltage Variation
PH	ΔiL > 0	I, II	ΔvC > 0
		III	ΔvC < 0
PL	ΔiL < 0	I	ΔvC > 0
		II, III	ΔvC < 0

, regardless of the selection of high or low power pulses, the inductor current could locate in any of these three regions divided by boundary currents i_b1 and i_b2.

At the beginning of each pulse period, the output voltage is sampled and compared to the reference voltage. If v_o ≤ V_ref, the high power pulse P_H will be selected to increase the input voltage of the inverter. Meanwhile, the inductor current of the front-stage converter can also be enlarged. However, when i_L(nT) is less than i_b1, i.e., located in Region III, the input voltage of the inverter will be reduced. Hence, in the following pulse period, high power pulse P_H will be still selected for driving the power switch, and the inductor current will be further increased and the input voltage of the inverter u₁ will keep decreasing until inductor current i_L(nT) is located outside Region III.

If v_o is greater than V_ref at the beginning of a sampling period, the low power pulse P_L will be selected and the inductor current will be reduced. However, when i_L(nT) locates in Region I, the input voltage of the inverter will be increased. Hence, in the following switching cycle, the low power pulse P_L will be chosen and the inductor current will keep decreasing. The input voltage of the inverter will be increased until the inductor current moves out of Region I. Therefore, the variation in the output voltage is decided not only by the sampled output voltage error but also by the inductor current, and low-frequency oscillations will appear in the input voltage of the inverter. These input low-frequency oscillations of the inverter will also lead to output voltage oscillations of the WPT system.

3. The Proposed CC&OV-PT Control

In order to suppress the low-frequency oscillations of the output voltage, a capacitor current and output voltage based pulse train, CC&OV-PT, control strategy is designed in this study. Note, that between the front-stage buck converter and the output load, there is one inverter, two contactless resonant LC circuits and one rectifier.

At the beginning of each switching cycle, the output voltage of the secondary side of the WPT system is sampled and then added to the capacitor current sampled from the output side of the front-stage power converter to obtain the feedback signal v_o + αi_C. If v_o + αi_C is less than V_ref, the high power pulse P_H will be selected to drive the power switch; if v_o + αi_C is greater than V_ref, the low power pulse P_L will be chosen to regulate the power switch. The output voltage, inductor current i_L and feedback signals are illustrated in Figure 6.

In Figure 6, without the capacitor current feedback, there is a phase difference of about π/2 between the load voltage and the inductor current in the front-stage Buck converter. By introducing the capacitor current, the phase difference between the inductor current and the feedback signal v_o + αi_C is much less than π/2, which is instrumental to the timely regulation of output voltage.

As listed in Table 1, when i_L(nT) sits in Region II, the inverter input voltage v₁ can be increased by P_H and reduced by P_L. The ripples of the inverter input voltage can be suppressed and hence the stability of the output voltage of the WPT system can be improved. In order to ensure that the inductor current is restricted in Region II, the control pulse must be correctly selected to adjust the inductor current. Namely, as the inductor current reaches the lower current boundary i_b1, the feedback signal v_o + αi_C should be less than V_ref, and hence P_H is selected to increase the inductor current. As the inductor current reaches the upper current boundary i_b2, v_o + αi_C is greater than V_ref, and hence P_L is selected to decrease the inductor current. Therefore, we have:

$$\left\{\begin{array}{l}{v}_{\mathrm{o}}+\alpha i_{\mathrm{C}}<V_{\mathrm{ref}}\hspace{1em}{i}_{\mathrm{L}}=i_{\mathrm{b}1}\\ v_{\mathrm{o}}+\alpha i_C>V_{\mathrm{ref}}\hspace{1em}{i}_{\mathrm{L}}=i_{\mathrm{b}2}\end{array}\right.$$

(21)

For the given output load voltage and current, the feedback coefficient α should be configured to satisfy the following condition:

$$\frac{v_{\mathrm{o}}-V_{\mathrm{ref}}}{i_{\mathrm{o}}-i_{\mathrm{b}2}}<\alpha <\frac{v_{\mathrm{o}}-V_{\mathrm{ref}}}{i_{\mathrm{o}}-i_{\mathrm{b}1}}$$

(22)

An example of t = nT when i_L(nT) > i_b2 and v_o(nT) < V_ref is shown in Figure 6. For the traditional PT controlled Buck converter, the high power pulse P_H will be selected since v_o(nT) < V_ref, and the inductor current will be increased to escape from Region II. When using the double-side feedback CC&OV-PT control, we have v_o(nT) + αi_C(nT) > V_ref, hence the low power pulse P_L is selected and the inductor current will be decreased. The inductor current will be dropped into region II. In this way, the inductor current can be restricted in Region II and the output voltage can be effectively regulated. Thus, the low-frequency oscillations of the output voltage are largely suppressed.

4. Dynamic Modeling of the Two-Stage WPT System

From (16), the output voltage variation Δv₁ and the inductor current variation Δi_L in one switching period are derived as:

$$\mathrm{\Delta }{v}_1=\frac{T_0^2}{2L{C}_1}\left(-V_{\mathrm{i}}{D}_v^2+2V_{\mathrm{i}}{D}_v-V_1\right)+\frac{T_0}{C_1}{i}_L(nT) - \frac{8m\left(R_2+R_{\mathrm{Lx}}\right)V_1}{{\omega }_{1}C\left[R_1(R_2+R_{\mathrm{Lx}})+{\omega }_1^2{M}^2{R}_1(R_2+R_{\mathrm{Lx}})+{\omega }_1^2{M}^2\right]}$$

(23)

$$\mathrm{\Delta }{i}_L=\frac{(V_{\mathrm{i}}{D}_v-V_1)T_0}{L}$$

(24)

According to (23), the discrete iterative model of the front-stage Buck converter of the WPT system is obtained, namely,

$$X_{n+1}=\left\{\begin{array}{l}A{X}_n+B^{\mathrm{H}}\hspace{1em}C{X}_n\le V_{\mathrm{ref}}\\ A{X}_n+B^{\mathrm{L}}\hspace{1em}C{X}_n>V_{\mathrm{ref}}\end{array}\right.$$

(25)

where:

$$\begin{gathered} X_{n+1}=\left[\begin{array}{l}{v}_1(nT+T)\\ i_{\mathrm{L}}(nT+T)\end{array}\right], X_n=\left[\begin{array}{l}{v}_1(nT)\\ i_{\mathrm{L}}(nT)\end{array}\right], \\ A=\left[\begin{array}{cc}1-\frac{T_0^2}{2L{C}_1}-\frac{8m(R_2+R_{\mathrm{Lx}}){\pi }^2}{{\omega }_{1}C[R_1(R_2+R_{\mathrm{Lx}})+{\omega }_1^2{M}^2]}& \frac{T_0}{C_1}\\ -\frac{T_0}{L}& 1\end{array}\right] \\ {B}^{\mathrm{H}}=\left[\begin{array}{c}\frac{T_0^2{V}_{\mathrm{i}}}{2L{C}_1}(-2D_{\mathrm{H}}^2+5D_{\mathrm{H}})\\ \frac{T_0{V}_{\mathrm{i}}{D}_{\mathrm{H}}}{L}\end{array}\right], B^{\mathrm{L}}=\left[\begin{array}{c}\frac{T_0^2{V}_{\mathrm{i}}}{2L{C}_1}(-2D_{\mathrm{L}}^2+5D_{\mathrm{L}})\\ \frac{T_0{V}_{\mathrm{i}}{D}_{\mathrm{L}}}{L}\end{array}\right], C=\left[\begin{array}{cc}1& \alpha \end{array}\right] \end{gathered}$$

With regards to the output voltage of the WPT system, we have [29],

$$v_{\mathrm{o}}(nT)=a{v}_1(nT-t_{\mathrm{b}})$$

(26)

where a is the amplitude ratio of the output to input voltages, and t_b is the delay time of the output voltage lagging the input voltage. The output voltage variation in a one switching cycle can be expressed as:

$$\mathrm{\Delta }{v}_{\mathrm{o}}(nT)=v_{\mathrm{o}}(nT+T)-v_{\mathrm{o}}(nT)=a\Delta v_1(nT-t_{\mathrm{b}})$$

(27)

Therefore, the discrete iterative model of the two-stage WPT system controlled by CC&OV-PT can be expressed by:

$$Y_{n+1}=A{Y}_n+B$$

(28)

where:

$$Y_{n+1}=\left[\begin{array}{l}{v}_{\mathrm{o}}(nT+T)\\ i_{\mathrm{L}}(nT+T-t_b)\end{array}\right],Y_n=\left[\begin{array}{l}{v}_{\mathrm{o}}(nT)\\ i_{\mathrm{L}}(nT-t_{\mathrm{b}})\end{array}\right]$$

5. Simulation Verification

In order to verify the operation performance of the two-stage WPT system controlled by the proposed double-side CC&OV-PT control strategy, the numerical models show the simulation diagram is built in MATLAB Simulink using the following simulation parameter specifications shown in

Table 2. The system parameters of numerical models in Simulink environment.

Parameter	Value
input voltage Vi	20 V
reference voltage Vref	10 V
switching frequency of Buck converter f1	20 kHz
switching frequency of Inverter f2	200 kHz
Inductance LP	200 uH
Inductance LS	200 uH
capacitance CP	3.2 nF
capacitance CS	3.2 nF
load resistance R	10 Ω/7 Ω
duty cycles of pulses PH and PL	0.7/0.3

. Simulation results are shown in Figure 7, Figure 8, Figure 9 and Figure 10.

The output voltage, inverter input voltage and inductor current under different control methods are also measured when shown in Figure 7 in terms of PWM control, traditional PT control, and CC&OV-PT control, respectively. As shown in Figure 7, the output voltage ripples of the PWM control, traditional PT control and CC&OV-PT control are 220 mV, 2 V and 25 mV, respectively. The input voltage ripples of the inverter are 330 mV, 4.5 V and 150 mV. According to these simulation results, the double-side CC&OV-PT control method has the best suppression effect on the system output oscillations, followed by the PWM control method.

The waveforms of output voltage and inverter input voltage are shown in Figure 8, Figure 9 and Figure 10 corresponding to the PWM control, traditional PT control and the proposed double-side CC&OV-PT control, when load and power supply voltage are changed abruptly. With regards to Figure 8a,b, Figure 9a,b and Figure 10a,b, the output load of the system is altered from 10 Ω to 7 Ω, and the input voltage is changed from 20 V to 15 V at t = 0.5 s, respectively. The transient time and the peak-to-peak values of the system output voltage under different control methods are shown in

Table 3. The regulation time and peak-to-peak values of system output voltage under different control methods and abrupt changes in different working conditions.

Working Conditions	Control Methods	Regulating Voltage Peak-to-Peak Value/V	Regulating Time/ms
Output load changed	PWM control	2.5	37.5
	Traditional PT control	3.1	3
	CC&OV-PT control	0.7	3
Input voltage changed	PWM control	3	30
	Traditional PT control	3.6	3
	CC&OV-PT control	0.09	4

. As shown in Figure 8, Figure 9 and Figure 10 and Table 3, as the operation condition is changed abruptly, the transient time of PT control is much shorter than that of the PWM control. Hence, the traditional PT control has a significant advantage in the recovery speed of the transient response. Regarding output voltage stabilization, the proposed CC&OV-PT control method can be effectively regulate the output voltage of this two-stage WPT system with a smaller oscillation amplitude than that of the PWM control and traditional PT control. Although the traditional PT control can achieve faster recovery, the low-frequency oscillations with high amplitudes greatly degrade the output voltage quality. Therefore, the traditional PT control method is not recommended to be used in this two-stage WPT system. In comparison, the proposed CC&OV-PT method has the best control performance from the simulation study.

6. Experimental Verification and Discussion

A prototype of the proposed CC&OV-PT controlled two-stage WPT system was designed and verified, as shown in Figure 11. The experimental system consists of a Buck converter, Inverter, coupling inductors, coupling capacitors, Rectifier, Resistor and Radio frequency (RF) module. The technical parameters are the same as those listed in Table 2. The vertical distance between the primary-side coil and the secondary-side coil is 30 cm. The experimental waveforms of the primary side voltage under the PT control and CC&OV-PT control methods are shown in Figure 12 and Figure 13, respectively. It can be seen from Figure 12 that, under the traditional PT control method, the output voltage, the inverter input voltage and the converter inductor current all have low-frequency oscillations with the frequency of about 120 Hz.

In a low-frequency oscillation cycle, 108 pulses can be observed, including 33 high power pulses and 75 low power pulses, namely the pulse combination of one oscillation cycle is 33 P_H-75 P_L. When the WPT system operates at a steady state, the oscillation ranges of output voltage v_o, inverter input voltage v₁ and the inductor current i_L are [8.8 V, 11.1 V], [8.3 V, 12.5 V] and [0 A, 2.1 A], respectively. These results show that the PT control method can result in unavoidable low-frequency oscillation in this two-stage WPT system.

As shown in Figure 13, when the CC&OV-PT control is applied, no low-frequency oscillation phenomena are observed in the waveforms of the output voltage v_o, the inverter input voltage v₁, and the converter inductor current i_L. It can be seen from the waveforms of the inductor current and power pulses, only 11 power pulses are measured in an oscillation cycle, and the combination of the power pulses is 1P_H-5P_L-1P_H-4P_L. When the WPT system operates at steady state, the output voltage v_o and inverter input voltage v₁ are stabilized at 10 V and 7 V, respectively. Meanwhile, the variation range of inductor current i_L is [1.1 A, 1.5 A]. These results show that the low-frequency oscillations have been eliminated by using the proposed CC&OV-PT control method.

The experimental waveforms corresponding to CC&OV-PT control and PWM control are presented in Figure 14a,b, respectively, when the load reduces from 10 Ω to 7 Ω. It can be seen that, for the CC&OV-PT controlled WPT system, the transient time is 5.6 ms and the overshoot of output voltage v_o is 1.1 V. However, when the conventional PWM method is applied, the transient time is 100 ms, and the overshoot of output voltage v_o is 1.3 V. Hence, the CC&OV-PT control is superior to the PWM control due to quicker transient response and smaller overshot of the output voltage.

The experimental waveforms with the CC&OV-PT and PWM methods are shown in Figure 15a,b, respectively, when the input voltage changes from 20 V to 15 V. It can be seen that, with the CC&OV-PT control method, the transient time is 5 ms and the output overshoot voltage v_o is 1.3 V. However, with the PWM method, the transient time is 170 ms and the output overshoot voltage v_o is 1.7 V. Therefore, the proposed CC&OV-PT method can provide better performance as compared to PWM method for two stage WPT system. These experimental results are consistent with the simulation verification conducted earlier. As mentioned in Section 5, the conventional PT control is not suitable for the two-stage coupled resonant WPT system. These have shown the superiority and applicability of the proposed double-side CC&OV-PT control approach.

7. Conclusions

In this paper, an improved double-sided CC&OV-PT control method for a two-stage coupled resonant WPT system is proposed to obtain a fast transient response and stable output voltage. Both simulation and experimental results have shown that, the proposed double-side CC&OV-PT control method has the minimum regulating voltage peak-to-peak value and regulating time as the operation condition is changed abruptly among the mentioned traditional PT control, PWM control and proposed CC&OV-PT control method in this work. It indicates that the double-side CC&OV-PT control has the best control performance in the three control methods, which can effectively suppress low-frequency oscillations and achieve fast voltage recovery against abnormalities.