Wireless Battery Charging Circuit Using Load Estimation without Wireless Communication

: 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 di ﬀ erence 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.


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 (Q 1 -Q 4 ), a transmitter coil (L 1 ), a full-bridge rectifier (D 1 -D 4 ), a receiver coil (L 2 ), and two capacitors (C 1 and C 2 ). L 1 forms a resonance circuit with C 1 , and L 2 forms a resonance circuit with C 2 . Both resonance circuits are designed to have the same resonance frequency ω o = 2π · f o = 1/ To charge a battery, the S-S WPT circuit should be operated in constant current (CC) output mode when the battery voltage Vbat is lower than predetermined limit voltage Vbat,cut, and in constant voltage (CV) mode when Vbat,cut ≤ Vbat < 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 ηe 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 ηe by using two intermediate coils that are placed between the transmitter and receiver coils, and uses f = fCC for CC output and f = fCV for CV output, where the frequencies fCC and fCV are determined by the coupling coefficients among the four coils. However, the values of fCC and fCV 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 fCC and fCV 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 RL 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, 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 k 12 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 M 12 . 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.

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 Q g2 lags behind that of Q g3 lags behind that of and Q g4 by an angle φ, so the bipolar output pulses of the full-bridge inverter (v 1 , Figure 2) have a dead phase angle φ between the pulses. The fundamental component of v 1 is given by and predicts the load resistance RL, output voltage Vbat, output current Ibat, 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 fo and auxiliary frequency fa,, 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.

Theoretical Models of the S-S WPT Circuit
The gate control pulses Qg1-Qg4 ( 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 (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 where θ and φ are phase angles, Vbat is the battery voltage, and Ibat is the averaged charging current of the battery. The S-S WPT circuit had an equivalent circuit ( Figure 3) for the fundamental component, where Rin, 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 Rbat can be modeled with an equivalent resistance RL,eq as: The current i 1 (t) of the transmitter coil, the current i 2 (t) of the receiver coil, and the input voltage v 2 (t) to the rectifier in Figure 1 can be expressed as 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 , R 1 , and R 2 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:  Then, the Kirchhoff's voltage law (KVL) gives where Z1 = R1 + jωL1 + 1/(jωC1) and Z2 = R2 + jωL2 + 1/(jωC2). Using Equations (6) and (7), the phase of the input impedance Zin (Figure 4a), the voltage conversion ratio Tv (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 Then, the Kirchhoff's voltage law (KVL) gives where Z 1 = R 1 + jωL 1 + 1/(jωC 1 ) and Z 2 = R 2 + jωL 2 + 1/(jωC 2 ). 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 i 1 (t), the peak current of i 1 (t) (I 1 ) (Figure 4c), the amplitude of i 2 (t), and the peak current of i 2 (t) (I 2 ) ( Figure 4d) are calculated as

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 I 1 as a DC value. The peak detection circuit is composed of a current sensor, an amplifier for the peak detection (A 1 ), an amplifier for the voltage follower (A 2 ), 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 A 2 (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 .

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 (Ri), a feedback loop resistance (Rf), a feedback loop diode (Df), a rectification diode (Do), an output capacitor (Co), and an output resistance (Ro). If the output voltage of the current sensor (Vs) is lower than the voltage of Co (Vo), Df remains on, and Do remains off. In this operating mode, the output voltage of A2 (Vout) is clamped to Vo, and Co is discharged by Ro. When Vo becomes smaller than Vs, Df is turned off and Do is turned on. In this operating mode, Co is charged to the new positive peak of Vs, so Vs = Vo = Vout.

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 (Ri), a feedback loop resistance (Rf), a feedback loop diode (Df), a rectification diode (Do), an output capacitor (Co), and an output resistance (Ro). If the output voltage of the current sensor (Vs) is lower than the voltage of Co (Vo), Df remains on, and Do remains off. In this operating mode, the output voltage of A2 (Vout) is clamped to Vo, and Co is discharged by Ro. When Vo becomes smaller than Vs, Df is turned off and Do is turned on. In this operating mode, Co is charged to the new positive peak of Vs, so Vs = Vo = Vout.  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, L 1 , C 1 and R 1 ) and receiver side (L 2 , C 2 and R 2 ) are assumed to be known, and the proposed method predicts M 12 , R L,eq , I bat , and V bat .
At first, the circuit operates at f = f o , and the M 12 can be expressed using detected I 1,o and Equation (8) as  (1), and the unknown parameters of Equation (12) are M 12 and R L,eq . Then, the circuit operates at f = f a , and the square of the absolute value of input impedance → Z in,a 2 = V 2 1,a /I 2 1,a can be expressed using Equation (8) as 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 φ)/π 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 αR 2 L,eq + βR L,eq + γ = 0, where α, β and γ are as follows: 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 Then, the estimated mutual inductance M 12,est can also be derived by applying Equation (17) to Equation (12) as: 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 V bat,est = π 2 8 · I bat,est · R L,eq,est .
Finally, the proposed method can predict R L,eq , M 12 , 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].

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 I 2 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 I 2 at ω = ω o as because Z 1 = R 1 and Z 2 = R 2 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 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 where 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 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:  The controller has a protection function for charging current limit (CCL), CVL, and coil alignment of the WPT circuit. When M12,est < M12,limit, the controller terminates the battery-charging operation, because the alignment of the coils is inappropriate for battery charging.

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 fo = 50 kHz. The input voltage VDC 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 fo. The values of circuit parameters are given in Table 1.  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.

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; L 1 = 202.49 µH, L 2 = 202.06 µH, and C 1 = C 2 = 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. 7) Repeat 5) -6) until Vbat,est[n] = CVL. 8) Change the operation of WPT circuit from the CC to CV mode, and maintain f = fo. 9) Using the PI controller, adjust φ[n] such that Vbat,est = CVL. 10) Repeat procedure 6 until Ibat,est[n] =Iend. 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 M12,est < M12,limit, the controller terminates the battery-charging operation, because the alignment of the coils is inappropriate for battery charging.

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 fo = 50 kHz. The input voltage VDC 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 fo. The values of circuit parameters are given in Table 1.   First, the load estimation was performed using the method in Section 2.2. R bat and M 12 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 (  (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 (Tables 2 and  3). 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 M 12 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 M 12 =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.
Controller TMS320F28335 First, the load estimation was performed using the method in Section 2.2. Rbat 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 (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 Rbat ( Table 2, Table 3). 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 Rbat was varied in the range of 15.06-25.17 Ω. As a result, the proposed method estimated the Rbat 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 fo and fa. A detailed error analysis is given in the next section.     In the CC mode of procedures 4-7, the waveform ( Figure 10) shows that v 1 and i 1 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 = fo, and that φ was compensated to regulate Vbat,est = CVL. When the circuit operated at Rbat = 54.10 Ω (Figure 11a), Vbat = 47.24 V (1.58% error) and Ibat = 0.88 A. In this CV mode, as Rbat increased, Tv increased ( Figure 4b); φ should be increased to maintain Vbat. Thus, Ibat gradually decreased until Ibat,est = Iend. The waveform of Figure 11b shows that Ibat decreased to 0.2 A at Rbat = 244 Ω, while Vbat = 47.92 V (0.16% error). When procedure 9 was used, the range of regulated Vbat 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 Rbat increased, and the range was 74.77-92.49% ( Figure 13).   The waveform of CV mode ( Figure 11) also shows that v 1 and i 1 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). 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 = fo, and that φ was compensated to regulate Vbat,est = CVL. When the circuit operated at Rbat = 54.10 Ω (Figure 11a), Vbat = 47.24 V (1.58% error) and Ibat = 0.88 A. In this CV mode, as Rbat increased, Tv increased ( Figure 4b); φ should be increased to maintain Vbat. Thus, Ibat gradually decreased until Ibat,est = Iend. The waveform of Figure 11b shows that Ibat decreased to 0.2 A at Rbat = 244 Ω, while Vbat = 47.92 V (0.16% error). When procedure 9 was used, the range of regulated Vbat 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 Rbat increased, and the range was 74.77-92.49% ( Figure 13).   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 = fo, and that φ was compensated to regulate Vbat,est = CVL. When the circuit operated at Rbat = 54.10 Ω (Figure 11a), Vbat = 47.24 V (1.58% error) and Ibat = 0.88 A. In this CV mode, as Rbat increased, Tv increased ( Figure 4b); φ should be increased to maintain Vbat. Thus, Ibat gradually decreased until Ibat,est = Iend. The waveform of Figure 11b shows that Ibat decreased to 0.2 A at Rbat = 244 Ω, while Vbat = 47.92 V (0.16% error). When procedure 9 was used, the range of regulated Vbat 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 Rbat 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 Ibat follow Iref in CC charging mode and to have Vbat follow CVL in CV charging mode.

Error Analysis
In the proposed estimation method, the errors of estimation results can be generated using the deviated inductance (Ldev) according to the coil alignment and measurement error of input impedance at fo and fa. 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 Ldev (Figure 14a,b) were calculated as where Rbat,est(  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.

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 where R bat,est (L dev ) and M 12,est (L dev ) are estimated R bat and M 12 in the L dev . The measurement errors of input impedance (Figure 14c-f) were calculated as    were set to zero in this analysis, and only Ldev was considered. As a result, the Error(Rbat,est,dev) and Error(M12,est,dev) increased as Rbat decreased at M12,max according to the L dev , the simulation parameters were set as L 1 = L 2 = 202.71 µH, C 1 = C 2 = 49.97 nF, R in = 12 mΩ, R 1 = 252 mΩ, and R 2 = 248 mΩ. Then, the L dev was equivalently set between L 1 and L 2 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 Error → Z in,a and Error → Z in,o 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 and Error(M 12,est,dev ) were set to zero in this analysis. In the Error → Z in,a , 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 Error → Z in,o 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 Error → Z in,a had more impact on the accuracy of proposed estimation method than the Error → Z in,o .
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 M 12 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.

Conclusions
This paper presents a wireless battery charging circuit that uses a new load estimation method. The proposed method estimates R L , M 12 , 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.

Conflicts of Interest:
The authors declare no conflict of interest. Nomenclature Q 1 -Q 4 Switches of full bridge inverter. D 1 -D 4 Diodes of full bridge rectifier. L 1 , L 2 Transmitter and receiver coil (H). C 1 , C 2 Resonant capacitors of transmitter and receiver coil (F). ω, f Angular switching frequency and switching frequency (rad/s), (Hz).
Resonance angular frequency and resonance frequency (rad/s), (Hz). f a auxiliary switching frequency (Hz). V DC DC input voltage of full-bridge inverter (V). V bat , V bat,cut Output voltage (battery voltage) and predetermined limit voltage of battery (V).

I bat
Output current (battery charging current) (A). η e Power transfer efficiency (%). k 12 , M 12 Coupling coefficient and mutual inductance (H).  Switching phase difference between lags of full-bridge inverter (rad). θ Phase difference between v 1 (t) and i 1 (t) (rad). φ Phase difference between i 1 (t) and v 2 (t), i 2 (t) (rad). R in , R 1 , R 2 Equivalent series resistance of full-bridge inverter, primary and secondary coil (Ω). R bat , R L,eq Resistance of battery and equivalent resistance of R bat (Ω).

R bat,est
Estimated R bat (Ω). → Z in Input impedance Vector.

T v
Voltage conversion ratio of V 2 /V 1 . A 1 , A 2 Amplifier for peak detection and voltage follower of peak detection circuit. R i , R f Input resistance of A 1 and feedback loop resistance of peak detection circuit (Ω).