Wireless Battery Charging Circuit Using Load Estimation without Wireless Communication

Abstract

A wireless battery charging circuit is proposed, along with a new load estimation method. The proposed estimation method can predict the load resistance, mutual inductance, output voltage, and output current without any wireless communication between the transmitter and receiver sides. Unlike other estimation methods that sense the high-frequency AC voltage and current of the transmitter coil, the proposed method only requires the DC output value of the peak current detection circuit at the transmitter coil. The proposed wireless power transfer (WPT) circuit uses the estimated parameters, and accurately controls the output current and voltage by adjusting the switching phase difference of the transmitter side. The WPT prototype circuit using a new load estimation method was tested under various coil alignment and load conditions. Finally, the circuit was operated in a constant current and constant voltage modes to charge a 48-V battery pack. These results show that the proposed WPT circuit that uses the new load estimation method is well suited for charging a battery pack.

1. Introduction

Wireless power transfer (WPT) technologies have been rapidly developed and widely applied to many industrial applications, such as biomedical devices, consumer electronics, manufacturing facilities, and electric vehicles (Evs), where direct contact between power supplies and applications is impossible or inconvenient [1,2,3,4]. To efficiently transfer power, most of the WPT circuits use electromagnetic coupling between coils. These WPT circuits use capacitors to reduce reactive power [5,6,7,8,9,10,11,12,13], and can be largely categorized into four types, depending on whether the capacitors are connected with the transmitter and receiver coils in series and series (S-S), series and parallel (S-P), parallel and parallel (P-P), or parallel and series (P-S) [5,6,7]. Among them, the S-S circuit has been widely used because the capacitances can be chosen independently of the load and coupling conditions [7,8,9,10].

A typical S-S WPT circuit (Figure 1) [7,9,10] consists of a full-bridge inverter (Q₁–Q₄), a transmitter coil (L₁), a full-bridge rectifier (D₁–D₄), a receiver coil (L₂), and two capacitors (C₁ and C₂). L₁ forms a resonance circuit with C₁, and L₂ forms a resonance circuit with C₂. Both resonance circuits are designed to have the same resonance frequency ${\omega }_o=2\pi \cdot f_o=1/\sqrt{L_1{C}_1}=1/\sqrt{L_2{C}_2}$. The transmitter and receiver coils have a mutual inductance, M₁₂. The input to the full-bridge inverter is a DC voltage V_DC.

To charge a battery, the S-S WPT circuit should be operated in constant current (CC) output mode when the battery voltage V_bat is lower than predetermined limit voltage V_bat,cut, and in constant voltage (CV) mode when V_bat,cut ≤ V_bat < charging voltage limit (CVL) [[9,10,11,12]. To support both modes, an additional DC–DC converter can be inserted between the S-S WPT circuit and the battery. However, the additional converter decreases the power transfer efficiency η_e and the power density [8,13]. To solve this problem, the battery can be directly connected to the S-S WPT circuit, as in Figure 1, and several control methods have been introduced [9,10,11,12,13].

The WPT circuit in [9] uses the same S-S WPT circuit (Figure 1) and adopts a pulse frequency modulation (PFM) method to obtain a CV output. In this circuit, the switching frequency range should be selected differently whenever the coupling coefficient is varied, so the range of the frequency limiter cannot be determined easily when the coupling coefficient k₁₂ varies widely. Also, wireless communication should be introduced to operate the PFM method. The circuit in [10] improves η_e by using two intermediate coils that are placed between the transmitter and receiver coils, and uses f = f_CC for CC output and f = f_CV for CV output, where the frequencies f_CC and f_CV are determined by the coupling coefficients among the four coils. However, the values of f_CC and f_CV vary in the manner that any coupling coefficient varies, and no method has been developed to date to measure the coupling coefficients, so accurate determination of f_CC and f_CV is a difficult task. The circuits in [11,12] use auxiliary switches and capacitors to change the output from CC to CV mode. However, this circuit needs wireless communication to change the operational mode, and additional components also decrease the power density. As mentioned above, most of control methods require wireless communication to know the load conditions and coupling state.

To eliminate the necessity for wireless communication, several load estimation methods have been presented [14,15,16,17,18,19]. The methods in [14,15,16] predict the load resistance R_L by using the information of the input voltage and current. However, these methods should know the value of the coupling state before estimating the load conditions, so they cannot be used for various coil alignments. The method in [17] adopts an additional capacitor in the S-S WPT circuit; this method operates the circuit in two modes for system identification, and analyzes the reflected impedance. However, the additional capacitor and bidirectional switch increase the circuit cost. The method in [18] measures the input voltage and current, and separates the imaginary part of the input impedance. To estimate the load conditions and coupling state, this method is implemented at one frequency, which is not a resonant frequency, so the impedance of the resonant tank slightly decreases the power transfer efficiency. The method in [19] injects a high frequency energy into the S-S WPT circuit, then detects the response of the circuit to estimate the load conditions. However, this method cannot follow the load conditions after initial energy injection. All of these methods [14,15,16,17,18,19] can estimate the load conditions well, so they should be able to sense the high-frequency AC input voltage and current. The resonant frequency of the WPT circuit can be up to several hundred kilohertz, so the sampling frequency should be much higher than the resonant frequency; as a result, the analog-to-digital conversion is difficult.

This paper proposes a wireless battery charging circuit along with a load estimation method. This circuit does not need any wireless communication between the transmitter and receiver sides, and predicts the load resistance R_L, output voltage V_bat, output current I_bat, and mutual inductance M₁₂. In addition, because the simple peak current detection circuit is applied at the transmitter coil, the proposed circuit only senses the DC value, and does not need a high sampling frequency. The proposed WPT circuit senses the peak current values of the transmitter coil at f_o and auxiliary frequency f_a, and calculates the load conditions by using these values. Then, the proposed WPT circuit operates in CC and CV modes, depending on the estimated load conditions and phase shift control of the full-bridge inverter. In Section 2, the analysis of the proposed WPT circuit with a load estimation method is given based on the fundamental harmonic approximation (FHA), experimental results are presented in Section 3, possible errors in the proposed estimation method are analyzed in Section 4, and a conclusion is given in Section 5.

2. Wireless Power Transfer Circuit for Battery Charging

2.1. Theoretical Models of the S-S WPT Circuit

The gate control pulses Q_g1–Q_g4 (Figure 2) for the full-bridge inverter have a switching frequency f = 1/T = ω/(2π). The switching phase of $Q_{g1}$ and ${\overline{Q}}_{g2}$ lags behind that of $Q_{g3}$ lags behind that of and ${\overline{Q}}_{g4}$ by an angle ϕ, so the bipolar output pulses of the full-bridge inverter (v₁, Figure 2) have a dead phase angle ϕ between the pulses. The fundamental component of v₁ is given by

$$v_1(t)=V_1\sin (\omega t)=\frac{2V_{DC}}{\pi }(1+\cos \varphi )\sin (\omega t).$$

(1)

The current i₁(t) of the transmitter coil, the current i₂(t) of the receiver coil, and the input voltage v₂(t) to the rectifier in Figure 1 can be expressed as

$$i_1=I_1\sin (\omega t+\theta )$$

(2)

$$v_2(t)=V_2\sin (\omega t+\theta +\varphi )=\frac{4V_{bat}}{\pi }\sin (\omega t+\theta +\varphi )$$

(3)

$$i_2(t)=I_2\sin (\omega t+\theta +\varphi )=\frac{I_{bat}\pi }{2}\sin (\omega t+\theta +\varphi ),$$

(4)

where θ and ϕ are phase angles, V_bat is the battery voltage, and I_bat is the averaged charging current of the battery.

The S-S WPT circuit had an equivalent circuit (Figure 3) for the fundamental component, where R_in, R₁, and R₂ are the equivalent series resistances (ESRs) of the full-bridge inverter, primary coil, and secondary coil, respectively. Using Equations (3) and (4), the equivalent resistance of the battery R_bat can be modeled with an equivalent resistance R_L,eq as:

$$R_{L,eq}=\frac{8}{{\pi }^2}\cdot R_{bat}=\frac{8}{{\pi }^2}\cdot \frac{V_{bat}}{I_{bat}}.$$

(5)

Then, the Kirchhoff’s voltage law (KVL) gives

$${\overrightarrow{V}}_1=(R_{in}+Z_1){\overrightarrow{I}}_1-j\omega M_{12}{\overrightarrow{I}}_2,$$

(6)

$$j\omega M_{12}{\overrightarrow{I}}_1=(Z_2+R_{L,eq}){\overrightarrow{I}}_2,$$

(7)

where Z₁ = R₁ + jωL₁ + 1/(jωC₁) and Z₂ = R₂ + jωL₂ + 1/(jωC₂). Using Equations (6) and (7), the phase of the input impedance Z_in (Figure 4a), the voltage conversion ratio T_v (Figure 4b), the amplitude of i₁(t), the peak current of i₁(t) (I₁) (Figure 4c), the amplitude of i₂(t), and the peak current of i₂(t) (I₂) (Figure 4d) are calculated as

$${\overrightarrow{Z}}_{in}=\frac{{\overrightarrow{V}}_1}{{\overrightarrow{I}}_1}=\frac{(R_{in}+Z_1)(Z_2+R_{L,eq})+{\omega }^2{M}_{12}^2}{Z_2+R_{L,eq}},$$

(8)

$$T_v=\left|\frac{{\overrightarrow{V}}_2}{{\overrightarrow{V}}_1}\right|=\left|\frac{R_{L,eq}{\overrightarrow{I}}_2}{{\overrightarrow{V}}_1}\right|=\left|\frac{j\omega M_{12}{R}_{L,eq}}{(R_{in}+Z_1)(Z_2+R_{L,eq})+{\omega }^2{M}_{12}^2}\right|,$$

(9)

$$I_1=\left|\frac{Z_2+R_{L,eq}}{(R_{in}+Z_1)(Z_2+R_{L,eq})+{\omega }^2{M}_{12}^2}\right|\cdot V_1,$$

(10)

$$I_2=\left|\frac{j\omega M_{12}}{(R_{in}+Z_1)(Z_2+R_{L,eq})+{\omega }^2{M}_{12}^2}\right|\cdot V_1.$$

(11)

2.2. Load Estimation Method Using the Magnitue of Input Impedance

The proposed circuit uses the simple peak detection circuit (Figure 5) in [20] to measure the peak current of the transmitter coil I₁ as a DC value. The peak detection circuit is composed of a current sensor, an amplifier for the peak detection (A₁), an amplifier for the voltage follower (A₂), an input resistance of peak detector (R_i), a feedback loop resistance (R_f), a feedback loop diode (D_f), a rectification diode (D_o), an output capacitor (C_o), and an output resistance (R_o). If the output voltage of the current sensor (V_s) is lower than the voltage of C_o (V_o), D_f remains on, and D_o remains off. In this operating mode, the output voltage of A₂ (V_out) is clamped to V_o, and C_o is discharged by R_o. When V_o becomes smaller than V_s, D_f is turned off and D_o is turned on. In this operating mode, C_o is charged to the new positive peak of V_s, so V_s = V_o = V_out.

To estimate the load conditions, the peak detection circuit measures the peak current I_1,o at f = f_o and I_1,a at f = f_a, respectively, and uses simple mathematical equations for the input impedance. The measurement time of I_1,o and I_1,a is short, so the load conditions are assumed to remain constant during the estimation process. Also, the system parameters of the transmitter side (V_DC, ϕ, L₁, C₁ and R₁) and receiver side (L₂, C₂ and R₂) are assumed to be known, and the proposed method predicts M₁₂, R_L,eq, I_bat, and V_bat.

At first, the circuit operates at f = f_o, and the M₁₂ can be expressed using detected I_1,o and Equation (8) as

$$M_{12}^2=\frac{[R_2+R_{L,eq}]\cdot \left[V_{1,o}-I_{1,o}(R_{in}+R_1)\right]}{{\omega }_o^2\cdot I_{1,o}}.$$

(12)

where V_1,o is the peak voltage of the transmitter coil at f = f_o, $V_{1,o}=2V_{DC}(1+\cos \varphi )/\pi$ from Equation (1), and the unknown parameters of Equation (12) are M₁₂ and R_L,eq.

Then, the circuit operates at f = f_a, and the square of the absolute value of input impedance ${\left|{\overrightarrow{Z}}_{in,a}\right|}^2=V_{1,a}^2/I_{1,a}^2$ can be expressed using Equation (8) as

$${\left|{\overrightarrow{Z}}_{in,a}\right|}^2={(R_1+R_{in})}^2+{({\omega }_a{L}_1-1/{\omega }_a{C}_1)}^2+\frac{2{\omega }_a^2{M}_{12}^2\left\{(R_1+R_{in})(R_2+R_{L,eq})-({\omega }_a{L}_1-1/{\omega }_a{C}_1)({\omega }_a{L}_2-1/{\omega }_a{C}_2)\right\}+{\omega }_a^4{M}_{12}^4}{{(R_2+R_{L,eq})}^2+{({\omega }_a{L}_2-1/{\omega }_a{C}_2)}^2},$$

(13)

where V_1,a, I_1,a is the peak voltage and current of transmitter coil at f = f_a, $V_{1,a}=2V_{DC}(1+\cos \varphi )/\pi$ from Equation (1). In this equation, the unknown parameters are the same as Equation (12).

If Equation (12) is applied to Equation (13), the R_L,eq can be arranged as $\alpha R_{L,eq}^2+\beta R_{L,eq}+\gamma =0$, where α, β and γ are as follows:

$$\begin{array}{l}\alpha ={\omega }_o^4{I}_{1,o}^2{\left|{\overrightarrow{Z}}_{in,a}\right|}^2-{\omega }_o^4{I}_{1,o}^2\left\{{(R_1+R_{in})}^2+{({\omega }_a{L}_1-1/{\omega }_a{C}_1)}^2\right\}-2{\omega }_a^2{\omega }_o^2{I}_{1,o}\left\{R_1+R_{in}\right\}\cdot \left\{V_{1,o}-I_{1,o}(R_{in}+R_1)\right\}\\ -{\omega }_a^4{\left\{V_{1,o}-I_{1,o}(R_{in}+R_1)\right\}}^2\end{array}$$

(14)

$$\begin{array}{l}\beta =2R_2{\omega }_o^4{I}_{1,o}^2{\left|Z_{in,a}\right|}^2-2R_2{\omega }_o^4{I}_{1,o}^2\left\{{(R_1+R_{in})}^2+{({\omega }_a{L}_1-1/{\omega }_a{C}_1)}^2\right\}\\ -2{\omega }_a^2{\omega }_o^2{I}_{1,o}\left\{2R_2(R_1+R_{in})-({\omega }_a{L}_1-1/{\omega }_a{C}_1)({\omega }_a{L}_2-1/{\omega }_a{C}_2)\right\}\cdot \left\{V_{1,o}-I_{1,o}(R_{in}+R_1)\right\}-2R_2{\omega }_a^4{\left\{V_{1,o}-I_{1,o}(R_{in}+R_1)\right\}}^2\end{array}$$

(15)

$$\begin{array}{l}\gamma ={\omega }_o^4{I}_{1,o}^2{\left|Z_{in,a}\right|}^2\left\{R_2^2+{({\omega }_a{L}_2-1/{\omega }_a{C}_2)}^2\right\}-{\omega }_o^4{I}_{1,o}^2\left\{{(R_1+R_{in})}^2+{({\omega }_a{L}_1-1/{\omega }_a{C}_1)}^2\right\}\cdot \left\{R_2^2+{({\omega }_a{L}_2-1/{\omega }_a{C}_2)}^2\right\}\\ -{\omega }_a^4{R}_2^2{\left\{V_{1,o}-I_{1,o}(R_{in}+R_1)\right\}}^2-2{\omega }_a^2{\omega }_o^2{I}_{1,o}\left\{R_2^2(R_1+R_{in})-R_2({\omega }_a{L}_1-1/{\omega }_a{C}_1)({\omega }_a{L}_2-1/{\omega }_a{C}_2)\right\}\cdot \left\{V_{1,o}-I_{1,o}(R_{in}+R_1)\right\}\end{array}.$$

(16)

This equation has two solutions for R_L,eq, and the smaller one is a reasonable value according to the calculation result, so estimated load resistance R_L,eq,est and estimated equivalent resistance of battery R_bat,est can be estimated as

$$R_{L,eq,est}=\frac{-\beta -\sqrt{{\beta }^2-4\alpha \cdot \gamma }}{2\alpha }=\frac{8}{{\pi }^2}\cdot R_{bat,est}.$$

(17)

Then, the estimated mutual inductance M_12,est can also be derived by applying Equation (17) to Equation (12) as:

$$M_{12,est}=\sqrt{\frac{[2\alpha \cdot R_2-\beta -\sqrt{{\beta }^2-4\alpha \cdot \gamma }]\cdot \left[V_{1,o}-I_{1,o}(R_{in}+R_1)\right]}{2\alpha \cdot {\omega }_o^2\cdot I_{1,o}}}.$$

(18)

Other important estimated load parameters I_bat,est and V_bat,est at f = f_o can be expressed using Equations (1)–(7), (17), and (18) as

$$I_{bat,est}=\frac{2I_{1,o}}{\pi }\cdot \frac{{\omega }_o{M}_{12,est}}{R_2+R_{L,eq,est}}$$

(19)

$$V_{bat,est}=\frac{{\pi }^2}{8}\cdot I_{bat,est}\cdot R_{L,eq,est}.$$

(20)

Finally, the proposed method can predict R_L,eq, M₁₂, I_bat, and V_bat, and does not need a high sampling frequency to measure AC voltage and current, similar to previous studies [14,15,16,17,18,19].

2.3. Control Method of the S-S WPT Circuit for Battery Charging

The battery should be charged in CC mode when V_bat ≤ V_bat,cut, and in CV mode when V_bat > V_bat,cut. In CV mode, I_bat decreases as V_bat increases, until I_bat reaches the end charging current I_end at which the charging operation stops [9,10,11,12].

T_v and I₂ in Equations (9) and (11) depend on R_L,eq, which varies as the charge state of battery varies. When all ESRs are negligibly small, Equation (11) gives I₂ at ω = ω_o as

$$I_2=\frac{{\omega }_o{M}_{12}{V}_1}{(R_{in}+R_1)(R_2+R_{L,eq})+{\omega }_o^2{M}_{12}^2}\approx \frac{V_1}{{\omega }_o{M}_{12}},$$

(21)

because Z₁ = R₁ and Z₂ = R₂ when ω = ω_o. This equation indicates that the WPT circuit can be operated in CC mode if all ESRs are ignored and ω = ω_o. However, ESRs affect the capability of CC regulation (Figure 4d), so a separate control method should be introduced to attain CC mode; the proposed WPT circuit applies phase shift control of the full-bridge inverter at f = f_o, and the ϕ to maintain the CC output is compensated by using the proportional integral (PI) controller, which can be calculated as

$$\varphi ={\cos }^{-1}\left\{\frac{{\pi }^2\cdot (I_{ref}/2)\cdot [(R_{in}+R_1)(R_2+R_{L,eq,est})+{\omega }_o^2{M}_{12,est}^2]}{2{\omega }_o{M}_{12,est}{V}_{DC}}-1\right\},$$

(22)

where I_ref is the predetermined charging current reference. If ESRs are very small in CC mode, the influence of R_L,eq in ϕ will also be very small.

To operate the WPT circuit in CV mode, T_v should not depend on R_L,eq. If all ESRs are negligible, Equation (9) can be approximated as

$$T_v\approx \left|[j\omega M_{12}]/[(j\omega L_1+1/j\omega C_1)+\kappa /R_{L,eq}]\right|,$$

(23)

where $\kappa ={\omega }^2(M_{12}^2-L_1{L}_2)-1/({\omega }^2{C}_1{C}_2)+L_1/C_2+L_2/C_1$. After setting κ = 0, the frequencies f_CV1 and f_CV2 for CV operation are obtained as $f_{CV}{}_1=2\pi \cdot f_o/\sqrt{1+k_{12}}$ and $f_{CV}{}_2=2\pi \cdot f_o/\sqrt{1-k_{12}}$, and T_v at f = f_CV1 or f = f_CV2 is calculated using Equation (23) and $M_{12}=k_{12}\sqrt{L_1{L}_2}$ as $T_v=\sqrt{L_2/L_1}$. However, ESRs in CV mode are also difficult to ignore, and if f_CV1 and f_CV2 deviate too much from f_o, the system efficiency also drastically decreases [14]. Therefore, the proposed WPT circuit still operates at f = f_o in CV mode, and the ϕ to maintain the CV output is compensated by using the PI controller, which can be calculated as

$$\varphi ={\cos }^{-1}\left\{\frac{4\mathrm{CVL}\cdot [(R_{in}+R_1)(R_2+R_{L,eq,est})+{\omega }_o^2{M}_{12,est}^2]}{2{\omega }_o{M}_{12,est}{V}_{DC}{R}_{L,eq,est}}-1\right\}.$$

(24)

The influence of R_L,eq in CV mode cannot be ignored, even if ESRs are very small. Thus, ϕ will increase as R_L,eq increases.

Finally, the proposed S-S WPT circuit applies the control algorithm (Figure 6) for battery charging, and it consists of the following procedures:

(1)

Modulate the WPT circuit at f = f_o and f_a; sense the I_1,o and I_1,a, respectively.

(2)

Using the I_1,o and I_1,a, estimate R_bat,est[1] = V_bat,est[1] / I_bat,est[1] and M_12,est.

(3)

If V_bat,est[1] < CVL, begin the control procedure. Otherwise, turn off the S-S WPT circuit.

(4)

Set f = f_o to operate the WPT circuit in the CC mode.

(5)

Using the PI controller, adjust ϕ[n] such that I_bat,est equals to I_ref.

(6)

Estimate the R_bat,est[n] = V_bat,est[n] / I_bat,est[n] by using I_1,o[n] and (12); R_bat,est[n] is continuously updated to follow the charging profile of battery.

(7)

Repeat (5)–(6) until V_bat,est[n] = CVL.

(8)

Change the operation of WPT circuit from the CC to CV mode, and maintain f = f_o.

(9)

Using the PI controller, adjust ϕ[n] such that V_bat,est = CVL.

(10)

Repeat procedure 6 until I_bat,est[n] =I_end.

(11)

Turn off the S-S WPT circuit.

The controller has a protection function for charging current limit (CCL), CVL, and coil alignment of the WPT circuit. When M_12,est < M_12,limit, the controller terminates the battery-charging operation, because the alignment of the coils is inappropriate for battery charging.

3. Experimental Results

The experimental S-S WPT circuit for battery charging (Figure 7a,b) was built and tested to prove the proposed control method. Two identical coils had an inner diameter of 100 mm and outer diameter of 200 mm; L₁ = 202.49 μH, L₂ = 202.06 μH, and C₁ = C₂ = 50 nF were chosen for f_o = 50 kHz. The input voltage V_DC was 50 V, and the sampling frequency to sense the output value of the peak detector was set as 50 kHz, which was simply synchronized to the f_o. The values of circuit parameters are given in

Table 1. Circuit parameters for the experimental circuit.

Component	Value (Model)
L1	202.49 μH
L2	202.06 μH
C1	49.97 nF
C2	50.09 nF
Rin	12 mΩ
R1	252 mΩ
R2	248 mΩ
Q1–Q4	FDP0715N15A
D1–D4	30ETH06
Controller	TMS320F28335

.

First, the load estimation was performed using the method in Section 2.2. R_bat and M₁₂ between the transmitter and receiver coils were measured and estimated using electrical load (DL1000H; NF, Co., Ltd.) and a inductance, capacitance and resistance (LCR) meter. The coil alignment was modulated on either the separation h in the axial direction of the coil or the misalignment v in the radial direction (Figure 8). At h = 6 cm and v = 0 cm, the WPT circuit was operated at f_o = 50 kHz and f_a = 55 kHz to estimate the load condition, and R_bat = 20.11 Ω and M₁₂ = 48.81 μH at ϕ = 0. The measured I_1,o = 4.21 A (Figure 9a) and I_1,a = 5.08 A (Figure 9b), and the estimated load conditions were R_bat,est = 20.49 Ω and M_12,est = 49.30 μH by using Equations (17) and (18). The errors of estimation results were −1.88% and −1.86%, respectively; other estimation results were obtained while varying h, v, and R_bat (

Table 2. Estimation results for R_bat.

h, v (cm)	Rbat (Ω)	Rbat,est (Ω) (Error (%))	Rbat (Ω)	Rbat,est (Ω) (Error (%))	Rbat (Ω)	Rbat,est (Ω) (Error (%))
5, 0	15.06	15.20 (−0.92)	20.11	19.61 (2.48)	25.17	24.22 (3.77)
6, 0		15.60 (−3.58)		20.49 (−3.58)		25.31 (−0.55)
7, 0		15.54 (−3.18)		20.54 (−2.08)		25.35 (−0.71)
5, 2		15.46 (−2.66)		19.72 (1.90)		24.19 (3.87)
5, 4		15.45 (−2.58)		20.37 (−1.29)		25.11 (0.23)
5, 6		15.60 (−3.58)		20.27 (−0.79)		24.84 (1.31)

and

Table 3. Estimation results for M₁₂.

h, v (cm)	M12 (μH)	M12,est (μH) @ 15.06 Ω (Error (%))	M12,est (μH) @ 20.11 Ω (Error (%))	M12,est (μH) @ 25.17 Ω (Error (%))
5, 0	59.18	58.41 (1.30)	58.7 (0.81)	59.03 (0.26)
6, 0	48.81	49.27 (−0.92)	49.73 (−1.86)	49.76 (−1.92)
7, 0	40.76	41.12 (−3.18)	41.59 (−2.08)	41.62 (−2.08)
5, 2	57.28	55.37 (3.38)	55.73 (2.71)	55.85 (2.50)
5, 4	49.54	49.28 (0.53)	49.68 (−0.26)	49.79 (−0.49)
5, 6	38.66	39.55 (−2.28)	39.49 (−2.12)	39.44 (−1.99)

). Here, h was varied in the range of 5–7 cm at v = 0 cm, v was varied in the range of 0–6 cm at h = 0 cm, and R_bat was varied in the range of 15.06–25.17 Ω. As a result, the proposed method estimated the R_bat and M₁₂ within absolute errors at <3.87% and <3.38%, respectively. These experimental results demonstrate the usefulness of the proposed load estimation method. The errors of estimation were caused by inductance variation according to the coil alignment conditions and measurement error at f_o and f_a. A detailed error analysis is given in the next section.

The current and voltage regulation for the battery charging were implemented using the controller proposed in Section 2.3. An electrical load was used to emulate the battery pack, which was assumed to have 30 V ≤ V_bat ≤ 48 V (corresponding to a pack of 12 serially connected Li-ion battery cells). The R_bat of the battery pack was 15 Ω ≤ R_bat ≤ 24 Ω for CC charging at I_ref = 2 A and 24 Ω ≤ R_bat ≤ 240 Ω for CV charging at CVL = 48 V and I_end = 200 mA. The transmitter and receiver coils were located at h = 5 cm and v = 0 cm. In procedures 1 and 2, the controller of the WPT circuit used f_o = 50 kHz and f_a = 55 kHz; R_bat,est (1) = 15.51 Ω at R_bat = 15.01 Ω (−3.33% error) and M_12,est = 59.78 μH at M₁₂ =59.18 μH (−1.01% error). Because V_bat,est (1) = 31.02 V < CVL in procedure 3, the controller began the charging control procedures in steps 4–11.

In the CC mode of procedures 4–7, the waveform (Figure 10) shows that v₁ and i₁ had the same phase because the S-S WPT circuit operated at f = f_o, and that ϕ was compensated to regulate I_bat,est = I_ref. When the circuit operated at R_bat = 15.01 Ω (Figure 10a), I_bat = 2.07 A (–3.5% error) and V_bat = 31.21 V. In this CC mode, V_bat increased as R_bat increased because I_bat,est tracked the predetermined I_ref = 2 A. The waveform of Figure 10b shows that V_bat increased to 46.55 V at R_bat = 22.47 Ω, while I_bat = 2.07 A (–3.5% error). When procedure 5 was used in the CC mode, the range of regulated I_bat was 2.074–2.079 A; the tracking absolute error was <3.95% (Figure 12). The power transfer efficiency of the CC mode gradually increased as R_bat increased, and the range of it was 88.81–92.05% (Figure 13). After V_bat,est reached CVL = 48 V, the charging mode was changed to CV mode in the procedures 8–11.

The waveform of CV mode (Figure 11) also shows that v₁ and i₁ had the same phase because the S-S WPT circuit operated at f = f_o, and that ϕ was compensated to regulate V_bat,est = CVL. When the circuit operated at R_bat = 54.10 Ω (Figure 11a), V_bat = 47.24 V (1.58% error) and I_bat = 0.88 A. In this CV mode, as R_bat increased, T_v increased (Figure 4b); ϕ should be increased to maintain V_bat. Thus, I_bat gradually decreased until I_bat,est = I_end. The waveform of Figure 11b shows that I_bat decreased to 0.2 A at R_bat = 244 Ω, while V_bat = 47.92 V (0.16% error). When procedure 9 was used, the range of regulated V_bat was 47.09–47.92 V; the tracking absolute error was <1.89% (Figure 12). The power transfer efficiency of the CV mode gradually decreased as R_bat increased, and the range was 74.77–92.49% (Figure 13).

These results show that the proposed load estimation method is suitable for use in battery charging, and that the adjustment of ϕ was crucial to have I_bat follow I_ref in CC charging mode and to have V_bat follow CVL in CV charging mode.

4. Error Analysis

In the proposed estimation method, the errors of estimation results can be generated using the deviated inductance (L_dev) according to the coil alignment and measurement error of input impedance at f_o and f_a. Therefore, these errors of the proposed method were analyzed by using MATLAB (R2015a, MathWorks, Massachusetts, USA) in this section.

In this section, the errors of estimation results due to the L_dev (Figure 14a,b) were calculated as

$$\mathrm{Error}(R_{bat,est,dev})=[(R_{bat}-R_{bat,est}(L_{dev}))/R_{bat}]\times 100,$$

(25)

$$\mathrm{Error}(M_{12,est,dev})=[(M_{12}-M_{12,est}(L_{dev}))/M_{12}]\times 100,$$

(26)

where R_bat,est(L_dev) and M_12,est(L_dev) are estimated R_bat and M₁₂ in the L_dev. The measurement errors of input impedance (Figure 14c–f) were calculated as

$$\mathrm{Error}\left|{\overrightarrow{Z}}_{in,a}\right|=[(\left|{\overrightarrow{Z}}_{in,a}\right|-\left|{\overrightarrow{Z}}_{in,a,measure}\right|)/\left|{\overrightarrow{Z}}_{in,a}\right|]\times 100,$$

(27)

$$\mathrm{Error}\left|{\overrightarrow{Z}}_{in,o}\right|=[(\left|{\overrightarrow{Z}}_{in,o}\right|-\left|{\overrightarrow{Z}}_{in,o,measure}\right|)/\left|{\overrightarrow{Z}}_{in,o}\right|]\times 100,$$

(28)

where $\left|{\overrightarrow{Z}}_{in,a,measure}\right|$ and $\left|{\overrightarrow{Z}}_{in,o,measure}\right|$ are measured values of $\left|{\overrightarrow{Z}}_{in,a}\right|$ and $\left|{\overrightarrow{Z}}_{in,o}\right|$ at f = f_a and f = f_o by using a peak detection circuit in Figure 5. Then, the errors of estimation results due to the $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,a}\right|$ and $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,o}\right|$ (Figure 14c–f) were calculated as

$$\mathrm{Error}(R_{bat,est,fa})=[(R_{bat}-R_{bat,est}(\left|{\overrightarrow{Z}}_{in,a,measure}\right|))/R_{bat}]\times 100,$$

(29)

$$\mathrm{Error}(M_{12,est,fa})=[(M_{12}-M_{12,est}(\left|{\overrightarrow{Z}}_{in,a,measure}\right|))/M_{12}]\times 100,$$

(30)

$$\mathrm{Error}(R_{bat,est,fo})=[(R_{bat}-R_{bat,est}(\left|{\overrightarrow{Z}}_{in,o,measure}\right|))/R_{bat}]\times 100,$$

(31)

$$\mathrm{Error}(M_{12,est,fo})=[(M_{12}-M_{12,est}(\left|{\overrightarrow{Z}}_{in,o,measure}\right|))/M_{12}]\times 100,$$

(32)

where R_bat,est($\left|{\overrightarrow{Z}}_{in,a,measure}\right|$) and M_12,est($\left|{\overrightarrow{Z}}_{in,a,measure}\right|$) are R_bat,est and M_12,est in the $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,a}\right|$, and R_bat,est($\left|{\overrightarrow{Z}}_{in,o,measure}\right|$) and M_12,est($\left|{\overrightarrow{Z}}_{in,o,measure}\right|$) are R_bat,est and M_12,est in the $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,o}\right|$.

At first, the coil alignment of the proposed estimation method was verified in the rage of h = 5–7 cm at v = 0 cm and v = 0–6 cm at v = 5 cm. In this misalignment range of coils, the self-inductance of L₁ and L₂ was changed according to the effect of the magnetic field between coils. The variation range of L₁ = 202.01–203.41 μH, and L₂ = 201.50–202.94 μH in M₁₂ = 38.66–59.18 μH. In this error analysis, according to the L_dev, the simulation parameters were set as L₁ = L₂ = 202.71 μH, C₁ = C₂ = 49.97 nF, R_in = 12 mΩ, R₁ = 252 mΩ, and R₂ = 248 mΩ. Then, the L_dev was equivalently set between L₁ and L₂ as L_dev = L_1,dev = L_2,dev = 202.02–203.41 μH, R_bat1 = 15 Ω, R_bat2 = 20 Ω, R_bat3 = 25 Ω, M_12,max = 59.18 μH, and M_12,min = 38.66 μH. The $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,a}\right|$ and $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,o}\right|$ were set to zero in this analysis, and only L_dev was considered. As a result, the Error(R_bat,est,dev) and Error(M_12,est,dev) increased as R_bat decreased at M_12,max and R_bat increased at M_12,min (Figure 14a,b). Also, the Error(R_bat,est,dev) and Error(M_12,est,dev) due to the variation of R_bat (R_bat1–R_bat3) were more sensitive at M_12,min than M_12,max.

Secondly, the proposed estimation method measures the $\left|{\overrightarrow{Z}}_{in,a,measure}\right|$ and $\left|{\overrightarrow{Z}}_{in,o,measure}\right|$, and the $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,a}\right|$ and $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,o}\right|$ have an effect on the accuracy of the estimation. In this error analysis, according to the $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,a}\right|$ and $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,o}\right|$, Error(R_bat,est,fa), Error(M_12,est,fa), Error(R_bat,est,fo), and Error(M_12,est,fo) were analyzed under the ±1% variation of $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,a}\right|$ and $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,o}\right|$, and the simulation parameters were equivalently set as the error analysis of Equations (25) and (26). The Error(R_bat,est,dev) and Error(M_12,est,dev) were set to zero in this analysis. In the $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,a}\right|$, the Error(R_bat,est,fa) and Error(M_12,est,fa) increased as R_bat increased, and were larger at M_12,min than M_12,max in the same R_bat (Figure 14c,d). In the $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,o}\right|$ at M_12,max, the Error(R_bat,est,fo) and Error(M_12,est,fo) increased as R_bat increased. At M_12,min, the Error(R_bat,est,fo) increased as R_bat decreased, and Error(M_12,est,fo) increased as R_bat increased (Figure 14e,f). Overall, the $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,a}\right|$ had more impact on the accuracy of proposed estimation method than the $\mathrm{Error}\left|{\overrightarrow{Z}}_{in,o}\right|$.

In conclusion, the errors of estimation results were <3% in Equations (25), (29), and (31), and <1.5% in Equations (26), (30) and (32). Also, Equations (25) and (26) (Figure 14a,b) were more sensitive to the variation of R_bat and M₁₂ than Equations (29)–(32) (Figure 14c–f). In the practical applications, the proposed controller (Figure 6) includes the protection function to limit the range of coil alignment as M_12,est < M_12,limit, and the auxiliary positioning device can be introduced to minimize the inductance deviation.

5. Conclusions

This paper presents a wireless battery charging circuit that uses a new load estimation method. The proposed method estimates R_L, M₁₂, V_bat, and I_bat without any wireless communication by using a simple peak detection circuit to sense the peak current of the transmitter coil; it samples this peak current as a DC value. After the peak current values are sampled at resonant frequency f_o and auxiliary frequency f_a, the estimation is performed by using the magnitude of the input impedance. Thus, this method does not need a high sampling frequency to detect the AC voltage and the current of the transmitter coil. When the proposed WPT circuit is operated to charge a battery pack, the circuit uses the proposed load estimation method and phase ϕ control of the full-bridge inverter to regulate the output current and voltage. A prototype circuit to charge a 48-V battery pack was tested under the various load resistance and coil alignment conditions. Then, the errors of estimation results due to the inductance variation and measurement error were analyzed. Finally, all experimental and simulation results indicated that the proposed method is well suited to control the WPT battery charging circuit efficiently.