*2.2.2 Near-field WPT system in mid-range*

When the distance between transmitter and receiver is smaller than the geometry of the transmitter, is indicated as a small-range WPT. As a practical rule, the mid-range refers to the situation when the gap is 2 to 3 times the size of either device involved in the power transfer.

Two coil systems are used for charging both portable and heavy power devices like powerbanks. An optimal alignment has the greatest coupling co-efficient where the coils are the same size and parallel to each other. The mutual inductance


#### **Table 1.**

*Comparison between the capacitive power transfer and inductive power transfer.*

declines as the ratio of the two coils' primary magnetic field decreases, particularly when there is a broad separation between the two coils.

Multiple coils in the transmitter, receiver, or in the middle are adopted essentially for two main reasons: (a) more degrees of freedom to maximise the efficiency and desensitise the link gain versus coupling factor; (b) highly coupled transmitterrepeater or repeater-receiver link work greatly as impedance matching elements at both sides. Although this last configuration requires four or more coils, it offers a better efficiency-distance than a three coils system [14]. For this reason, the three coil WPT is not very popular, unless the application has no space for additional coils.

Let us consider a four-coil resonator system with two intermediate repeaters coils called "2" and "3" where an impedance (capacitor) compensation *Z*<sup>2</sup> and *Z*<sup>2</sup> are connected to form LC resonators. As shown in **Figure 7**, the transmitter and receiver are referred as "1" and "4" respectively. It has been considered the transmitter *RT* and the load impedance *ZLoad* having relatively low quality factor of *QT* ¼ *Q*<sup>1</sup> and *QR* ¼ *Q*4. Considering only the parasitic resistance, much higher quality factor *Q*<sup>2</sup> and *Q*<sup>3</sup> can be achieved. With this new nomination, *k*<sup>23</sup> will be much lower than *k*<sup>12</sup> and *k*<sup>34</sup> because the distance between the intermediate coils are usually larger than the geometry of the coils. In this way the cross coupling effect could be neglected because of either the low quality factor Q and the small coupling coefficient depicted in the **Figure 7** in yellowish green. Similar to the two-coil system, the figure of merit could be written as a generic Δ*i j* for any two of the four coils:

$$
\Delta\_{ij} = \kappa\_{ij}^2 Q\_i Q\_j \tag{7}
$$

calling *i* and *j* the number of the referred coils. An important equation to notice in design of a multi-coils system comes from the impedance reflected from the all coils to the primary transmitter. Considering the Eq. (4) introduced in a two coil system, it is possible to write for each coil the reflected impedance:

$$\begin{cases} Z\_{\text{ref},3} = \frac{\alpha^2 k\_{34}^2 L\_3 L\_4}{Z\_4 + Z\_{Load}} \\\\ Z\_{\text{ref},2} = \frac{\alpha^2 k\_{23}^2 L\_2 L\_3}{Z\_3} \\\\ Z\_{\text{ref},1} = \frac{\alpha^2 k\_{12}^2 L\_1 L\_2}{Z\_2} \end{cases} \tag{8}$$

#### **Figure 7.**

*Four coils WPT system with the coupling factors. The couplings are marked following their value.κ*13*, κ*<sup>14</sup> *and κ*<sup>24</sup> *are not visible because their intensity values are negligible.*

Combining these equations in the Eq. (8)c, it is possible to obtain the impedance reflected in the primary transmitter:

$$Z\_{r\!f^{1},1} = \frac{\alpha^2 k\_{12}^2 L\_1 L\_2}{\frac{\alpha^2 k\_{34}^2 L\_2 L\_3}{\frac{\alpha^2 k\_{34}^2 L\_3 L\_4}{Z\_4 + Z\_{\text{Load}}} + Z\_3}} + Z\_2 \tag{9}$$

where simplifying we obtain:

$$Z\_{r\text{cf},1} = \frac{\alpha^2 \left(\frac{k\_{12}k\_{34}}{k\_{23}}\right)^2 L\_1 L\_4}{Z\_4 + Z\_{\text{Load}}} \quad = \frac{\alpha^2 k\_{\text{TOT}}^2 L\_1 L\_4}{Z\_4 + Z\_{\text{Load}}} \tag{10}$$

In this equation we can notice that the reflected impedance of the all system depends directly only by the total coupling coefficient and the value of receiver impedance. Moreover, the WPT system can be seen as an equivalent total coupling coefficient defined by:

$$k\_{TOT} = \frac{k\_{12}k\_{34}}{k\_{23}}\tag{11}$$

It is a design rule making sure that the following condition can be met:

$$k\_{TOT} = \frac{k\_{12}k\_{34}}{k\_{23}} = 1\tag{12}$$

the reflected load will be matched and we will have the maximum power transferred. In such a way, the four coils system creates a possibility to extend the distance from primary to the load using more and more coils. In order to maximise the transmission distance, the mutual coupling coefficient between the repeaters could be minimised. Additional intermediate coils with still be loose coupled between them but they will increase the total coupling coefficient of the system. For example, even if the coefficient *κ*<sup>23</sup> between the intermediate coils is loosely coupled to 0.01 because of the long transmission distance, the equivalent coupling coefficient *κTOT* of the whole system can still be adjusted to 1 when both *κ*<sup>12</sup> and *κ*<sup>34</sup> are considered strongly coupled set to 0.1.

However, the impedance matching such a system is not endowed with a high overall efficiency because it is restricted by the merit factor given by the Eq. (9) Nonetheless, the four-coil system still offers (in terms of efficiency-distance) a better solution rather the two-coil systems when the distance is much bigger than the coil size.
