### I. Introduction

*h*

_{2}) in polarized coils is significantly higher than non-polarized coils (

*h*

_{1}) for a similar size. As a result, the coupling coefficient is higher in the polarized coils and is also more robust to misalignment in the horizontal direction [1], which makes the coils more appropriate for their application in electric vehicle charging in both SWC and DWC methods. Among polarized coils, the double-D (DD) design is preferred due to its simple structure, high efficiency, and low sensitivity in misalignment conditions [5]. To optimize a WPT system or estimate its operating parameters (such as received power, efficiency, and gain), the designer should accurately compute the self- and mutual inductances [6]. These computations could be carried out using analytical, numerical (finite element method [FEM]) [7], or hybrid analytical-numerical methods. The analytical method has remarkable advantages over the other methods due to its time-saving analysis and cost effectiveness. Though many studies have been carried out to compute self- and mutual inductances of WPT system coils for circular [8–11] and square structures [12–14], few studies have been published regarding rectangular [15, 16] and DD power coils [17]. An analytical method to compute self- and mutual inductances of spiral rectangular coils is presented in [18]. However, it is rather complicated due to its dependence on computers and the need to set initial burdensome values [6]. A numerical-analytical method based on infinite integral, infinite Bessel series, and trigonometric functions is proposed in [19] to calculate the mutual inductance between two rectangular coils, which must be resolved numerically [20]. An analytical method to calculate inductance for multi-loop spiral coils is proposed in [21] based on Grover equations [22], which for each piece of conductor, the self- and mutual inductances must be computed and added together. Applying this method to compute the inductance of multi-loop coils is time-consuming, even with fast computers [16]. In [17], the Grover-based method is used to calculate self-inductance, and the Neumann-based method is employed for mutual inductance. This method is not ideal for the computation of DD power coils in terms of computational errors. Also, since no experimental setup has been implemented, the analytical results were not validated by any experimental results.

### II. Analytical Modeling of DD Coils

### 1. Computation of Self-inductance

*w*is the wire diameter and

*s*is the distance between two adjacent wires. Variables in Fig. 2(d), including the length and width of the loops, are

*a*,

*b*. The dimensions of the

*i*

^{th}loop, including

*a*

*and*

_{i}*b*

*can be obtained by*

_{i}*i*= 1 is the outermost loop.

*i*

^{th}loop is obtained according to

*B*

*is the field density, and*

_{i}*S*

*is the cross-section of*

_{i}*i*

^{th}loop. According to right-hand law, the field density is perpendicular to the plane. Therefore, given the perpendicularity of the unit vector on the

*yz*plane, according to Fig. 2(c), the internal multiplication is eliminated. Moreover, since, in this case, the largest physical dimension of coil a, relative to wavelength λ, is electrically very small (i.e.,

*a*≪ λ), the frequency effect of the source could be neglected [23]. Hence, Eq. (2) can be used to compute the self- and mutual inductances.

*C*

*, in Fig. 2(d), is obtained using the Biot-Savart law according to (3). In the following, Eqs. (3)–(6) are applied to calculate the flux density and flux caused by conductor*

_{I}*C*

*, as well as the total flux of the loop, respectively [24].*

_{I}*R*is the distance between the element and the point at which we are computing the magnetic field

*B*, which can be defined as

*Z*is the distance of the given point to the

*xy*plane,

*r*is the perpendicular distance of the given point to the

*z*axis,

*μ*

_{0}is the vacuum permeability coefficient, and

*I*is the loop current, the value of which is considered a unit. Loop flux, caused by

*C*

*conductor current in Fig. 2(c), is computed by*

_{I}*C*

*,*

_{I}*C*

*,*

_{II}*C*

*,*

_{III}*C*

*), flux caused by all four conductors should be added. Thus, the total flux of loop*

_{IV}*i*caused by the current of the loop itself, is as follows:

##### (6)

*Φ*

_{i}_{−}

*=*

_{i}*L*

*. In the following self-inductance computation of DD coils, the mutual inductance of the loops of sub-coil 1 together, the mutual inductance of the loops of sub-coil 2 together, and the mutual inductance between the loops of sub-coils 1 and 2 together should be obtained and added. Given that the coils of WPT systems are wound with Litz wire and normally operate at a working frequency of less than 100 kHz, the skin effect is negligible. Consequently, the filamentary is logical for the Litz wire [14]. Fig. 3 shows loop*

_{i}*i*as part of sub-coil 1 through which current

*I*

_{1}flows, and the objective is to compute mutual inductance in loop

*j*or

*h*. According to the DD coil structure, if the other loop is part of sub-coil 2 (loop

*h*), it would be on the right side of loop

*i*(right-side dashed-line loop), in which case

*S*

*is negative and*

_{a}*S*

*is positive. If the other loop (loop*

_{b}*j*) is part of sub-coil 1, it would be inside loop

*i*(in-loop

*i*dashed-line loop), in which case

*S*

*and*

_{a}*S*

*are negative.*

_{b}*h*or

*j*, the flux enclosed to loops

*h*or

*j*should be divided on current

*I*

_{1}(as part of loop

*i*):

*Φ*

_{i}_{−}

_{j}_{,}

*is the enclosed flux to loops*

_{h}*h*or

*j*, and

*I*

_{1}is the current of loop

*i*. The enclosed flux to loops

*h*or

*j*is obtained according to the following equation:

*S*

_{j}_{,}

*is the plane of loop*

_{h}*j*or

*h*, and

*B*

_{i}_{−}

_{j}_{,}

*is the field density. According to the right-hand law, the field density is perpendicular to the plane. As a result, internal multiplication could be eliminated. The formula for field density is obtained through the Biot-Savart law, which is similar to Eq. (3). The difference is that it is shifted toward the positive part of the*

_{h}*z-*axis (Fig. 3) as much as

*h*or

*j*, caused by the current of conductor

*C*

*, is brought below. In the following, Eqs. (9) and (10) are applied to compute the flux of loops*

_{I}*h*or

*j*, caused by the current of conductor

*C*

*and loop*

_{I}*i*[24].

##### (9a)

*C*

*and*

_{I}*C*

*create imported flux, and conductors*

_{IV}*C*

*and*

_{II}*C*

*create exported flux on the plane of loops*

_{III}*h*or

*j*. Therefore, the mutual inductance equation per four inductors is as follows:

##### (10)

*L*

_{sc}_{1}, inductance of a DD coil is obtained and is represented by

*L*

*.*

_{DD}*L*

*self-inductance of a DD coil is analytically obtained as*

_{DD}*L*

_{sc}_{1},

*M*

_{sc}_{1}, and

*M*

_{sc}_{1,2}are the self-inductance of sub-coil 1, the mutual inductance between loops of sub-coil 1, and the mutual inductance between loops of sub-coils 1 and 2. Using the results of Section IV, it can be inferred that section

*M*

_{sc}_{1,2}of self-inductance calculations can be neglected due to its low effect on the intended response and to reduce the complexity of the calculations. Therefore, Eq. (11a) can be simplified as follows:

*μ*

*in Eqs. (6) and (10) [9, 13].*

_{r}*i*,

*j*, and

*n*are symbols of loop numbers of sub-coil 1. The sum of the self-inductance of sub-coil 1 is named sum 1, and the mutual inductance between loops of sub-coil 1 is named sum 2. The total sum of sum 1 and sum 2 composes the self-inductance of sub-coil 1 and is presented by

*L*

_{sc}_{1}in the relevant flowchart. Sub-coils 1 and 2 are similar; thus, by doubling

*L*

_{sc}_{1}, the inductance of a DD coil is obtained and is represented by

*L*

*.*

_{DD}### 2. Computation of Mutual Inductance

*M*

_{sc}_{1,3},

*M*

_{sc}_{2,4}, and

*M*

_{sc}_{1,4},

*M*

_{sc}_{2,3}are mutual inductances of the facing sub-coils and mutual inductances of non-facing sub-coils in DD coils, respectively. According to the investigation presented in Section IV, since the mutual inductance value is insignificant in non-facing sub-coils, they are neglected to avoid computational complications. Hence, Eq. (12) is simplified as

*i*

^{th}single-loop (transmitter) and

*u*

^{th}single-loop (receiver), according to Fig. 5(b), is obtained from the division of flux of

*u*

^{th}loop to the current of

*i*

^{th}loop as

*u*

^{th}loop is calculated from

*B*

*is the field density caused by the conductor*

_{iu}*C*

*, and*

_{II}*S*

*is the plane of*

_{u}*u*

^{th}loop. Also, the point here represents internal multiplication. θ is the angle between the unit vector of the plane of

*u*

^{th}loop and the vector of

*B*

*field density (Fig. 5(b)). According to the Biot-Savart law, the equation of field density along the*

_{iu}*z*-axis is as below. Eqs. (16) and (17) are the flux density and flux (or mutual inductance) of conductor

*C*

*[20].*

_{II}*C*

*along the*

_{II}*z*-axis, which is the effective component for computing mutual flux, is obtained by

##### (17)

*a*

_{1},

*b*

_{1},

*a*

_{2}, and

*b*

_{2}are the dimensions of the transmitter’s and receiver’s loops, and

*z*is the distance between two loops along the

*z*-axis. Given the conductors’ symmetry,

*Φ*

_{C}_{I}=

*Φ*

_{C}_{III}and

*Φ*

_{C}_{II}=

*Φ*

_{C}_{IV}are equal. To obtain the magnetic field of the conductor

*Φ*

_{C}_{II}in Eq. (17)

*a*

_{1}and

*a*

_{2}should be replaced with

*b*

_{1}with

*b*

_{2}, respectively. According to right-hand law, the general formula of the magnetic field is equal to

*Φ*

*=*

_{iu}*Φ*

_{C}_{I}+

*Φ*

_{C}_{II}+

*Φ*

_{C}_{III}+

*Φ*

_{C}_{IV}. The general mutual inductance is obtained as

*i*and

*n*are loops of sub-coil 1;

*h*and

*m*are loops of sub-coil 2;

*u*and

*v*are loops of sub-coil 3; and

*w*and

*x*are loops of sub-coil 4. This formula holds true when ferromagnetic materials are not used. The proposed model can be easily developed for this case by inserting a scaling factor for

*μ*

*in Eqs. (18) [9, 13].*

_{r}### III. Finite Element Modelling of DD Coils

*L*

*,*

_{DDT}*L*

*,*

_{DDR}*M*, and

*k*. The model is designed based on the parameters given in Table 1.

*L*

*,*

_{DDT}*L*

*, and*

_{DDR}*M*, are 0.25%, 0.5%, and 0.77%, respectively.

### 1. Coupling Coefficient Optimization

*k*. Also, at a distance of z > 150 mm, the transmitter loop with 600 × 300 mm dimensions has the highest

*k*.

### IV. Experimental Results and Coupling Coefficient Optimization

### 1. Self-inductance Measurement

*M*

_{sc}_{1,2}or

*M*

_{sc}_{3,4}) proves that the occurred errors are very small, such that the maximum result error is 4.38% (Fig. 9(b)). According to the analytical results marked by a star in Fig. 9, it could be observed that by neglecting mutual inductance between sub-coils 1 and 2 in the transmitter coil (

*M*

_{sc}_{1,2}) or between sub-coils 3 and 4 in the receiver coil (

*M*

_{sc}_{3,4}) for the self-inductance calculation (for simplification), the maximum error between the analytical-experimental results is 6.08% and between the analytical-simulation is 7.93%, which is acceptable.

*M*

*. To measure mutual inductance, one can act as shown in Fig. 8(b).*

_{DD}*M*

*is caused by an approximation of the neglected mutual inductance in non-facing sub-coils, error measurement, and errors caused by considering a spiral coil in several concentric filamentary single-loop forms.*

_{DD}### 2. Comparison of the Simulation and Proposed Analytical Model

*M*

_{sc}_{1,2}and

*M*

_{sc}_{3,4}, the number of iterations of the equations (calculate

*L*

*) will decrease from 484 to 242; as a result, the computation time will be reduced to 0.13 seconds.*

_{DDT}### V. Conclusion

*M*

_{sc}_{1,2}and

*M*

_{sc}_{3,4}in the calculation of self-inductances and

*M*

_{sc}_{1,4}and

*M*

_{sc}_{2,3}in the mutual inductance calculation reduces the complexity of analytical calculations but does not considerably affect the proposed method accuracy. The optimization of the coupling coefficient on the transmitter coil was performed in applications where the receiver coil had a size or weight limit. Table 3 also proves the accuracy of the results of the proposed method compared to the existing literature. An analytical model could be applied to design and optimize the DD coils of a WPT system; nonetheless, it does not have the problems of 3D-FEM, such as being time-consuming and costly. The proposed method could be extended to calculate self- and mutual inductance for the computation of various chargers’ coils, such as DD, DDQ, BP, QD, and extended DD, which are appropriate for DWC application.