# Analysis of the Influence of Compensation Capacitance Errors of a Wireless Power Transfer System with SS Topology

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analysis of SS Compensation Topology

_{in}and U

_{O}is the input and output direct current (DC) voltage, U

_{1}and U

_{2}are the root-mean-square (RMS) values of the output voltage of the high frequency inverter and input voltage of the diode rectifier filter circuit, I

_{1}and I

_{2}are the RMS values of the output current of the high frequency inverter and input current of the diode rectifier filter circuit. By the Fourier decomposition, only the fundamental harmonic is considered, because high-order harmonic amplitudes are small and these harmonics can hardly be transmitted. L

_{1}and L

_{2}are the self-inductances of the loosely coupled transformer, and M is the coil mutual inductance. R

_{1}and R

_{2}are the internal resistances of the primary coil and secondary coil, and R

_{L}and R

_{E}are the load resistance and equivalent resistance, respectively.

_{1}and U

_{2}are expressed as:

_{E}) and load resistance (R

_{L}) is expressed as follows [23]:

_{Dth}is the threshold voltage, which has a constant value, and r

_{D}is the equivalent resistance of the diode. And the loss on the diodes is usually ignored in the theoretical analysis, so ${R}_{E}=\frac{8}{{\pi}^{2}}{R}_{L}$ is considered in the following.

_{1}and output current I

_{2}can be calculated:

_{IN}is pure resistive. Keeping the secondary under resonance can enhance the power transfer capability and efficiency, while keeping the primary under resonance can decrease the voltage-ampere (VA) rating of the source under the same output power [18]. It can be seen from Equation (8) that, when the coil self-inductance and system operating frequency are determined, the values of the compensation capacitors are only related to the operating frequency and self-inductance of the coils, and not to the load resistance and mutual inductance between the coils.

_{IN}] represents the real part of the input impedance. When the system is fully compensated, then ${\theta}_{{Z}_{\mathrm{IN}}}={0}^{\xb0}$, and $\lambda =1$.

## 3. Analysis of Influence of Compensation Errors

_{10}and C

_{20}, while the actual values of the compensation capacitances are expressed by C

_{1}and C

_{2}, and the ratio between the actual values and exact values the compensation capacitors are represented by k

_{1}and k

_{2}, thus C

_{1}= k

_{1}C

_{10}and C

_{2}= k

_{2}C

_{20}are given.

#### 3.1. Influence on Power Factor

_{1}and e

_{2}, are defined as follows:

_{1}and e

_{2}can be considered as the relative errors of the two capacitors. Then Equation (14) can be simplified as follows:

_{1}and e

_{2}are both real values close to zero, so that the Taylor formula can be used to expand Equation (16) at e

_{1}= 0 and e

_{2}= 0, and the result is as follows:

_{1}, L

_{2}, M, R

_{E}, e

_{1}and e

_{2}are all important factors which affect the power factor. Once the system is determined, there will be an optimal load calculated by Equation (21), in which the maximum coil transfer efficiency is obtained:

#### 3.2. Influence on Output Power

#### 3.3. Influence on Coil Transfer Efficiency

#### 3.4. Influence on Capacitors Voltage Stress

_{1}is as follows:

_{1}is:

_{2}is:

_{2}is:

_{1}and C

_{2}can be calculated using Equations (27) and (29) when there are errors in the compensation capacitances, which is important for determining the voltage stress class of the compensation capacitors.

## 4. System Performance Analysis

_{1}= 10% and e

_{2}= −10%, the error between the original and simplified formula is larger than 10%, but the power factor in these regions is low, which will be abandoned. And the error between the original and simplified formula is smaller than 4% when the capacitance errors are within the allowable range calculated using the simplified formula. Thus, it is acceptable to use the Taylor formula to expand the original formula and simplify it. Besides, it can be seen from Figure 3a that, when two compensation capacitances are both larger and smaller than the exact values, the power factor remains high. The reason is that when the compensation capacitance of the secondary is larger (smaller), the secondary is in a capacitive (inductive) state, that is, the imaginary part of Z

_{2}is negative (positive), while the reflection impedance reflected to the primary is inductive (capacitive), that is, the imaginary part of $\left(\frac{{\omega}^{2}{M}^{2}}{{R}_{E}+{Z}_{2}}\right)$ is positive (negative), as shown in Equation (7). Therefore, the equivalent load of the inverter may be turned into pure resistance if the compensation capacitance of the primary is larger (smaller) too, and the high power factor is maintained as a result.

_{1}and C

_{2}, respectively. The red region is the intersection of all the above allowable range.

_{1}= 10%, e

_{2}must be between 3% and 7%, are abandoned. And it can be seen from the red area, e

_{1}= ±3%, e

_{2}= ±7%, and e

_{1}= ±4%, e

_{2}= ±5% and other selections are all acceptable. Obviously, the price of a capacitor rises sharply with the increase of precision. Therefore, e

_{1}= ±4%, e

_{2}= ±5% may a better choice, finally, just like the purple region shows.

## 5. Experimental Verification

_{1}and output voltage U

_{2}are measured using a THDP0200 (Tektronix, OH, USA), and the input current I

_{1}and output voltage I

_{2}are measured using a TCP0030A (Tektronix, OH, USA). The phase between the voltage and current waveforms are measured using the phase measurement function of a Tektronix dpo3034 digital phosphor oscilloscope (Tektronix, OH, USA), and the other data are calculated using the measured output voltage and current of the inverter and load. And the system uses the parameters listed in Table 1 for experimental verification. The actual capacitances values, the mutual inductance, and the self-inductance of coils are measured by a HM8118 LCR bridge (HAMEG, Mainhausen, Germany), and the actual capacitance values of C

_{1}and C

_{2}and the calculated values of e

_{1}and e

_{2}are listed in Table 2.

_{1}are −10.1%, −5.2%, −0.3%, +4.8%, and +9.8%, respectively, while those of e

_{2}are −10.0%, −5.3%, +0.3%, +4.4%, and +10.2%. These capacitors are employed, respectively, thus 25 sets of experiments are carried out. When the experiment is carried out, and there is no rectifier in the secondary. Limited by the parameters of the coils, the power level of the confirmatory experiment is very small, only a few dozen watts. And it can be seen from Equation (5) that, the equivalent resistance of coupling coils R

_{E}can be considered to be equal to $\frac{8}{{\pi}^{2}}{R}_{L}$ only when $\left(\frac{4\sqrt{2}}{\pi}\frac{{V}_{Dth}}{{I}_{2}}+2{r}_{D}\right)$ is smaller than $\text{}\frac{8}{{\pi}^{2}}{R}_{L}$ and can be ignored, and the loss on the rectifier diodes will not significantly affect the system’s output characteristics. When there is a rectifier in the secondary, the output voltage U

_{2}will be a square wave due to the existence of the filter capacitance, which is a sinusoidal wave when there is no rectifier. But high-order harmonic amplitudes are small and these harmonics can hardly be transmitted. So it is feasible to use the resistor without parasitic inductance to be connected to the compensated coil in the experiment. And the output voltage and current waveforms of the inverter and load under the different compensation capacitances errors are shown in Figure 7, in which the annotation U

_{1}is the RMS value of the fundamental harmonic of the inverter’s output voltage.

## 6. Conclusions

_{1}and ±5% of C

_{2}, can be obtained according to the requirements of the system on the power factor, output power, and capacitors’ voltage stress. Especially when the product with the WPT technology is mass-produced, the cost could go down. Thus, the proposed method has a certain guiding role in practical engineering applications.

## Acknowledgements

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Typical structure of wireless power transfer (WPT) system with series-series (SS) compensation topology.

**Figure 3.**The variation of output characteristics with capacitance errors calculated using the original and simplified formulas, and allowable capacitance error ranges calculated using the simplified formulas: (

**a**) The power factor varying with capacitance errors and allowable capacitance error range when the power factor is greater than 0.9; (

**b**) The output power varying with capacitance errors and allowable capacitance error range when the output power change rate is smaller than 0.1; (

**c**) The coil transfer efficiency varying with capacitance errors.

**Figure 4.**The variation of the capacitors voltage stress with capacitance errors calculated using the original and simplified formulas, and allowable capacitance error ranges calculated using the simplified formulas: (

**a**) The voltage stress of C

_{1}varying with capacitance errors and allowable capacitance error range when the rise rate of the voltage stress of C

_{1}is smaller than 0.1; (

**b**) The voltage stress of C

_{2}varying with capacitance errors and allowable capacitance error range when the rise rate of the voltage stress of C

_{2}is smaller than 0.1.

**Figure 5.**Intersection of allowable capacitance errors satisfying the system requirements on the output characteristics.

**Figure 7.**Experiment waveforms: (

**a**) e

_{1}= −0.3%, e

_{2}= +0.3%; (

**b**) e

_{1}= −10.1%, e

_{2}= −10.0%; (

**c**) e

_{1}= −5.2%, e

_{2}= +0.3%; and (

**d**) e

_{1}= +9.8%, e

_{2}= +4.4%.

**Figure 8.**The variation of output characteristics with capacitance errors and the variation of output characteristics fitted by the experimental data: (

**a**) The power factor varying with capacitance errors; (

**b**) the output power varying with capacitance errors; (

**c**) The coil transfer efficiency varying with capacitance errors; (

**d**) The voltage stress of C

_{1}varying with capacitance errors; (

**e**) The voltage stress of C

_{2}varying with capacitance errors.

Parameter | Value |
---|---|

Resonant frequency f (KHz) | 21.9 |

Inverter output voltage U_{1} (V) | 8.0 |

Primary coil inductance L_{1} (μH) | 47.1 |

Primary coil resistance R_{1} (Ω) | 0.09 |

Secondary coil inductance L_{2} (μH) | 45.7 |

Secondary coil resistance R_{2} (Ω) | 0.12 |

Mutual inductance M (μH) | 10.6 |

Equivalent resistance R_{E} (Ω) | 1.4 |

Parameter | Value | Value | Value | Value | Value |
---|---|---|---|---|---|

C_{1} | 1.014 μF | 1.059 μF | 1.113 μF | 1.171 μF | 1.235 μF |

e_{1} | −10.1% | −5.2% | −0.3% | +4.8% | +9.8% |

C_{2} | 1.045 μF | 1.090 μF | 1.145 μF | 1.202 μF | 1.279 μF |

e_{2} | −10.0% | −5.3% | +0.3% | +4.4% | +10.2% |

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**MDPI and ACS Style**

Wang, Y.; Lin, F.; Yang, Z.; Liu, Z.
Analysis of the Influence of Compensation Capacitance Errors of a Wireless Power Transfer System with SS Topology. *Energies* **2017**, *10*, 2177.
https://doi.org/10.3390/en10122177

**AMA Style**

Wang Y, Lin F, Yang Z, Liu Z.
Analysis of the Influence of Compensation Capacitance Errors of a Wireless Power Transfer System with SS Topology. *Energies*. 2017; 10(12):2177.
https://doi.org/10.3390/en10122177

**Chicago/Turabian Style**

Wang, Yi, Fei Lin, Zhongping Yang, and Zhiyuan Liu.
2017. "Analysis of the Influence of Compensation Capacitance Errors of a Wireless Power Transfer System with SS Topology" *Energies* 10, no. 12: 2177.
https://doi.org/10.3390/en10122177