Multi-Parameter Estimation for an S/S Compensated IPT Converter Based on the Phase Difference between Tx and Rx Currents

Abstract

This paper proposes a multi-parameter estimation method based on the phase difference between primary and secondary currents, for a series/series (S/S) compensated contactless converter. To achieve secondary current sensing in the transmitter side, a primary sensing coil is added into the coupler. By introducing the phase difference between primary and secondary currents, a straightforward, multi-parameter estimation model is derived, significantly reducing the multi-parameter solving difficulty. Based on the derived model, a method combining pre-parameter identification based on frequency sweeping, with fast online parameter identification is proposed, offering a general, accurate, and rapid parameter estimation solution. Detailed implementation of the parameter identification method and the asymmetrical configuration of the coupler are also presented. The proposed method is verified with a 1 kW S/S compensated converter. Experimental results show that the estimated values agree well with the theoretical ones. Based on the estimated results, the transmitter-side, closed-loop control can also be achieved.

1. Introduction

Due to its advantages of isolation, safety, convenience, and reliability, inductive power transfer (IPT) technology has been used in many applications, such as underwater devices, implantable devices, electric vehicles, and high-voltage isolated power supplies [1,2,3,4]. Different from conventional power converters, IPT converters have the problem of a wide range of parameter variations, which include not only the load change, but also the variations of the self-inductances and mutual inductance of the coupler, due to the inevitable change of the clearance and misalignment. Hence, it greatly increases the difficulty of the control for IPT converters [5,6].

To achieve tight output control against the parameter variations, a post regulator in the receiver side is often adopted in an IPT system, by introducing a cascaded DC/DC converter, a secondary dynamic-tuned resonant tank [7,8], or an active rectifier [9,10]. It dramatically complexes the system’s structure. By comparison, a transmitter-side (Tx-side) controller has the advantages of simplicity, lightness, and cost-effectiveness, especially for low- and medium-power applications. Numerous Tx-side control strategies, for instance, variable frequency control, phase shift control, pulse density modulation [11] and DC/DC conversion, have been applied in IPT systems. To realize closed-loop control, the output voltage/current should be sensed and fed back to the Tx side accurately and timely. Hence, real-time wireless communication approaches, such as Wi-Fi, Bluetooth, and ZigBee, are used to send the information of output voltage/current to the Tx side [12,13,14]. However, they suffer from the issues of disconnection, high latency, desynchrony, and interference. As a more reliable alternative, Tx-side parameter estimation methods have been proposed and investigated in recent years [15,16,17,18,19,20,21,22,23,24].

As shown in

Table 1. Main transmitter-side parameter estimation methods.

Parameter Identification Methods		Compensations	Identified Variables	Operating Frequency	Ref.
Transient model	Free oscillation damping model	S/S	RL (>50 Ω)	Self-oscillating frequency	[15]
			RL (<25 Ω)	Self-oscillating frequency	[16]
Steady-state model	Phasor analysis	LCL/P	Vo and M	Fixed at resonant frequency	[17]
		S/S	RL and M	Tracking primary resonant frequency	[18]
			M	Frequency with zero input phase angle	[19]
			M, r2	Fixed at resonant frequency	[20]
			RL and M	Different from ω2	[21]
	Energy conservation	S/S	Reflected impedance	Fixed at resonant frequency	[22]
	Curve fitting	S/S	M and equivalent load	Fixed at resonant frequency	[23]
			Q1, Q2, ω1, ω2, L1, L2, and k	/	[24]

Note: Vo is the output voltage and R_L is the load resistance. M and k represent the mutual inductance and the coupling coefficient between the transmitting and receiving coils. r₂ is the parasitic resistance of the secondary coil. L₁, L₂, Q₁, and Q₂ represent the self-inductances and quality factors of the two coils of the coupler, respectively. ω₁ and ω₂ represent the resonant frequencies of the transmitter and the receiver, respectively.

, the parameter identification methods for the IPT system can be classified into two categories. One is based on a steady-state circuit model, and the other is based on the transient model, i.e., to construct the identification equations according to the dynamic response. Based on the amplitude decay rate of the voltage or current in the transient process, [15,16] presented the load resistance estimations for series/series (S/S) compensated converters operating at the self-oscillating frequency. However, the identification of the variable coil inductance and the mutual inductance, which affect the output voltage/current gain and the estimation of V_o and I_o, was not mentioned in them.

Steady-state models are widely used in IPT systems for parameter estimation. Among them, an identification algorithm derived with phasor analysis is the normally used one [17,18,19,20,21]. In [17], identification of the mutual inductance and the output voltage for a secondary parallel compensated converter was conducted by analyzing the real and imaginary part of the reflected impedance under resonance. According to the phasor analysis under the conditions of switched compensation capacitors [18], shorted secondary circuit [19], special operating frequency [17,18,19,20] or startup process with variable duty cycle [19,20], and mutual inductance or load can be estimated. Reference [22] presents another method. It established the identification equations based on energy conservation and the characteristics of the zero-crossing point of the current in the transmitting (Tx) coil. However, most methods have special a requirement for the operating frequency, as shown in Table 1. Moreover, aforementioned methods are performed with the assumption of a constant self-inductance for the receiving (Rx) coil. Considering that the self-inductances and the resistances of the Tx and Rx coils vary with different displacements, it is necessary to perform a multi-parameter estimation including them, for achieving an accurate transmitter-side output control. Thus, [24] proposed a multi-parameter estimation method, including coil inductances and quality factors based on the curve fitting method, by using an optimization algorithm to iteratively solve the estimated parameters. However, due to a large number of iterations in the data processing, the estimation time is too long to be applied in real-time control.

The purpose of this article is to propose a multi-parameter estimation method for S/S compensated converters, which features generality, i.e., without operating frequency limitation, and with high accuracy and identification efficiency, suitable for Tx-side real-time control. This paper is organized as follows. In Section 2, the parameter estimation model of the S/S compensated converter, based on the phase difference between Tx and Rx currents, is proposed, as well as the estimation method combining pre-parameter identification based on frequency-sweeping, with fast online parameter identification. Detailed implementation of the proposed parameter estimation method is described in Section 3. Then, Section 4 proposes an asymmetric structure of the coupler, suitable for parameter estimation. Using a 1 kW S/S compensated converter, the proposed identification method is experimentally verified in Section 5. In Section 6, a conclusion of this article is conducted.

2. Parameter Estimation Method Based on the Phase Difference between Tx and Rx Current

2.1. Multi-Parameter Estimation Model

Figure 1 shows the schematic diagram of the S/S compensated converter, in which Q₁~Q₄ are power MOSFETs constituting an inverter, D₁~D₄ are secondary rectifier diodes, and R_L is the load resistor. Powered by a DC input voltage, V_in, the inverter produces an AC voltage, v₁ (current i₁), driving a resonant tank consisting of two compensation capacitors of C₁ and C₂, two equivalent series resistances of r₁, r₂, and a coupler. The coupler has a primary inductance, L₁, secondary inductance, L₂, and mutual inductance, M₁₂. The resonant tank delivers an AC current, i₂, which, after being rectified and filtered, gives the DC voltage Vo and current Io. It should be noted that r₁ and r₂ are the total equivalent series resistances in the primary and secondary circuits, which include the parasitic resistances of L₁, C₁, L₂, C₂, and the MOSFETs.

Using the fundamental harmonic approximation, the equivalent circuit of the S/S compensated converter can be obtained, as shown in Figure 2. In the figure, all voltage and current variables are represented by fundamental phasors, and the rectifier is represented by an equivalent resistance R_E.

With the help of Fourier analysis, we have

$$V_1=\frac{2\sqrt{2}}{\pi }{V}_{in},V_2=\frac{2\sqrt{2}}{\pi }\left(V_o+2V_F\right),I_2=\frac{\pi }{2\sqrt{2}}{I}_o$$

(1)

where V₁, V₂, and I₂ stand for the root-mean-square values of the fundamental components of v₁, v₂, and i₂, and V_F is the forward voltage drop of the rectifier diode.

Then, R_E can be obtained as

$$R_E=\frac{V_2}{I_2}=\frac{8}{{\pi }^2}\frac{V_o+2V_F}{I_o}=\frac{8}{{\pi }^2}\left(R_L+\frac{2V_F}{I_o}\right)\approx \frac{8}{{\pi }^2}{R}_L$$

(2)

Since 2V_F/(IoR_L) is normally less than 5% for ensuring a high efficiency, the effect of V_F on R_E can be neglected.

By using Kirchhoff’s voltage law, we have

$$\left[\begin{array}{l}{V}_1\angle 0°\\ 0\end{array}\right]=\left[\begin{array}{cc}j{X}_1+r_1& j\omega M_{12}\\ j\omega M_{12}& j{X}_2+r_2+R_E\end{array}\right]\left[\begin{array}{l}{I}_1\angle \phi \\ I_2\angle \theta \end{array}\right]$$

(3)

where

$$X_1=\omega L_1-\frac{1}{\omega C_1}=\frac{1}{\omega C_1}\left(\frac{{\omega }^2}{{\omega }_p^2}-1\right),X_2=\omega L_2-\frac{1}{\omega C_2}=\frac{1}{\omega C_2}\left(\frac{{\omega }^2}{{\omega }_s^2}-1\right)$$

(4)

ω represents the operating angular frequency; ω_p and ω_s are the resonant angular frequencies of Tx and Rx sides, equaling to $\frac{1}{\sqrt{L_1{C}_1}}$, and $\frac{1}{\sqrt{L_2{C}_2}}$, respectively; φ and θ are the phase angles for ${\dot{I}}_1$ and ${\dot{I}}_2$, referring to ${\dot{V}}_1$.

Expanding the real parts and imaginary parts of (3), yields

$$\left\{\begin{array}{l}{X}_1{I}_1\cos \phi +r_1{I}_1\sin \phi -\omega M_{12}{I}_2\cos \theta =0\\ -X_1{I}_1\sin \phi +r_1{I}_1\cos \phi +\omega M_{12}{I}_2\sin \theta =V_1\\ \omega M_{12}{I}_1\cos \phi -X_2{I}_2\cos \theta -(r_2+R_E)I_2\sin \theta =0\\ -\omega M_{12}{I}_1\sin \phi +X_2{I}_2\sin \theta -(r_2+R_E)I_2\cos \theta =0\end{array}\right.$$

(5)

Equations of (2), (4), and (5) provide the fundamental parameter estimation model for the S/S converter. Where, ω is a known parameter determined by the controller of the IPT system; φ, V₁, and I₁ can be directly measured on the Tx side; C₁, C₂, and V_F can be known prior to charging; L₁, L₂, M₁₂, r₁, V_o, I_o, and R_L are the parameters that need to be estimated. To facilitate the parameter identification, a primary sensing coil is introduced into the coupler for the detection of θ. As shown in Figure 3, the sensing coil has a self-inductance, L₃, mutual inductance, M₂₃, coupling with Rx coil, and mutual inductance, M₁₃, coupling with Tx coil. Adopting a decoupled configuration between Tx and sensing coils, M₁₃ can approximate to zero. Hence, the phase θ of ${\dot{I}}_2$ can be obtained in Tx side by the open-circuit voltage, ${\dot{V}}_3$, across the sensing coil, which satisfies

$${\dot{V}}_3=j\omega M_{23}{\dot{I}}_2-j\omega M_{13}{\dot{I}}_1\approx {\left.j\omega M_{23}{\dot{I}}_2\right|}_{M_{13}\approx 0}$$

(6)

Further, the phase difference, γ, between i₁ and i₂ can be obtained. Substituting γ = φ − θ into (5), yields

$$r_1{I}_1+\omega M_{12}{I}_2\sin \gamma =V_1\cos \phi$$

(7)

$$X_1{I}_1-\omega M_{12}{I}_2\cos \gamma =-V_1\sin \phi$$

(8)

$$X_1{I}_1\sin \gamma +r_1{I}_1\cos \gamma =V_1\cos \theta$$

(9)

$$X_2\sin \gamma -(r_2+R_E)\cos \gamma =0$$

(10)

$$\omega M_{12}{I}_1\sin \gamma -(r_2+R_E)I_2=0$$

(11)

$$\omega M_{12}{I}_1\cos \gamma -X_2{I}_2=0$$

(12)

The deriving procedure is provided in Appendix A.

Combining (7) and (12) with (4), and eliminating I₂ and X₂, gives

$$\frac{{\omega }^3\sin 2\gamma C_2}{2\left[\left({\omega }^2/{\omega }_s^2\right)-1\right]}{M}_{12}^2+r_1=\frac{V_1\cos \phi }{I_1}$$

(13)

In (13), M₁₂ and r₁ are independent of other parameters to be estimated. So, (13) gives a simple identification equation for M₁₂ and r₁. Substituting the identified value of M₁₂ and r₁ into (7) and (11), the values of I₂ and (r₂ + R_E) can be obtained. Then, according to (2), V_o, I_o, and R_E can be identified easily. It can be seen that, by introducing the phase difference between i₁ and i₂, a straightforward, multi-parameter estimation method can be derived.

However, the determination of ω_s and the efficient solving of (13) are still two important issues. In order to avoid any limitations on the operating frequency, we propose a frequency-sweeping method so as to obtain ω_s and other required information for the solving of (13). By selecting some special points with simplified equations for parameter identification, the solving efficiency can be improved. Thus, the process of the multi-parameter identification comprises two steps. Before charging, a frequency sweeping is performed for the IPT system under weak excitation. As a result, M₁₂, L₁, L₂, and r₁ can be determined prior to charging. During charging, V_o, I_o, and R_E can be identified rapidly online, based on the obtained parameters during frequency sweeping. At the same time, a Tx-side close-loop control for the IPT system can be realized. The proposed two-step parameter estimation method is suitable for static wireless charging applications, since M₁₂, L₁, L₂, and r₁ do not change during charging.

Next, detailed identifications of the parameters will be described.

2.2. Parameter Estimation during Frequency Sweeping

Firstly, we will discuss how to determine ω_s through frequency sweeping.

It can be seen from (6) that ${\dot{V}}_3$ is 90° ahead of ${\dot{I}}_2$ when M₁₃ approximates to zero. By frequency sweeping, it is easy to find a frequency at which ${\dot{V}}_3$ is in the phase of ${\dot{I}}_1$, meaning γ = 90°. Substituting γ = 90° into (9) and (10), and combining (4), we have

$$\gamma =90°:X_1=\frac{1}{\omega C_1}\left(\frac{{\omega }^2}{{\omega }_p^2}-1\right)=\frac{V_1\cos \theta }{I_1}, X_2=\frac{1}{\omega C_2}\left(\frac{{\omega }^2}{{\omega }_s^2}-1\right)=0$$

(14)

It indicates that the angular frequency, ω, corresponding to γ = 90° is just the secondary resonance frequency, ωs.

Substituting ωs = ω into (14) yields

$$\gamma =90°:\left\{\begin{array}{l}{\omega }_s=\omega \\ {\omega }_p=\omega \sqrt{\frac{I_1}{V_1\omega C_1\cos \theta +I_1}}\end{array}\right.$$

(15)

That means, by detecting the values of ω, V_in, I₁, and θ, corresponding to γ = 90°, both ω_s and ω_p can be obtained.

Then, substituting ω_s and ω_p into ${\omega }_p=\frac{1}{\sqrt{L_1{C}_1}}$ and ${\omega }_s=\frac{1}{\sqrt{L_2{C}_2}}$, L₁ and L₂ can be calculated.

At last, we will discuss how to solve (13) to identify M₁₂ and r₁.

As can be seen from (13), there are two unknown parameters (M₁₂ and r₁), but only one equation. During frequency sweeping, multiple sets of variables (ω, γ, φ, V₁, and I₁) can be measured, ensuring that (13) can be easily solved.

Defining

$$x_1\left(\omega \right)=\frac{{\omega }^2\sin 2\gamma }{2X_2}=\frac{{\omega }^3{C}_2\sin 2\gamma }{2\left({\omega }^2/{\omega }_s^2-1\right)},x_2\left(\omega \right)=\frac{V_1\cos \phi }{I_1}$$

(16)

Equation (13) can be rewritten as

$$x_1\left(\omega \right)M_{12}^2+r_1-x_2\left(\omega \right)=0$$

(17)

Equation (17) can be iteratively solved using an optimization algorithm, or directly solved adopting two special cases. We choose the latter for efficient solving. By using two sets of measured variables (ω, γ, φ, V₁, and I₁) at frequencies ω₁ and ω₂, the values of M₁₂ and r₁ can be solved as

$$\begin{array}{l}{M}_{12}=\sqrt{\frac{x_2\left({\omega }_1\right)-x_2\left({\omega }_2\right)}{x_1\left({\omega }_1\right)-x_1\left({\omega }_1\right)}}\\ r_1=\frac{x_1\left({\omega }_2\right)x_2\left({\omega }_1\right)-x_2\left({\omega }_2\right)x_1\left({\omega }_1\right)}{x_1\left({\omega }_2\right)-x_1\left({\omega }_1\right)}\end{array}$$

(18)

To facilitate the signal sensing and equation solving, we chose ω₁ as the corresponding frequency when v₁ is in-phase with i₂, i.e., θ = 0, and ω₂ as (ω₁ + ω_s) / 2. Other frequency selections are also feasible. However, the selection of ω₁ and ω₂ should be avoided around the frequency corresponding to γ = 90° or φ = 90°, near which the slopes of the functions sin2γ and cosφ are relatively large, having a large phase detection error.

2.3. Online Parameter Estimation

Substituting the identified value of M₁₂ and r₁ into (7) and (11), we have.

$$I_2=\left(V_1\cos \phi -r_1{I}_1\right)/\left(\omega M_{12}\sin \gamma \right)$$

(19)

$$V_2=R_E{I}_2=\frac{\omega M_{12}{I}_1\sin \gamma }{1+r_2/R_E}\approx \omega M_{12}{I}_1\sin \gamma$$

(20)

Normally, r₂/R_E is less than 3% for achieving a high efficiency. Thus, 1 + r₂/R_E ≈ 1 in (20).

Substituting (1) into (19) and (20), V_o and I_o can be derived

$$I_o=\frac{0.81V_{in}\cos \phi -0.9r_1{I}_1}{\omega M_{12}\sin \gamma }$$

(21)

$$V_o\approx 1.11\omega M_{12}{I}_1\sin \gamma -2V_F$$

(22)

Since M₁₂, r₁, and V_F are known parameters, and V_in, I₁, φ, γ, and ω can be measured online, the real-time identification of V_o and I_o can be realized.

Combining (11) and (12) with (4), gives

$$R_E+r_2=X_2\tan \gamma =\frac{1}{\omega C_2}\left(\frac{{\omega }^2}{{\omega }_s^2}-1\right)\tan \gamma$$

(23)

Combining (23) with (2), R_L can be obtained as

$$R_L\approx \frac{{\pi }^2}{8}{R}_E=\frac{{\pi }^2}{8}\frac{1}{\omega C_2}\left(\frac{{\omega }^2}{{\omega }_s^2}-1\right)\tan \gamma$$

(24)

Besides, according to (6), we can calculate the value of M₁₂ online, denoted as ${\widehat{M}}_{12}$, satisfying

$${\widehat{M}}_{12}=\sqrt{\frac{2}{\sin 2\gamma }\frac{1}{{\omega }^3{C}_2}\left(\frac{{\omega }^2}{{\omega }_s^2}-1\right)\left(\frac{V_1\cos \phi }{I_1}-r_1\right)}$$

(25)

According to the value of ${\widehat{M}}_{12}$ and the difference between ${\widehat{M}}_{12}$ and M₁₂, we can judge if the Rx pad is still in the charging region, thereby improving the reliability of the IPT system.

3. Implementation of the Proposed Estimation Method

3.1. Diagram of the IPT System

According to the proposed estimation method in Section 2, the diagram of the IPT system with the Tx-side circuit can be illustrated, as shown in Figure 4, where S₁ is used for operating state switching. When S₁ is connected with R_O, the IPT system will operate in a frequency-sweeping state under weak excitation, for pre-parameter-identification. When S₁ is connected with R_L, the IPT system will operate in a normal charging state, and the online parameter identification is achieved. As shown in Figure 4, v₃ and i₃ are converted to square wave signals by the zero detector, to facilitate the detection of φ, θ, and γ. After the parameter estimation, the values of L₁, L₂, M₁₂, r₁, Vo, Io, and R_L can be obtained. Based on them, a Tx-side, closed-loop control can be conducted.

3.2. Flowchart of the Parameter Estimation

Figure 5 and Figure 6 illustrate the flowcharts of the proposed parameter estimation algorithm, where the values of γ, φ, θ, V_in, and I₁ are measured by the sensors, ω can be obtained by the controller, and V₁ can be calculated by (2).

First, the offline parameter estimation is conducted. As shown in Figure 5, ω_s and ω₁ are obtained by sweeping the operating frequency f from f_min to f_max. The controller measures γ and θ at each frequency, and compares them with 90° and 0°, respectively. When γ = 90°, the corresponding operating frequency is recorded as ω_s; when θ = 90°, the corresponding frequency is recorded as ω₁. ε is the allowable error. Then, based on ω_s and ω₁, the operating angular frequency is adjusted, and the three groups of variables (ω, γ, φ, θ, V_in, and I₁) are measured at different frequencies to calculate ω_p, M₁₂, and r₁.

After finishing the offline parameter estimation, the online real-time identification of load variables (V_o, I_o, and R_L) can be realized, as shown in Figure 6. During the operation, ${\widehat{M}}_{12}$ is estimated and compared with M₁₂ to judge if the Rx pad is in the allowable charging region.

It is worth noting that the proposed estimation method has no operating frequency limitations. Besides, the proposed identification model has considered the effect of r₁ and V_F, offering high accuracy.

4. Configuration of the Magnetic Coupler

For the IPT system as shown in Figure 4, the contactless transformer is a crucial part, not only for power transfer but also for phase detection of the secondary current. In this section, we will discuss how to determine its configuration.

4.1. Proposed Asymmetrical Configuration

As indicated in (6), to realize an accurate phase detection, the contactless transformer is required to have a nearly zero M₁₃ and nonzero M₂₃. That means the sensing coil should always be decoupled with Tx coil and coupled with Rx coil. In previous studies, [25] has presented several decoupled configurations for achieving a nearly zero M₁₃. However, they also make M₂₃ zero when Rx coil is exactly aligned with Tx coil. To satisfy the requirements, an asymmetric structure of the coupler is proposed. As shown in Figure 7, the coupler adopts different sizes for Tx and Rx pads, and the sensing coil is divided into two asymmetric segments, L_C1 and L_C2.

To obtain the zero coupling (M₁₃ = 0), two sub coils of the sensing coil, L_C1 and L_C2, are concentrically placed on the Tx pad, and reversely connected in a series. Thus, the equivalent mutual inductance, M₁₃, can be expressed as

$$M_{13}=\left(N_{C1}{\varphi }_{C1}-N_{C2}{\varphi }_{C2}\right)/i_1$$

(26)

where N_C1 and N_C2 are the number of turns of L_C1 and L_C2, and ϕ_C1 and ϕ_C2 represent the magnetic flux coupled by L_C1 and L_C2, respectively. Both ϕ_C1 and ϕ_C2 are produced by the Tx coil, and they are the same sign. Then, by adjusting L_C1 and L_C2, M₁₃ can approach zero.

The asymmetric structure for L_C1 and L_C2, combining with the different size for Tx and Rx pads, ensures the nonzero M₂₃ under both aligned and misaligned conditions.

4.2. Simulation Results

We choose the magnetic coupler recommended by SAE J2954-2017 as a design example. Detailed dimensions are as shown in Figure 7, and the clearance, ∆Z, ranges from 10–16 cm.

As shown in Figure 7a, L_c1 is placed around the center of the Tx pad, with six turns for N_C1; L_c2 is placed around the outer border of the Tx pad with N_C2 turns. By measuring the mutual inductance with an LCR meter, we can easily find a proper value for N_C2 to achieve a zero M₁₂. The practical configuration of the sensing coil is shown in Figure 7c. As seen, N_C2 is about 4.7 since the outermost turn of L_c2 is not fully wound.

We also use the software COMSOL to assist the design of the sensing coil. Figure 8 shows the curves of M₁₃ and M₂₃ versus N_C2. It can be seen that zero coupling (M₁₃ = 0) occurs at the point of N_C2 = 4.7, which also guarantees a nonzero M₂₃. When Tx and Rx pads are aligned, due to the asymmetrical configurations of the coupler and the sensing coil, the couplings between L_c1/ L_c2 and Rx coil will not be completely offset, yielding a nonzero M₂₃. Besides, the simulated value of N_C2 (4.7) for M₁₃ = 0 matches the measured one very well.

Using N_C1 = 6 and N_C2 = 4.7, M₁₃ and M₂₃ are further simulated in the case of misalignments, as shown in Figure 9. It can be seen that M₁₃ is close to zero and M₂₃ is larger than zero within a wide range of misalignments, meeting the coupling requirements. Therefore, the proposed asymmetric configuration is suitable for phase detection.

Since the sensing coil is mounted on the Tx pad, M₁₃ is a constant, while M₂₃ is a parameter sensitive to the relative position between Tx coil and Rx coil. As shown in Figure 9, M₂₃ gradually decreases with increasing ∆X and ∆Y. When M₂₃ is reduced to zero, the sensing coil will lose the function for the phase detection, which should be avoided. Figure 10 illustrates the simulated zero coupling position of M₂₃. The radius of the effective coupling area is approximately 10 cm, satisfying the misalignment requirements (∆X ≤ ±7.5 cm and ∆Y ≤ ±10 cm) by the SAE J2954 standard.

5. Experimental Evaluation and Discussion

5.1. Experimental Prototype

To verify the above analysis, a 1 kW prototype was built and tested in the laboratory. Figure 11 shows the experimental setup for the built prototype. The system parameters are listed in

Table 2. IPT prototype parameters.

Components	Value	Components	Value
Input voltage	80 V	Output power	1 kW
Resonant frequency	85 kHz	Compensation	C1 = 88.1 nF, rC1 = 41.2 mΩ
Clearance range	10–16 cm		C2 = 94.5 nF, rC1 = 27.4 mΩ

. The variable frequency range is from 85 kHz to 100 kHz. Q₁~Q₄ are IXTV200N10T (R_DS(ON) = 5.5 mΩ) and D₁~D₄ are DSEI120-06A (V_F = 0.7 V). The parameter estimation method and the closed-loop control are implemented in the DSP TMS320F28335 (150 MHz).

The experimental verifications include three parts: (1) the verification of the phase-detection function; (2) the feasibility and accuracy of the proposed parameter estimation method; (3) the transmitter-side, closed-loop control with the estimated results.

5.2. Phase Detection of the Secondary Current

The practical inductances and coil resistances (r_L1, r_L2, and r_L3) of the proposed magnetic coupler are listed in

Table 3. Measured parameters of the magnetic coupler.

∆Z	L1/μH	L2/μH	L3/μH	M12/μH	M13/μH	M23/μH	rL1/Ω	rL2/Ω	rL3/Ω
10 cm	42.56	38.66	47.26	10.62	0.165	8.36	0.065	0.06	0.28
13 cm	44.07	37.52	46.5	8.455	0.06	4.92
16 cm	44.97	36.7	46.25	6.635	0.015	2.915

. Figure 12 illustrates the variation of M₁₂, M₂₃, and M₁₃ under misalignments and different clearance conditions. As expected, M₁₃ is approximately zero within the required clearance range and a certain range of misalignments, while M₂₃ is greater than zero, satisfying the coupling requirements discussed in Section 4.

To verify the phase-detection function of the sensing coil, the experimental waveforms of v₃, its filtered output, v_{3_s}, and i₂, are tested with different misalignments. As shown in Figure 13, v₃ is distorted. To correct this distortion, a low-frequency filter with a 90° phase shift is adopted. In the waveforms, v_{3_s} and i₂ are always in-phase, and lag, v₃, is nearly 90°, agreeing with the theoretical analysis in (6). Therefore, the presented asymmetrical configuration of the coupler can accurately detect the phase information of i₂.

5.3. Parameter Estimation Results

5.3.1. Offline Parameter Estimation

Offline parameter estimation is conducted under weak excitation with V_in = 30 V. Relay S₁ in Figure 4 is switched to the test resistor Ro, and Ro is fixed at 8 Ω during the estimation process. Following the procedure provided in Figure 5, M₁₂, L₁, L₂, and r₁ are obtained, as illustrated in

Table 4. Estimated results of M₁₂, L₁, L₂, and r₁ at different clearances.

∆Z	Estimated Results				Estimation Error ε
	L1/μH	L2/μH	M12/μH	r1/Ω	L1/μH	L2/μH	M12/μH	r1/Ω
10 cm	41.98	37.48	10.2	0.156	−1.3%	−3.05%	−3.95%	33.1%
13 cm	43.61	37.08	8.162	0.15	−1.04%	−1.17%	−3.47%	27.9%
16 cm	44.52	36.41	6.709	0.147	−1%	−0.08%	−1.12%	25.4%

and Figure 14.

Note that r₁ represents the total equivalent resistance in the transmitter circuit, and its measured value is calculated as the sum of r_L1, r_C1, and R_DS(ON).

$$r_{1\_\mathrm{mea}}\approx 2R_{DS\left(ON\right)}+r_{L1}+r_{C1}=117.2 \mathrm{m}\mathsf{\Omega }$$

(27)

To evaluate the accuracy of the estimation, the estimation error is defined as

$$\epsilon =\left(\mathrm{Mea}.-\mathrm{Cal}.\right)/\mathrm{Mea}.\times 100\%$$

(28)

where Mea. and Cal. represent the measured and calculated values, respectively.

It can be seen from Table 4 that the ε for M₁₂, L₁, and L₂ are all less than 4%, indicating the high accuracy of the estimation method. However, the ε for r₁ is much larger. From (27), it can be noted that r₁ is only 117.2 mΩ. Thus, a small deviation will lead to a large error.

Figure 14 shows that the estimation accuracy is lower when the misaligned distance is relatively large. Hence, the misalignment should be set within a suitable range, to ensure estimation accuracy. Within the misaligned range of ∆X ≤ 6 cm and ∆Y ≤ 6 cm, the maximum identification error of M₁₂ is 5.5%, the maximum identification error of L₁ is 3.9%, and the maximum identification error of L₂ is 3.6%.

5.3.2. Online Parameter Estimation

With the estimated M₁₂, L₁, L₂, and r₁, the online parameter estimation is carried out based on the estimation method in Figure 6. The relative position of the Tx coil and the Rx coil is fixed at (∆X = 0 cm, ∆Y = 0 cm, and ∆Z = 10 cm).

Figure 15 shows the measured and estimated values of V_o and I_o at various frequencies and load resistances. The frequency is changed from 85 kHz to 105 kHz, and R_o is varied from 6 Ω to 16 Ω. All the identification results are in good agreement with the measured ones, and the maximum ε is less than 8%. The feasibility and accuracy of the proposed identification method are well verified.

The online estimated values of M₁₂ with different Ro are illustrated in Figure 16. The measured value (10.62 μH) and the offline estimated value (10.2 μH) are also marked in Figure 16 for comparison. As seen, the estimated ${\widehat{M}}_{12}$ varies around M₁₂. Therefore, by comparing ${\widehat{M}}_{12}$ with M₁₂, it can be detected whether there is abnormal movement during charging.

Figure 17 shows the identification results of R_L. The switching frequency is fixed at 85 kHz, and R_L is changed from 4 to 20 Ω. The estimated results coincide with the theoretical results, verifying the effectiveness of the proposed estimation method.

5.4. Closed-Loop Control Results

Based on the dynamically estimated values of V_o and I_o, constant current (CC) and constant voltage (CV) charging for S/S compensation are also achieved by employing only a transmitter controller. Here, the variable frequency control is employed. The operating frequency range is set as 85 kHz–100 kHz. The calculation time for I_o and V_o are listed in

Table 5. Calculation time for I_o and V_o (TMS320F28335 150 MHz).

	Execution Cycle	Calculation Time
Vo	346	2.30667 μs
Io	178	1.18667 μs

. As seen, the calculation time is smaller than the switching period. So, V_o and I_o can be identified rapidly during charging.

In the prototype, a bidirectional, programmable DC power supply of IT6006C-500-40 is used to imitate the characteristics of the battery. The parameters of the battery emulator are set as follows: an empty voltage of 56 V, a full voltage of 80 V, a negative/positive current limit value of 20 A, and an inner resistance of 10 mΩ. The target charging current for CC mode is set as 13 A. The battery-charging waveforms of V_o and I_o, and the measured efficiency, are shown in Figure 18. As seen in Figure 18a, both CC charging and CV charging are achieved. It demonstrates the effectiveness of the proposed parameter estimation method. The maximum efficiency is 88.6%.

6. Conclusions

In this paper, a multi-parameter estimation method that utilizes the phase difference between primary and secondary currents is proposed for an S/S compensated contactless converter. This method has the advantages of generality, i.e., being without operating frequency limitation, and having high accuracy and identification efficiency, which is suitable for transmitter-side, real-time control. The detailed implementation of the parameter estimation method is studied. For realizing precise phase detection of the secondary current, a novel structure, which divides the sensing coil into two asymmetrical segments, is presented. A 1 kW wireless charger prototype is built for verification. The estimated results and the actual values are in close agreement, and the maximum identification error is less than 8%. Also, a closed-loop control is performed based on the estimated results.