**1. Introduction**

Near-field wireless power transfer (WPT) represents a promising solution for wirelessly providing power to electronic devices. Two main technologies exist: inductive and capacitive WPT, based on resonant magnetic or electric coupling, respectively. The simplest setup consists of two resonators: a transmitting resonator, powered by an input supply, which transfers power wirelessly to a receiving resonator connected to the load (SISO: single-input single-output).

For certain applications, WPT from multiple transmitters to a single receiver (MISO: multiple-input single-output) can be beneficial compared to the single-transmitter configuration:


In this work, focus will lie on capacitive power transfer (CPT), which utilizes a high-frequency electric field as a medium to transfer energy wirelessly. Main advantages compared to inductive WPT are low-weight and cost, a minimal eddy-current loss, and a larger robustness against misalignment. A typical CPT coupler consists of four metal plates: two plates at the transmitter side, and two at the receiver side, resulting in a return path for the current [1]. CPT has been demonstrated in low-power applications such as portable electronics [2,3], integrated circuits [4], drones [5], and biomedical devices [6,7]. However, also at higher power levels, up to several kW [8], CPT can be applied—e.g., in automatic guided vehicles [9] and electric vehicles [10–12].

Maximizing the efficiency of an *inductive* WPT system with multiple transmitters has already been solved, e.g., [13–18]. Additionally, for the general WPT system, this problem has been solved by reducing the entire system to an impedance matrix of a multiport network [19–21]. However, this methodology loses the internal structure of the WPT system, e.g., the coupling strengths between transmitters and receiver. Moreover, a CPT system can be more easily described by its admittance matrix instead of its impedance matrix.

Efficiency maximization for CPT was already solved for a single transmitter with multiple receivers (SIMO: single-input multiple-output) [22,23], but to date, an analysis specifically for a CPT system with multiple (coupled) transmitters (MISO) is lacking.

In this work, a CPT transfer system with *any* number of transmitters and a single receiver is considered. Varying the receiver's loads (e.g., via impedance matching) and/or the input currents results in different values for the power-transfer efficiency (also called power gain) of the CPT system. In this work, the load and input currents that maximize the power-transfer efficiency are determined while taking into account, among others, the coupling strengths between transmitters and receiver. More specifically, the contributions are the following:


#### **2. Problem Description**

Figure 1 depicts a CPT system with *N* transmitters (on the bottom, subscripts 1 to *N*) and a single receiver (on top, subscript 0). The transmitters are powered by a power supply control circuit. Each transmitter can operate at a different voltage and phase, but the operating angular frequency *ω*0 is the

same for all transmitters. The resistive and reactive components within each transmitter are described by the conductances *gnn* and susceptances *bnn* (*n* = 1, ... , *N*), respectively. In the remainder of this work, the subscript *n* always counts from 1 to *N*.

**Figure 1.** A general capacitive wireless power transfer system with *N* transmitters (bottom) and a single receiver (top). The (desired) electric couplings between transmitters and receiver are depicted by the full arrows. The undesired cross-couplings between the transmitters themselves are indicated by the dashed arrows.

Energy is transferred wirelessly to the load of the receiver, represented by the admittance *YL*0 (including a possible compensation circuit). The conductance *g*00 and susceptance *b*00 correspond to the resistive and reactive part of the receiving resonator, respectively.

The strength of the electric coupling between each transmitter and the receiver is given by the coupling factor *k*0*<sup>n</sup>*, a dimensionless number which can vary from zero (no coupling) to unity (maximum coupling). In a practical CPT system, the electric coupling between the transmitters and receiver is desired to realize wireless power transmission. However, an undesired (nonzero) electric cross-coupling can be present between the transmitters themselves, represented by *knm* (*<sup>n</sup>*, *m* = 1, . . . , *N*; *n* = *m*). The coupling factor is defined as [24,25]

$$k\_{ij} = \frac{\mathbf{C}\_{ij}}{\sqrt{\mathbf{C}\_i \mathbf{C}\_j}},\tag{1}$$

for *i*, *j* = 0, ... , *N*; *i* = *j*, where *Cn* is the transmitter capacitance of the *n*-th transmitter, *C*0 is the receiver capacitance, and *Cij* is the mutual capacitance, corresponding to the electric coupling. Note that *C*0 and *Cn* do not correspond to the capacitance between the physical transmitter and receiver plate, but to an equivalent circuit representation of electric coupling [25]. The measurement procedure to determine the value of these capacitances is described in [24].

The CPT system can be considered as a multiport with *N* input ports (the *N* transmitters) and one output port (the receiver). The multiport is indicated by the dashed rectangle in Figure 1. Notice that this (*N* + 1)-port network is linear and reciprocal due to the passive components it is constructed from. The currents through and voltages at the (*N* + 1) ports are given by the peak current phasors *Ij* and peak voltage phasors *Vj*, as defined in Figure 1 (*j* = 0, . . . , *N*).

The problem description is the following: given the network of Figure 1 (with given and fixed values for the components of the CPT network and coupling factors), determine the values of the load admittance *YL*0 and input currents *In* that maximize the power-transfer efficiency *η*. The problem can be reduced to finding the current and voltage phasors at the ports, corresponding to the optimal efficiency configuration. Therefore, any remote electronics external to the wireless link (e.g., rectifiers, matching networks, actual passive loads,. . . ) can be ignored, since it can be taken into account once the optimal current–voltage relationship at the ports are determined.

#### **3. Power and Efficiency of the CPT System**

First, the (normalized) admittance matrix, input power, output power, and efficiency will be expressed as functions of the characteristics of the network. The efficiency is not ye<sup>t</sup> maximized in this section.

#### *3.1. Admittance Matrix*

The CPT system can be fully characterized by its admittance matrix *Y*. The admittance matrix *Y* describes the relation between the port currents and port voltages:

$$I = \mathbf{Y} \cdot \mathbf{V}\_{\prime} \tag{2}$$

with the (*N* + 1)×1 matrices *V* and *I* defined as

$$\mathbf{V} = \begin{bmatrix} V\_1 \\ V\_2 \\ \vdots \\ V\_N \\ V\_0 \end{bmatrix}, I = \begin{bmatrix} I\_1 \\ I\_2 \\ \vdots \\ I\_N \\ I\_0 \end{bmatrix}. \tag{3}$$

The admittance matrix *Y* of the CPT system can be written as [25]

$$\mathbf{Y} = \begin{bmatrix} g\_{11} - jb\_{11} & -jb\_{12} & \dots & -jb\_{1N} & -jb\_{10} \\ -jb\_{21} & g\_{22} - jb\_{22} & \dots & -jb\_{2N} & -jb\_{20} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ -jb\_{N1} & -jb\_{N2} & \dots & g\_{NN} - jb\_{NN} & -jb\_{N0} \\ -jb\_{01} & -jb\_{02} & \dots & -jb\_{0N} & g\_{00} - jb\_{00} \end{bmatrix},\tag{4}$$

with *bij* = *<sup>ω</sup>*0*Cij*, (*i*, *j* = 0, ... , *N*; *i* = *j*). Since the network is reciprocal, *Y* is symmetric: *bij* = *bji*. In practical applications, the admittance matrix *Y* can be measured. Note that each transmitter and the receiver have a self-susceptance expressed by <sup>−</sup>*bjj*.

A normalization matrix *n* is defined:

$$m = \begin{bmatrix} \frac{1}{\sqrt{\omega\_0 \mathbb{C}\_1}} & \dots & 0 & 0\\ \vdots & \ddots & \vdots & \vdots\\ 0 & \dots & \frac{1}{\sqrt{\omega\_0 \mathbb{C}\_N}} & 0\\ 0 & \dots & 0 & \frac{1}{\sqrt{\omega\_0 \mathbb{C}\_0}} \end{bmatrix},\tag{5}$$

in order to normalize the admittance matrix:

$$\mathbf{y} = \mathbf{n} \cdot \mathbf{Y} \cdot \mathbf{n} = \begin{bmatrix} \frac{1}{Q\_1} - jk\_{11} & \dots & -jk\_{1N} & \dots & jk\_{10} \\ \vdots & \ddots & \vdots & \vdots \\ -jk\_{N1} & \dots & \frac{1}{Q\_N} - jk\_{NN} & -jk\_{N0} \\ -jk\_{01} & \dots & -jk\_{0N} & \frac{1}{Q\_0} - jk\_{00} \end{bmatrix} \tag{6}$$

with quality factor *Qi* of the coupled resonators (*i*, *j* = 0, . . . , *N*):

$$Q\_i = \frac{\omega\_0 C\_i}{\mathcal{g}\_{ii}} \tag{7}$$

and

$$k\_{ij} = \frac{b\_{ij}}{\omega \alpha \sqrt{\mathbf{C}\_i \mathbf{C}\_j}}.\tag{8}$$

For *i* = *j*, the parameter *kij* corresponds to the coupling factor between circuits *i* and *j*. The voltages and currents are normalized as follows:

$$\mathbf{i} = \mathbf{n} \cdot \mathbf{I},\tag{9}$$

$$
\sigma = \mathfrak{n}^{-1} \cdot \mathcal{V}.\tag{10}
$$

The normalized current–voltage relationship is thus given by

$$
\dot{a} = \mathbf{y} \cdot \mathbf{v}.\tag{11}
$$

The real and imaginary parts of the (normalized) current and voltage phasors can be explicitly written out: *in* = *iren* + *jiimn* and *vn* = *vren* + *jvimn* . Without loss of generality, we choose *v*0 as the reference phasor, i.e., *<sup>v</sup>re*0 = *v*0, *<sup>v</sup>im*0 = 0.

#### *3.2. Input Power*

The input power *Pn* (*n* = 1, . . . , *N*) for the *n*-th transmitter system is given by

$$P\_n = \frac{1}{2} \Re(\upsilon\_n i\_n^\*)\_\prime \tag{12}$$

where *i*∗*n* is the complex conjugate of *in*, and (*vni*<sup>∗</sup>*n*) is the real part of *vni*<sup>∗</sup>*n*. This result for *Pn* is

$$P\_n = \frac{1}{2} (\upsilon\_n^{rc} i\_n^{rc} + \upsilon\_n^{im} i\_n^{im}).\tag{13}$$

The total input power *Pin* of the entire CPT system is

$$P\_{\rm in} = \sum\_{n=1}^{N} P\_n.\tag{14}$$

Substituting the currents from Equation (11) into the above equation results in the total input power *Pin*:

$$P\_{in} = \frac{1}{2} \sum\_{n=1}^{N} \frac{1}{Q\_n} [(\upsilon\_n^{I\varepsilon})^2 + (\upsilon\_n^{im})^2] + \frac{1}{2} \sum\_{n=1}^{N} k\_{n0} v\_0 \upsilon\_n^{im}.\tag{15}$$

The input power *Pin* is expressed as function of the parameters of the network and the port voltages.

#### *3.3. Output Power*

Analogously, the output power can be determined as a function of the network variables and port voltages.

Applying the passive sign convention, the output power *Pout* can be written as

$$P\_{out} = -\frac{1}{2}\mathfrak{R}(v\_0 i\_0^\*) = -\frac{1}{2}v\_0 i\_0^{rc}.\tag{16}$$

Substituting the currents from Equation (11) into the above equation, the normalized output power *Pout* is determined:

$$P\_{\rm out} = -\frac{1}{2Q\_0} \upsilon\_0^2 - \frac{1}{2} \upsilon\_0 \sum\_{n=1}^{N} k\_{\rm Out} \upsilon\_n^{im}. \tag{17}$$
