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

: In this paper was analysed a wireless power transfer system (WPT) with multiple resonators supplying, for example, sensors or LED lighting. Energy is transferred simultaneously using a group of identical planar spiral circular coils acting as transmitters and receivers. These coils were arranged to form transmitting and receiving planes. The receivers were connected to independent power supply circuits of each, e.g., sensor or LED lighting. Higher power reliability and ﬂexibility can be achieved by isolating these circuits. The proposed system was described and discussed. Taking into account the skin effect and mutual couplings, a theoretical analysis was made. A detailed analysis was made at the resonant frequency of the system. The system was modeled using a matrix equation and appropriate formulas. The calculations were veriﬁed experimentally for different loads and two distances between transmitters and receivers. The efﬁciency and receiver power were compared and discussed. The maximum efﬁciency was about 45% at the small distance between the planes. The maximum efﬁciency of the WPT system decreased more than two times to less than 20% when the distance between the coils was doubled. The results and discussion of the conducted analysis may provide valuable knowledge when designing this type of system.


Introduction
Electricity generated in a conventional power plant is very expensive, and the overall transmission efficiency from the power plant to the end user can be very low (about 30%), in particular due to energy losses in parasitic resistances.A power plant running on coal, gas or nuclear fuel requires many resources to produce electricity and faces many environmental problems (e.g., pollution).Additionally, insufficient compensation of reactive power, voltage asymmetry, and current distortions reduce the efficiency and quality of electricity transmitted to consumers [1].
Charging a device using cables is very lossy, difficult to install and often ends in failure.The number of portable devices is growing, but wired chargers prevent these devices from being fully portable.Taking the above into account, alternative solutions for charging devices, mainly home ones, were sought.The solution to eliminate the indicated problems is Wireless Power Transfer (WPT) technology [2].The WPT system can efficiently transfer energy from the source to the device using the phenomenon of electromagnetic induction.The advantage of this system is the mobility of the devices and the lack of radiation.
Wireless power transfer (WPT) systems are used in, for example, electric vehicles (EVs) [2][3][4][5], data transmission [6], and consumer electronics [7,8].WPT technology has also found its application in medical devices [9,10], robotic systems [11], and buildings equipped with sensors [12].In the publications [13,14], various shapes of coils and the influence of their geometry on the system efficiency are analysed.These factors have a major influence on the quality of the transmission.In the articles [15,16], the authors analysed laser cut and printed textile coils.The authors showed that the resistance of the coil affects the misalignment.The authors of [30] showed various types of array coils, e.g., domino.The analysis was made only for the series configuration.Based on the available literature, it was noticed that a series-parallel system with planar coils has not been analysed so far.
Although WPT technology has many applications, the increasing power transmission distance often results in a decrease in transmission efficiency, as well as an urgent need to address safety concerns.Metamaterials offer a way to improve efficiency and reduce flux density in WPT systems [31,32].The authors of [31] present an overview of the status and technological challenges facing metamaterial-based WPT systems.The paper reviews a metamaterial-based wireless energy transmission system from three perspectives: transmission efficiency, misalignment tolerance, and electromagnetic shielding.
This article describes the power supply system for low-power devices.This solution applies to receivers such as sensors, LED lighting, wearable electronics, etc.A power supply method based on multiple resonators was proposed.Then, the energy is transferred simultaneously and independently to the target devices.This solution increases the reliability of the power supply and the operation of receiving devices by using its own WPT system and galvanic separation of individual circuits from each receiver.For this reason, the occurrence of damage, short circuit, or overvoltage of the resonators, or connecting cables of one of the receivers, will not have a significant impact on the functionality of the others.The proposed multi-coil system with more than one load requires a preliminary analysis of its electrical parameters, such as power and efficiency.
There are four sections in this paper.The methodology is presented in Section 2, which includes the concept of the proposed WPT system in Section 2.1, the equivalent electrical circuit in Section 2.2, and finally the experimental study for verification of theoretical research in Section 2.3.A comparison and discussion of the results, obtained during the analysis, is presented in Section 3. In Section 4, the summary is presented, and the most important conclusions are discussed.

System Description
The presented WPT system has two planes: transmitting (Tx) and receiving (Rx).Eighteen coils were used to create these planes, and each plane contained nine coils (Figure 1).They were identical to simplify analysis.Each of them had a self-inductance (L 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.
The results obtained from the experimental study were compared with those obtained from the circuit model.The parameters used in the experiment and calculations are presented below in Tables 1 and 2.  The results obtained from the experimental study were compared with those obtained from the circuit model.The parameters used in the experiment and calculations are presented below in Tables 1 and 2.  Results from the experimental study are presented as solid blue lines (marked as Experimental results in the legend).Results from the circuit model are presented as solid red lines (marked as Analytical results in the legend).

Equivalent Circuit
The WPT system was created from two square planes (Figure 1).Each plane consisted of nine coils (resonators).Each coil was wound with a wire with a diameter wc and an insulation thickness wi.The radius of the coil was r and the number of turns nt.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 Uz, the internal resistance of which was Rz.This caused an equal voltage drop Uin at the input terminals of each resonator.The receiving coils (Rx) directly powered loads R10-R18.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  Results from the experimental study are presented as solid blue lines (marked as Experimental results in the legend).Results from the circuit model are presented as solid red lines (marked as Analytical results in the legend).

Equivalent Circuit
The WPT system was created from two square planes (Figure 1).Each plane consisted of nine coils (resonators).Each coil was wound with a wire with a diameter w 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.presented, where each coil was magnetically coupled with the other coil through the mutual inductance Ma,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.The mutual inductance (Ma,b) of an arbitrary coil and any planar coil (a and b were the numbers of these coils, respectively) was presented below [33]: where s = (wc + wi)/(2π) in [m] is the screw pitch; Φi = [r − (wc + wi)nt]/s is the starting angle of the spiral; Φo = r/s is the ending angle of the spiral; φ1, φ2-is the angles of rotation along the edge of the spiral.Moreover, dz is the vertical distance between coils a and b [m]; dx, dy 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: ) where ΦN = (Φo − Φi)/N is the discrete step; Φs = Φi/ΦN 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).Equation (2) can be used to find the self-inductance (Lc) of the coil, because it can be interpreted as the mutual inductance of the considered inductor and itself.To calculate Lc, substitute dz = dx = dy = 0 and simplify the final formula to the equation: The mutual inductance (M a,b ) of an arbitrary coil and any planar coil (a and b were the numbers of these coils, respectively) was presented below [33]: where 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: where With a higher number of N, the accuracy of Equation ( 2) tends to be the exact solution for Equation (1).Equation ( 2) can be used to find the self-inductance (L 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: For identical resonators, L 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 Energies 2023, 16, 7651 6 of 15 capacitance (C cp ) of the compensation capacitor at the assumed desired frequency f c can be calculated below: In real applications, the compensation capacitor has its own internal resistance.This equivalent series resistance (ESR) can be calculated from equation [34]: where DF is the dissipation factor; f is the current frequency [Hz].
The resistance of the coil (R 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 ): Therefore, the resistance of the coil is given below: where σ 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]: where δ eff is the effective skin depth [m], and it is given below: The equivalent resistance of the resonator [Ω] is presented below: where k 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.
To calculate the currents in each resonator, lumped parameters must be calculated from Equations ( 2)-( 5), (7), and (10) at the desired frequency f c .The simplest system of equations to solve is obtained from Kirchhoff's voltage law: where Z is the impedance of the resonator; Z = R 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: . . .
Then, the unknown vector of currents (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: where P 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 R n is the resistance of the n-th load [Ω]; cosϕ is the power factor.

Experimental Study
The experimental study allowed for the verification of the model presented in Section 2.2.Therefore, an experimental stand was created.The coil carcasses were made of a nonconductive filament using a 3D printer.Then, 18 identical coils were created (circular planar coils were wound).Their self-inductance and resistance were measured.The compensation capacitor was then selected separately for each coil to obtain one desired resonant frequency f 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.
In the measurements, two planes were mounted on a gripper placed on a horizontal slide.This made it possible to adjust the distance between them.The transmitting plane was connected to a Rigol DG4062 (Beijing, China) signal generator, and then both planes were connected to an oscilloscope.A signal generator was used as a source (U 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)).
During measurements, measurement errors related to the method or device may occur.The accuracy of the measurements was certainly influenced by the accuracy of the coil winding.A total of 18 coils were made, which consisted of the transmitting and receiving surfaces.The accuracy of the coil manufacturing influenced the coil's self-inductance as well as their mutual inductances.In order to obtain the resonance state, a compensation capacitor had to be selected separately for each coil.The capacitors were selected with a tolerance of no more than 0.1 nF.For these reasons, it is difficult to obtain a resonance state for each coil at the same frequency.Measuring probes were used for measurements.The electronic components of the tested system were assembled on a PCB, which introduces losses (including solder pads and copper tracks).Additionally, near the coils, there are various metal elements used to build the experimental stand, i.e., wires, screws, mountings, the rail on which the measuring system moves, etc.All this introduces resistances and parasitic capacitances that affect the analyzed system, disturbing the measurements.In the measurements, two planes were mounted on a gripper placed on a horizontal slide.This made it possible to adjust the distance between them.The transmitting plane was connected to a Rigol DG4062 (Beijing, China) signal generator, and then both planes were connected to an oscilloscope.A signal generator was used as a source (Uz = 20 p − pV, Rz = 50 Ω).Probes connected to a Rigol DS2072 (Beijing, China) oscilloscope were used to measure voltages and currents.The measured values were Uin (RMS input voltage), Iz (RMS source current), and UB/2 + 1 ÷ UB (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)).
During measurements, measurement errors related to the method or device may occur.The accuracy of the measurements was certainly influenced by the accuracy of the coil winding.A total of 18 coils were made, which consisted of the transmitting and receiving surfaces.The accuracy of the coil manufacturing influenced the coil's self-inductance as well as their mutual inductances.In order to obtain the resonance state, a compensation capacitor had to be selected separately for each coil.The capacitors were selected with a tolerance of no more than 0.1 nF.For these reasons, it is difficult to obtain a resonance state for each coil at the same frequency.Measuring probes were used for measurements.The electronic components of the tested system were assembled on a PCB, which introduces losses (including solder pads and copper tracks).Additionally, near the coils, there are various metal elements used to build the experimental stand, i.e., wires, screws, mountings, the rail on which the measuring system moves, etc.All this introduces resistances and parasitic capacitances that affect the analyzed system, disturbing the measurements.

Discussion of the Results
The results obtained from the experimental study were compared with those obtained from the circuit model (Equation ( 12)).The parameters used in both analyses are presented in Tables 1 and 2. The circuit model was solved in Matlab R2013b (MathWorks, Natick, MA, USA).Calculations were made for N = 1000 subintervals.In the numerical considerations, it was necessary to determine one self-inductance and B(B − 1)/2 = 153 mutual inductances.

Discussion of the Results
The results obtained from the experimental study were compared with those obtained from the circuit model (Equation ( 12)).The parameters used in both analyses are presented in Tables 1 and 2. The circuit model was solved in Matlab R2013b (MathWorks, Natick, MA, USA).Calculations were made for N = 1000 subintervals.In the numerical considerations, it was necessary to determine one self-inductance and B(B − 1)/2 = 153 mutual inductances.The mutual inductances (M 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.
At the design frequency, the influence of the load resistance (R) on the efficiency (η) and powers of the WPT system was examined.To adjust the load resistance and examine its influence on efficiency and powers, variable resistors connected to the receiving coils as energy receivers were used.Analytical and experimental results (efficiency, receiver, and transmitter powers) are presented in Section 3.1.Results from the experimental study are presented as solid blue lines (marked as Experimental results in the legend).Results from the circuit model are presented as solid red lines (marked as Analytical results in the legend).
The lumped parameters of the circuit model were calculated using Matlab R2013b software and obtained L 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 fre-Energies 2023, 16, 7651 9 of 15 quency.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.

Results
It should be crucial to analyse the system in order to find the resonance point and determine the optimal load resistance (i.e., maximum efficiency or receiver power).The influence of load resistance on the efficiency and powers of the WPT system was analysed.The calculation and measurement results were obtained at the resonant frequency, at the distance d 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.ware and obtained Lc = 13.84 µH, Rco = 753 mΩ, and Ccp = 7.32 nF, which gave fc = 500 kHz.The average measured parameters of the coils and capacitors used in the experiment (Lc, Rco and Ccp) 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.

Results
It should be crucial to analyse the system in order to find the resonance point and determine the optimal load resistance (i.e., maximum efficiency or receiver power).The influence of load resistance on the efficiency and powers of the WPT system was analysed.The calculation and measurement results were obtained at the resonant frequency, at the distance dz = 5 mm (Section 3.1.1)and dz = 10 mm (Section 3.1.2).In the experimental study, the resonant frequency at the distance dz = 5 mm was 576 kHz and 550 kHz at the distance dz = 10 mm, while the design frequency was 500 kHz.The efficiency of the WPT system increased with the increase in load resistance, and then after reaching the maximum, it began to decrease (Figure 4).The maximum efficiency obtained from calculations was 44.13% with R = 12.5 Ω and from measurements 36.83% with R = 9 Ω.With low load resistance, the efficiency of the system obtained from calculations and measurements was comparable.Additionally, with low load resistance, the efficiency increased very quickly until it reached the maximum value.The greatest difference in efficiency obtained from calculations and measurements was approximately 15% with R = 40 Ω.This difference decreases as the load resistance increases and does not exceed 8% with R = 200 Ω.However, the smallest difference in efficiency was approximately 5% with R = 7.5 Ω.The shape of the characteristics obtained from calculations and measurements was maintained.
The active transmitter (Pz) and active receiver (Po) powers varied depending on the load resistance (Figures 5 and 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 The efficiency of the WPT system increased with the increase in load resistance, and then after reaching the maximum, it began to decrease (Figure 4).The maximum efficiency obtained from calculations was 44.13% with R = 12.5 Ω and from measurements 36.83% with R = 9 Ω.With low load resistance, the efficiency of the system obtained from calculations and measurements was comparable.Additionally, with low load resistance, the efficiency increased very quickly until it reached the maximum value.The greatest difference in efficiency obtained from calculations and measurements was approximately 15% with R = 40 Ω.This difference decreases as the load resistance increases and does not exceed 8% with R = 200 Ω.However, the smallest difference in efficiency was approximately 5% with R = 7.5 Ω.The shape of the characteristics obtained from calculations and measurements was maintained.
The active transmitter (P z ) and active receiver (P o ) powers varied depending on the load resistance (Figures 5 and 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.

Results at the Distance d z = r = 10 mm
Efficiency, transmitter, and receiver powers diagrams are presented in Figures 7-9.
(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 dz = r = 10 mm Efficiency, transmitter, and receiver powers diagrams are presented in Figures 7-9.(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.The maximum efficiency of the WPT system decreased more than two times to less than 20% when the distance between the coils (dz) 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 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 (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.
The receiver and transmitter powers varied depending on the load resistance (Figures 8 and 9).In order to determine the variability of the receiver and transmitter powers along with the load resistance, characteristics from measurements and calculations were presented.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 in the experimental study and the circuit model (7 Ω).The smallest difference in the receiver power, obtained from calculations and measurements, occurred with a load resistance of about 7 Ω.The greatest difference in the receiver power, obtained from calculations and measurements, occurred for R = 3-4 Ω.This difference decreases as the load resistance increases.Then, after exceeding the maximum receiver power, this difference begins to increase again.The transmitter power obtained from the measurements increased with the increase in load resistance (about 9 Ω), and then, after reaching the maximum, it began to decrease (Figure 9).Then, as the load resistance increased, the transmitter power decrease was relatively small.The transmitter power obtained from the calculations decreases as the load resistance increases.As the load resistance increased, the transmitter power obtained from calculations and measurements changed more and more.
In all analyzed cases and for small and large distances between planes, theoretical calculations adequately predicted these relationships and can be helpful in determining the power, efficiency, and optimal load resistance already at the system design stage.

Comparison of Results at Both Distances between Planes
In this section was presented a comparison of the efficiency and receiver power results, depending on the distance d z between the transmitting and receiving planes and taking into account the load resistance, obtained experimentally and analytically (Figures 10 and 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).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).Efficiency and receiver power diagrams are presented in Figures 10 and 11.As the distance dz 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 (dz) 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 dz.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 dz.The maximum receiver power was obtained with the same load resistance from measurements and calculations (12.5 Ω) at the small distance dz (Figure 11).Doubling the distance dz resulted in the maximum receiver power being obtained with R = 7 Ω for both measurements and calculations.
Results obtained for the efficiency of the WPT system are presented numerically in Table 3. Results from measurements and calculations are presented for small and large distance dz.Efficiency and receiver power diagrams are presented in Figures 10 and 11.
As the distance d 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.
Results obtained for the efficiency of the WPT system are presented numerically in Table 3. Results from measurements and calculations are presented for small and large distance d z .The difference between the experimental and computational results presented in Table 3 is larger for the system at the small distance (d 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.

Conclusions
In this paper was presented the analysis of a multi-coil Wireless Power Transfer system.The WPT system consisted of identical spiral circular coils and provided power to lowpower devices.These coils were arranged to form transmitting and receiving planes.Taking into account the skin effect and mutual couplings, this system was described.A detailed analysis of the system at the resonant frequency was made.The calculations were verified experimentally.The efficiency and receiver power were obtained for various loads and at two distances between the planes.The results were compared and discussed.
The maximum efficiency obtained from calculations was about 45% at the small distance between the planes and decreased more than twice to less than 20% when the distance between the coils was doubled.The maximum efficiency obtained from measurements was over 7% lower than from calculations at the distance d 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.

Figure 1 .
Figure 1.The transmitting and receiving planes forming the analyzed WPT system.

Figure 1 .
Figure 1.The transmitting and receiving planes forming the analyzed WPT system.

Figure 2 .
Figure 2. The transmitting and receiving planes: the electric circuits.

Figure 2 .
Figure 2. The transmitting and receiving planes: the electric circuits.
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].

Figure 3 .
Figure 3. Experimental stand containing transmitting and receiving planes.

Figure 5 .
Figure 5. Results of the active power of the receiver for different load resistances (dz = r/2 = 5 mm).

Figure 6 .
Figure 6.Results of the active power of the transmitter for different load resistances (dz = r/2 = 5 mm).

Figure 6 .
Figure 6.Results of the active power of the transmitter for different load resistances (d z = r/2 = 5 mm).

Figure 7 .
Figure 7. Results of the efficiency for different load resistances (dz = r = 10 mm).

Figure 8 .
Figure 8. Results of the active power of the receiver for different load resistances (dz = r = 10 mm).

Figure 7 .
Figure 7. Results of the efficiency for different load resistances (d z = r = 10 mm).

Figure 7 .
Figure 7. Results of the efficiency for different load resistances (dz = r = 10 mm).

Figure 8 .
Figure 8. Results of the active power of the receiver for different load resistances (dz = r = 10 mm).

Figure 8 . 16 Figure 9 .
Figure 8. Results of the active power of the receiver for different load resistances (d z = r = 10 mm).Energies 2023, 16, x FOR PEER REVIEW 12 of 16

Figure 9 .
Figure 9. Results of the active power of the transmitter for different load resistances (d z = r = 10 mm).

Figure 10 .Figure 10 .
Figure 10.Results of the efficiency for different load resistances.

Figure 10 .
Figure 10.Results of the efficiency for different load resistances.

Figure 11 .
Figure 11.Results of the active power of the receiver for different load resistances.

Figure 11 .
Figure 11.Results of the active power of the receiver for different load resistances.

Table 1 .
Parameters of the WPT system model.

Table 1 .
Parameters of the WPT system model.

Table 2 .
Parameters used in the analysis.

Table 2 .
Parameters used in the analysis.

Table 3 .
Comparison of results at both distances between planes.