A Design Method for Magnetically Coupled Resonant Coils Considering Transmission Objectives and Dimension Constraints

This paper proposes a coil design method for the magnetically coupled resonant wireless power transfer (MCR-WPT) system. Based on the Biot–Savart law, the magnetic flux density at the observation point was derived, and the magnetic flux of the observation plane generated by the exciting coil was deduced to build the calculation model of power transfer efficiency (PTE) and power delivered to the load (PDL). The PTE and PDL curves via coil parameters could be fitted in minutes using numerical calculation. The coil was designed according to transmission objectives and dimension constraints. In addition, the calculated PTE and PDL were compared with those from finite element analysis to verify the credibility of the method. Finally, the actual curves of PTE and PDL were achieved, which showed a strong positive correlation with the corresponding curves from the calculation model. The relative average deviations of PDL curves were less than 6.11%. Meanwhile, coils designed with the numerical calculation could realize 309.80 W and 88.51%, which achieved the objectives under the constraints. The results demonstrate that the proposed method can realize a rapid and accurate coil design under constraints. It can also be applied to other coil structures or circuit topologies with strong universality.


Introduction
Owing to numerous advantages, such as midrange transfer, good effect on transmission, and little influence on nonmagnetic obstacles, the magnetically coupled resonant wireless power transfer (MCR-WPT) system has been extensively applied in daily life [1][2][3][4]. Faced with different applications, the system needs to be designed based on actual demands. The demands usually refer to objectives and constraints, such as power transfer efficiency (PTE), power delivered to the load (PDL), and coil dimension. Besides the circuit topology, resonant coils are also of vital importance for realizing power transfer. The shape and size of coils can directly influence PTE and PDL [5][6][7][8]. However, studies on coil design, especially the direct relationship between coil parameters and PTE and PDL, are lacking. In addition, constraint conditions for the coil dimension are seldom considered [9]. All the above In Figure 1, u1 is the input voltage of the primary side; RL indicates the purely resistive load; i1 and i2 are the currents; R1 and R2 signify the equivalent series resistances (ESRs) of coils; L1 and L2 are the self-inductances of coils. The coils have two main parameters including turns and radius. N1 and N2 represent the turns per coil; r1 and r2 are the radii of coils. The subscripts 1 and 2 represent parameters of the primary side and secondary side, respectively. ϕ12 is the magnetic flux through the transmitting coil produced by i2, which produces the induced voltage u12. ϕ21 is the magnetic flux through the receiving coil produced by i1, which produces the induced voltage u2. To simply describe the proposed method, the follow-up analysis is made under the preconditions of N1 = N2 = N, r1 = r2 = r, and R1 = R2 = R.
The wavelength of the electromagnetic wave in the spatial magnetic field of the WPT system is much larger than the transfer distance. It meets analytic conditions of the near-zone field and magnetoquasistatic field. The model can ignore the effect of the time-varying electric field. The magnetic field of the model can be approximately considered to be caused by a time-varying current.
The mathematical relationship between electrical parameters of the system and the magnetic flux of coils according to Faraday's law of electromagnetic induction is shown as Equation (1) when the MCR-WPT system is on a resonant state. Because coils operate at kHz and MHz, the radiative resistance can be ignored [17,18]. The ohmic resistance RΩ is considered as the main source of power dissipation of the coils. The formula is expressed as Equation (2) with the resonant angular frequency ω, vacuum permeability μ0, radius of the wire a, and conductivity of the wire σ [17,19,20].

The PTE and PDL Calculation Model Applying Simpson Quadrature
Flat spiral coils are equivalent to a series of concentric circles [21], and cylindrical spiral coils can also be equivalent to a series of equally superposed circular coils. The relative error of the induced voltage between the coil and the equivalent coils at the middle cross section is within 5% according to the preliminary finite element simulation. Furthermore, for high-power MCR-WPT systems, a << r. Therefore, for the convenience of calculation, the spiral coil is simplified into N closely arranged single-turn coils with the same current.
First, the magnetic field analysis model of cylindrical spiral coils is established under the condition of magnetoquasistatic field, as shown in Figure 2. The Z axis is perpendicular to the cross section of the exciting coil. The origin of the coordinate system is the center of the exciting coil's In Figure 1, u 1 is the input voltage of the primary side; R L indicates the purely resistive load; i 1 and i 2 are the currents; R 1 and R 2 signify the equivalent series resistances (ESRs) of coils; L 1 and L 2 are the self-inductances of coils. The coils have two main parameters including turns and radius. N 1 and N 2 represent the turns per coil; r 1 and r 2 are the radii of coils. The subscripts 1 and 2 represent parameters of the primary side and secondary side, respectively. φ 12 is the magnetic flux through the transmitting coil produced by i 2 , which produces the induced voltage u 12 . φ 21 is the magnetic flux through the receiving coil produced by i 1 , which produces the induced voltage u 2 . To simply describe the proposed method, the follow-up analysis is made under the preconditions of N 1 = N 2 = N, r 1 = r 2 = r, and R 1 = R 2 = R.
The wavelength of the electromagnetic wave in the spatial magnetic field of the WPT system is much larger than the transfer distance. It meets analytic conditions of the near-zone field and magnetoquasistatic field. The model can ignore the effect of the time-varying electric field. The magnetic field of the model can be approximately considered to be caused by a time-varying current.
The mathematical relationship between electrical parameters of the system and the magnetic flux of coils according to Faraday's law of electromagnetic induction is shown as Equation (1) when the MCR-WPT system is on a resonant state. Because coils operate at kHz and MHz, the radiative resistance can be ignored [17,18]. The ohmic resistance R Ω is considered as the main source of power dissipation of the coils. The formula is expressed as Equation (2) with the resonant angular frequency ω, vacuum permeability µ 0 , radius of the wire a, and conductivity of the wire σ [17,19,20].

The PTE and PDL Calculation Model Applying Simpson Quadrature
Flat spiral coils are equivalent to a series of concentric circles [21], and cylindrical spiral coils can also be equivalent to a series of equally superposed circular coils. The relative error of the induced voltage between the coil and the equivalent coils at the middle cross section is within 5% according to the preliminary finite element simulation. Furthermore, for high-power MCR-WPT systems, a << r. Therefore, for the convenience of calculation, the spiral coil is simplified into N closely arranged single-turn coils with the same current.
First, the magnetic field analysis model of cylindrical spiral coils is established under the condition of magnetoquasistatic field, as shown in Figure 2. The Z axis is perpendicular to the cross section of the exciting coil. The origin of the coordinate system is the center of the exciting coil's middle cross section. In addition, the middle cross section of the exciting coil is the plane XOY, and the other coil's middle cross section is the observation plane. middle cross section. In addition, the middle cross section of the exciting coil is the plane XOY, and the other coil's middle cross section is the observation plane. ym, zm) The observation plane

H'
The observation unit To calculate u2, the transmitting coil is set as the exciting coil. Each turn of the exciting coil is divided into Nh segments. The coordinates of the starting point, ending point, and midpoint in the jth turn and i-th segment of the exciting coil are denoted as respectively. Then, the observation plane is the middle cross section of the receiving coil. The observation plane is divided into numerous observation units, that is, annular sectors. When the observation units are small enough, the magnetic field inside each observation unit at any time can be approximated to a uniform magnetic field. Then, any point close to the center of a discrete unit can be chosen as the equivalent point of the unit's magnetic flux density. The magnetic flux density B at the equivalent point can represent B inside the corresponding observation unit. The total accumulation of the magnetic flux from each observation unit is fairly close to the real magnetic flux of the entire observation plane.
The division of the observation plane is shown in Figure 3. The radius r is uniformly divided into n segments. With the origin of axes as the center and r•k/n as the radius, n concentric circles are formed, in which k = 1, 2, 3, …, 2n−1. The concentric circles are shown in black dotted lines in Figure  3. Meanwhile, the full angle is also uniformly divided into s parts. Then, each part owns the same angle α with the value of 2π/s. The dividing lines are shown in dotted gray lines in Figure 3    To calculate u 2 , the transmitting coil is set as the exciting coil. Each turn of the exciting coil is divided into N h segments. The coordinates of the starting point, ending point, and midpoint in the j-th turn and i-th segment of the exciting coil are denoted as (x T ji1 , y T ji1 , z T ji1 ), (x T ji2 , y T ji2 , z T ji2 ) and (x m , y m , z m ) respectively.
Then, the observation plane is the middle cross section of the receiving coil. The observation plane is divided into numerous observation units, that is, annular sectors. When the observation units are small enough, the magnetic field inside each observation unit at any time can be approximated to a uniform magnetic field. Then, any point close to the center of a discrete unit can be chosen as the equivalent point of the unit's magnetic flux density. The magnetic flux density B at the equivalent point can represent B inside the corresponding observation unit. The total accumulation of the magnetic flux from each observation unit is fairly close to the real magnetic flux of the entire observation plane.
The division of the observation plane is shown in Figure 3. The radius r is uniformly divided into n segments. With the origin of axes as the center and r·k/n as the radius, n concentric circles are formed, in which k = 1, 2, 3, . . . , 2n−1. The concentric circles are shown in black dotted lines in Figure 3. Meanwhile, the full angle is also uniformly divided into s parts. Then, each part owns the same angle α with the value of 2π/s. The dividing lines are shown in dotted gray lines in Figure 3 middle cross section. In addition, the middle cross section of the exciting coil is the plane XOY, and the other coil's middle cross section is the observation plane.

H'
The observation unit To calculate u2, the transmitting coil is set as the exciting coil. Each turn of the exciting coil is divided into Nh segments. The coordinates of the starting point, ending point, and midpoint in the jth turn and i-th segment of the exciting coil are denoted as The division of the observation plane is shown in Figure 3. The radius r is uniformly divided into n segments. With the origin of axes as the center and r•k/n as the radius, n concentric circles are formed, in which k = 1, 2, 3, …, 2n−1. The concentric circles are shown in black dotted lines in Figure  3. Meanwhile, the full angle is also uniformly divided into s parts. Then, each part owns the same angle α with the value of 2π/s. The dividing lines are shown in dotted gray lines in Figure 3   Second, the current i 1 = I 1m sin(ωt + ϕ 1 ) is regarded as the excitation. The magnetic flux density B at the center H caused by the current i 1 is analyzed as follows. The induced voltage originates from the variation over time of the magnetic flux density in Z-axis. That is, the component perpendicular to the observation plane takes effect. Hence, according to Biot-Savart Law and line integral, the Z-axis component of the B at an equivalent point caused by the j-th turn and i-th segment of the exciting coil can be deduced as B z hji = . The r T is the directed line segment from the source to the field point H.
To calculate numerical integral with a higher accuracy, the integral interval is equally divided into several segments, and the Simpson quadrature is used in each subinterval, which is the compound Simpson quadrature [22]. Applying compound Simpson quadrature, the starting point, midpoint, and ending point of the i-th subinterval are substituted into the equation of the B z hji . The corresponding integral parts after substituting the coordinates are recorded as f (i), f (i+0.5), and f (i+1), respectively. The Equation is shown as (3). where: The same calculation is conducted for the remaining N h -1 subintervals of this turn. And the Z-axis component of the magnetic flux density B at the point H caused by the j-th turn of the coil is: Then the Z-axis component of B from N single-turn coils at the point H is And the magnetic flux of the observation plane is dθr dr through surface integral. However, the above integral is hard to gain through elementary functions. To increase the computational efficiency, the observation plane is discretized according to the division shown in Figure 3. When observation units are small enough, φ RT is approximately expressed as Equation (5).
According to the definition of mutual inductance, the mutual inductance M of coils can be derived by Nφ RT = Mi 1 . And M is a function of coils' dimensions and position which is shown in Equation (6).
Sequentially, u 2 and i 2 are derived according to the model of the system in Equation (1) with φ RT ≈φ 21 . Then using mutual inductance, the induced voltage u 12 can be deduced through Then, the relationship between PTE and PDL and coil parameters (N, r) is derived, as shown in Equation (7).
So far, the PTE and PDL calculation model of the MCR-WPT system has been achieved. The relationship between PTE and PDL and parameters like turns N and radius r can be achieved via numerical calculation. Because the independent variables are coil parameters (N, r), the design range of parameters can be directly defined. Thus, according to the design objectives, such as PTE optimization, PDL optimization, and PTE and PDL product optimization, or specific PTE and PDL requirements, the optimal coil parameters under dimension constraints can be determined.

Design Demands
Due to numerous parameters of the MCR-WPT system, we used a set of fixed values for all system parameters, except the coil parameters, as shown in Table 1. According to the practical design requirements of another subject (an ultrasonic generator), the operating frequency f was set as 32.47 kHz. The detailed design demands were as follows. Transmission objectives: PDL needs to be 300-350 W and PTE is above 80%. Dimension constraints: the coil diameter is 20-40 cm and the height is 50-75 mm.
According to the dimension constraints, the ranges of radius and turns were 10-20 cm and 14-20 cm, respectively. The following numerical calculation was based on the above parameters and constraints.

Hyperparameters Setting of the PTE and PDL Calculation Model
In order to increase computational efficiency of the method, the optimal values of three hyperparameters, namely, N h , n, and s, were determined through numerical calculation. N h is the number of dividing segments of the exciting coil, and n and s are the number of dividing segments of the radius and full angle, respectively. Three resonant coil models with (N, r) of (16, 16 cm), (22, 16 cm) and (16, 18 cm) were randomly selected as analytic objects. The following analysis was based on the coil model of (16, 16 cm). The analytic results based on the coil models of (22, 16 cm) and (16,18 cm) were the same.
Because the influence on the calculated results of the value of hyperparameter N h and hyperparameters n and s are independent, their optimal values were explored separately.
When the coils were coaxial, the calculated values of PTE and PDL remained unchanged when only the value of s varied. Figure 4a shows the calculated values of PTE and PDL when n varied in the range 2-20. With the increase of n, the calculated values of PTE and PDL tended to become gradually stable. Therefore, considering the accuracy of the calculated values and the computing time, n was Energies 2020, 13, 4144 7 of 15 set as 10 and s was set as 1. Figure 4b shows calculated values of PTE and PDL when N h varied in the range 0-600 and n = 10, s = 1. The variation of N h in 75-600 results had little influence on PTE and PDL. Therefore, considering computing accuracy, N h was set as 400. The computing time was about 0.78 s when the algorithm took these hyperparameters and operated on the computer with CPU Intel(R) Core(TM) i5-7400 @3.00 GHz, RAM 8 GB (Intel Corporation, Portland, Oregon, USA). The annotations in Figure 4a are the mean and standard deviation of the calculated results when n varied in the range 6-20. Because the influence on the calculated results of the value of hyperparameter Nh and hyperparameters n and s are independent, their optimal values were explored separately.
When the coils were coaxial, the calculated values of PTE and PDL remained unchanged when only the value of s varied. Figure 4a shows the calculated values of PTE and PDL when n varied in the range 2-20. With the increase of n, the calculated values of PTE and PDL tended to become gradually stable. Therefore, considering the accuracy of the calculated values and the computing time, n was set as 10 and s was set as 1. Figure 4b shows calculated values of PTE and PDL when Nh varied in the range 0-600 and n = 10, s = 1. The variation of Nh in 75-600 results had little influence on PTE and PDL. Therefore, considering computing accuracy, Nh was set as 400. The computing time was about 0.78 s when the algorithm took these hyperparameters and operated on the computer with CPU Intel(R) Core(TM) i5-7400 @3.00 GHz, RAM 8 GB (Intel Corporation, Portland, Oregon, USA). The annotations in Figure 4a are the mean and standard deviation of the calculated results when n varied in the range 6-20.

Model Verification
In order to verify the accuracy of the proposed method, results from the proposed model and FEA were compared. Additionally, a noncoaxial situation was considered to illustrate the universality of the method. The noncoaxial situation here refers to the radial offset.
The comparison was conducted considering coaxial coils. The results from FEA needed to obtain mutual inductance through 3D finite element simulation, and the PDL and PTE could then be calculated based on the Kirchhoff voltage laws. The equations of PDL and PTE with respect to M are shown as Equation (8) [21], where U 1 is the root mean square (RMS) value of u 1 and U 1 = 0.9U 1, dc . The 3D finite element simulation and the PTE and PDL calculation model were based on the same coil models and the same parameters of the system as f and u 1 . The solution type of the simulation was eddy current. The simulation took nearly 3 h in modeling and model solving. The PTE and PDL calculation model set the RMS value of the full-bridge inverter output voltage U 1 = 0.9U 1, dc as the given value.
The partial key data of two methods are shown in Table 2, while the complete results are shown in Figure 5. When results from FEA were treated as true values, the maximum relative errors of PDL and PTE were 9.40% and 3.68%, respectively. The relative average deviations (RADs), which are the average values of relative deviations in the same set, were lower than 3% and 1%, respectively. In addition, the variational trend of PDL from the two methods above was alike, as well as that of PTE, as shown in Figure 5. With the increase of radius and turns, the PDL dropped, the PTE increased, and curves tended to become gradually constant. Therefore, the quantitative relationship between PDL and PTE and coil parameters could be rapidly acquired via the proposed model with relatively high accuracy.  In addition, the method relating to mutual inductance evaluation proposed in Ref. [23] was compared with the proposed method. The method in Ref. [23] is based on the numerical form of the Neumann formula. The calculated results of mutual inductance M through FEA, the method in Ref. [23], and the proposed method are shown in Table 3. The coils were divided into 400 segments in the method in Ref. [23] and the proposed method. The symbol MN is the calculated results of M using the method in Ref. [23]. The symbol εr-MN is the relative error between MN and MFEA. It can be seen from the relative error that the proposed method is more accurate for most coil models. In addition, the method relating to mutual inductance evaluation proposed in Ref. [23] was compared with the proposed method. The method in Ref. [23] is based on the numerical form of the Neumann formula. The calculated results of mutual inductance M through FEA, the method in Ref. [23], and the proposed method are shown in Table 3. The coils were divided into 400 segments in the method in Ref. [23] and the proposed method. The symbol M N is the calculated results of M using the method in Ref. [23]. The symbol ε r-MN is the relative error between M N and M FEA . It can be seen from the relative error that the proposed method is more accurate for most coil models. When the coil model has a radial offset, the proposed model is also applicable. If the receiving coil has a radial offset along the x axis, the proposed model can also be conducted just by adding the deviation to the original x coordinate of the point H.
The coil with the parameters of N = 16, r = 16 cm was taken as an example. The offset was set as 0, 1 cm, 3 cm, and 5 cm. The key data and the comparison between the numerical and experimental results are both shown in Table 4.

Resonant Coils Design
The numerical calculation was conducted according to the design demands. The numerical results are shown in Figure 6, where the red dashed lines are the lower and upper limits of the PDL target P out * , while the green one is the lower limit of PTE target η * . Due to the existence of the wire loss, PTE was considered with a margin. Then, the parameters were determined as N = 16 and r = 16 cm considering PDL, the coil dimension, material consumption for coil winding, and difficulty of coil winding. Note that other coil parameters (N, r) whose numerical results satisfies the design demands can also be selected. The designed coils are verified in Figure 5, where the critical values of PTE and PDL are marked with annotations of P out * , η * and dotted lines, respectively. This proves that the design of resonant coils can be realized according to the design demands. The quantitative relationship can also be acquired by the PDL and PTE calculation model, thus verifying the credibility of the proposed method.

Experimental Method
The main purpose of this section is to verify the credibility of the proposed method and the influence of coil parameters on PDL and PTE gained in the numerical calculation. Taking the same system parameters as Table 1, an experimental platform of MCR-WPT system was built. Furthermore, it was ensured that the parameters of the spiral coils met the dimension constraints. Before model validation, the resonant point was matched first to guarantee normal power transfer of the system. Then, the influence of coil radius r and turns N on PDL and PTE was explored under this frequency. The coil could then be designed based on the results. The noncoaxial situation was also explored. The flow chart is shown in Figure 7.

Verification of the Resonance point
The MCR-WPT

Experimental Method
The main purpose of this section is to verify the credibility of the proposed method and the influence of coil parameters on PDL and PTE gained in the numerical calculation. Taking the same system parameters as Table 1, an experimental platform of MCR-WPT system was built. Furthermore, it was ensured that the parameters of the spiral coils met the dimension constraints. Before model validation, the resonant point was matched first to guarantee normal power transfer of the system. Then, the influence of coil radius r and turns N on PDL and PTE was explored under this frequency. The coil could then be designed based on the results. The noncoaxial situation was also explored. The flow chart is shown in Figure 7.
The main purpose of this section is to verify the credibility of the proposed method and the influence of coil parameters on PDL and PTE gained in the numerical calculation. Taking the same system parameters as Table 1, an experimental platform of MCR-WPT system was built. Furthermore, it was ensured that the parameters of the spiral coils met the dimension constraints. Before model validation, the resonant point was matched first to guarantee normal power transfer of the system. Then, the influence of coil radius r and turns N on PDL and PTE was explored under this frequency. The coil could then be designed based on the results. The noncoaxial situation was also explored. The flow chart is shown in Figure 7.

The Experimental Platform of MCR-WPT System
The experimental platform of SS topological MCR-WPT system built as per Figure 1 is presented in Figure 8. The full-bridge inverter module consisted of IGBT half-bridge module FF100R12KS4 (Infineon, Munich, Germany), which could realize the operating frequency f = 32.47 kHz. Coils were winded by Litz wire to reduce the skin effect [24]. The wires were all compactly winded to reduce the influence of distributed capacitance between turns and conform to the magnetic field analysis model of coils. We used plastic material for both coil formers and supporting frames to minimize the influence of the platform on magnetic field distribution. A long straight ruler supported two coils to make it convenient to measure the distance. The resistance and inductance of the circuits were measured by AGILENT 4294A (Agilent Technologies Inc., Santa Clara, California, USA) impedance analyzer to select the most suitable compensating capacitor. Its basic impedance accuracy reaches

The Experimental Platform of MCR-WPT System
The experimental platform of SS topological MCR-WPT system built as per Figure 1 is presented in Figure 8. The full-bridge inverter module consisted of IGBT half-bridge module FF100R12KS4 (Infineon, Munich, Germany), which could realize the operating frequency f = 32.47 kHz. Coils were winded by Litz wire to reduce the skin effect [24]. The wires were all compactly winded to reduce the influence of distributed capacitance between turns and conform to the magnetic field analysis model of coils. We used plastic material for both coil formers and supporting frames to minimize the influence of the platform on magnetic field distribution. A long straight ruler supported two coils to make it convenient to measure the distance. The resistance and inductance of the circuits were measured by AGILENT 4294A (Agilent Technologies Inc., Santa Clara, California, USA) impedance analyzer to select the most suitable compensating capacitor. Its basic impedance accuracy reaches ±0.08%. The operating frequency of Pearson 4100 (Pearson Electronics, Palo Alto, California, USA) current probe when measuring current is 140 Hz-35 MHz and its sensitivity can reach 1V/A+1/−0%. TEKTRONIX TBS1104 (Tektronix, Portland, Oregon, USA) digital storage oscilloscope has 8-digit vertical resolution and ±3% DC gain accuracy. Its sample rate on each channel can reach 1 GS/s. The limit of the bandwidth is 20 MHz, and it is compatible with the chosen current probe. Thus, it was selected to sample the voltage and signals of the current probe.

Resonant Point Matching
By matching the resonant capacitance, the precondition that both sides are in a resonant state was satisfied. After both sides met the resonant condition, the follow-up model validation was conducted.

Resonant Point Matching
By matching the resonant capacitance, the precondition that both sides are in a resonant state was satisfied. After both sides met the resonant condition, the follow-up model validation was conducted.
The coil model with r = 16 cm and N = 12 was taken as an example. The capacitor with a suitable capacitance was selected for the compensation till the waveforms of u 1 , i 1 and voltage u L across the load reached the states shown in Figure 9. At this time, u 1 and i 1 were in phase, and i 1 lagged behind u L for about π/2. The phase relationship between i 1 and u 2 met the theoretical derivation. Therefore, both sides were considered to reach the resonant state.
Oscilloscope Figure 8. The experimental platform of the MCR-WPT system.

Resonant Point Matching
By matching the resonant capacitance, the precondition that both sides are in a resonant state was satisfied. After both sides met the resonant condition, the follow-up model validation was conducted.
The coil model with r = 16 cm and N = 12 was taken as an example. The capacitor with a suitable capacitance was selected for the compensation till the waveforms of u1, i1 and voltage uL across the load reached the states shown in Figure 9. At this time, u1 and i1 were in phase, and i1 lagged behind uL for about π/2. The phase relationship between i1 and u2 met the theoretical derivation. Therefore, both sides were considered to reach the resonant state.

Result Analysis
The results in the coaxial situation was analyzed first. Keeping turns N = 16 unchanged and taking several values between 12.5 cm and 20 cm as the coil radii, the coil models were formed. The voltage U1,dc was kept constant. After the power stably transferred, the primary side current I1, Figure 9. The phase relationship between voltage and current.

Result Analysis
The results in the coaxial situation was analyzed first. Keeping turns N = 16 unchanged and taking several values between 12.5 cm and 20 cm as the coil radii, the coil models were formed. The voltage U 1,dc was kept constant. After the power stably transferred, the primary side current I 1 , inverter output voltage U 1 , and load voltage U L were measured to calculate the PTE and PDL. The above data points were compared with the numerical results listed in Table 2, as shown in Table 4. The fitted curves are shown in Figure 10a. inverter output voltage U1, and load voltage UL were measured to calculate the PTE and PDL. The above data points were compared with the numerical results listed in Table 2, as shown in Table 4.
The fitted curves are shown in Figure 10a. Then, keeping the coil radius r = 16 cm unchanged and taking integers in 14 and 20 as coil turns, the same experiment and measurements were conducted. The comparison is shown in Figure 10b.

PDL
Variational trends of PDL curves from the experiment and numerical calculation based on the proposed model showed a strong positive correlation, and the numerical values were close. The Pearson's correlation coefficient (PCC) can describe the level of similarity between two vectors. The value of PCC in the range 0.8-1 means two vectors present a strong positive correlation. The curve can also be expressed as a vector, which is formed by the data points on the curve. For the PDL curves gained through the experiment and numerical calculation, the PCCs of curves over r and N were 0.993 and 0.996, respectively, while the RADs were 6.11% and 2.17%, respectively. As for the PDL Then, keeping the coil radius r = 16 cm unchanged and taking integers in 14 and 20 as coil turns, the same experiment and measurements were conducted. The comparison is shown in Figure 10b.

PDL
Variational trends of PDL curves from the experiment and numerical calculation based on the proposed model showed a strong positive correlation, and the numerical values were close. The Pearson's correlation coefficient (PCC) can describe the level of similarity between two vectors. The value of PCC in the range 0.8-1 means two vectors present a strong positive correlation. The curve can also be expressed as a vector, which is formed by the data points on the curve. For the PDL curves gained through the experiment and numerical calculation, the PCCs of curves over r and N were 0.993 and 0.996, respectively, while the RADs were 6.11% and 2.17%, respectively. As for the PDL curves gained through the experiment and FEA, the PCCs of curves over r and N were 0.987 and 0.985, respectively, while the RADs were 14.33% and 6.46%, respectively. However, when the PDL data within 12 cm and 14 cm of radius were excluded from the RAD calculation, the RAD between PDL curves over r gained through the experiment and numerical calculation was only 3.57%. The larger deviation in this range might have resulted from two reasons, namely, the larger leakage magnetic flux [25] and the larger ratio of circuital resistance in the input resistance. Because the circuital resistance excluding the ESR of the coil was not calculated in the derivation, the larger ratio of circuital resistance made the deviation of the value of i 1 larger. This led to a further considerable effect on the value of i 2 and PDL.

PTE
Variational trends of PTE curves from the experiment and numerical calculation based on the proposed method also showed a strong positive correlation. However, possibly owing to the wire loss in the experiment, numerical values had a certain deviation, and error in the PDL numerical calculation made the deviation larger. The quantity mean ± standard deviation (mean ± SD) describes the dispersion degree of the deviation between curves. A lower dispersion degree means the cause of the deviation is more likely to be always present in all coil models. For the PTE curves gained through the experiment and numerical calculation, the PCCs of curves over r and N were 0.987 and 0.994, respectively, while the mean ± SD of the deviation were 5.90 ± 2.18% and 4.03 ± 1.21%, respectively. As for the PTE curves gained through the experiment and FEA, the PCCs of curves over r and N were 0.980 and 0.998, respectively, while the mean ±SD of the deviation were 8.74 ± 3.48% and 4.63 ± 1.73%, respectively.

Actual Transmission Effect of the System with Designed Coils
For the design demands, the optimal coil parameters confirmed by the experiment were consistent with those confirmed by numerical calculation, that is, N = 16 and r = 16 cm. With this coil model, the PDL of the system reached 309.80 W and the PTE was about 88.51%, which was about 8% better than the target PTE.
Then, the results in the noncoaxial situation were analyzed. The parameters of the system were the same as those in Table 1. The coil parameters were N = 16 and r = 16 cm. The offset along the x axis was also set as 0, 1 cm, 3 cm, and 5 cm. The experimental results are listed in the Table 5. The values of the PDL were similar when the offset was within 3 cm. This was probably due to the system being close to the state of critical coupling. The values of PTE presented a continuous descent. The numerical and experimental results both had the same trend. In summary, the variational trends of PTE and PDL curves from the proposed model had a strong correlation with those from the experiment, and the numerical values were also approximately the same as the calculated values from the experiment. The designed coils based on this relationship reached transmission objectives in the experiment. Additionally, the noncoaxial situation, which refers to the radial offset, could also be analyzed by the proposed method. The proposed method can therefore be applied to practical resonant coils design.

Conclusions
Based on the magnetic resonance principle of the MCR-WPT system, a magnetic field analysis model is proposed in this work. First, the PTE and PDL calculation model was obtained. The quantitative relationship between PTE and PDL and coil parameters was acquired through numerical calculation. Thus, the coil design could be realized according to the actual demands. Experiments proved that the designed coils reached the actual transmission objectives. The PTE and PDL calculation model could rapidly and accurately realize the design of resonant coils. The proposed method has credibility as well as strong universality for its possible application to other circuit topologies or coil structures like planar coils. It provides a theoretical basis for research on the effect of coil parameters on PTE and PDL. It also provides an idea for coil design under dimension constraints and practical engineering applications in MCR-WPT systems.