Single-Layer Transmitter Array Coil Pattern Evaluation toward a Uniform Vertical Magnetic Field Distribution

: A uniform magnetic ﬁeld distribution is a critical aspect in the transmitter array coil design process for achieving a homogenous vertical magnetic ﬁeld distribution. The free position and orientation features can thus be implemented in the wireless power charging system. This paper presents vertical magnetic ﬁeld distribution generated by a single-layer circular ﬂat spiral air core transmitter array coil model analysis and evaluation using a numerical analysis method. This method is developed based on the o ﬀ -symmetry axis magnetic ﬁeld distribution due to a circular current loop derived from the Biot-Savart law. The proposed evaluation criteria are used to obtain the vertical magnetic ﬁeld distribution characteristic of the evaluated array coil model. The vertical magnetic ﬁeld distribution of several circular ﬂat spiral air core coils, in both single and array coil models with di ﬀ erent coil geometries were investigated to obtain the relation between the coil parameters and the distance between the adjoining coil centers to generate uniform vertical magnetic ﬁeld distribution. A case study was also conducted to analyze and evaluate several array coil model patterns (1 × 2 array coil, 1 × 3 array coil, 2 × 2 array coil, 2 × 3 array coil) to meet uniform vertical magnetic ﬁeld distribution. The array coil model is composed of an identical single circular ﬂat spiral air-core coil. Every single coil has inner coil diameter (Di), outer coil diameter (Do), wire diameter (W), pitch (P) and a number of turns (N) at 25 mm, 47.8 mm, 0.643 mm, 0.03 mm, 17 respectively. The study and evaluation of several array coil pattern models show that the distance between the adjoining coil centers should be deﬁned close to the half of coil outer diameter (1 / 2Do) to generate close to uniform vertical magnetic ﬁeld distribution. The vertical magnetic ﬁeld distribution average and magnetic ﬁeld e ﬀ ective transmitting areas array coil model with the given coil parameters changing as the e ﬀ ect in variation in distances between the adjoining coil centers.


Introduction
Wireless power transmission (WPT) systems have been become fascinating re-emerging technology in recent years. By increasing the overall system efficiency, this technology is appealing to various applications because of its convenience and better user experience. Among those applications, coil geometry and structure study in portable electronic charger applications is attractive to achieve uniform magnetic field distribution.

Biot-Savart Law
The magnetic field source comes from currents which arise because of the charge motion. Figure  1 shows that when current flows through a wire, a magnetic field will exist around the wire. The total number of magnetic field (dB) contributions from all small wire segments (dl) at any point (Q) around the wire is the magnetic field at point Q [18]. The Biot-Savart law expresses the magnetic field contribution as follows: where dB is the magnetic field contribution in Tesla (T), idl is the segment current in Ampere (A), a is the distance from the current source to the field point Q in meter (m), the corresponding unit vector and µo is the permeability of free space: µ = 4π × 10 T · m/A (2) The total magnetic field contribution at the point Q is a vector integral of the magnetic field contribution (dB) as follows:

The Off-symmetrical Axis Magnetic Field Formulation due to a Circular Current Loop
The off-symmetrical axis magnetic field due to a circular current loop is illustrated in Figure 2. Using (1), the magnetic field contribution of the current element idl at Q can be expressed as: The Biot-Savart law expresses the magnetic field contribution as follows: idl ×â a 2 (1) where dB is the magnetic field contribution in Tesla (T), idl is the segment current in Ampere (A), a is the distance from the current source to the field point Q in meter (m),â the corresponding unit vector and µ o is the permeability of free space: The total magnetic field contribution at the point Q is a vector integral of the magnetic field contribution (dB) as follows:

The Off-Symmetrical Axis Magnetic Field Formulation due to a Circular Current Loop
The off-symmetrical axis magnetic field due to a circular current loop is illustrated in Figure 2.
Energies 2019, 12, x 3 of 21 for evaluating their magnetic field distribution to achieve uniform magnetic field vertical distribution.

Biot-Savart Law
The magnetic field source comes from currents which arise because of the charge motion. Figure  1 shows that when current flows through a wire, a magnetic field will exist around the wire. The total number of magnetic field (dB) contributions from all small wire segments (dl) at any point (Q) around the wire is the magnetic field at point Q [18]. The Biot-Savart law expresses the magnetic field contribution as follows: where dB is the magnetic field contribution in Tesla (T), idl is the segment current in Ampere (A), a is the distance from the current source to the field point Q in meter (m), the corresponding unit vector and µo is the permeability of free space: µ = 4π × 10 T · m/A (2) The total magnetic field contribution at the point Q is a vector integral of the magnetic field contribution (dB) as follows:

The Off-symmetrical Axis Magnetic Field Formulation due to a Circular Current Loop
The off-symmetrical axis magnetic field due to a circular current loop is illustrated in Figure 2. Using (1), the magnetic field contribution of the current element idl at Q can be expressed as: a Q = yĵ + zk a = a Q − a = −r cos∅ î +(y ∓ r sin ∅ )ĵ +k (7) a = |a|= (−r cos ∅ ) 2 + (y − r sin ∅ ) 2 + z 2 = r 2 + y 2 + z 2 − 2yrsin∅ (8) dl ×â= |a|= r d∅ (− sin ∅ î + cos ∅ ĵ ) × −r cos ∅ î + (y − r sin∅ )ĵ + zk] (10) = r d∅ z cos∅ î + z sin∅ ĵ + (r − y sin∅ )k ·µ (11) Thus, magnetic field at Q is: (r 2 + y 2 + z 2 − 2yr sin∅ ) 3/2 = 0 (13) with the changing of the variable r 2 + y 2 + z 2 − 2yr sin∅', the y and the z components of B can be expressed as follows: B z = µ o ir 4π 2π 0 (r − y sin ∅ )d∅ (r 2 + y 2 + z 2 − 2yr sin∅ ) 3/2 (15) From the symmetry, every point (Q) in any-plane (see Figure 2) with the same z and a Q around the circular loop coil has equal Bz value. With this principle, Bz can be calculated for any arbitrary point at x 0 using the equivalent value of Bz for arbitrary points in the yz-plane. Therefore, the magnetic field from a circular coil at an arbitrary point can be obtained. In case of array coils, the magnetic field at an arbitrary point from an array coil is the total number of magnetic field contributions from all single circular coils in the array coil pattern.
In [15,19,20], the magnetic field at an arbitrary point from a circular coil also can be obtained with a different approach. In the Smythe's textbook, the magnetic field is derived from the vector potential. The vector potential A of the circular loop current at point S ( Figure 3) in the cylindrical coordinate system is given by where r = the coil radius, p = displacement in horizontal direction and z = displacement in vertical direction in cylindrical coordinate system. By substituting ∅ by π + 2θ with complete elliptic integrals of K and E, magnetic field component in vertical direction can be presented as: The vertical direction magnetic field can be calculated from the vector potential:

Analysis and Evaluation Method
The vertical magnetic field distribution is simulated and analyzed using a numerical method based on the off-symmetrical axis magnetic field formulation due to a circular current loop. The numerical method is derived from the Bio-Savart law as proposed in Subsection 2.2. The evaluated coil geometry is a circular flat spiral air-core coil shown in Figure 4. Two kind evaluation scenarios are applied to the proposed analysis and evaluation method to evaluate the array coil model vertical magnetic field distribution. The scenarios are described as follows: By substituting ∅ by π + 2θ with complete elliptic integrals of K and E, magnetic field component in vertical direction can be presented as: The vertical direction magnetic field can be calculated from the vector potential:

Analysis and Evaluation Method
The vertical magnetic field distribution is simulated and analyzed using a numerical method based on the off-symmetrical axis magnetic field formulation due to a circular current loop. The numerical method is derived from the Bio-Savart law as proposed in Section 2.2. The evaluated coil geometry is a circular flat spiral air-core coil shown in Figure 4. By substituting ∅ by π + 2θ with complete elliptic integrals of K and E, magnetic field component in vertical direction can be presented as: The vertical direction magnetic field can be calculated from the vector potential:

Analysis and Evaluation Method
The vertical magnetic field distribution is simulated and analyzed using a numerical method based on the off-symmetrical axis magnetic field formulation due to a circular current loop. The numerical method is derived from the Bio-Savart law as proposed in Subsection 2.2. The evaluated coil geometry is a circular flat spiral air-core coil shown in Figure 4. Two kind evaluation scenarios are applied to the proposed analysis and evaluation method to evaluate the array coil model vertical magnetic field distribution. The scenarios are described as follows: (1) In the first scenario, the vertical magnetic field distribution of single coil model with different  Two kind evaluation scenarios are applied to the proposed analysis and evaluation method to evaluate the array coil model vertical magnetic field distribution. The scenarios are described as follows: (1) In the first scenario, the vertical magnetic field distribution of single coil model with different coil parameter is analysed and evaluated. The vertical magnetic field distribution characteristic of single coil evaluation result is used to define the evaluation criteria for analyses and evaluation of the array coil models vertical magnetic field distribution. The vertical direction evaluation distance is defined at 2.5 mm from the coil surface with a unit current at 1 Ampere. The array coil model vertical magnetic field distribution is analyzed and evaluated with a variation in distances between the adjoining coil centers to meet the vertical magnetic field uniform distribution with the transmitting area as wide as possible. The evaluation criteria used in this paper is proposed as follows: (a) Average vertical magnetic field resultant is not less than 60% single coil vertical magnetic field peak, and (b) Magnetic field effective transmitting area is as wide as possible.
The optimal array coil pattern must comply with all criteria variables above. The array coil model evaluation is applied with different coil parameters (number of turns, inner diameter, outer diameter, pitch and wire diameter) and variety in the distance between the adjoining coil centers.
(2) The second scenario is a case study. The general investigation result from the first evaluation scenario is applied in the array coil model design. In this evaluation scenario, different array coil model patterns which are composed of 2, 3, 4, and 6 coils are constructed from identical circular flat spiral air-core coils with given coil parameters. The vertical magnetic field distribution is analyzed and evaluated using the proposed method as described in first evaluation scenario with variety in the distance between the adjoining coil centers. The goal evaluation in this case study is recommended array coil model with close to uniform vertical magnetic field distribution and widely effective transmitting area. The array coil model patterns are shown in Figure 5.
Energies 2019, 12, x 6 of 21 evaluation distance is defined at 2.5 mm from the coil surface with a unit current at 1 Ampere. The array coil model vertical magnetic field distribution is analyzed and evaluated with a variation in distances between the adjoining coil centers to meet the vertical magnetic field uniform distribution with the transmitting area as wide as possible. The evaluation criteria used in this paper is proposed as follows: (a) Average vertical magnetic field resultant is not less than 60% single coil vertical magnetic field peak, and (b) Magnetic field effective transmitting area is as wide as possible.
The optimal array coil pattern must comply with all criteria variables above. The array coil model evaluation is applied with different coil parameters (number of turns, inner diameter, outer diameter, pitch and wire diameter) and variety in the distance between the adjoining coil centers.
(2) The second scenario is a case study. The general investigation result from the first evaluation scenario is applied in the array coil model design. In this evaluation scenario, different array coil model patterns which are composed of 2, 3, 4, and 6 coils are constructed from identical circular flat spiral air-core coils with given coil parameters. The vertical magnetic field distribution is analyzed and evaluated using the proposed method as described in first evaluation scenario with variety in the distance between the adjoining coil centers. The goal evaluation in this case study is recommended array coil model with close to uniform vertical magnetic field distribution and widely effective transmitting area. The array coil model patterns are shown in Figure 5.

The Coil with Different Coil Parameter
Two circular flat spiral air-core coil models were considered with coil parameters as shown in

The Coil with Different Coil Parameter
Two circular flat spiral air-core coil models were considered with coil parameters as shown in Table 1. These parameters were used to compute vertical magnetic field distribution in both single and array coil models. Two coil model have same outer diameter, distance between coil wire (pitch), and wire diameter but different in number of turns and inner diameter. The single coil magnetic field distribution calculation and evaluation was first addressed and then followed with array coil models. The vertical magnetic field distribution of single coil is numerically calculated along the y-axis to obtain the magnetic field off-symmetry axis due to the circular current loop. The evaluation path in the y-axis direction is shown in Figure 6.  The single coil magnetic field distribution calculation and evaluation was first addressed and then followed with array coil models. The vertical magnetic field distribution of single coil is numerically calculated along the y-axis to obtain the magnetic field off-symmetry axis due to the circular current loop. The evaluation path in the y-axis direction is shown in Figure 6. The vertical magnetic field (Bz) distribution calculation and characteristic of single coil (Coil_1.1) with Di = 10 mm and N = 8 is presented in Figure 7a. As shown in Figure 7a, the vertical magnetic field peak (Bz max) = 1.214 mT, and 60% Bz max = 0.728 mT. While the horizontal length of the vertical magnetic field effective transmitting area = 20 mm.
The array coil model magnetic field distribution calculation and evaluation were conducted with The vertical magnetic field (Bz) distribution calculation and characteristic of single coil (Coil_1.1) with Di = 10 mm and N = 8 is presented in Figure 7a.  The single coil magnetic field distribution calculation and evaluation was first addressed and then followed with array coil models. The vertical magnetic field distribution of single coil is numerically calculated along the y-axis to obtain the magnetic field off-symmetry axis due to the circular current loop. The evaluation path in the y-axis direction is shown in Figure 6. The vertical magnetic field (Bz) distribution calculation and characteristic of single coil (Coil_1.1) with Di = 10 mm and N = 8 is presented in Figure 7a. As shown in Figure 7a, the vertical magnetic field peak (Bz max) = 1.214 mT, and 60% Bz max = 0.728 mT. While the horizontal length of the vertical magnetic field effective transmitting area = 20 mm.
The array coil model magnetic field distribution calculation and evaluation were conducted with the number of coils at 2 (the 1 × 2 array coil pattern). The coil model and the evaluation path in the yaxis direction is shown in Figure 8.  As shown in Figure 7a, the vertical magnetic field peak (Bz max) = 1.214 mT, and 60% Bz max = 0.728 mT. While the horizontal length of the vertical magnetic field effective transmitting area = 20 mm. The array coil model magnetic field distribution calculation and evaluation were conducted with the number of coils at 2 (the 1 × 2 array coil pattern). The coil model and the evaluation path in the y-axis direction is shown in Figure 8.  Figure 7b presents the vertical magnetic field (Bz) distribution calculation and characteristic of the 1 × 2 array coil model composed of two identical coils with Di = 10 mm and N = 23. The evaluation was performed with variety of distances between the adjoining coil centers (d) at 22 mm, 23 mm, 24 mm, and 25 mm. The magnetic field threshold criteria were derived from magnetic field distribution characteristic in Figure 7a. It is equal to 60% Bz max = 0.728 mT. From the evaluation result as shown in Figure 7b, the characteristic curve has the vertical magnetic field average (Bz Avg) = 0.748 mT, the horizontal length of the vertical magnetic field effective transmitting area = 50 mm, and the distance between the adjoining coil centers (d) at 23 mm was a recommended vertical magnetic field distribution for 1 × 2 array coil model with coil Di = 10 mm, coil Do = 40 mm, and N = 23 turns.
The second coil (Coil_1.2) with Di = 30 mm and N = 8 also was evaluated using the same method described in the first coil (Coil_1.1) evaluation. The evaluation result is shown in Figure 9.  The vertical magnetic field (Bz) distribution calculation and characteristic of the 1 × 2 array coil model composed of Coil_1.2 is presented in Figure 9b. The evaluation was performed with a variety of distance between the adjoining coil centers (d) at 20 mm, 21 mm, 22 mm, and 23 mm. The magnetic field threshold criteria were derived from the magnetic field distribution characteristic in Figure 9a. It is equal with 60% Bz max = 0.244 mT. The characteristic curve has the vertical magnetic field average (Bz Avg) = 0.248 mT, the horizontal length of the vertical magnetic field effective transmitting area = 55 mm, and the distance between the adjoining coil centers (d) at 22 mm was a recommended vertical magnetic field distribution for 1 × 2 array coil model with coil Di = 30 mm, coil Do = 40 mm, and N = 8 turns.   Figure 7b, the characteristic curve has the vertical magnetic field average (Bz Avg) = 0.748 mT, the horizontal length of the vertical magnetic field effective transmitting area = 50 mm, and the distance between the adjoining coil centers (d) at 23 mm was a recommended vertical magnetic field distribution for 1 × 2 array coil model with coil Di = 10 mm, coil Do = 40 mm, and N = 23 turns.
The second coil (Coil_1.2) with Di = 30 mm and N = 8 also was evaluated using the same method described in the first coil (Coil_1.1) evaluation. The evaluation result is shown in Figure 9.   Figure 7b, the characteristic curve has the vertical magnetic field average (Bz Avg) = 0.748 mT, the horizontal length of the vertical magnetic field effective transmitting area = 50 mm, and the distance between the adjoining coil centers (d) at 23 mm was a recommended vertical magnetic field distribution for 1 × 2 array coil model with coil Di = 10 mm, coil Do = 40 mm, and N = 23 turns.
The second coil (Coil_1.2) with Di = 30 mm and N = 8 also was evaluated using the same method described in the first coil (Coil_1.1) evaluation. The evaluation result is shown in Figure 9.  The vertical magnetic field (Bz) distribution calculation and characteristic of the 1 × 2 array coil model composed of Coil_1.2 is presented in Figure 9b. The evaluation was performed with a variety of distance between the adjoining coil centers (d) at 20 mm, 21 mm, 22 mm, and 23 mm. The magnetic field threshold criteria were derived from the magnetic field distribution characteristic in Figure 9a. It is equal with 60% Bz max = 0.244 mT. The characteristic curve has the vertical magnetic field average (Bz Avg) = 0.248 mT, the horizontal length of the vertical magnetic field effective transmitting area = 55 mm, and the distance between the adjoining coil centers (d) at 22 mm was a recommended vertical magnetic field distribution for 1 × 2 array coil model with coil Di = 30 mm, coil Do = 40 mm, and N = 8 turns.  equal with 60% Bz max = 0.244 mT. The characteristic curve has the vertical magnetic field average (Bz Avg) = 0.248 mT, the horizontal length of the vertical magnetic field effective transmitting area = 55 mm, and the distance between the adjoining coil centers (d) at 22 mm was a recommended vertical magnetic field distribution for 1 × 2 array coil model with coil Di = 30 mm, coil Do = 40 mm, and N = 8 turns.
The evaluation was continued with different coils that have coil parameters as shown in Table 2. These coils have the outer diameter larger than the two previous coils (Coil_1.1 and Coil_1.2). Coil_2.1 and Coil_2.2 have the same outer diameter, distance between coil wire (pitch), and wire diameter but different in number of turns and inner diameter. The second coil group (Coil_2.1 and Coil_2.2) was also evaluated using the same method described in the first coil group evaluation. The evaluation results are shown in Figures 10 and 11. The evaluation was continued with different coils that have coil parameters as shown in Table  2. These coils have the outer diameter larger than the two previous coils (Coil_1.1 and Coil_1.2). Coil_2.1 and Coil_2.2 have the same outer diameter, distance between coil wire (pitch), and wire diameter but different in number of turns and inner diameter. The second coil group (Coil_2.1 and Coil_2.2) was also evaluated using the same method described in the first coil group evaluation. The evaluation results are shown in Figures 10 and 11.  The vertical magnetic field (Bz) distribution calculation and characteristic of the 1 × 2 array coil pattern which composed of Coil_2.1 is presented in Figure 10b. The evaluation was performed with variety of distance between the adjoining coil centers (d) at 36, 37, 38 and 39 mm. The magnetic field threshold criteria is 0.244 mT. The characteristic curve has vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.805 mT, the horizontal length of the vertical magnetic field effective transmitting area = 72 mm, and the distance between the adjoining coil centers (d) at   While Figure 11b describes the vertical magnetic field (Bz) distribution calculation and characteristic of the 1 × 2 array coil model composed of Coil_2.2. The evaluation was performed with variety of distances between the adjoining coil centers (d) at 25 mm, 27 mm, 28 mm, and 29 mm. The magnetic field threshold criteria = 0.244 mT. The characteristic curve has vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.382 mT, the horizontal length of the vertical magnetic field effective transmitting area = 75 mm, and the distance between the adjoining coil centers (d) at 28 mm was a recommended vertical magnetic field distribution for 1 × 2 array coil model with coil Di = 40 mm, coil Do = 60 mm, and N = 15 turns.
The next evaluation was the coils in the third coil group (Coil_3.1 and Coil_3.2). The evaluation also used the same method described in the first and second group coil evaluations. The coil parameters of the third coil group is presented in Table 3. The evaluation results are shown in Figures  12 and 13.   While Figure 11b describes the vertical magnetic field (Bz) distribution calculation and characteristic of the 1 × 2 array coil model composed of Coil_2.2. The evaluation was performed with variety of distances between the adjoining coil centers (d) at 25 mm, 27 mm, 28 mm, and 29 mm. The magnetic field threshold criteria = 0.244 mT. The characteristic curve has vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.382 mT, the horizontal length of the vertical magnetic field effective transmitting area = 75 mm, and the distance between the adjoining coil centers (d) at 28 mm was a recommended vertical magnetic field distribution for 1 × 2 array coil model with coil Di = 40 mm, coil Do = 60 mm, and N = 15 turns.
The next evaluation was the coils in the third coil group (Coil_3.1 and Coil_3.2). The evaluation also used the same method described in the first and second group coil evaluations. The coil parameters of the third coil group is presented in Table 3. The evaluation results are shown in Figures 12 and 13.
The calculation and evaluation result for single coil (Coil_3.1) with Di = 30 mm and Do = 80 mm is described in Figure 12a. The vertical magnetic field peak (Bz max) = 1.222 mT, and 60% Bz max = 0.733 mT. The horizontal length of the vertical magnetic field effective transmitting area = 45 mm.
The vertical magnetic field (Bz) distribution calculation and characteristic of the 1 × 2 array coil pattern composed of Coil_3.1 is presented in Figure 12b. The evaluation was performed with variety of distance between the adjoining coil centers (d) at 40 mm, 41 mm, 42 mm, and 43 mm. The magnetic field threshold criteria = 0.733 mT. The vertical magnetic field distribution characteristic curve with the vertical magnetic field average (Bz Avg) = 0.731 mT, the horizontal length of the vertical magnetic field effective transmitting area = 85 mm, and the distance between the adjoining coil centers (d) at 42 mm was recommended vertical magnetic field distribution for 1x2 array coil model with coil Di = 30 mm, coil Do = 80 mm, and N = 38 turns. Figure 13a describes the calculation and evaluation result for single coil (Coil_3.2) with Di = 45 mm and Do = 80mm. The vertical magnetic field peak (Bz max) = 0.916 mT, and 60% Bz max = 0.549 mT. The horizontal length of the vertical magnetic field effective transmitting area = 50 mm.
While the vertical magnetic field (Bz) distribution calculation and characteristic of the 1 × 2 array coil pattern composed of Coil_3.2 is presented in Figure 13b. The evaluation was performed with a variety of distance between the adjoining coil centers (d) at 34 mm, 35 mm, 36 mm, and 37 mm. The magnetic field threshold criteria = 0.549 mT. The vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.567 mT, the horizontal length of the vertical magnetic field effective transmitting area = 90 mm, and the distance between the adjoining coil centers (d) at 36 mm was recommended vertical magnetic field distribution for 1 × 2 array coil model with coil Di = 45 mm, coil Do = 80 mm, and N = 23 turns.
The magnetic field distribution numerical calculation and evaluation was also conducted on other single and array coils with a variety of inner diameters, outer diameters, number of turns, and different array coil model patterns (2 × 2, 2 × 3, 2 × 6, 4 × 4).  The calculation and evaluation result for single coil (Coil_3.1) with Di = 30 mm and Do = 80 mm is described in Figure 12a. The vertical magnetic field peak (Bz max) = 1.222 mT, and 60% Bz max = 0.733 mT. The horizontal length of the vertical magnetic field effective transmitting area = 45 mm.
The vertical magnetic field (Bz) distribution calculation and characteristic of the 1 × 2 array coil pattern composed of Coil_3.1 is presented in Figure 12b. The evaluation was performed with variety of distance between the adjoining coil centers (d) at 40 mm, 41 mm, 42 mm, and 43 mm. The magnetic field threshold criteria = 0.733 mT. The vertical magnetic field distribution characteristic curve with the vertical magnetic field average (Bz Avg) = 0.731 mT, the horizontal length of the vertical magnetic field effective transmitting area = 85 mm, and the distance between the adjoining coil centers (d) at 42 mm was recommended vertical magnetic field distribution for 1x2 array coil model with coil Di = 30 mm, coil Do = 80 mm, and N = 38 turns. Figure 13a describes the calculation and evaluation result for single coil (Coil_3.2) with Di = 45 mm and Do = 80mm. The vertical magnetic field peak (Bz max) = 0.916 mT, and 60% Bz max = 0.549 mT. The horizontal length of the vertical magnetic field effective transmitting area = 50 mm.
While the vertical magnetic field (Bz) distribution calculation and characteristic of the 1 × 2 array coil pattern composed of Coil_3.2 is presented in Figure 13b. The evaluation was performed with a variety of distance between the adjoining coil centers (d) at 34 mm, 35 mm, 36 mm, and 37 mm. The magnetic field threshold criteria = 0.549 mT. The vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.567 mT, the horizontal length of the vertical magnetic field effective transmitting area = 90 mm, and the distance between the adjoining coil centers (d) at 36 mm was recommended vertical magnetic field distribution for 1 × 2 array coil model with coil Di = 45 mm, coil Do = 80 mm, and N = 23 turns. The magnetic field distribution numerical calculation and evaluation was also conducted on other single and array coils with a variety of inner diameters, outer diameters, number of turns, and different array coil model patterns (2 × 2, 2 × 3, 2 × 6, 4 × 4).

Discussion
The vertical magnetic field distribution and effective transmitting area of the evaluated single or array coils were examined following the evaluation criteria is described in Subsection 2.3. In the case of the coil which has the coil parameters as presented in Tables 1-3, the array coil models generate a

Discussion
The vertical magnetic field distribution and effective transmitting area of the evaluated single or array coils were examined following the evaluation criteria is described in Section 2.3. In the case of the coil which has the coil parameters as presented in Tables 1-3, the array coil models generate a wider horizontal transmitting area than the single coil along the y-axis direction. At the evaluated (recommended) distance between the adjoining coil centers (d), the array coil models also generate close to uniform vertical magnetic field distribution average at the level which the single coil generates a magnetic field at 60% of its magnetic field peak.
From the magnetic field distribution characteristic curve evaluation result, the distance between the adjoining coil centers (d) and outer coil diameter (Do) impact significant effect upon the evaluated array coil model vertical magnetic field distribution. The evaluation result is shown in Figures 7  and 9-13. The comparison evaluation result is summarized in Table 4. The evaluation and investigation of other single coils and array coil model patterns with different coil parameters also presented a similar result pattern to that shown in Table 4.
From Table 4, the distance between the adjoining coil centers (d) in the array coil model can be defined close to the half coil outer diameter (1/2Do) to generate close to uniform vertical magnetic field distribution.

Case Study
The circular flat spiral air-core coil with coil parameters as described in Table 5 was examined in order to generate uniform vertical magnetic field distribution in the form of array coil model. In this case study, the coil geometry was chosen with considering the coil usage in wireless power charger applications. The numerical calculation and evaluation results are presented in the following subsections.

Single Coil Evaluation
The characteristic curve of a single coil vertical magnetic field is shown in Figure 14a. The vertical magnetic field peak (Bz max) along the y-axis is 0.804 mT and the 60% vertical magnetic field peak is 0.482 mT. The 60% single coil vertical magnetic field peak is the proposed value to obtain the vertical magnetic field effective transmitting area to meet the uniform vertical magnetic field distribution. The horizontal length of the vertical magnetic field effective transmitting area = 45 mm. The overall single coil vertical magnetic field distribution is shown in Figure 14b. The magnetic field effective transmitting area is plotted on that distribution based on Figure 14a. In this case study, the coil geometry was chosen with considering the coil usage in wireless power charger applications. The numerical calculation and evaluation results are presented in the following subsections.

Single Coil Evaluation
The characteristic curve of a single coil vertical magnetic field is shown in Figure 14a. The vertical magnetic field peak (Bz max) along the y-axis is 0.804 mT and the 60% vertical magnetic field peak is 0.482 mT. The 60% single coil vertical magnetic field peak is the proposed value to obtain the vertical magnetic field effective transmitting area to meet the uniform vertical magnetic field distribution. The horizontal length of the vertical magnetic field effective transmitting area = 45 mm. The overall single coil vertical magnetic field distribution is shown in Figure 14b. The magnetic field effective transmitting area is plotted on that distribution based on Figure 14a.

The Two Coils Pattern Evaluation
The evaluation path in the y-axis direction of the 1 × 2 array coil pattern is shown in Figure 15.

The Two Coils Pattern Evaluation
The evaluation path in the y-axis direction of the 1 × 2 array coil pattern is shown in Figure 15. The distance between the adjoining coil centers (d) of the 1 × 2 array coil is arranged at 50 mm based on the single coil characteristic curve in Figure 14a. The 1 × 2 array coil vertical magnetic field characteristic curve is shown in Figure 16a, and the vertical magnetic field average (Bz Avg) resultant for this pattern along the y-axis is 0.291 mT.
The magnetic field evaluation of this pattern was performed using the difference in the distance between the adjoining coil centers (d) at 20 mm, 25 mm, 26 mm, 27 mm, 28 mm, and 30 mm respectively. The vertical magnetic field distribution numerical calculation results in the characteristic curve are shown in Figure 16b. The characteristic curve shows that the vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.494 mT, the horizontal length of the vertical magnetic field effective transmitting area = 57 mm, and the distance between the adjoining coil centers (d) at 26 mm was the recommended vertical magnetic field distribution for the 1 × 2 array coil model. The distance between the adjoining coil centers (d) of the 1 × 2 array coil is arranged at 50 mm based on the single coil characteristic curve in Figure 14a. The 1 × 2 array coil vertical magnetic field characteristic curve is shown in Figure 16a, and the vertical magnetic field average (Bz Avg) resultant for this pattern along the y-axis is 0.291 mT.
The magnetic field evaluation of this pattern was performed using the difference in the distance between the adjoining coil centers (d) at 20 mm, 25 mm, 26 mm, 27 mm, 28 mm, and 30 mm respectively. The vertical magnetic field distribution numerical calculation results in the characteristic curve are shown in Figure 16b. The characteristic curve shows that the vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.494 mT, the horizontal length of the vertical magnetic field effective transmitting area = 57 mm, and the distance between the adjoining coil centers (d) at 26 mm was the recommended vertical magnetic field distribution for the 1 × 2 array coil model. respectively. The vertical magnetic field distribution numerical calculation results in the characteristic curve are shown in Figure 16b. The characteristic curve shows that the vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.494 mT, the horizontal length of the vertical magnetic field effective transmitting area = 57 mm, and the distance between the adjoining coil centers (d) at 26 mm was the recommended vertical magnetic field distribution for the 1 × 2 array coil model.

The Three Coils Pattern Evaluation
The evaluation path in the y-axis direction of the 1 × 3 array coil pattern is shown in Figure 17. In this analysis, the distance between the adjoining coil centers (d) of the 1 × 3 array coil was arranged at 26 mm. This refers to the 1 × 2 array coil pattern which generates the vertical magnetic field average (Bz Avg) resultant more than magnetic field threshold criteria (0.482 mT). The 1 × 3 array coil vertical magnetic field characteristic curve is shown in Figure 18a and the vertical magnetic field average (Bz Avg) resultant for this pattern at x = 0 is 0.543 mT. This array coil model was also evaluated with the difference in the distance between the adjoining coil centers (d) at 26 mm, 28 mm, 30 mm, 31 mm, and 32 mm respectively. The vertical magnetic field distribution numerical calculation results in the characteristic curve as shown in Figure 18b.

The Three Coils Pattern Evaluation
The evaluation path in the y-axis direction of the 1 × 3 array coil pattern is shown in Figure 17. In this analysis, the distance between the adjoining coil centers (d) of the 1 × 3 array coil was arranged at 26 mm. This refers to the 1 × 2 array coil pattern which generates the vertical magnetic field average (Bz Avg) resultant more than magnetic field threshold criteria (0.482 mT). The 1 × 3 array coil vertical magnetic field characteristic curve is shown in Figure 18a and the vertical magnetic adjoining coil centers (d) at 26 mm, 28 mm, 30 mm, 31 mm, and 32 mm respectively. The vertical magnetic field distribution numerical calculation results in the characteristic curve as shown in Figure  18b.
The characteristic curve shows that the vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.493 mT, the horizontal length of the vertical magnetic field effective transmitting area = 90 mm, and the distance between the adjoining coil centers (d) at 30 mm was the recommended vertical magnetic field distribution for 1 × 3 array coil model.

The Four Coils Pattern
The evaluation path in the y-axis direction for the 2 × 2 array coil vertical magnetic field distribution is shown in Figure 19. The distance between the adjoining coil centers (d) of the 2 × 2 array coil is arranged at 26 mm. This refers to the 1 × 2 array coil pattern which generates the vertical magnetic field average (Bz Avg) resultant more than the magnetic field threshold criteria (0.482 mT). The 2 × 2 array coil vertical magnetic field characteristic curve is shown in Figure 20a, and the vertical magnetic field average (Bz Avg) resultant for this pattern is 0.599 mT.
This pattern evaluated with the difference in the distance between the adjoining coil centers (d) at 26 mm, 26.5 mm, 27 mm, 27.5 mm, and 28 mm respectively. The vertical magnetic field distribution numerical calculation results in the characteristic curve are shown in Figure 20b. The characteristic The characteristic curve shows that the vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.493 mT, the horizontal length of the vertical magnetic field effective transmitting area = 90 mm, and the distance between the adjoining coil centers (d) at 30 mm was the recommended vertical magnetic field distribution for 1 × 3 array coil model.

The Four Coils Pattern
The evaluation path in the y-axis direction for the 2 × 2 array coil vertical magnetic field distribution is shown in Figure 19.
The distance between the adjoining coil centers (d) of the 2 × 2 array coil is arranged at 26 mm. This refers to the 1 × 2 array coil pattern which generates the vertical magnetic field average (Bz Avg) resultant more than the magnetic field threshold criteria (0.482 mT). The 2 × 2 array coil vertical magnetic field characteristic curve is shown in Figure 20a, and the vertical magnetic field average (Bz Avg) resultant for this pattern is 0.599 mT.
This pattern evaluated with the difference in the distance between the adjoining coil centers (d) at 26 mm, 26.5 mm, 27 mm, 27.5 mm, and 28 mm respectively. The vertical magnetic field distribution numerical calculation results in the characteristic curve are shown in Figure 20b. The characteristic curve shows that the vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.487 mT, the horizontal length of the vertical magnetic field effective transmitting area = 52 mm, and the distance between the adjoining coil centers (d) at 27.5 mm was the recommended vertical magnetic field distribution for 2 × 2 array coil model. y-axis: (a) with the distance between the adjoining coil centers (d) at 26 mm, (b) with a variation in the distances between the adjoining coil centers.

The Four Coils Pattern
The evaluation path in the y-axis direction for the 2 × 2 array coil vertical magnetic field distribution is shown in Figure 19. The distance between the adjoining coil centers (d) of the 2 × 2 array coil is arranged at 26 mm. This refers to the 1 × 2 array coil pattern which generates the vertical magnetic field average (Bz Avg) resultant more than the magnetic field threshold criteria (0.482 mT). The 2 × 2 array coil vertical magnetic field characteristic curve is shown in Figure 20a, and the vertical magnetic field average (Bz Avg) resultant for this pattern is 0.599 mT.
This pattern evaluated with the difference in the distance between the adjoining coil centers (d) at 26 mm, 26.5 mm, 27 mm, 27.5 mm, and 28 mm respectively. The vertical magnetic field distribution numerical calculation results in the characteristic curve are shown in Figure 20b. The characteristic curve shows that the vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.487 mT, the horizontal length of the vertical magnetic field effective transmitting area = 52 mm, and the distance between the adjoining coil centers (d) at 27.5 mm was the recommended vertical magnetic field distribution for 2 × 2 array coil model.

The Six Coils Pattern
The evaluation path in the y-axis direction of the 2 × 3 array coil pattern is shown in Figure 21.

The Six Coils Pattern
The evaluation path in the y-axis direction of the 2 × 3 array coil pattern is shown in Figure 21. Based on the 1 × 2 array coil pattern, the distance between the adjoining coil centers (d) of this array coil pattern is arranged at 26 mm. The vertical magnetic field characteristic curve is shown in Figure 22a with the vertical magnetic field average (Bz Avg) resultant along the y-axis direction is 0.640 mT. This pattern evaluated with the difference in the distance between the adjoining coil centers (d) at 26 mm, 27 mm, 27.5 mm, and 28 mm respectively.
The numerical calculation results of vertical magnetic field distribution in the characteristic curve is shown in Figure 22b. The characteristic curve shows that the vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.506 mT, the horizontal length of the vertical magnetic field effective transmitting area = 79 mm, and the distance between the adjoining coil centers (d) at 27.5 mm is the recommend array coil.
(b) Figure 20. The characteristic curve of 2 × 2 array coil vertical magnetic field (Bz) distribution along the y-axis direction at the center of the distance between the adjoining coil centers (d): (a) with the distance between the adjoining coil centers at 26 mm, (b) with a variation in the distances between the adjoining coil centers.

The Six Coils Pattern
The evaluation path in the y-axis direction of the 2 × 3 array coil pattern is shown in Figure 21. Based on the 1 × 2 array coil pattern, the distance between the adjoining coil centers (d) of this array coil pattern is arranged at 26 mm. The vertical magnetic field characteristic curve is shown in Figure 22a with the vertical magnetic field average (Bz Avg) resultant along the y-axis direction is The numerical calculation results of vertical magnetic field distribution in the characteristic curve is shown in Figure 22b. The characteristic curve shows that the vertical magnetic field distribution with the vertical magnetic field average (Bz Avg) = 0.506 mT, the horizontal length of the vertical magnetic field effective transmitting area = 79 mm, and the distance between the adjoining coil centers (d) at 27.5 mm is the recommend array coil.

Discussion
According the result investigation in Section 3, the array coil model patterns were analyzed and evaluated with the variety of distance between the adjoining coil centers close to 24 mm (outer coil diameter is 47.8 mm).
The numerical analysis and evaluation results of 1 × 2, 1 × 3, 2 × 2, and 2 × 3 array coil model patterns are shown in Figures 16b, 18b, 20b, and 22b. Based on the analysis method presented in Subsections 2.2 and 2.3, only the array coil model vertical magnetic field distribution that meet the evaluation criteria are selected to provide close to uniform vertical magnetic field distribution. Therefore, with reference to the characteristic curves in Figures 16b, 18b, 20b, and 22b, the 1 × 2 array coil model with the distance between the adjoining coil centers (d) at 26 mm, the 1 × 3 array coil model with d = 30 mm, the 2 × 2 array coil model with d = 27.5 mm, and the 2 × 3 array coil model with d = Figure 22. The characteristic curve of 2 × 3 array coil vertical magnetic field distribution along the y-axis direction at the center of the distance between the adjoining coil centers: (a) with the distance between the adjoining coil centers (d) at 26 mm, (b) with a variation in the distances between the adjoining coil centers.

Discussion
According the result investigation in Section 3, the array coil model patterns were analyzed and evaluated with the variety of distance between the adjoining coil centers close to 24 mm (outer coil diameter is 47.8 mm).
The numerical analysis and evaluation results of 1 × 2, 1 × 3, 2 × 2, and 2 × 3 array coil model patterns are shown in Figure 16b, Figure 18b, Figure 20b, and Figure 22b. Based on the analysis method presented in Sections 2.2 and 2.3, only the array coil model vertical magnetic field distribution that meet the evaluation criteria are selected to provide close to uniform vertical magnetic field distribution. Therefore, with reference to the characteristic curves in Figure 16b, Figure 18b, Figure 20b, and Figure 22b, the 1 × 2 array coil model with the distance between the adjoining coil centers (d) at 26 mm, the 1 × 3 array coil model with d = 30 mm, the 2 × 2 array coil model with d = 27.5 mm, and the 2 × 3 array coil model with d = 27.5 mm are recommended array coil models that provide close to uniform vertical magnetic field distribution. Moreover, these array coil models also provide wider vertical magnetic field effective transmitting area than other array coil models with other the distance between the adjoining coil centers. The other vertical magnetic field distributions that also provide the vertical magnetic field average (Bz Avg) resultant more than the magnetic field threshold criteria are not considered because they have the vertical magnetic field effective transmitting area less than the vertical magnetic field effective transmitting area of the recommended array coil model.
The overall recommended vertical magnetic field distribution of array coil pattern models for the case study is plotted in Figure 23 including their magnetic field effective transmitting area plotting. The magnetic field effective transmitting area is plotted based on the magnetic field distribution numerical evaluation results as shown in Figure 16b, Figure 18b, Figure 20b, and Figure 22b.
Single circular flat spiral air-core array coil with the proper distance between the adjoining coil centers can provide close to uniform vertical magnetic field distribution and the wider magnetic field effective transmitting area. 27.5 mm are recommended array coil models that provide close to uniform vertical magnetic field distribution. Moreover, these array coil models also provide wider vertical magnetic field effective transmitting area than other array coil models with other the distance between the adjoining coil centers. The other vertical magnetic field distributions that also provide the vertical magnetic field average (Bz Avg) resultant more than the magnetic field threshold criteria are not considered because they have the vertical magnetic field effective transmitting area less than the vertical magnetic field effective transmitting area of the recommended array coil model. The overall recommended vertical magnetic field distribution of array coil pattern models for the case study is plotted in Figure 23 including their magnetic field effective transmitting area plotting. The magnetic field effective transmitting area is plotted based on the magnetic field distribution numerical evaluation results as shown in Figures 16b, 18b, 20b, and 22b. Single circular flat spiral air-core array coil with the proper distance between the adjoining coil centers can provide close to uniform vertical magnetic field distribution and the wider magnetic field effective transmitting area.

Conclusions
With the proposed magnetic field calculation and evaluation, the characteristic array coil pattern model vertical magnetic field distribution can be observed and investigated to generate close to uniform vertical magnetic field distribution.
The vertical magnetic field average (Bz Avg) resultant and vertical magnetic field effective transmitting area of array coil model with given coil parameters (inner coil diameter (Di), outer coil diameter (Do), wire diameter (W), pitch (P) and a number of turns (N)) changing as the effect in variation in distances between the adjoining coil centers. The proper distance between the adjoining coil centers can provide uniform vertical magnetic field distribution and the wider magnetic field effective transmitting area. The distance between the adjoining coil centers should be defined close to the half coil outer diameter (1/2Do) to generate close to uniform vertical magnetic field distribution.
This evaluation method provides an easier and faster the vertical magnetic field distribution evaluation for single-layer circular flat spiral air-core n × m array coil pattern models. Moreover, it can be extended with a verifying step in practical implementation.