Research ArticleDetecting Load Resistance and Mutual Inductance inSeries-Parallel Compensated Wireless Power Transfer SystemBased on Input-Side Measurement

Longzhao Sun ,1,2 Mingui Sun,2 Dianguang Ma,1 and Houjun Tang1

1Department of Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China2Department of Electrical and Computer Engineering, University of Pittsburgh, Pittsburgh, PA 15213, USA

Correspondence should be addressed to Longzhao Sun; sunlongzhao@sjtu.edu.cn

Received 12 October 2017; Revised 21 February 2018; Accepted 4 March 2018; Published 25 June 2018

Academic Editor: Gino Sorbello

Copyright © 2018 Longzhao Sun et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In wireless power transfer (WPT) system, the variations in load resistance and mutual inductance influence the output voltage andoutput current, making the system deviate from its desirable operating condition; hence, it is essential to monitor load resistanceand 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 sideand eliminate communication system feeding back the load measurements. This paper investigates load resistance and mutualinductance detection method in series-parallel compensated WPT system. By measuring input current and input voltage, theequation for calculating load resistance is deduced; when the operating frequency is lower than or equal to the receiving-sideresonant frequency, the rigorous mathematical derivations prove that load resistance can be uniquely determined by using onlyone measurement of input current and input voltage. Furthermore, the analytical expressions for identifying load resistance andmutual inductance are deduced. Experiments are conducted to verify the proposed method.

1. Introduction

Through magnetic coupling between transmitting andreceiving coils, wireless power transfer (WPT) techniquecan deliver electrical energy from a source to a load withoutany electric wire [1, 2]. To improve system efficiency,compensation is needed to nullify inductive reactance. Ifinput is voltage source, series compensation is required ontransmitting side, and the receiving-side capacitor can beconnected in parallel or series with the receiving coil. There-fore, voltage-fed WPT system can be categorized into series-parallel and series-series types [3].

Recently, WPT has been extensively applied in householdappliances, robotics, portable electronic products, underwa-ter equipment, implantable medical devices, and electricvehicle charging [4–8] because of its convenience, flexibility,reliability, and safety.

One challenge in WPT applications is the dynamic char-acteristics caused by load and mutual inductance variations.

Different devices can be charged by WPT system, and theload values of these devices are different. Even for chargingthe same device, its battery equivalent resistance changesdrastically as the charging process goes from constantcurrent mode to constant voltage mode [9]. To provide moreconvenience for users, a receiver can be located with spatialfreedom, which means the relative location of transmitterand receiver changes, thus resulting in mutual inductancevariations. In dynamic WPT system (or roadway-poweredelectric vehicle), a vehicle that is running on a road can bewirelessly powered; in this case, the mutual inductance istime varying. These inevitable variations in load and mutualinductance will cause fluctuation in output voltage andoutput current, thus making WPT system deviate from itsdesirable operating condition. Therefore, monitoring loadresistance and mutual inductance is essential for controllingand optimizing WPT system.

Generally, current and voltage sensors are installed onboth transmitting and receiving sides, and a communication

HindawiInternational Journal of Antennas and PropagationVolume 2018, Article ID 2094637, 6 pageshttps://doi.org/10.1155/2018/2094637

module is utilized to implement a feedback control betweenthe transmitting and receiving sides. However, recent studies[10–16] show that load resistance and/or mutual inductancein series-series compensated WPT system can be estimatedby measuring the input current and input voltage, eliminat-ing sensors on receiving side and communication module;therefore it can significantly simplify circuit, reduce cost,and improve reliability. Paper [10] deduced the differentialequation of the transmitting-side current; by investigatingthe variation rate of the approximate positive envelope of thiscurrent under different loading conditions, a load identifica-tion method was proposed. However, the effective detectableregion was limited, and the accuracy in [10] got worse whenthe value of resistor was lower than 100Ω. By root locusmethod, the whole detection region was categorized intotwo subregions; the mathematical equations of transient pro-cess in each subregion were established, and [11] presented aload identification model to increase the detection range.However, the load detection techniques in [10, 11] did notconsider mutual inductance variations. By adding an extracapacitor into the transmitting side, WPT system could oper-ate in two compensation modes via switching this capacitor;paper [12] combines the reflected impedance model andsteady-state properties and presents a steady-state loaddetection method. The authors of [13] put forward a 1-portmeasurement approach, and the coupling factor and receiverquality factor could be monitored from the transmitter.However, these studies [10–16] did not investigate how todetect load resistance and mutual inductance in series-parallel compensated WPT system.

As discussed in [17, 18], series-parallel compensatedtopology plays an important role in WPT technology. More-over, it is highlighted by the researchers [18] from Oak RidgeNational Laboratory that they evaluated various compensa-tion schemes and selected the series-parallel topology as themost appropriate one for voltage source operation.

This paper explores load resistance and mutual induc-tance detection in series-parallel compensated WPT system.The derivations in this paper are completely different fromprevious studies. Moreover, the proposed method is signifi-cantly intuitive and simple. To the best of our knowledge, thispaper is the first work to prove that load resistance andmutual inductance in series-parallel compensated WPTsystem can be uniquely determined by using only one mea-surement of input current and input voltage. Furthermore,the analytical expressions for identifying load resistance andmutual inductance are deduced.

2. Modelling and Analysis

Figure 1 is the equivalent model of series-parallel compen-sated WPT system. VS is the voltage source. The self-inductances of transmitting and receiving coils are L1 andL2, respectively; R1 and R2 denote the equivalent resistancesof transmitting and receiving coils. Resonant capacitor C1 isconnected in series with transmitting coil, C2 in parallel withreceiving coil, forming series-parallel compensation topol-ogy. M is the mutual inductance between transmitting andreceiving coils, and RL is the load resistance.

Using Kirchhoff Voltage Law, the mathematical equa-tions for series-parallel compensated WPT system inFigure 1 can be expressed as

VS = Z1 × I1 + jωM × I2

0 = jωM × I1 + Z2 + ZL × I2, 1

where ZL = RL/ 1 + jωC2RL , Z1 = R1 + jωL1 + 1/jωC1 ,Z2 = R2 + jωL2.

Since the input current I1 and input voltage VS aremeasurable, the input impedance Zin =VS/I1 is known. Bycalculating I2 from (1) and substituting Zin =VS/I1, thefollowing equation can be obtained

Z2 + ZL = R2 + jωL2 +RL

1 + jωC2RL= ω2M2

Zin − Z12

By separating the complex number equation (2) into thereal part and imaginary part, (2) can be divided into two realnumber equations

R2 +RL

1 + ω2C22R

2L

= ω2M2 Re 1Zin − Z1

,

ωL2 +−ωC2R

2L

1 + ω2C22R

2L= ω2M2 Im 1

Zin − Z1

3

To facilitate the derivation, a factor λ is introduced, and λis defined as

λ = Re 1/ Zin − Z1Im 1/ Zin − Z1

4

By calculatingM from (3) and substituting (4), (3) can besimplified as

ωC2 ωR2C2 + λ 1 − ω2L2C2 R2L + RL + R2 − λωL2 = 0 5

The coefficients of (5) are

a = ωC2 ωR2C2 + λ 1 − ω2L2C2 , b = 1, c = R2 − λωL2

6When the operating frequency ω is higher than the

receiving-side resonant frequency ω2 (i.e., ω2 = 1/ L2C2),the numerical analysis indicates that there are two sets ofpositive roots in (5). For example, the same circuit param-eters as listed in Section 4 are used, and the receiving-sideresonant frequency is 209.6 kHz. When the WPT systemoperates at 215.0 kHz, if the input impedance Zin is

R1 R2

RLL1 L2

C1

C2

I1 I2M

VS

Z1 Z2 ZL

Figure 1: Equivalent circuit for series-parallel compensated WPTsystem.

2 International Journal of Antennas and Propagation

26 76∠−25 04°, there are two sets of possible load resistanceRL and mutual inductance M combinations: RL = 311 3Ωand M = 8 7 μH, RL = 96 0Ω and M = 14 2 μH, which canlead to the identical input impedance Zin.

The amplitude-frequency plot of input impedance Zin inFigure 2(a) and phase-frequency plot in Figure 2(b) furtherclarify that two curves intersect at the same input impedanceZin when operating at 215.0 kHz. This is the reason why loadresistance RL and mutual inductance M cannot be exactlyidentified based on one sample of input impedance Zin whenthe operating frequency is higher than the receiving-sideresonant frequency.

3. Load Resistance and Mutual InductanceDetection Method

Note that (5) has one positive root at least, corresponding tothe actual value of the load resistance in series-parallelcompensated WPT system. This paper aims to calculatetheoretically and then validate experimentally the uniquevalue of load resistance RL (i.e., the solution of (5)).

When the operating frequency ω is lower than thereceiving-side resonant frequency ω2, that is, ω < 1/ L2C2,it is found that there is only one positive root for load resis-tance RL by solving (5), which means that load resistanceRL and mutual inductance M can be uniquely determinedbased on one measurement of input impedance Zin only,and the rigorous derivations are as follows.

When ω < 1/ L2C2, that is, 1 − ω2L2C2 > 0, how to findthe root of (5) is discussed under four cases:

Case 1. Assuming a = ωC2 ωR2C2 + λ 1 − ω2L2C2 = 0,λ = −ωR2C2/ 1 − ω2L2C2 < 0 and c = R2 − λωL2 > 0 canbe obtained. Solving (5), there is only one negative root; thusthis case does not exist.

Case 2. Assuming c = R2 − λωL2 = 0, λ = R2/ωL2 > 0 anda = ωC2 ωR2C2 + λ 1 − ω2L2C2 > 0 can be obtained.Solving (5), there are one zero root and one negativeroot for RL; therefore this case is not existing.

Case 3. If a = ωC2 ωR2C2 + λ 1 − ω2L2C2 < 0, λ < −ωR2C2/ 1 − ω2L2C2 < 0 can be obtained, therefore c = R2 −λωL2 > 0.

Case 4. If c = R2 − λωL2 < 0, λ > R2/ωL2 > 0 can be obtained,therefore a = ωC2 ωR2C2 + λ 1 − ω2L2C2 > 0.

Based on the four cases discussed above, we conclude thata and c have opposite signs when ω < 1/ L2C2. Accordingto Vieta theorem, one and only one positive root exists bysolving (5).

If a > 0, the detected load resistance is

RL =−1 + 1 − 4ωC2 ωR2C2 + λ 1 − ω2L2C2 R2 − λωL2

2ωC2 ωR2C2 + λ 1 − ω2L2C27a

If a < 0,

RL =−1 − 1 − 4ωC2 ωR2C2 + λ 1 − ω2L2C2 R2 − λωL2

2ωC2 ωR2C2 + λ 1 − ω2L2C27b

Substituting ((7a) and (7b)) into (3), the mutual induc-tance can be calculated by

M = R2 + RL + ω2C22R2RL

2

ω2 1 + ω2C22R

2L Re 1/ Zin − Z1

8

When the operating frequency ω is equal to the receiving-side resonant frequency ω2, we have ω = ω2 = 1/ L2C2.According to (3) and (4), λ becomes

λ = R2 + ω2R2C22R

2L + RL

ωL2 + ω3L2C22R

2L − ωC2R

2L

9

Substituting ω = ω2 = 1/ L2C2, (9) can be simplified as

λ = R2 + ω2R2C22R

2L + RL

ωL210

Substituting ω = ω2 = 1/ L2C2 and (10) into (5), thecoefficients of (5) are

a = ωC2 ωR2C2 + λ 1 − ω2L2C2 = ω2C22R2 > 0, b = 1,

c = R2 − λωL2 = − ω2R2C22R

2L + RL < 0

11

When ω = ω2 = 1/ L2C2, the analysis shows that a > 0and c < 0. According to Vieta theorem, one and only one pos-itive root exists by solving (5). The detected value of loadresistance is

RL =−1 + 1 − 4ω2C2

2R2 R2 − λωL2

2ω2C22R2

12

Substituting (12) into (3), the mutual inductance can becalculated by

M = R2 + RL + ω2C22R2R

2L

ω2 1 + ω2C22R

2L Re 1/ Zin − Z1

13

The input current I1 and input voltage VS can be mea-sured at an operating frequency ω, thus the input impedanceZin =VS/I1 is known; the coil parameters (L1, R1, C1, L2, R2,and C2) are given. When the operating frequency is lowerthan the receiving-side resonant frequency, the detectedvalues of load resistance RL and mutual inductance M canbe obtained by ((7a) and (7b)) and (8), respectively; whenthe operating frequency is equal to the receiving-side reso-nant frequency, the detected values of load resistance RLand mutual inductance M can be obtained by (12) and(13), respectively.

3International Journal of Antennas and Propagation

4. Experiment Analysis

To verify the proposed method for detecting load resistanceand mutual inductance, a series-parallel compensated WPTprototype is implemented in this study, as shown inFigure 3. High-frequency input voltage drives the transmit-ting coil, and the transmitting coil is connected in series withthe compensating capacitor. Through the inductive couplingbetween the transmitting coil and receiving coil, wirelesspower can be delivered to the receiving coil. The receivingcoil is connected in parallel with the compensating capacitor,and the output voltage feeds the load. The parameters oftransmitting coil are L1 = 27 3 μH, R1 = 0 425Ω, and C1 =20 53 nF. The parameters of receiving coil are L2 = 27 9 μH,R2 = 0 531Ω, and C2 = 20 67 nF, and the receiving-side reso-nant frequency is 209.6 kHz.

When the WPT system operates at a specific frequencythat is lower than or equal to the receiving-side resonantfrequency, the input current I1 and input voltage VS aremeasured, and then the input impedance Zin = VS/I1 can beobtained. These known information are utilized by theproposed detection method in Section 3 to obtain the loadresistance and mutual inductance.

When a resistor of 55Ω is utilized as the load, and themutual inductance varies from 6.8 μH to 13.4μH, theproposed method for detecting the load resistance and themutual inductance is implemented, and the results arepresented in Figure 4. In Figure 4(a), the dots and the dia-monds denote the actual load resistance RL and the detectedload resistance RL

∗, respectively. Figure 4(b) shows howdetected mutual inductanceM∗ varies with the actual mutualinductance M.

Using the same method, when mutual inductance is10.7 μH and the load varies from 12Ω to 82Ω, the exper-iments have been conducted to detect the mutual induc-tance and load resistance, and the results are shown inFigure 5. In Figure 5(a), the dots and the diamonds repre-sent the actual mutual inductance M and the detectedmutual inductance M∗, respectively. Figure 5(b) presentsthe plot of detected load resistance RL

∗ when actual loadresistance RL varies.

Based on the analysis of Figures 4 and 5, using themeasured value of input current and input voltage, the loadresistance and mutual inductance can be uniquely deter-mined by the proposed method. Furthermore, it can beobserved that the detected results of load resistance andmutual inductance agree well with their actual values, whichvalidates the proposed detection method.

5. Conclusion

This paper thoroughly investigates load resistance andmutual inductance detection method in series-parallel com-pensated WPT system. The derivations and conclusions pre-sented in this paper are brand new compared with previousstudies. Moreover, the proposed method is significantly intu-itive and simple.

When the operating frequency is lower than or equalto the receiving-side resonant frequency, the detailedmathematical derivations prove that the load resistanceand mutual inductance can be uniquely determined basedon one measurement of input current and input voltageonly. And the analytical expressions for identifying loadresistance and mutual inductance are deduced for the firsttime. Furthermore, a series-parallel compensated WPT

Transmittingcoil

Receivingcoil

Parallelcompensation

Seriescompensation

Input voltageLoad

Zoom in

Figure 3: Photograph of the series-parallel compensated WPTexperimental setup.

190 200 210 220 230 2400

5

10

15

20

25

30A

mpl

itude

(ohm

)

Frequency (kHz)

RL = 311.3 𝛺M = 8.7 𝜇H

RL = 96.0 𝛺M = 14.2 𝜇H

(a)

190 200 210 220 230 240Frequency (kHz)

−60

−40

−20

0

Phas

e (de

gree

)

20

RL = 311.3 𝛺M = 8.7 𝜇H

RL = 96.0 𝛺M = 14.2 𝜇H

(b)

Figure 2: Input impedance spectral characteristics for two sets of load resistance RL and mutual inductance M: (a) amplitude versusfrequency; (b) phase versus frequency.

4 International Journal of Antennas and Propagation

prototype is built, and the experimental results confirmthe proposed method.

This work can be applied to dynamically monitor theload status and the coupling relationship between the trans-mitting coil and receiving coil in series-parallel compensatedWPT system; moreover, only one measurement of input cur-rent and input voltage is needed, thus making the proposedmethod straightforward.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This work was supported by China Scholarship Council andChinese National Natural Science Foundation (U1604136).

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Detected M⁎

Actual M

0 20 40 60 80

4

6

8

10

Mut

ual i

nduc

tanc

e (𝜇

H)

12

14

16

Actual load resistance (𝛺)

(a)

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ecte

d lo

ad re

sista

nce (

𝛺)

0 20 40 60 80Actual load resistance (𝛺)

0

20

40

60

80

(b)

Figure 5: Detected results of (a) mutual inductance and (b) load resistance when actual load resistance varies.

5 7 9 11 13 15

Load

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)

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Actual mutual inductance (𝜇H)

Detected RL⁎

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(a)

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Det

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d m

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𝜇H

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(b)

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