# Detecting Load Resistance and Mutual Inductance in Series-Parallel Compensated Wireless Power Transfer System Based on Input-Side Measurement

Authors: Longzhao Sun; Mingui Sun; Dianguang Ma; Houjun Tang
Journal: International Journal of Antennas and Propagation
 (2018)

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## Abstract

In wireless power transfer (WPT) system, the variations in load resistance and mutual inductance influence the output voltage and output current, making the system deviate from its desirable operating condition; hence, it is essential to monitor load resistance and mutual inductance. Using input-side measurement to detect load resistance and mutual inductance has great advantages, because it does not need any direct measurements on the receiving side. Therefore, it can remove sensors on the receiving side and eliminate communication system feeding back the load measurements. This paper investigates load resistance and mutual inductance detection method in series-parallel compensated WPT system. By measuring input current and input voltage, the equation for calculating load resistance is deduced; when the operating frequency is lower than or equal to the receiving-side resonant frequency, the rigorous mathematical derivations prove that load resistance can be uniquely determined by using only one measurement of input current and input voltage. Furthermore, the analytical expressions for identifying load resistance and mutual inductance are deduced. Experiments are conducted to verify the proposed method.


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## Body

## 1. Introduction

Through magnetic coupling between transmitting and receiving coils, wireless power transfer (WPT) technique can deliver electrical energy from a source to a load without any electric wire [1, 2]. To improve system efficiency, compensation is needed to nullify inductive reactance. If input is voltage source, series compensation is required on transmitting side, and the receiving-side capacitor can be connected in parallel or series with the receiving coil. Therefore, voltage-fed WPT system can be categorized into series-parallel and series-series types [3].Recently, WPT has been extensively applied in household appliances, robotics, portable electronic products, underwater equipment, implantable medical devices, and electric vehicle charging [4–8] because of its convenience, flexibility, reliability, and safety.One challenge in WPT applications is the dynamic characteristics caused by load and mutual inductance variations. Different devices can be charged by WPT system, and the load values of these devices are different. Even for charging the same device, its battery equivalent resistance changes drastically as the charging process goes from constant current mode to constant voltage mode [9]. To provide more convenience for users, a receiver can be located with spatial freedom, which means the relative location of transmitter and receiver changes, thus resulting in mutual inductance variations. In dynamic WPT system (or roadway-powered electric vehicle), a vehicle that is running on a road can be wirelessly powered; in this case, the mutual inductance is time varying. These inevitable variations in load and mutual inductance will cause fluctuation in output voltage and output current, thus making WPT system deviate from its desirable operating condition. Therefore, monitoring load resistance and mutual inductance is essential for controlling and optimizing WPT system.Generally, current and voltage sensors are installed on both transmitting and receiving sides, and a communication module is utilized to implement a feedback control between the transmitting and receiving sides. However, recent studies [10–16] show that load resistance and/or mutual inductance in series-series compensated WPT system can be estimated by measuring the input current and input voltage, eliminating sensors on receiving side and communication module; therefore it can significantly simplify circuit, reduce cost, and improve reliability. Paper [10] deduced the differential equation of the transmitting-side current; by investigating the variation rate of the approximate positive envelope of this current under different loading conditions, a load identification method was proposed. However, the effective detectable region was limited, and the accuracy in [10] got worse when the value of resistor was lower than 100 Ω. By root locus method, the whole detection region was categorized into two subregions; the mathematical equations of transient process in each subregion were established, and [11] presented a load identification model to increase the detection range. However, the load detection techniques in [10, 11] did not consider mutual inductance variations. By adding an extra capacitor into the transmitting side, WPT system could operate in two compensation modes via switching this capacitor; paper [12] combines the reflected impedance model and steady-state properties and presents a steady-state load detection method. The authors of [13] put forward a 1-port measurement approach, and the coupling factor and receiver quality factor could be monitored from the transmitter. However, these studies [10–16] did not investigate how to detect load resistance and mutual inductance in series-parallel compensated WPT system.As discussed in [17, 18], series-parallel compensated topology plays an important role in WPT technology. Moreover, it is highlighted by the researchers [18] from Oak Ridge National Laboratory that they evaluated various compensation schemes and selected the series-parallel topology as the most appropriate one for voltage source operation.This paper explores load resistance and mutual inductance detection in series-parallel compensated WPT system. The derivations in this paper are completely different from previous studies. Moreover, the proposed method is significantly intuitive and simple. To the best of our knowledge, this paper is the first work to prove that load resistance and mutual inductance in series-parallel compensated WPT system can be uniquely determined by using only one measurement of input current and input voltage. Furthermore, the analytical expressions for identifying load resistance and mutual inductance are deduced.


## 2. Modelling and Analysis

Figure1 is the equivalent model of series-parallel compensated WPT system. VS is the voltage source. The self-inductances of transmitting and receiving coils are L1 and L2, respectively; R1 and R2 denote the equivalent resistances of transmitting and receiving coils. Resonant capacitor C1 is connected in series with transmitting coil, C2 in parallel with receiving coil, forming series-parallel compensation topology. M is the mutual inductance between transmitting and receiving coils, and RL is the load resistance.Figure 1
Equivalent circuit for series-parallel compensated WPT system.Using Kirchhoff Voltage Law, the mathematical equations for series-parallel compensated WPT system in Figure1 can be expressed as
(1)VS=Z1×I1+jωM×I20=jωM×I1+Z2+ZL×I2,where ZL=RL/1+jωC2RL, Z1=R1+jωL1+1/jωC1, Z2=R2+jωL2.Since the input currentI1 and input voltage VS are measurable, the input impedance Zin=VS/I1 is known. By calculating I2 from (1) and substituting Zin=VS/I1, the following equation can be obtained
(2)Z2+ZL=R2+jωL2+RL1+jωC2RL=ω2M2Zin−Z1.By separating the complex number equation (2) into the real part and imaginary part, (2) can be divided into two real number equations
(3)R2+RL1+ω2C22RL2=ω2M2Re1Zin−Z1,ωL2+−ωC2RL21+ω2C22RL2=ω2M2Im1Zin−Z1.To facilitate the derivation, a factorλ is introduced, and λ is defined as
(4)λ=Re1/Zin−Z1Im1/Zin−Z1.By calculatingM from (3) and substituting (4), (3) can be simplified as
(5)ωC2ωR2C2+λ1−ω2L2C2RL2+RL+R2−λωL2=0.The coefficients of (5) are
(6)a=ωC2ωR2C2+λ1−ω2L2C2,b=1,c=R2−λωL2.When the operating frequencyω is higher than the receiving-side resonant frequency ω2 (i.e., ω2=1/L2C2), the numerical analysis indicates that there are two sets of positive roots in (5). For example, the same circuit parameters as listed in Section 4 are used, and the receiving-side resonant frequency is 209.6 kHz. When the WPT system operates at 215.0 kHz, if the input impedance Zin is 26.76∠−25.04°, there are two sets of possible load resistance RL and mutual inductance M combinations: RL=311.3Ω and M=8.7μH, RL=96.0Ω and M=14.2μH, which can lead to the identical input impedance Zin.The amplitude-frequency plot of input impedanceZin in Figure 2(a) and phase-frequency plot in Figure 2(b) further clarify that two curves intersect at the same input impedance Zin when operating at 215.0 kHz. This is the reason why load resistance RL and mutual inductance M cannot be exactly identified based on one sample of input impedance Zin when the operating frequency is higher than the receiving-side resonant frequency.Figure 2
Input impedance spectral characteristics for two sets of load resistanceRL and mutual inductance M: (a) amplitude versus frequency; (b) phase versus frequency.
(a)
(b)


## 3. Load Resistance and Mutual Inductance Detection Method

Note that (5) has one positive root at least, corresponding to the actual value of the load resistance in series-parallel compensated WPT system. This paper aims to calculate theoretically and then validate experimentally the unique value of load resistance RL (i.e., the solution of (5)).When the operating frequencyω is lower than the receiving-side resonant frequency ω2, that is, ω<1/L2C2, it is found that there is only one positive root for load resistance RL by solving (5), which means that load resistance RL and mutual inductance M can be uniquely determined based on one measurement of input impedance Zin only, and the rigorous derivations are as follows.Whenω<1/L2C2, that is, 1−ω2L2C2>0, how to find the root of (5) is discussed under four cases:Case 1.
Assuminga=ωC2ωR2C2+λ1−ω2L2C2=0, λ=−ωR2C2/1−ω2L2C2<0 and c=R2−λωL2>0 can be obtained. Solving (5), there is only one negative root; thus this case does not exist.Case 2.
Assumingc=R2−λωL2=0, λ=R2/ωL2>0 and a=ωC2ωR2C2+λ1−ω2L2C2>0 can be obtained. Solving (5), there are one zero root and one negative root for RL; therefore this case is not existing.Case 3.
Ifa=ωC2ωR2C2+λ1−ω2L2C2<0, λ<−ωR2C2/1−ω2L2C2<0 can be obtained, therefore c=R2−λωL2>0.Case 4.
Ifc=R2−λωL2<0, λ>R2/ωL2>0 can be obtained, therefore a=ωC2ωR2C2+λ1−ω2L2C2>0.Based on the four cases discussed above, we conclude thata and c have opposite signs when ω<1/L2C2. According to Vieta theorem, one and only one positive root exists by solving (5).Ifa>0, the detected load resistance is
(7a)RL=−1+1−4ωC2ωR2C2+λ1−ω2L2C2R2−λωL22ωC2ωR2C2+λ1−ω2L2C2.Ifa<0,
(7b)RL=−1−1−4ωC2ωR2C2+λ1−ω2L2C2R2−λωL22ωC2ωR2C2+λ1−ω2L2C2.Substituting ((7a) and (7b)) into (3), the mutual inductance can be calculated by
(8)M=R2+RL+ω2C22R2RL2ω21+ω2C22RL2Re1/Zin−Z1.When the operating frequencyω is equal to the receiving-side resonant frequency ω2, we have ω=ω2=1/L2C2. According to (3) and (4), λ becomes
(9)λ=R2+ω2R2C22RL2+RLωL2+ω3L2C22RL2−ωC2RL2.Substitutingω=ω2=1/L2C2, (9) can be simplified as
(10)λ=R2+ω2R2C22RL2+RLωL2.Substitutingω=ω2=1/L2C2 and (10) into (5), the coefficients of (5) are
(11)a=ωC2ωR2C2+λ1−ω2L2C2=ω2C22R2>0,b=1,c=R2−λωL2=−ω2R2C22RL2+RL<0.Whenω=ω2=1/L2C2, the analysis shows that a>0 and c<0. According to Vieta theorem, one and only one positive root exists by solving (5). The detected value of load resistance is
(12)RL=−1+1−4ω2C22R2R2−λωL22ω2C22R2.Substituting (12) into (3), the mutual inductance can be calculated by
(13)M=R2+RL+ω2C22R2RL2ω21+ω2C22RL2Re1/Zin−Z1The input currentI1 and input voltage VS can be measured at an operating frequency ω, thus the input impedance Zin=VS/I1 is known; the coil parameters (L1, R1, C1, L2, R2, and C2) are given. When the operating frequency is lower than the receiving-side resonant frequency, the detected values of load resistance RL and mutual inductance M can be obtained by ((7a) and (7b)) and (8), respectively; when the operating frequency is equal to the receiving-side resonant frequency, the detected values of load resistance RL and mutual inductance M can be obtained by (12) and (13), respectively.


## 4. Experiment Analysis

To verify the proposed method for detecting load resistance and mutual inductance, a series-parallel compensated WPT prototype is implemented in this study, as shown in Figure3. High-frequency input voltage drives the transmitting coil, and the transmitting coil is connected in series with the compensating capacitor. Through the inductive coupling between the transmitting coil and receiving coil, wireless power can be delivered to the receiving coil. The receiving coil is connected in parallel with the compensating capacitor, and the output voltage feeds the load. The parameters of transmitting coil are L1=27.3μH, R1=0.425Ω, and C1=20.53nF. The parameters of receiving coil are L2=27.9μH, R2=0.531Ω, and C2=20.67nF, and the receiving-side resonant frequency is 209.6 kHz.Figure 3
Photograph of the series-parallel compensated WPT experimental setup.When the WPT system operates at a specific frequency that is lower than or equal to the receiving-side resonant frequency, the input currentI1 and input voltage VS are measured, and then the input impedance Zin=VS/I1 can be obtained. These known information are utilized by the proposed detection method in Section 3 to obtain the load resistance and mutual inductance.When a resistor of 55 Ω is utilized as the load, and the mutual inductance varies from 6.8μH to 13.4 μH, the proposed method for detecting the load resistance and the mutual inductance is implemented, and the results are presented in Figure 4. In Figure 4(a), the dots and the diamonds denote the actual load resistance RL and the detected load resistance RL∗, respectively. Figure 4(b) shows how detected mutual inductance M∗ varies with the actual mutual inductance M.Figure 4
Detected results of (a) load resistance and (b) mutual inductance when actual mutual inductance varies.
(a)
(b)Using the same method, when mutual inductance is 10.7μH and the load varies from 12 Ω to 82 Ω, the experiments have been conducted to detect the mutual inductance and load resistance, and the results are shown in Figure 5. In Figure 5(a), the dots and the diamonds represent the actual mutual inductance M and the detected mutual inductance M∗, respectively. Figure 5(b) presents the plot of detected load resistance RL∗ when actual load resistance RL varies.Figure 5
Detected results of (a) mutual inductance and (b) load resistance when actual load resistance varies.
(a)
(b)Based on the analysis of Figures4 and 5, using the measured value of input current and input voltage, the load resistance and mutual inductance can be uniquely determined by the proposed method. Furthermore, it can be observed that the detected results of load resistance and mutual inductance agree well with their actual values, which validates the proposed detection method.


## 5. Conclusion

This paper thoroughly investigates load resistance and mutual inductance detection method in series-parallel compensated WPT system. The derivations and conclusions presented in this paper are brand new compared with previous studies. Moreover, the proposed method is significantly intuitive and simple.When the operating frequency is lower than or equal to the receiving-side resonant frequency, the detailed mathematical derivations prove that the load resistance and mutual inductance can be uniquely determined based on one measurement of input current and input voltage only. And the analytical expressions for identifying load resistance and mutual inductance are deduced for the first time. Furthermore, a series-parallel compensated WPT prototype is built, and the experimental results confirm the proposed method.This work can be applied to dynamically monitor the load status and the coupling relationship between the transmitting coil and receiving coil in series-parallel compensated WPT system; moreover, only one measurement of input current and input voltage is needed, thus making the proposed method straightforward.



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