# A Load-Independent Output Current Method for Wireless Power Transfer Systems with Optimal Parameter Tuning

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Contribution of this study
- -
- This study elaborates a mathematical method for achieving a Load-Independent Output Current.
- -
- This study proposes a parameter-design method by adjusting the proportion of active power to reactive power ratios as an index to optimize the topology size as well as reduction of voltage/current stresses on their elements. Due to preventing the increase in power losses and reducing the damage to the devices, the analysis of compensating topologies from the perspective of voltage and current stresses is important.
- -
- According to the above-mentioned findings, this is the first paper that improves a systematic investigation to identify and explore an immittance conditions-qualified family of PRN, which is associated with a more appropriate topological description in WPT applications. Moreover, their conditions are specified with their topological superiority description.

- Outlines

## 2. Theoretical Analysis of the IPRN

#### 2.1. Analysis of an RN

#### 2.2. Analysis of an IPRN

_{0}as the base characteristic impedance of the network, it is realized that the output of the network depends merely on its input and is independent of its output impedance. This is very useful for power supply sources with a constant output voltage/current. It is well known that a quarter-wave distributed constant line exhibits immittance properties. While the length of the distributed constant line is manageable for operation in the megahertz range, it becomes prohibitively long for power converters operating in the kilohertz range. So, some lumped-element IRPN topologies based on the transmission line approximation competed using discrete inductors/capacitors have been discussed [41]. In addition, by expanding (1), the relationship between the input and output impedance in these networks is shown by the following relation:

## 3. Design Conditions of IPRN Topologies in WPT System

#### 3.1. Analysis of the Proposed IPRN

_{1}and L

_{2}, which are in series with two capacitors C

_{1}and C

_{2}, respectively. Moreover, L

_{3}is a parallel inductor with a T-composite. The selected topology is the most adaptable structure in terms of complete modeling of transformers instead of L

_{1}, L

_{2}, and L

_{3}. The leakage inductances of transformer are applied as elements of RN (L

_{1}, L

_{2}) and primary-side magnetizing inductance (Lm) as L

_{3}. Therefore, leakage inductors and parasitic elements are reduced, resulting in minimal losses and higher output efficiency.

#### 3.2. Achieving Mathematical Equations

_{i}and ($\frac{{V}_{i}}{{Z}_{0}})$, respectively. Additionally, $({Z}_{0}=\sqrt{\frac{{L}_{1}}{{C}_{1}}}$) is defined as the following [40], the quality factor (Q) and equivalent AC resistance seen from the primary side of transformer (R

_{L}) can be expressed as:

_{1}, and C

_{1}is summarized as the following through regulated RMS voltage/current as follows:

_{omax}is the maximum load resistance and f

_{s}is the switching frequency.

#### 3.3. Parameter-Tuning Process

_{2}, L

_{3}, and C

_{2}can be calculated. These parameters are a function of ($\alpha $, $\beta $, and $\gamma )$. Furthermore, all transformer parameters are expressed as functions of K [42]. Therefore, the relations governing a transformer with a similar magnetic structure within the two initial and secondary parts can be achieved as the following

_{p}, L

_{s}, and M are the leakage-inductance primary side, leakage-inductance secondary side, and mutual inductance of the transformer, respectively. The Primary-side magnetizing inductance, i.e., Lm and M can be estimated as follows:

_{0}as the proper coupling coefficient at the designed operating point with the purpose of achieving suitable $\alpha $, $\beta $, and $\gamma $ values with respect to K

_{0}, the significant points are:

- The minimum voltage and current stresses can be realized with well-designed parameters. Table 2 shows that selecting the proper $\beta $ can considerably decrease the voltage/current stresses on the components. The stresses of voltage/current parameters are directly related to ($\beta $). Due to the relationship between L
_{1}and L_{2}with L_{p}and L_{s}(based on Section 3.1), therefore, the possible value range of $\beta $ should be selected according to the rated Lp and Ls and subsequently, is related to the volumes and dimensions of primary and secondary coils. According to practical limitations, the values obtained at the 30 mm air gap for the proposed ferrite core in the laboratory, the value of 0.14 is appropriate for ($\beta $). It is noteworthy that regulated voltage/current stresses on T1 components under different ($\beta $) values, which are within immittance mode under the steady-state output circumstances, were presented and compared in Figure 7. These diagrams are extracted from Table 3. A significant observation from Figure 7 is that the current and voltage stresses decrease with the $\beta $ reduction. - K
_{0}selection must be carried out to decrease the stresses resulting from voltage/current. Therefore, the regulated voltage/current values of components with respect to K_{0}and Q_{optimum}are summarized in Table 3. According to (17), the Q_{optimum}is the function of $\alpha $ and $\beta $. Using Figure 8, proper values for $\mathsf{\alpha}$ and $\beta $ can be achieved. Figure 8 indicates the relationship between MPTB and Q of various quantities of $\beta $ and $\alpha $. It can be inferred that higher and lower values of $\beta $ and $\mathsf{\alpha}$ must be considered, respectively, while performing the designing procedure. Also, to achieve a reduction in the physical size of the circuit, it is necessary to select the lower Q_{optimum}value. Since $\beta $ is proportional to K_{0}, the regulation procedure must be conducted with the purpose of achieving the appropriate coupling coefficient.

**Figure 7.**Normalized voltage and current stresses on the proposed IPRN elements versus $\beta $. (

**a**) Currents; (

**b**) voltages.

**Figure 8.**Quality factor diagram in terms of the physical size of elements for different values of $\mathsf{\beta}$ and $\mathsf{\alpha}$.

_{i}) = 20 V, I

_{o}= 1.4 A, R

_{o,max}(full load) = 20 Ω, f

_{s}= 100 kHz. Ten simulations have been performed for the different $\alpha $, $\beta $, and $\gamma $ values to achieve proper perspective in the selection of parameters. With the help of simulation results, it can be found that the optimal values for $\alpha $, $\beta ,$ and $\gamma $ are determined to be 1, 0.14, and 1, respectively. It should be noted that in practice, the input current is required to slightly lag the input voltage and consequently, achieve ZVS. The required phase lag is achieved through a slight reduction of $\gamma $.

_{on}), as well as the phase angle between IPRN input current and voltage (θ); also, it indicates that there is an independent output current, and the input voltage is nearly in phase with the input current. In other words, the phase angle of the input current and voltage is zero under immittance conditions (SOF and SRE) in various loads. Therefore, the proposed resonant compensation tank has eliminated most of the reactive power.

## 4. Experimental Verification

#### 4.1. Practical Result

_{1}, L

_{2}, and L

_{3}as the values of primary and secondary and magnetizing inductances of the transformer, the number of elements used in the construction is reduced. By using an RLC meter, the inductance and internal resistance of the coils are measured as follows: L

_{P}= 103 µH, L

_{S}= 183 µH, R

_{Lp}(Primary coil DC ESR) = 27.8 mΩ, and R

_{Ls}(Primary coil DC ESR) = 53.2 mΩ. The coupling coefficient is also practically adjusted using the air gap. With the 30 mm air gap and the windings 23 and 35 that turn onto the initial and secondary cores, desired values of the proposed circuit are achieved. The M

_{1}~ M

_{4}MOSFETs with a frequency of about 100 kHz are switched on by the HCPL-3120 and Micro ATmega32A drivers and a square wave with the amplitude of V

_{i}is supplied to the RN input. The values and types of elements applied to the proposed IWPT can be observed in Table 4.

_{3}is also observed after the decrease of load and input current, which suggests the ability to adjust the load and independent load. Similarly, the output current seems to be almost unchanged at different loads; thus, it proves the characteristics of the current resource. However, the current load is mathematically fixed regardless of the load change; therefore, it experimentally falls with output due to the IPRN component losses.

_{out}, and efficiency, η, are calculated versus load variations by using:

_{1}through D

_{4}was estimated to be 0.75 w. Finally, most of it was wasted by around 1 w in the core and winding. Therefore, this figure shows the full-bridge inverter losses, magnetic coupler losses, and rectifier losses account for a large proportion of the total system losses. These results are also similar to the literature [42]. Therefore, the proper design of the transformer, inverter, and rectifier should be considered to improve the efficiency of the system.

#### 4.2. Discussion and Comparison

_{m}, which is obtained by deriving efficiency from operating frequency. The expresses have indicated that w

_{m}is dependent on the load. Therefore, in a conventional mode for achieving maximum efficiency, it is assumed that proper frequency tracking is performed by complex control methods. However, in the method discussed in this paper, obtaining the Q

_{optimum}, reducing the physical dimensions of the circuit, and increasing the efficiency have been achieved. (3) According to [43], the current gain of the S/S compensation is inversely proportional to K, therefore when K varies, the output current also varies. However, due to the current gain, and Table 3, can be observed:

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Liu, H.; Huang, X.; Tan, L.; Guo, J.; Wang, W.; Yan, C.; Xu, C. Dynamic Wireless Charging for Inspection Robots Based on Decentralized Energy Pickup Structure. IEEE Trans. Ind. Inform.
**2018**, 14, 1786–1797. [Google Scholar] [CrossRef] - Zhang, Y.; Tian, G.; Lu, J.; Zhang, M.; Zhang, S. Efficient Dynamic Object Search in Home Environment by Mobile Robot: A Priori Knowledge-Based Approach. IEEE Trans. Veh. Technol.
**2019**, 68, 9466–9477. [Google Scholar] [CrossRef] - Huang, S.; Lee, T.; Li, W.; Chen, R. Modular On-Road AGV Wireless Charging Systems Via Interoperable Power Adjustment. IEEE Trans. Ind. Electron.
**2019**, 66, 5918–5928. [Google Scholar] [CrossRef] - Siroos, A.; Sedighizadeh, M.; Afjei, E.; Fini, A.S. Comparison of different controllers for wireless charging system in AUVs. In Proceedings of the 2022 13th Power Electronics, Drive Systems, and Technologies Conference (PEDSTC), Tehran, Iran, 1–3 February 2022; pp. 155–160. [Google Scholar]
- Tang, Y.; Chen, Y.; Madawala, U.K.; Thrimawithana, D.J.; Ma, H. A New Controller for Bidirectional Wireless Power Transfer Systems. IEEE Trans. Power Electron.
**2018**, 33, 9076–9087. [Google Scholar] [CrossRef] - Esmaeil, J.; Salehizadeh, M.R.; Rahimikian, A. Optimal control of the power ramp rate with flicker mitigation for directly grid connected wind turbines. Simulation
**2020**, 96, 141–150. [Google Scholar] - Chen, X.; Huang, D.; Li, Q.; Lee, F.C. Multichannel LED driver with CLL resonant converter. IEEE J. Emerg. Sel. Top. Power Electron.
**2015**, 3, 589–598. [Google Scholar] [CrossRef] - Haga, H.; Kurokaw, F. Modulation method of a full-bridge three-level LLC resonant converter for battery charger of electrical vehicles. IEEE Trans. Power Electron.
**2017**, 32, 2498–2505. [Google Scholar] [CrossRef] - Hou, J.; Chen, Q.; Zhang, Z.; Wong, S.; Tse, C.K. Analysis of Output Current Characteristics for Higher-Order Primary Compensation in Inductive Power Transfer Systems. IEEE Trans. Power Electron.
**2018**, 33, 6807–6821. [Google Scholar] [CrossRef] - Li, Y.L.; Sun, Y.; Dai, X. μ-Synthesis for Frequency Uncertainty of the ICPT System. IEEE Trans. Ind. Electron.
**2013**, 60, 291–300. [Google Scholar] [CrossRef] - Huang, Z.; Lam, C.; Mak, P.; Martins, R.P.d.S.; Wong, S.; Tse, C.K. A Single-Stage Inductive-Power-Transfer Converter for Constant-Power and Maximum-Efficiency Battery Charging. IEEE Trans. Power Electron.
**2020**, 35, 8973–8984. [Google Scholar] [CrossRef] - Lourdusami, S.S.; Viaramani, R. Analysis, design and experimentation of series-parallel LCC resonant converter for constant current source. IEICE Electron. Exp.
**2014**, 11, 20140711. [Google Scholar] [CrossRef] [Green Version] - Duan, F.; Xu, M.; Yang, X.; Yao, Y. Canonical model and design methodology for LLC DC/DC converter with constant current operation capability under shorted load. IEEE Trans. Power Electron.
**2016**, 31, 6870–6883. [Google Scholar] [CrossRef] - Li, Y.; Xu, Q.; Lin, T.; Hu, J.; He, Z.; Mai, R. Analysis and Design of Load-Independent Output Current or Output Voltage of a Three-Coil Wireless Power Transfer System. IEEE Trans. Transp. Electrif.
**2018**, 4, 364–375. [Google Scholar] [CrossRef] - Salehizadeh, M.R.; Koohbijari, M.A.; Nouri, H.; Taşcıkaraoğlu, A.; Erdinç, O.; Catalao, J.P. Bi-objective optimization model for optimal placement of thyristor-controlled series compensator devices. Energies
**2019**, 12, 2601. [Google Scholar] [CrossRef] [Green Version] - Naghash, R.; Alavi, S.M.M.; Afjei, S.E. Robust Control of Wireless Power Transfer Despite Load and Data Communications Uncertainties. IEEE J. Emerg. Sel. Top. Power Electron.
**2021**, 9, 4897–4905. [Google Scholar] [CrossRef] - Jamakani, B.E.; Afjei, E.; Mosallanejad, A. A Novel Triple Quadrature Pad for Inductive Power Transfer Systems for Electric Vehicle Charging. In Proceedings of the 2019 10th International Power Electronics, Drive Systems and Technologies Conference (PEDSTC), Shiraz, Iran, 12–14 February 2019; pp. 618–623. [Google Scholar]
- Zhang, W.; Mi, C.C. Compensation topologies of high-power wireless power transfer systems. IEEE Trans. Vehicular Tech.
**2016**, 65, 4768–4778. [Google Scholar] [CrossRef] - Mahdizadeh, A.H.; Afjei, E. LLC Resonant Converter Utilizing Parallel-Series Transformers Connection. In Proceedings of the 2019 International Power System Conference (PSC), Tehran, Iran, 9–11 December 2019; pp. 472–477. [Google Scholar]
- Ramezani, A.; Farhangi, S.; Iman-Eini, H.; Farhangi, B.; Rahimi, R.; Moradi, G.R. Optimized LCC-Series Compensated Resonant Network for Stationary Wireless EV Chargers. IEEE Trans. Ind. Electron.
**2019**, 66, 2756–2765. [Google Scholar] [CrossRef] - Nagatsuka, Y.; Ehara, N.; Kaneko, Y.; Abe, S.; Yasuda, T. Compact contactless power transfer system for electric vehicles. In Proceedings of the The 2010 International Power Electronics Conference-ECCE ASIA-, Sapporo, Japan, 21–24 June 2010; pp. 807–813. [Google Scholar]
- Wang, C.-S.; Covic, G.A.; Stielau, O.H. Power transfer capability and bifurcation phenomena of loosely coupled inductive power transfer systems. IEEE Trans. Ind. Electron.
**2004**, 51, 148–157. [Google Scholar] [CrossRef] - Zhang, W.; Wong, S.C.; Tse, C.K.; Chen, Q. Analysis and Comparison of Secondary Series- and Parallel-Compensated Inductive Power Transfer Systems Operating for Optimal Efficiency and Load-Independent Voltage-Transfer Ratio. IEEE Trans. Power Electron.
**2014**, 29, 2979–2990. [Google Scholar] [CrossRef] - Qu, X.; Han, H.; Wong, S.C.; Tse, C.K.; Chen, W. Hybrid IPT Topol-ogies with Constant Current or Constant Voltage Output for Battery Charging Applications. IEEE Trans. Power Electron.
**2015**, 30, 6329–6337. [Google Scholar] [CrossRef] - Borage, M.; Nagesh, K.V.; Bhatia, M.S.; Tiwari, S. Resonant immittance converter topologies. IEEE Trans. Ind. Electron.
**2011**, 58, 771–778. [Google Scholar] [CrossRef] - Irie, H.; Yamana, H. Immittance converters suitable for power electronics. Electr. Eng. Jpn.
**1998**, 124, 53–62. [Google Scholar] [CrossRef] - Sakamoto, Y.; Wada, K.; Shimizu, T. A 13.565 MHz current output type inverter utilizing an immittance conversion element. In Proceedings of the 2008 13th International Power Electronics and Motion, Poznan, Poland, 1–3 September 2008; pp. 288–294. [Google Scholar]
- Zhang, E. Inverter design shines in photovoltaic systems. Power Electron. Technol.
**2008**, 34, 20–25. [Google Scholar] - Kimura, N.; Morizane, T.; Taniguchi, K.; Irie, H. Dynamic performance of current sourced forced commutation HVDC converter with immitance conversion link. In Proceedings of the IEEE/PES Transmission and Distribution Conference and Exhibition, Yokohama, Japan, 6–10 October 2002; Volume 3, pp. 1937–1942. [Google Scholar]
- Irie, H.; Minami, N.; Minami, H.; Kitayoshi, H. Non-contact energy transfer system using immittance converter. Elect. Eng. Jpn.
**2001**, 136, 58–64. [Google Scholar] [CrossRef] - Borage, M.; Tiwari, S.; Kotaiah, S. Analysis and design of LCL-T resonant converter as a constant-current power supply. IEEE Trans. Ind. Electron.
**2005**, 52, 1547–1554. [Google Scholar] [CrossRef] - Borage, M.; Tiwari, S.; Kotaiah, S. LCL-T resonant converter with clamp diodes: A novel constant-current power supply with inherent constant-voltage limit. IEEE Trans. Ind. Electron.
**2007**, 54, 741–746. [Google Scholar] [CrossRef] - Borage, M.; Tiwari, S.; Kotaiah, S. A constant-current, constant voltage half-bridge resonant power supply for capacitor charging. Proc. Inst. Elect. Eng. Elect. Power. Appl.
**2006**, 153, 343–347. [Google Scholar] [CrossRef] - Borage, M.; Nagesh, K.V.; Bhatia, M.S.; Tiwari, S. Characteristics and design of an asymmetrical duty-cycle controlled LCL-T resonant converter. IEEE Trans. Power Electron.
**2009**, 24, 2268–2275. [Google Scholar] [CrossRef] - Borage, M.; Nagesh, K.V.; Bhatia, M.S.; Tiwari, S. Design of LCL T resonant converter including the effect of transformer winding capacitance. IEEE Trans. Ind. Electron.
**2009**, 56, 1420–1427. [Google Scholar] [CrossRef] - Tamate, M.; Ohguchi, H.; Hayashi, M.; Takagi, H.; Ito, M. A novel approach of power converter topology based on immittance conversion theory. In Proceedings of the ISIE’2000. Proceedings of the 2000 IEEE International Symposium on Industrial Electronics (Cat. No.00TH8543), Cholula, Puebla, Mexico, 4–8 December 2000; pp. 482–487. [Google Scholar]
- Braun, W.D.; Perreault, D.J. A High-Frequency Inverter for Variable-Load Operation. IEEE J. Emerg. Sel. Top. Power Electron.
**2019**, 7, 706–721. [Google Scholar] [CrossRef] [Green Version] - Sinha, S.; Kumar, A.; Afridi, K.K. Optimized Design of High-Efficiency Immittance Matching Networks for Capacitive Wireless Power Transfer Systems. In Proceedings of the 2021 IEEE PELS Workshop on Emerging Technologies: Wireless Power Transfer (WoW), San Diego, CA, USA, 1–4 June 2021; pp. 1–6. [Google Scholar]
- Batarseh, I. Resonant converter topologies with three and four energy storage elements. IEEE Trans. Power Electron.
**1994**, 9, 64–73. [Google Scholar] [CrossRef] - Kim, M.; Kim, J.; Lee, B. Adjustable frequency–duty-cycle hybrid control strategy for full-bridge series resonant converters in electric vehicle chargers. IEEE Trans. Ind. Electron.
**2014**, 61, 5354–5361. [Google Scholar] - Irie, H.; Yamana, H. Immittance converter suitable for power electronics. Trans. Inst. Elect. Eng. Jpn.
**1997**, 117, 962–969. [Google Scholar] [CrossRef] [Green Version] - Yang, L.; Li, X.; Liu, S.; Xu, Z.; Cai, C. Analysis and Design of an LCCC/S-Compensated WPT System with Constant Output Characteristics for Battery Charging Applications. IEEE J. Emerg. Sel. Top. Power Electron.
**2021**, 9, 1169–1180. [Google Scholar] [CrossRef] - Zhang, W.; Wong, S.C.; Tse, C.K.; Chen, Q. Load-independent duality of current and voltage outputs of a series- or parallel-compensated inductive power transfer converter with optimized efficiency. IEEE J. Emerg. Sel. Topics in Power Electron.
**2015**, 3, 137–146. [Google Scholar]

**Figure 2.**The RN block diagram based on the source and sink types; (

**a**) voltage-type source, (

**b**) current-type source, (

**c**) voltage-type sink, and (

**d**) current-type sink.

**Figure 6.**Circuit diagrams of proposed T1-IPRN with (

**a**) reactive components, and (

**b**) integrated magnetic components.

**Figure 9.**The constant current in the design condition. (

**a**) Output current (I

_{RL}); (

**b**) output voltage (V

_{RL}); (

**c**) phase angle between the input current and voltage (θ).

**Figure 11.**IPRN waveforms in the maximum load. (

**a**) Transformer secondary current and voltage; (

**b**) waveforms of voltage across switches and current through it.

**Figure 12.**IPRN waveforms in 5% of the maximum load. (

**a**) Transformer secondary current and voltage; (

**b**) waveforms of voltage across switches, as well as including current.

Name | SOF | SRE |
---|---|---|

T1 | ${\omega}_{n,s}=\frac{1}{\sqrt{1+\beta}}$ | $\alpha =\frac{1+\beta -\beta \gamma}{\gamma}$ |

T2 | ${\omega}_{n,s}=\sqrt{\frac{1+\beta}{\beta}}$ | $\alpha =\frac{\beta \left(1-\beta \gamma -\gamma \right)}{\gamma \left(1+\beta \right)}$ |

T3 | ${\omega}_{n,s}=\sqrt{\frac{1+\beta}{\beta}}$ | $\alpha =\frac{\beta}{\beta \gamma +\gamma -1}$ |

T4 | ${\omega}_{n,s}=\frac{1}{\sqrt{1+\beta}}$ | $\alpha =\frac{\beta \left(1+\beta \right)}{\beta \gamma -\beta -1}$ |

T7 | ${\omega}_{n,s}=\sqrt{\frac{\gamma -\tau}{\gamma -\tau -\tau \gamma}}$ | $a=\frac{\gamma -\tau -\tau \gamma}{\tau \left(\gamma -\tau \right)}$ |

T8 | ${\omega}_{n,s}=\sqrt{\frac{1}{\beta \gamma}}$ | $\alpha =\frac{\beta \left(\beta \gamma +\gamma -1\right)}{1-\beta \gamma}$ |

T11 | ${\omega}_{n,s}=\sqrt{\frac{\gamma -\tau}{\gamma -\tau -\tau \gamma}}$ | $\alpha =\frac{\gamma -\tau -\tau \gamma}{\tau \left(\gamma -\tau \right)}$ |

T12 | ${\omega}_{n,s}=\sqrt{\frac{1}{\beta \gamma}}$ | $\alpha =\frac{\beta \left(\beta \gamma +\gamma -1\right)}{1-\beta \gamma}$ |

Normalized I & V of Elements | Conventional Mode | Immittance Mode |
---|---|---|

${I}_{nL1}={I}_{nC1}=\frac{{I}_{L1}}{\frac{{V}_{i}}{{Z}_{0}}}=\frac{{I}_{C1}}{\frac{{V}_{i}}{{Z}_{0}}}$ | $\frac{2\sqrt{2}}{\pi}\frac{[\frac{-8\gamma {\omega}_{n}^{2}}{{\pi}^{2}Q}\left]+j\right[{\omega}_{n}-\gamma \left(\beta +\alpha \right){\omega}_{n}^{3}]}{{A}_{1}+j{A}_{2}}$ | $\frac{16\sqrt{2}\left(1-\beta \right)}{{\pi}^{3}Q{\beta}^{2}}$ |

${I}_{nL2}={I}_{nC2}=\frac{{I}_{L2}}{\frac{{V}_{i}}{{Z}_{0}}}=\frac{{I}_{C2}}{\frac{{V}_{i}}{{Z}_{0}}}$ | $\frac{2\sqrt{2}}{\pi}\frac{-j\gamma \beta {\omega}_{n}^{3}}{{A}_{1}+j{A}_{2}}$ | $\frac{2\sqrt{2}\sqrt{\left(1+\beta \right)}}{\pi \beta}$ |

${I}_{nL3}=\frac{{I}_{L3}}{\frac{{V}_{i}}{{Z}_{0}}}$ | $\frac{2\sqrt{2}}{\pi}\frac{[\frac{-8\gamma {\omega}_{n}^{2}}{{\pi}^{2}Q}\left]+j\right[{\omega}_{n}-\gamma \alpha {\omega}_{n}^{3}]}{{A}_{1}+j{A}_{2}}$ | $\frac{2\sqrt{2}\sqrt{1+\beta}}{\pi \beta}\sqrt{{\left(\frac{8\sqrt{1+\beta}}{{\pi}^{2}Q\beta}\right)}^{2}+1}$ |

${V}_{nL1}=\frac{{V}_{L1}}{{V}_{i}}$ | $\frac{2\sqrt{2}}{\pi}\frac{\left(\gamma \left(\beta +\alpha \right){\omega}_{n}^{4}-{\omega}_{n}^{2}\right)-j\frac{8\gamma {\omega}_{n}^{3}}{{\pi}^{2}Q}}{{A}_{1}+j{A}_{2}}$ | $\frac{16\sqrt{2}\sqrt{1+\beta}}{{\pi}^{3}Q{\beta}^{2}}$ |

${V}_{nL2}=\frac{{V}_{L2}}{{V}_{i}}$ | $\frac{2\sqrt{2}}{\pi}\frac{\left(\gamma \beta \alpha {\omega}_{n}^{4}\right)}{{A}_{1}+j{A}_{2}}$ | $\frac{2\sqrt{2}\alpha}{\pi \beta}$ |

${V}_{nL3}=\frac{{V}_{L3}}{{V}_{i}}$ | $\frac{2\sqrt{2}}{\pi}\frac{\left(\gamma \beta \alpha {\omega}_{n}^{4}-\beta {\omega}_{n}^{2}\right)-j\frac{8\gamma \beta {\omega}_{n}^{3}}{{\pi}^{2}Q}}{{A}_{1}+j{A}_{2}}$ | $\frac{2\sqrt{2}}{\pi}\sqrt{{\left(\frac{8\sqrt{1+\beta}}{{\pi}^{2}Q\beta}\right)}^{2}+1}$ |

${V}_{nC1}=\frac{{V}_{C1}}{{V}_{i}}$ | $\frac{2\sqrt{2}}{\pi}\frac{\left(1-\gamma \left(\alpha +\beta \right){\omega}_{n}^{2}\right)+j\frac{8\gamma {\omega}_{n}}{{\pi}^{2}Q}}{{A}_{1}+j{A}_{2}}$ | $\frac{16\sqrt{2}}{{\pi}^{3}Q}\frac{\left(1+\beta \right)\sqrt{1+\beta}}{{\beta}^{2}}$ |

${V}_{nC2}=\frac{{V}_{C2}}{{V}_{i}}$ | $\frac{2\sqrt{2}}{\pi}\frac{\left(\beta {\omega}_{n}^{2}\right)}{{A}_{1}+j{A}_{2}}$ | $\frac{2\sqrt{2}\left(\beta +\alpha \right)}{\pi \beta}$ |

Parameters | Values According to Different Coupling Coefficients in IPRN Conditions |
---|---|

${I}_{nL1}={I}_{nC1}$ | $\frac{2\sqrt{2}\left(1-{K}_{0}\right)\left(1+{K}_{0}\right)}{{\pi}^{2}{K}_{0}\left(1+{K}_{0}\right)}$ |

${I}_{nL2}={I}_{nC2}$ | $\frac{2\sqrt{2}\sqrt{1-{K}_{0}}}{\pi {K}_{0}}$ |

${I}_{nL3}$ | $\frac{2\sqrt{2}\sqrt{1-{K}_{0}}}{\pi {K}_{0}}\sqrt{{{K}_{0}}^{3}+{{K}_{0}}^{2}+1}$ |

${V}_{nL1}$ | $\frac{16\sqrt{2}{\left(1-{K}_{0}\right)}^{2}}{{\pi}^{3}Q{{K}_{0}}^{2}\sqrt{1-{K}_{0}}}$ |

${V}_{nL2}$ | $\frac{2\sqrt{2}}{\pi {K}_{0}}$ |

${V}_{nL3}$ | $\frac{4}{\pi}$ |

${V}_{nC1}$ | $\frac{2\sqrt{2}}{\pi {K}_{0}}\sqrt{1-{{K}_{0}}^{2}}$ |

${V}_{nC2}$ | $\frac{2\sqrt{2}\left(1+{K}_{0}\right)}{\pi {K}_{0}}$ |

Elements | Designed Value by the IPRN Method |
---|---|

L_{1}(uH) | 103.69 |

L_{2}(uH) | 183.42 |

L_{3}(uF) | 14.5 |

C_{1}(nF) | 21.43, (MKP type) |

C_{2}(nF) | 12.11, (MKP type) |

n | 1.33 |

M_{1}~M_{4} | IRFP150 |

D_{1}~D_{4} | MBR20100 |

R_{0ut}(Ω) | 20 |

K | 0.14 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yarmohammadi, L.; Hosseini, S.M.H.; Olamaei, J.; Mozafari, B.
A Load-Independent Output Current Method for Wireless Power Transfer Systems with Optimal Parameter Tuning. *Sustainability* **2022**, *14*, 9391.
https://doi.org/10.3390/su14159391

**AMA Style**

Yarmohammadi L, Hosseini SMH, Olamaei J, Mozafari B.
A Load-Independent Output Current Method for Wireless Power Transfer Systems with Optimal Parameter Tuning. *Sustainability*. 2022; 14(15):9391.
https://doi.org/10.3390/su14159391

**Chicago/Turabian Style**

Yarmohammadi, Leila, S. Mohammad Hassan Hosseini, Javad Olamaei, and Babak Mozafari.
2022. "A Load-Independent Output Current Method for Wireless Power Transfer Systems with Optimal Parameter Tuning" *Sustainability* 14, no. 15: 9391.
https://doi.org/10.3390/su14159391