# Analysis of the Wireless Power Transfer System Using a Finite Grid of Planar Circular Coils

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. System Description

_{c}), a coil resistance (R

_{co}), and a lumped capacitor (C

_{cp}). The receiver can be modeled as an equivalent resistance (R). The transmitters were connected in parallel, while the receivers were isolated from each other. The dimensions of the coil were 2r × 2r. Each coil has the same radius (r) and number of turns (n

_{t}). The distance between the planes was (d

_{z}). The distance between the centers of adjacent coils was (d

_{s}). The coils were wound on a non-conductive carcass. Each coil has a compensation capacitor connected in series with the coil to achieve a resonance state. The Tx coils were created as the transmitting plane and the Rx coils as the receiving plane. Wireless energy transfer took place between these planes. Each Tx coil was connected in parallel to the power source (U

_{z}). Each Rx coil directly powered its load. Therefore, many receivers can be powered at the same time.

#### 2.2. Equivalent Circuit

_{c}and an insulation thickness w

_{i}. The radius of the coil was r and the number of turns n

_{t}. Numbers 1 to 9 were assigned to the transmitting plane (circuits), while numbers 10 to 18 were assigned to the receiving plane (circuits). The transmitting coils (Tx) were connected in parallel to the power source U

_{z}, the internal resistance of which was R

_{z}. This caused an equal voltage drop U

_{in}at the input terminals of each resonator. The receiving coils (Rx) directly powered loads R

_{10}—R

_{18}. The Tx and Rx sides were magnetically coupled. Energy transfer occurred not only between two vertically directed coils, but also occurred horizontally and diagonally, i.e., between any coil and the other 17 coils. The electrical equivalent circuit of this system is presented in Figure 2. The transmitting and receiving planes with magnetic couplings are presented, where each coil was magnetically coupled with the other coil through the mutual inductance M

_{a,b}. In order to not confuse the drawing, the magnetic couplings for coil number five were marked. For the remaining coils, the magnetic couplings will be identical.

_{a,b}) of an arbitrary coil and any planar coil (a and b were the numbers of these coils, respectively) was presented below [33]:

_{c}+ w

_{i})/(2π) in [m] is the screw pitch; Φ

_{i}= [r − (w

_{c}+ w

_{i})n

_{t}]/s is the starting angle of the spiral; Φ

_{o}= r/s is the ending angle of the spiral; φ

_{1}, φ

_{2}—is the angles of rotation along the edge of the spiral. Moreover, d

_{z}is the vertical distance between coils a and b [m]; d

_{x}, d

_{y}are the horizontal distances between coils a and b along the x and y axis [m]. All mutual inductances transfer power from the source to the receivers and, as a result, generate redundant transmission paths, even in the case of the Tx coil failure. The above equation can be solved numerically by using summation instead of double integral and dividing the angles into discrete steps:

_{N}= (Φ

_{o}− Φ

_{i})/N is the discrete step; Φ

_{s}= Φ

_{i}/Φ

_{N}is the starting step; N is the number of subintervals; N > 2(r/s). With a higher number of N, the accuracy of Equation (2) tends to be the exact solution for Equation (1).

_{c}) of the coil, because it can be interpreted as the mutual inductance of the considered inductor and itself. To calculate L

_{c}, substitute d

_{z}= d

_{x}= d

_{y}= 0 and simplify the final formula to the equation:

_{c}has to be calculated only once, unlike M

_{a,b}, which has to be calculated for all pairs of coils in the system. When the self-inductance is known, the capacitance (C

_{cp}) of the compensation capacitor at the assumed desired frequency f

_{c}can be calculated below:

_{co}) will be calculated. Then, the length (l

_{co}) of the spiral has to be calculated. The formula for the finite length straight conductor was found by multiplying the average coil circumference by the number of turns (n

_{t}):

_{w}is the electrical conductivity of the wire [S/m]; a

_{eff}is the effective cross-section of the wire [m

^{2}]. A skin effect occurs in a high-frequency electromagnetic field; therefore, the effective cross-section of the wire is presented below [35]:

_{eff}is the effective skin depth [m], and it is given below:

_{R}is the coefficient responsible for the undesirable increase in resistance in real applications. The resulting equivalent resistance R

_{eq}> (R

_{ESR}+ R

_{co}) is due to the appearance of contact and solder resistance, impedance of the connecting wires, temperature rise in the presence of a positive temperature coefficient, capacitor leakage resistance, other parasitic resistances in the system (e.g., solder pads and printed copper paths), etc.

_{c}. The simplest system of equations to solve is obtained from Kirchhoff’s voltage law:

_{eq}+ jωL

_{c}+ 1/jωC

_{cp}[Ω]; I

_{a}is the current in a-th resonator [A]; B is the number of all resonators. The matrix equation

**A**·

**I**=

**U**, where

**A**is the impedance matrix, can be calculated from Equation (11) for B resonators and different loads:

**I**=

**A**

^{−1}·

**U**) will be calculated. Equation (12) allows for a multi-parameter analysis of the designed system, e.g., for various coils, compensation capacitors, and loads. Finally, the efficiency of the WPT system is given below:

_{o}is the output active power (sum of real powers of loads) [W]; P

_{z}is the input active power (sum of real powers at the input terminals) [W]; I

_{n}is the complex current of the n-th resonator [A]; I

_{z}is the RMS source current [A]; U

_{n}is the RMS voltage of the n-th load [V]; R

_{n}is the resistance of the n-th load [Ω]; cosφ is the power factor.

#### 2.3. Experimental Study

_{c}using Equation (4). The coils were then connected to the capacitors on the PCB. Variable resistors were used as energy receivers (R

_{B}

_{/2 + 1}÷ R

_{B}). This helped to smoothly adjust the load resistances and examine their influence on the powers and efficiency of the WPT system. At the small distance between the planes, the influence of load resistance on the results was tested up to 200 Ω and at the large distance between the planes up to 70 Ω. Then, transmitting and receiving planes were made, each containing nine identical coils (resonators). The experimental stand is presented in Figure 3.

_{z}= 20 p − pV, R

_{z}= 50 Ω). Probes connected to a Rigol DS2072 (Beijing, China) oscilloscope were used to measure voltages and currents. The measured values were U

_{in}(RMS input voltage), I

_{z}(RMS source current), and U

_{B}

_{/2 + 1}÷ U

_{B}(RMS load voltage). All these measured values were saved on the oscilloscope in separate files for each analyzed case. Then, the efficiency and powers were calculated (Equation (13)).

## 3. Discussion of the Results

_{a,b}) were calculated numerically from Equation (2) and the self-inductance of the coils from Equation (3). Eleven mutual inductances were estimated due to the internal and external symmetry of the transmitting and receiving planes. This resulted in a significant reduction in computation time.

_{c}= 13.84 μH, R

_{co}= 753 mΩ, and C

_{cp}= 7.32 nF, which gave f

_{c}= 500 kHz. The average measured parameters of the coils and capacitors used in the experiment (L

_{c}, R

_{co}and C

_{cp}) were, respectively, 14.99 μH, 866 mΩ, and 6.80 nF, which gave a resonant frequency equal to almost 499 kHz. Therefore, the obtained frequency differed by only 1 kHz. Each coil used in the experimental stand had a different self-inductance because it was handmade. In practice, the resonance point may vary slightly for each resonator. This therefore required the use of appropriate capacitors to obtain the desired resonant frequency. In real applications, it is also difficult to use coils with identical parameters because the coils are made with a certain accuracy. Additionally, each coil must be attached to the PCB with additional pins, which increases its self-inductance and resistance. Therefore, it is very important to make all coils as precisely as possible. Capacitors selected for individual coils should therefore be selected with the greatest possible precision.

#### 3.1. Results

_{z}= 5 mm (Section 3.1.1) and d

_{z}= 10 mm (Section 3.1.2). In the experimental study, the resonant frequency at the distance d

_{z}= 5 mm was 576 kHz and 550 kHz at the distance d

_{z}= 10 mm, while the design frequency was 500 kHz.

#### 3.1.1. Results at the Distance d_{z} = r/2 = 5 mm

_{z}) and active receiver (P

_{o}) powers varied depending on the load resistance (Figure 5 and Figure 6). The theoretical analysis of the multi-resonator WPT system was complex due to its multi-coupling nature, in which finding analytically the optimal load resistance (maximizing the efficiency or the receiver power) was significantly difficult. In order to determine the variability of the receiver and transmitter powers depending on the load resistance, characteristics from measurements and calculations were presented. The dependence of the receiver power on the load resistance (Figure 5) was almost identical to that in terms of efficiency (Figure 4). The receiver power increased as the load resistance increased and then began to decrease after reaching the maximum. The maximum receiver power was obtained with the same load resistance in the experimental study and the circuit model (12.5 Ω). The smallest difference in the receiver power, obtained from calculations and measurements, occurred with a load resistance below 12.5 Ω. The greatest difference in the receiver power, obtained from calculations and measurements, occurred for R = 70 Ω. This difference decreases as the load resistance increases. The transmitter power increased as the load resistance increased, and then after reaching the maximum, it began to decrease (Figure 6). As the load resistance increased, the transmitter power decrease was relatively small, and the results obtained from calculations and measurements differed slightly. The maximum transmitter power obtained from calculations was obtained with a lower load resistance than that obtained from measurements.

#### 3.1.2. Results at the Distance d_{z} = r = 10 mm

_{z}) was doubled (Figure 7). The maximum efficiency obtained from calculations was 18.80% with R = 9 Ω and from measurements 13.25% with R = 7 Ω. The distance between the transmitting and receiving coils influenced the matched load resistance. The larger the distance between them, the lower the load resistance. The efficiency increased as the load resistance increased and then began to decrease after reaching the maximum. With low load resistance, efficiency increased very quickly. Then, the efficiency obtained from calculations was much higher than from measurements. The greatest difference in efficiency obtained from calculations and measurements was less than 8% with R = 20 Ω. This difference decreases as the load resistance increases and slightly exceeds 4% with R = 70 Ω. The shape of the characteristics obtained from calculations and measurements was maintained.

#### 3.2. Comparison of Results at Both Distances between Planes

_{z}between the transmitting and receiving planes and taking into account the load resistance, obtained experimentally and analytically (Figure 10 and Figure 11). Results from the experimental study are presented as solid blue and light blue lines at the small and large distances, respectively (marked as Experimental results in the legend). Results from the circuit model are presented as solid red and green lines at the small and large distances, respectively (marked as Analytical results in the legend).

_{z}between the transmitting and receiving planes increases, the efficiency of the WPT system decreases (Figure 10). The maximum efficiency of the WPT system decreased more than two times to less than 20% when the distance between the coils (d

_{z}) was doubled. The maximum efficiency obtained from calculations was 44.13% with R = 12.5 Ω and from measurements 36.83% with R = 9 Ω at the small distance d

_{z}. The maximum efficiency obtained from calculations was 18.80% with R = 9 Ω and from measurements 13.25% with R = 7 Ω at the large distance d

_{z}. The maximum receiver power was obtained with the same load resistance from measurements and calculations (12.5 Ω) at the small distance d

_{z}(Figure 11). Doubling the distance d

_{z}resulted in the maximum receiver power being obtained with R = 7 Ω for both measurements and calculations.

_{z}.

_{z}) than for the system at the large distance. The greatest difference in efficiency obtained from calculations and measurements was approximately 15% with R = 40 Ω at the small distance. The greatest difference in efficiency obtained from calculations and measurements was approximately 8% with R = 20 Ω at the large distance.

## 4. Conclusions

_{z}= 5 mm and almost 6% lower at the distance d

_{z}= 10 mm. The load resistance changed depending on the distance between the transmitting and receiving planes, i.e., the larger the distance, the lower the load resistance at which maximum efficiency occurred. The maximum efficiency of the WPT system decreased more than two times when the distance between the coils was doubled. In all analyzed cases, the shape of the characteristics obtained from calculations and measurements was maintained. Theoretical calculations can be helpful in estimating the load resistance in order to obtain the highest possible efficiency of the WPT system or the receiver power already at the stage of preliminary calculations.

## Funding

**ZIREG project—Integrated Program of the Bialystok University of Technology for Regional Development**, contract no. POWR.03.05.00-00-ZR22/18. Project co-financed by the European Union from the European Social Fund under the Knowledge Education Development Operational Program 2014–2020.

## Data Availability Statement

## Conflicts of Interest

## References

- Sołjan, Z.; Hołdyński, G.; Zajkowski, M. The mathematical concept of the currents’ asymmetrical components in three-phase four-wire systems with sinusoidal and asymmetric voltage supply. Bull. Pol. Acad. Sci. Tech. Sci.
**2019**, 67, 271–278. [Google Scholar] [CrossRef] - Sun, L.; Ma, D.; Tang, H. A review of recent trends in wireless power transfer technology and its applications in electric vehicle wireless charging. Renew. Sustain. Energy Rev.
**2018**, 91, 490–503. [Google Scholar] [CrossRef] - Okasili, I.; Elkhateb, A.; Littler, T. A Review of Wireless Power Transfer Systems for Electric Vehicle Battery Charging with a Focus on Inductive Coupling. Electronics
**2022**, 11, 1355. [Google Scholar] [CrossRef] - Batra, T.; Schaltz, E.; Ahn, S. Effect of ferrite addition above the base ferrite on the coupling factor of wireless power transfer for vehicle applications. J. Appl. Phys.
**2015**, 117, 17D517. [Google Scholar] [CrossRef] - Inoue, K.; Kusaka, K.; Itoh, J.I. Reduction in radiation noise level for inductive power transfer systems using spread spectrum techniques. IEEE Trans. Power Electron.
**2018**, 33, 3076–3085. [Google Scholar] [CrossRef] - Li, X.; Tang, C.; Dai, X.; Deng, P.; Su, Y. An inductive and capacitive combined parallel transmission of power and data forwireless power transfer systems. IEEE Trans. Power Electron.
**2018**, 33, 4980–4991. [Google Scholar] [CrossRef] - Kang, S.H.; Choi, J.H.; Jung, C.W. Magnetic resonance wireless power transfer using three-coil system with single planar receiver for laptop applications. IEEE Trans. Consum. Electron.
**2015**, 61, 160–166. [Google Scholar] - Barman, S.D.; Reza, A.W.; Kumar, N.N.; Karim, M.E.; Munir, A.B. Wireless powering by magnetic resonant coupling: Recent trends in wireless power transfer system and its applications. Renew. Sust. Energy Rev.
**2015**, 51, 1525–1552. [Google Scholar] [CrossRef] - Nithiyanandam, V.; Sampath, V. Approach-Based Analysis on Wireless Power Transmission for Bio-Implantable Devices. Appl. Sci.
**2023**, 13, 415. [Google Scholar] [CrossRef] - Fitzpatrick, D.C. Implantable Electronic Medical Devices; Academic Press: San Diego, CA, USA, 2014; pp. 7–35. [Google Scholar]
- Sugino, M.; Kondo, H.; Takeda, S. Linear motion type transfer robot using the wireless power transfer system. In Proceedings of the 2016 International Symposium on Antennas and Propagation (ISAP), Okinawa, Japan, 24–28 October 2016; pp. 508–509. [Google Scholar]
- Matetić, I.; Štajduhar, I.; Wolf, I.; Ljubic, S. Improving the Efficiency of Fan Coil Units in Hotel Buildings through Deep-Learning-Based Fault Detection. Sensors
**2023**, 23, 6717. [Google Scholar] [CrossRef] - Stankiewicz, J.M. Comparison of the efficiency of the WPT system using circular or square planar coils. Przegląd Elektrotechniczny
**2021**, 97, 38–43. [Google Scholar] [CrossRef] - Stankiewicz, J.M. Evaluation of the Influence of the Load Resistance on Power and Efficiency in the Square and Circular Periodic WPT Systems. Energies
**2023**, 16, 2950. [Google Scholar] [CrossRef] - Micus, S.; Padani, L.; Haupt, M.; Gresser, G.T. Textile-Based Coils for Inductive Wireless Power Transmission. Appl. Sci.
**2021**, 11, 4309. [Google Scholar] [CrossRef] - Sun, D.; Chen, M.; Podilchak, S.; Georgiadis, A.; Abdullahi, Q.S.; Joshi, R.; Yasin, S.; Rooney, J.; Rooney, J. Investigating flexible textile-based coils for wireless charging wearable electronics. J. Ind. Text.
**2020**, 50, 333–345. [Google Scholar] [CrossRef] - Stankiewicz, J.M. Analysis of the Influence of the Skin Effect on the Efficiency and Power of the Receiver in the Periodic WPT System. Energies
**2023**, 16, 2009. [Google Scholar] [CrossRef] - Lee, S.-H.; Lorenz, R.D. Development and validation of model for 95%-efficiency 220-W wireless power transfer over a 30-cm air gap. IEEE Trans. Ind. Appl.
**2011**, 47, 2495–2504. [Google Scholar] [CrossRef] - Choroszucho, A.; Pieńkowski, C.; Jordan, A. Electromagnetic wave propagation into building constructions. Przegląd Elektrotech.
**2008**, 84, 44–49. [Google Scholar] - Solouma, N.H.; Kassahun, H.B.; Alsharafi, A.S.; Syed, A.; Gardner, M.R.; Alsharafi, S.S. An Efficient Design of Inductive Transmitter and Receiver Coils for Wireless Power Transmission. Electronics
**2023**, 12, 564. [Google Scholar] [CrossRef] - Sampath, J.P.K.; Alphones, A.; Shimasaki, H. Coil design guidelines for high efficiency of wireless power transfer (WPT). In Proceedings of the 2016 IEEE Region 10 Conference (TENCON), Singapore, 22–25 November 2016; pp. 726–729. [Google Scholar]
- Prengel, S.; Helwig, M.; Modler, N. Lightweight coil for efficient wireless power transfer: Optimization of weight and efficiency for WPT coils. In Proceedings of the 2014 IEEE Wireless Power Transfer Conference, Jeju, Republic of Korea, 8–9 May 2014; pp. 96–99. [Google Scholar]
- Zhang, Y.; Lu, T.; Zhao, Z. Reducing the impact of source internal resistance by source coil in resonant wireless power transfer. In Proceedings of the 2014 IEEE Energy Conversion Congress and Exposition (ECCE), Pittsburgh, PA, USA, 14–18 September 2014; pp. 845–850. [Google Scholar]
- Stankiewicz, J.M. Estimation of the Influence of the Coil Resistance on the Power and Efficiency of the WPT System. Energies
**2023**, 16, 6210. [Google Scholar] [CrossRef] - Choroszucho, A. Analysis of the influence of electrical parameters of concrete and reinforcement inside concrete walls on the values of the electric field intensity. Przegląd Elektrotechniczny
**2022**, 98, 111–117. [Google Scholar] [CrossRef] - Stankiewicz, J.M.; Choroszucho, A. Comparison of the Efficiency and Load Power in Periodic Wireless Power Transfer Systems with Circular and Square Planar Coils. Energies
**2021**, 14, 4975. [Google Scholar] [CrossRef] - Li, J.; Huang, X.; Tan, L.; Wang, R. Resistance optimization of a coil with substrate and design of a high-power-density coupler for wireless power transfer. ISA Trans.
**2022**, 137, 692–705. [Google Scholar] [CrossRef] [PubMed] - Stankiewicz, J.M. The analysis of the influence of the plane coils geometry configuration on the efficiency of WPT system. Przegląd Elektrotechniczny
**2020**, 96, 174–178. [Google Scholar] [CrossRef] - Yuan, Z.; Yang, Q.; Zhang, X.; Ma, X.; Wang, R.; Xue, M.; Zhang, P. A Misalignment Tolerate Integrated S-S-S-Compensated WPT System with Constant Current Output. Energies
**2023**, 16, 2798. [Google Scholar] [CrossRef] - Wu, J.; Wang, Z.; Dai, X. Constant Output-Voltage Design for Bi-Directional Wireless Power Transfer System with Multiple Stages. Energies
**2020**, 13, 3739. [Google Scholar] [CrossRef] - Rong, C.; Yan, L.; Li, L.; Li, Y.; Liu, M. A Review of Metamaterials in Wireless Power Transfer. Materials
**2023**, 16, 6008. [Google Scholar] [CrossRef] - Steckiewicz, A. Efficient Transfer of the Medium Frequency Magnetic Field Using Anisotropic Metamaterials. Energies
**2023**, 16, 334. [Google Scholar] [CrossRef] - Liu, S.; Su, J.; Lai, J. Accurate Expressions of Mutual Inductance and Their Calculation of Archimedean Spiral Coils. Energies
**2019**, 12, 2017. [Google Scholar] [CrossRef] - Torki, J.; Joubert, C.; Sari, A. Electrolytic capacitor: Properties and operation. J. Energy Storage
**2023**, 58, 106330. [Google Scholar] [CrossRef] - Knight, D.W. Practical Continuous Functions for the Internal Impedance of Solid Cylindrical Conductors; G3YNH: Southport, UK, 2016. [Google Scholar] [CrossRef]

**Figure 5.**Results of the active power of the receiver for different load resistances (d

_{z}= r/2 = 5 mm).

**Figure 6.**Results of the active power of the transmitter for different load resistances (d

_{z}= r/2 = 5 mm).

**Figure 8.**Results of the active power of the receiver for different load resistances (d

_{z}= r = 10 mm).

**Figure 9.**Results of the active power of the transmitter for different load resistances (d

_{z}= r = 10 mm).

r (mm) | n_{t} | d_{z} (mm) | |
---|---|---|---|

10 | 30 | 5 | 10 |

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

wire diameter | w_{c} | 200 µm |

wire insulation thickness | w_{i} | 5 µm |

wire conductivity | σ_{w} | 5.6 10^{7} S/m |

resistance increase coefficient | k_{R} | 1.25 |

dissipation factor | DF | 0.05 |

separation between coils | d_{s} | 0.025 m |

design frequency | f_{c} | 500 kHz |

Load Resistance | η (%) | |||
---|---|---|---|---|

Calculations | Measurements | |||

d_{z} = 5 mm | d_{z} = 10 mm | d_{z} = 5 mm | d_{z} = 10 mm | |

3 | 28.89 | 13.73 | 18.40 | 6.70 |

4 | 33.47 | 15.79 | 24.73 | 9.71 |

5 | 36.78 | 17.14 | 28.74 | 11.72 |

7 | 40.91 | 18.50 | 35.26 | 13.25 |

7.5 | 41.58 | 18.68 | 36.18 | 13.19 |

8 | 42.18 | 18.74 | 36.68 | 13.14 |

9 | 43.02 | 18.80 | 36.83 | 12.92 |

10 | 43.57 | 18.73 | 36.53 | 12.60 |

12.5 | 44.13 | 18.14 | 35.80 | 11.39 |

15 | 43.90 | 17.42 | 33.66 | 10.20 |

20 | 42.36 | 15.72 | 29.96 | 8.37 |

25 | 40.29 | 14.14 | 27.01 | 6.94 |

30 | 38.13 | 12.77 | 23.93 | 5.92 |

40 | 34.12 | 10.63 | 19.18 | 4.54 |

50 | 30.72 | 9.07 | 16.16 | 3.69 |

60 | 27.88 | 7.89 | 13.78 | 3.10 |

70 | 25.50 | 6.98 | 12.02 | 2.67 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. 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**

Stankiewicz, J.M.
Analysis of the Wireless Power Transfer System Using a Finite Grid of Planar Circular Coils. *Energies* **2023**, *16*, 7651.
https://doi.org/10.3390/en16227651

**AMA Style**

Stankiewicz JM.
Analysis of the Wireless Power Transfer System Using a Finite Grid of Planar Circular Coils. *Energies*. 2023; 16(22):7651.
https://doi.org/10.3390/en16227651

**Chicago/Turabian Style**

Stankiewicz, Jacek Maciej.
2023. "Analysis of the Wireless Power Transfer System Using a Finite Grid of Planar Circular Coils" *Energies* 16, no. 22: 7651.
https://doi.org/10.3390/en16227651