#### 3.1. WPT Equivalent Circuit

The WPT technology allows to transfer electrical energy by means of magnetic resonant coupling between two coils that act as a loosely coupled transformer. The complete electrical circuit of a WPT system is reported in

Figure 4. The two coupled coils are characterized by self-inductances

L_{1} and

L_{2}, mutual inductance

M and self-resistances

R_{1} and

R_{2} that model the losses in the coils [

20]. To reduce the AC losses due to skin and proximity effects the coils are made by a copper Litz wire. The coupling factor

k is given by:

Capacitance compensation networks are added on transmitting and receiving sides to obtain resonance condition [

21] and to improve the electrical performances of the system. There are several compensation network configurations that can be used in a WPT system. The most commonly used are the capacitive Series-Series (SS) and Series-Parallel (SP) topologies.

In real applications, the source and the load are composed by complex electronic blocks. At the transmitting side, a full bridge inverter made by MOSFETs (

M_{1},

M_{2},

M_{3},

M_{4}) is adopted to generate a high frequency square wave voltage

V_{1} from the DC source

V_{IN}. At the receiving side the high frequency voltage is firstly rectified by a full bridge rectifier, composed by diodes

D_{1},

D_{2},

D_{3},

D_{4}. The DC output voltage is filtered by a low pass filter composed by a shunt capacitance

C_{0} and a series inductance

L_{0}. Finally, it is connected to the battery. However, to extract the lumped parameters of the system, a simplified circuit is adopted as shown in

Figure 5. The primary circuit can be modeled by a simple sinusoidal voltage source

V_{g} with a very small internal resistance

R_{g}. In addition, the load can be simplified and modeled by an equivalent resistance

R_{L} [

9].

The lumped inductances (

L_{1},

L_{2} and

M) are numerically extracted at the frequency of interest solving the magneto quasi-static (MQS) field equations by the commercial software COMSOL [

22], while the AC resistance of the Litz wire is obtained from wire datasheet [

23]. After the evaluation of the circuit lumped parameters, the compensation capacitors

C_{1} and

C_{2} can be obtained at the given resonance frequency for the selected compensation topology [

20]. The resonant frequency of 150 kHz [

14] is here adopted for this kind of application since it is commonly used for the medium power application (e.g., mobile device). This frequency allows the transfer of an adequate amount of power with good efficiency at short range.

Higher operational frequency should permit an improvement of the power transfer, but it could require a more complex electronic system and could generate electromagnetic interference (EMI) issues [

24,

25].

The electrical performances of the WPT system, i.e., efficiency and transferred power, are obtained by the analysis of the equivalent simplified circuit. The transferred power

P_{L} = R_{L}|I_{2}|

^{2} is the power dissipated on the load resistance

R_{L}, while the efficiency

η =

P_{L}/

P_{1} is calculated as the ratio between the output real power

P_{L} and the input real power at port 1-1′ [

20].

In order to choose the configuration with the best weight to performance ratio, a comparison between SS and SP compensation topologies is carried out. Two planar stacked coils of rectangular shaped are considered. The

Tx coil has outer dimensions

L_{p} = 400 mm,

W_{p} = 150 mm while the

Rx coil has outer dimensions

L_{s} = 20 mm,

W_{s} = 50 mm. The intraturn spacing is negligible. The separation distance between parallel coils is fixed at

D = 10 mm. The Litz wire used for both

Tx and

Rx coils has an external diameter of

d_{wire} = 3 mm. The load is modeled by a resistor

R_{L} = 3 Ω. The efficiency is then calculated for the two examined compensation topologies (SS and SP) varying the number of turns,

N_{1} and

N_{2}, of the

Tx and

Rx coils, respectively, while keeping fixed the outer dimensions of the coils. The obtained results, reported in

Figure 6, show that the maximum efficiency of the SP compensation is similar to that of the SS compensation when assuming

N_{1} = 8, but with a different number of turns of the secondary coil. For the SP configuration, the maximum efficiency is obtained for

N_{2} = 2 while for the SS compensation it is obtained for

N_{2} = 8. It is therefore evident the great advantage of the SP compensation topology in respect of the SS compensation topology for the minor number of the secondary turns.

This configuration leads also to a considerable weight reduction of the on-board components. The efficiency and weight of the secondary coil vs.

N_{2} for

N_{1} = 8 are shown in

Figure 7 demonstrating that the SP compensation is better than the SS compensation for the considered application. Thus, the SP compensation topology is adopted and applied to a WPT system for a commercial drone, as described in the following.

#### 3.2. Coil Design

The goal of the primary coil array design is to ensure an adequate efficiency of the WPT system at any point in the landing area, and at the same time to minimize the number of the independent primary coils in the array. It means that the WPT system efficiency

η must be greater than a preset value

η_{min} at any point of the ground pad where the drone can land. The dimensions and number of the primary coils are obtained by the procedure schematized in

Figure 8. The procedure starts by fixing the system requirements in terms of electrical and geometrical constraints. The electrical constraints are the output voltage and power, and the resonant frequency. The geometrical constraints are the dimensions, the shape and the weight of the on-board secondary coil (they are set in advance to fit the landing pad, to guarantee mechanical robustness and aerodynamics, and to avoid any obstruction to the vision of on-board sensors or cameras). Another geometrical constraint is the size of the square landing area with side length

L where the array of primary coils is installed.

The secondary coil of small size and narrow rectangular shape is placed on one skid of the landing gear. Since the autopilot of the drone can ensure a near perfect angular alignment during the landing, it is possible to orient the drone in such way that the landing skids are always parallel to one long side of the square landing pad.

The battery charging area is smaller than the landing area. Indeed, the small receiving coil will never be positioned in the area covered by the drone (i.e., area between the two skids and depicted as dotted blue area in

Figure 2), otherwise part of the landing gear should be outside the landing area and this is clearly not acceptable. In conclusion, the charging area has a size smaller than the landing area and the narrow rectangular secondary coil is always well oriented. The variables of the optimization procedure are the primary coil dimensions

W_{p} and

L_{p}, the overlapping distance between adjacent coils

O_{W} and

O_{L}, and the array dimension

N_{Cx} and

N_{Cy} of the primary coils. The rectangular primary coils are assumed to be identical to each other. The ratio of the coil sides

L_{p}/

W_{p} is kept fixed to permit an adequate coupling with the secondary coil. In addition, the ratio

O_{L}/

O_{W} and the number of turns

N_{1} and

N_{2} are considered fixed. Only the misalignments along the

x- and

y-directions are variable in the proposed application, while the vertical separation of the coils along

z (i.e., air gap) is kept fixed. It should be noted that the intraturn spacing could also be a design variable. However, for the considered frequency, the gain in terms of efficiency varying the spacing is very small due to the use of Litz wire, thus for the sake of simplicity it is considered fixed in this procedure [

9].

The WPT efficiency

η is calculated for any possible landing point by the analysis of the equivalent circuit, whose lumped parameters are extracted numerically for any mutual position between primary and secondary coils [

26]. The algorithm is based on the iterative reduction of the single primary coil dimensions and the increase of the overlapping distance. Smaller dimensions of the primary coils permit to reduce the flux leakage improving the coupling factor with the secondary coil (and then the efficiency). The optimization of the overlapping distances

O_{L} and

O_{W} is important to obtain an efficiency

η >

η_{min} also at the points where two or more coils are overlapped.

The procedure starts considering as initial parameters:

If the calculated minimum efficiency is lower than a preset value,

η >

η_{min}, the overlapping distance

O_{L} is increased and the procedure is iterated. If it reaches the maximum possible value

O_{Lmax} (

O_{L} >

O_{Lmax}), the algorithm restarts decreasing the coil dimension

L_{p}. The procedure is repeated until the calculated efficiency is higher than

η_{min} at any point inside the charging area. At the end of the procedure the optimum primary coil dimensions

L_{p}_{_opt} and

W_{p}_{_opt}. are obtained, as well as the optimum overlapping factors

O_{L_opt} and

O_{W_opt}. The numbers of the primary coils

N_{Cx} in

x-direction and

N_{Cy} in

y- direction are finally calculated to entirely cover the charging area by:

where

$\lfloor \hspace{0.17em}\rfloor $ represents the greatest integer floor. Thus, the overall dimensions of the landing pad are found as as (

L_{p_opt} −

O_{L_opt})

N_{Cx} × (

W_{p_opt} −

O_{W_opt})

N_{Cy} If this size is much larger than the preset value

L ×

L, all the dimensions can be resized, otherwise they remain unaltered.