Qiaowei Yuan
S-parameters are members of a family of circuit network parameters. Other members are Z-parameters, Y-parameters, T-parameters, and ABCD-parameters [1]. The first published description of S-parameters was in the thesis of Vitold Belevitch, in 1945 [2]. Kaneyuki Kurokawa popularized the S-parameters referred to as power waves [3], making S-parameters much easier to apply to microwave engineering when port voltages and currents are difficult to define and measure.
Scattering parameters, or S-parameters (the elements of a scattering matrix, or S-matrix), describe the relationships between the ports’ incident power waves and reflected power waves; thus, S-parameters are very useful in describing the electrical behavior of linear electrical networks and can be applied to electronics, communication systems design, and, especially, microwave engineering, including the design of amplifiers, filters, couplers, and so on.
The S-matrix is a mathematical construct that quantifies how RF energy propagates through a multiport network. The S-matrix can accurately represent and describe the properties of incredibly complicated networks as simple “black boxes.” Therefore, the S-matrix has also been successfully used to analyze the power transfer efficiency (PTE) and maximum PTE (MPTE) of various wireless power transfer (WPT) systems [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. In 2009, Yuan et al [4], [5], [6], [7], [8] formulated the PTE and MPTE for one transmitter and one receiver, the most popular WPT system, which is also called a single-input, single-output (SISO) WPT system. The proposed approach for calculating the PTE and MPTE can be applied not only to coil transmitters and receivers but also to any type of transmitter and receiver for SISO-WPT systems; moreover, the effect from surrounding objects, such as human bodies, can also be easily taken into account.
With the spread of WPT applications and resurgence of research due to Kurs’ publication in 2007 [16], multiple-input, multiple-output (MIMO)-WPT technology has attracted a great deal of attention. In 2017, Wiedmann and Weber discussed the MPTE of MIMO-WPT systems by using the Rayleigh quotient [9]. Yuan et al. [10], [11] formulated the generalized Rayleigh quotient problem of the PTE of MIMO-WPT systems, using S-parameters and Z-parameters. According to the generalized Rayleigh quotient problem formulated in [10], the optimal currents and optimal load impedance of MIMO-WPT systems for maximizing the PTE were explicitly obtained. On the other hand, Wen et al. published the MPTE of MIMO-WPT systems based on the Rayleigh quotient [12], [13], [14], including the case when the receiving power of each antenna was individually controlled, which they called the weighted method of maximum power transmission efficiency. Wen’s group also provides several experimental results to demonstrate practical applications using its proposed method. The fundamental concept of Wen’s approach is similar to those of the former two groups, but the unknown vector in Wen’s generalized Rayleigh quotient includes only incident power waves at transmitting ports. As a result, the formulas for calculating the MPTE are suitable only under the condition that all receiving antennas are loaded by 50 Ω instead of the optimal load impedance (50 Ω is the popular reference, or characteristic, impedance in the RF field).
In this article, the MPTEs of SISO-WPT and multiple-transmitter, single-receiver MISO-WPT systems are highlighted. The approach to calculating the PTE and MPTE of MIMO-WPT systems is based on the Rayleigh quotient (M-RQ), while the approach [6] to achieve the MPTE of SISO-WPT uses matching circuit conditions (M-MC). Numerical MPTE examples of typical SISO-WPT systems are presented in the “MPTEs of SISO-WPT Examples” section. MPTEs obtained using M-RQ are compared with those obtained by M-MC to demonstrate the effectiveness of M-RQ and reveal the physical meaning of the condition for achieving the MPTE by using M-RQ in the SISO-WPT case. In the MISO case, MISO-WPT systems’ MPTEs versus the distance between multiple transmitters and the receiver are compared to show how the receiver’s location affects the MPTE, in the “MPTEs of MISO-WPT Systems” section. The MPTEs versus the number of transmitters is presented to see how the number of transmitters affects the MPTE. Moreover, the MPTEs of those examples are compared with MPTEs obtained by Wen’s approach to confirm the importance of using the optimal load impedance.
According to the preceding description, the applicable WPT systems of our M-RQ and M-MC and other approaches are compared in Table 1. In the table, a circle signifies that a method is applicable without any conditions, a triangle means that a method is applicable with conditions, and a cross means that a method is not applicable. Both our M-RQ and M-MC utilize S-parameters to calculate the PTE and MPTE and are applicable to arbitrary transmitter and receiver structures. The difference between these two approaches is that M-MC can be applied only to SISO-WPT systems, while M-RQ can be applied to any MIMO-WPT; therefore, M-RQ is the most universal approach. In the table, rows 1–3 are the methods for obtaining the MPTE of MIMO-WPT systems, and rows 4–6 are the methods to calculate the MPTE of SISO-WPT. Rows 5 and 6 will be further reviewed at the end of the “PTE and MPTE of MIMO-WPT Systems” section.
Table 1. The applicable WPT systems using different approaches.
This article is organized as follows. M-RQ to calculate the PTE and MPTE of MIMO-WPT systems by using an S-matrix is briefly introduced in the “PTE and MPTE of MIMO-WPT Systems” section. Then, in the “MPTE of SISO-WPT Systems” section, M-MC for calculating the MPTE of SISO systems is reviewed, and the MPTEs of a dipole pair and loop pair are presented. Finally, in the “MPTEs of MISO-WPT Systems” section, the MPTEs of several MISO-WPT systems are demonstrated, and the features and applicable fields of M-RQ are described in detail, as well.
For an ${M}\times{N}$ MIMO system with M transmitters and N receivers, as shown in Figure 1, the relationship between the incident power waves and their reflected power waves at all ports can be described by the $\left({{M} + {N}}\right)\left({{M} + {N}}\right)$ S-matrix as \begin{align*}\begin{array}{c}{\left[{\begin{array}{c}{\begin{array}{c}{{b}_{1}}\\{{b}_{2}}\\{\vdots}\end{array}}\\{{b}_{{M} + {N}}}\end{array}}\right] = \left[{\begin{array}{cccc}{{s}_{11}}&{{s}_{12}}&{\cdots}&{{s}_{{1},{M} + {N}}}\\{{s}_{21}}&{{s}_{22}}&{\cdots}&{{s}_{{2},{M} + {N}}}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{{s}_{{M} + {N},{1}}}&{{s}_{{M} + {N},{2}}}&{\cdots}&{{s}_{{M} + {N},{M} + {N}}}\end{array}}\right]\left[{\begin{array}{c}{\begin{array}{c}{{a}_{1}}\\{{a}_{2}}\\{\vdots}\end{array}}\\{{a}_{{M} + {N}}}\end{array}}\right]}\end{array} \tag{1} \end{align*}
Figure 1. The (a) $M$ transmitters and (b) $N$ receivers of MIMO-WPT.
where ${a}_{i}\left({{i} = {1},\cdots{, }{M} + {N}}\right)$ represents the incident power wave and ${b}_{i}\left({{i} = {1},\cdots{, }{M} + {N}}\right)$ represents the reflected power wave at the ith port, respectively. Parameter ${s}_{ij}\left({{i} = {1},\ldots{, }{M} + {N}}\right)$ along the diagonal of the S-matrix is referred to as the self-reflection coefficient because it refers only to what happens at the ith port, while off-diagonal S-parameter ${s}_{ij}\left({{i} = {1},\ldots{, }{M} + {N},{j} = {1},\ldots{, }{M} + {N}}\right)$ is referred to the transmission coefficient because it refers to what happens at the jth port when it is excited by a signal incident wave at the ith port.
To calculate the total transmitting power at all transmitting ports and total receiving power at all receiving ports in an ${M}\times{N}$ MIMO system, the following equation is used by grouping the transmitters as one block matrix and the receivers as another block matrix. Then, the incident power wave vector ${A}_{T}$ with M elements at the transmitting ports, ${A}_{R}$ with N elements at the receiving ports, reflected power wave vector ${B}_{T}$ with M elements at the transmitting ports, and ${B}_{R}$ with N elements at the receiving ports have the following relationship: \begin{align*}\begin{array}{c}{\left[{\begin{array}{c}{{B}_{T}}\\{{B}_{R}}\end{array}}\right] = \left[{\begin{array}{cc}{{S}_{TT}}&{{S}_{TR}}\\{{S}_{RT}}&{{S}_{RR}}\end{array}}\right]\left[{\begin{array}{c}{{A}_{T}}\\{{A}_{R}}\end{array}}\right]}\end{array} \tag{2} \end{align*} where ${S}_{TT},\,{S}_{TR},\,{S}_{RT,}$ and ${S}_{RR}$ are the ${M}\times{M}$ block scattering matrix among the $M$ transmitters, ${M}\times{N}$ is the transfer coefficient block matrix from the $N$ receivers to the $M$transmitters, ${N}\times{M}$ is the transfer coefficient block matrix from the $M$ transmitters to the $N$ receivers, and ${N}\times{N}$ is the block scattering matrix of the $N$ receivers, respectively.
The PTE denoted by ${\eta}$ is the ratio of the output power ${P}_{\text{out}}$ over the input power ${P}_{\text{in}}$; that is, \[{\eta} = \frac{{P}_{\text{out}}}{{P}_{\text{in}}} \tag{3} \] where ${P}_{\text{out}}$ and ${P}_{\text{in}}$ represent the total power consumed at all receiving ports and total transmitting power at all transmitting ports, respectively. Here, ${P}_{\text{out}}$ and ${P}_{\text{in}}$ can be obtained from the port incident power wave vector and port reflected power wave vector, which are \begin{align*}{P}_{\text{in}} & = \frac{1}{2}\left({{\Vert}{{A}_{T}}{\Vert}^{2}{-}{\Vert}{{B}_{T}}{\Vert}^{2}}\right) \tag{4a} \\ {P}_{\text{out}} & = \frac{1}{2}\left({{\Vert}{{B}_{R}}{\Vert}^{2}{-}{\Vert}{{A}_{R}}{\Vert}^{2}}\right) \tag{4b} \end{align*} where ${\Vert}\,\,{\Vert}$ is the Euclidean norm of a vector.
Combining (2) and (4) into (3), the PTE ${\eta}$ will be expressed by S-parameters, and incident power wave vector ${A}_{T}$ at the transmitting ports and ${A}_{R}$ at the receiving ports will be expressed as \begin{align*}{\eta} = {-}\frac{{\left[{\begin{array}{c}{{A}_{T}}\\{{A}_{R}}\end{array}}\right]}^{H}\left[{\begin{array}{cc}{{S}_{TR}^{*}{S}_{RT}}&{{S}_{TR}^{*}{S}_{RR}}\\{{S}_{RR}^{*}{S}_{RT}}&{{S}_{RR}^{*}{S}_{RR}{-}{E}_{R}}\end{array}}\right]\left[{\begin{array}{c}{{A}_{T}}\\{{A}_{R}}\end{array}}\right]}{{\left[{\begin{array}{c}{{A}_{T}}\\{{A}_{R}}\end{array}}\right]}^{H}\left[{\begin{array}{cc}{{S}_{TT}^{*}{S}_{TT}{-}{E}_{T}}&{{S}_{TT}^{*}{S}_{TR}}\\{{S}_{RT}^{*}{S}_{TT}}&{{S}_{RT}^{*}{S}_{TR}}\end{array}}\right]\left[{\begin{array}{c}{{A}_{T}}\\{{A}_{R}}\end{array}}\right]}{.} \tag{5} \end{align*}
In (5), ${E}_{T}$ and ${E}_{R}$ are the ${M}\times{M}$ unit matrix and ${N}\times{N}$ unit matrix, respectively. In all equations and formulations in this article, the superscript ${\left({}\right)}^{T}$ means the matrix transpose, ${\left({}\right)}^{*}$ means the complex conjugate, and ${\left({}\right)}^{H}$ means the complex conjugate transpose, respectively.
Equation (5) can be further transformed into the following simple expression: \[{\eta} = {-}\frac{{A}^{H}{CA}}{{A}^{H}{DA}} \tag{6} \] where vector $A$ is the power incident waves at all ports. Matrix ${C}$ and matrix ${D}$ are expressed, respectively, as \begin{align*}{C} & = \left[{\begin{array}{cc}{{S}_{TR}^{*}{S}_{RT}}&{{S}_{TR}^{*}{S}_{RR}}\\{{S}_{RR}^{*}{S}_{RT}}&{{S}_{RR}^{*}{S}_{RR}{-}{E}_{R}}\end{array}}\right] \tag{7a} \\ {D} & = \left[{\begin{array}{cc}{{S}_{TT}^{*}{S}_{TT}{-}{E}_{T}}&{{S}_{TT}^{*}{S}_{TR}}\\{{S}_{RT}^{*}{S}_{TT}}&{{S}_{RT}^{*}{S}_{TR}}\end{array}}\right]{.} \tag{7b} \end{align*}
The efficiency expressed in (6) is the generalized Rayleigh quotient and has the maximum value ${\eta}_{\text{max}}$ because both matrix $C$ and matrix $D$ are Hermitian matrices. From the generalized Rayleigh quotient, (6) has the maximum value of the Rayleigh quotient, which is equivalent to the maximum among the following matrix’s generalized eigenvalues: \[{CX} = {\gamma}{DX} \tag{8} \] where ${\gamma}$ is the eigenvalue that has ${M} + {N}$ values in total and $X$ is the ${M} + {N}$ eigenvector. However, the power efficiency defined in (3) must be less than or equal to one, so ${\eta}_{\text{max}}$ is the largest among the eigenvalues whose value is less than or equal to one.
Without any approximation and additional hypotheses for the S-matrix and derivation processes of the preceding efficiency ${\eta}$ and ${\eta}_{\text{max}},$ the proposed formulation is universal and exact and can be applied to any type of MIMO-WPT system, which can have various transmitter and receiver geometries. Also, the preceding approach has no limitation on the operating frequency, distance between the transmitters and receivers, and power coupling methods, as indicated in Table 1.
The corresponding eigenvector to the eigenvalue of ${\eta}_{\text{max}}$ is the optimal incident power wave vector. If this optimal incident power wave vector is denoted by ${A}^{\text{opt}},$ then the optimal incident power wave vector ${B}^{\text{opt}}$ can be obtained by ${SA}^{\text{opt}}.$ Finally, the optimal excitation voltage for each transmitter port and optimal load impedance at each receiver’s port can be easily obtained by ${A}^{\text{opt}}$ and ${B}^{\text{opt}}$ according to the relationship between the incident/reflected power wave and voltage/current at each port, as described in [11].
When ${M} = {N} = {1}$ in Figure 1, the MIMO-WPT system is a SISO-WPT system focused as the fundamental WPT system. The most popular commercial WPT charging case uses coils as the transmitter and receiver. Of course, the approach described in the preceding section can be applied to calculate the PTE and MPTE of any SISO-WPT system. Here, another approach, M-MC [4], [5], [6], [7], [8], based on the matching conditions at the transmitting and receiving ports, is briefly introduced. MPTEs of typical WPT systems consisting of two dipoles and two loops are calculated and compared.
A SISO-WPT system with one arbitrary transmitter and one receiver can be equivalent to a two-port network, as demonstrated in Figure 2, where ${a}_{i}$ and ${b}_{i}$ represent the incident power wave and reflected power wave at port ${i}\left({{i} = {1},{2}}\right),$ respectively. Port 1 represents the transmitting port, and port 2 represents the receiving port. The S-matrix describes the relationship between ${a}_{i}$ and ${b}_{i}$ and can be obtained by either numerical electromagnetic simulation or measurement using a vector network analyzer (VNA).
Figure 2. The equivalent circuit of an arbitrary transmitter and receiver.
If port 2 is loaded with impedance ${Z}_{l},$ as in Figure 2, the PTE ${\eta}$ between the arbitrary transmitting and receiving antennas is formulated using S-parameters: \[{\eta} = \frac{{P}_{2}}{{P}_{1}} = \frac{{\left|{{b}_{2}}\right|}^{2}{-}{\left|{{a}_{2}}\right|}^{2}}{{\left|{{a}_{1}}\right|}^{2}{-}{\left|{{b}_{2}}\right|}^{2}} = \frac{\left({{1}{-}{\left|{{\Gamma}_{l}}\right|}^{2}}\right){\left|{{s}_{21}}\right|}^{2}}{{\left|{{1}{-}{s}_{22}{\Gamma}_{l}}\right|}^{2}{-}{\left|{{s}_{11}{-}\bigtriangleup{\Gamma}_{l}}\right|}^{2}} \tag{9} \] where \[{\bigtriangleup} = {s}_{11}{s}_{22}{-}{s}_{12}{s}_{21} \tag{10} \] \[{\Gamma}_{l} = \frac{{Z}_{l}{-}{Z}_{0}}{{Z}_{l} + {Z}_{0}}{.} \tag{11} \]
Here, ${Z}_{0}$ is the referenced impedance; usually, its value is ${50}\Omega{.}$ MPTE ${\eta}_{\text{max}}$ is obtained by applying the following matching conditions at port 1 and port 2: \[{Z}_{s} = {Z}_{\text{in}}^{*},\,{Z}_{l} = {Z}_{\text{out}}^{*}{.} \tag{12} \]
As shown in Figure 2, ${Z}_{\text{in}}$ is the input impedance, looking into the right side from port 1, and includes load impedance ${Z}_{l},$ and ${Z}_{\text{out}}$ is the output impedance, looking into the left side from port 2, and includes source impedance ${Z}_{s}.$ Applying the matching condition of (12), the MPTE of (9) will be \[{\eta}_{\text{max}} = \frac{\left({{1}{-}{\left|{{\Gamma}_{l}^{\text{opt}}}\right|}^{2}}\right){\left|{{s}_{21}}\right|}^{2}}{{\left|{{1}{-}{s}_{22}{\Gamma}_{l}^{\text{opt}}}\right|}^{2}{-}{\left|{{s}_{11}{-}\bigtriangleup{\Gamma}_{l}^{\text{opt}}}\right|}^{2}}{.} \tag{13} \]
Compared with the PTE in (9), ${\Gamma}_{l}$ is replaced with ${\Gamma}_{l}^{\text{opt}}$ in (13), while ${\Gamma}_{l}^{\text{opt}}$ and ${\Gamma}_{s}^{\text{opt}}$ are obtained by the following formulations, respectively: \begin{align*}{\Gamma}^{\text{opt}}_{s} & = \frac{{B}_{1}\pm\sqrt{{B}_{1}^{2}{-}{4}{\left|{{C}_{1}}\right|}^{2}}}{2{C}_{1}} \tag{14a} \\ {\Gamma}_{l}^{\text{opt}} & = \frac{{B}_{2}\pm\sqrt{{B}_{2}^{2}{-}{4}{\left|{{C}_{2}}\right|}^{2}}}{2{C}_{2}} \tag{14b} \end{align*} where \begin{align*}{B}_{1} & = {1} + {\left|{{s}_{11}}\right|}^{2}{-}{\left|{{s}_{22}}\right|}^{2}{-}{\left|{\bigtriangleup}\right|}^{2} \tag{15a} \\ {B}_{2} & = {1} + {\left|{{s}_{22}}\right|}^{2}{-}{\left|{{s}_{11}}\right|}^{2}{-}{\left|{\bigtriangleup}\right|}^{2} \tag{15b} \end{align*} \begin{align*}{C}_{1} & = {s}_{11}{-}\Delta{s}_{22}^{*} \tag{16a} \\ {C}_{2} & = {s}_{22}{-}\Delta{s}_{11}^{*} \tag{16b} \end{align*} \[{\Delta} = {s}_{11}{s}_{22}{-}{s}_{12}{s}_{21}{.} \tag{17} \]
Therefore, once the ${2}\times{2}$ S-matrix of the SISO-WPT system is known, the MPTE can be calculated, and both the optimal source impedance and load impedance can be obtained.
As mentioned before, two coils are the most popular transmitter and receiver used in SISO-WPT systems. The MPTE of two tightly coupling coils can be estimated conventionally using the kQ product based on circuit theory [8] as \[{\eta}_{\text{max}} = {1}{-}\frac{2}{{1} + \sqrt{{1} + {k}^{2}{Q}_{1}{Q}_{2}}} \tag{18} \] where ${k} = {M}{/}{{L}_{1}{L}_{2}}$ is the coupling coefficient between two coils, $M$ is the mutual inductance, and ${L}_{1}$ and ${L}_{2}$ are the self-inductances of the transmitting coil and receiving coil, respectively; ${Q}_{1}$ is the quality factor of the transmitting coil, while ${Q}_{2}$ is the quality factor of the receiving coil. However, this approach is effective only when the k and Q factors can be separately measured; it cannot be applied to general cases where the kQ product cannot be separately measured and estimated, and it is also not correct for weak coupling cases when two coils are separated too far, as explained in [8]. Although the kQ product was extended to a general SISO system based on Z-parameters in [15], this extended kQ product is the intermediate parameter to obtain the MPTE in WPT systems and further needed to calculate the MPTE, which is essentially the same, derived using S-parameters in (13). The MPTE based on Z-parameters can also be found in [8].
To calculate the PTE and MPTE based on (9) and (13) quickly and efficiently, a C#-based application, E-WPT, presented in Figure 3, has been developed in our laboratory. E-WPT is capable of calculating the PTE and MPTE between arbitrary transmitters and receivers when the S-parameters of the equivalent two-port networks are known. The MPTE based on formula (13) is achieved when both the source side and load side relate to their optimal impedances, respectively. E-WPT can also calculate the optimum impedances for achieving the MPTE.
Figure 3. The (a) E-WPT software and (b) its flowchart.
As illustrated in Figure 4, E-WPT was successfully used to calculate MPTEs of various transmitters and receivers prototyped by the competitors at a 2016 WPT contest organized by the Institute of Electronics, Information, and Communication Engineers Technical Committee on Wireless Power Transfer [18]. In Figure 4, there is a dipole pair, loop pair, dipole and loop pair, horn pair, and so on. E-WPT is open for free use [18], [19] and will soon be developed to include the MIMO-WPT system.
Figure 4. Various transmitters/receivers (prototypes at the 2016 WPT contest).
As described in the flowchart of Figure 3(b), first, E-WPT requires loading an S-parameter file from either data measured by a VNA or data from simulated electromagnetic software. Second, the characteristic impedance, source impedance, and load impedance are set. Finally, E-WPT calculates the PTE, MPTE, and optimal source and load impedances simultaneously. Moreover, E-WPT is capable of designing a matching circuit to realize the optimal source and load impedances.
The structures in Figures 5 and 6 are two typical SISO systems with one transmitter and one receiver, as used in [8]. The dipole–dipole pair represents the capacitive coupling, while the loop–loop pair represents the inductive coupling. In Figure 5, both the transmitter and receiver are composed of a dipole with a length of $2l,$ which will be set to $0.1$ and ${0}{.}{5}{\lambda}$ for calculating the MPTEs, while in Figure 6, both the transmitter and receiver are composed of a loop with a circumference of $0.1$ and ${0}{.}{5}{\lambda}{.}$ All wires of the transmitters and receivers are made of copper with a conductivity of 5.783 × 107 (S/m) and have the same radius of 0.65 mm. The operating frequency is set to 6.78 MHz. In this article, ${\lambda}$ represents the wavelength.
Figure 5. The dipole pair (SISO system).
Figure 6. The loop pair (SISO system).
The MPTE ${\eta}_{\text{max}}$versus the distance d between the transmitter and receiver is provided in Figure 7. Solid lines represent the results obtained by M-RQ, while circles represent the results obtained by M-MC. As shown in Figure 7, the two MPTEs agree with each other completely, confirming that the condition for achieving the maximum eigenvalue of the generalized Rayleigh quotient problem of (8) is nothing but the matching conditions at the transmitting and receiving ports in the SISO case. From Figure 7, the MPTE decreases when the receiver moves away from the transmitter along the x direction for both the dipole and loop cases. For the dipole pair with a dipole length of ${2}{l} = {0}{.}{1}{\lambda}$ and loop pair with a loop circumference of ${0}{.}{1}{\lambda},$ the MPTEs of the two dipoles are larger than those of the loop pair when the distance d is the same, and almost the same performance is observed when the dipole length and loop circumference are ${0}{.}{5}{\lambda}$ except the case when the receiver is located very near the transmitter. In both the dipole case and loop case, the longer the antenna element, the lower the MPTE in the near-field region, which indicates that the electrically small element is much more suitable for near-field power transfer, while the electrically large element is much more suitable for far-field power transfer.
Figure 7. The ${\eta}_{\max}$ versus $\mathbf{d}$ using M-RQ (solid lines) and M-MC (circles).
In this section, two types of MISO-WPT systems, included in Figure 8(b) and (c), are used as numerical models to calculate the MPTE versus the distance $d$ by using M-RQ. For comparison, the ${1}\times{1}$ SISO system in Figure 8(a) is also included, which is a special case of a MISO system, where the transmitter number is one. Figure 8(b) is a ${4}\times{1}$ MISO system, and Figure 8(c) is a ${16}\times{1}$ MISO system.
Figure 8. A SISO system and two types of MISO-WPT systems: (a) a 1 × 1 SISO, (b) a 4 × 1 MISO, and (c) a 16 × 1 MISO.
As detailed in Figure 8, the transmitting antenna element is a half-wavelength dipole antenna, while the receiving antenna is a 0.1-wavelength dipole antenna; $d$ represents the distance between the center of the transmitting antenna array and receiving antenna along the $Z$ direction, and $s$ represents the distance between two neighbor elements in the transmitting array antenna along $X$ and $Y$ directions. In the following simulation results, $s$ is set to ${0}{.}{65}{\lambda}{.}$ The models in Figure 8(b) and (c) were used in [17] to define the boundary between the near-field region and far-field region of different array antennas, where the ${M}\left({{M} = {1},{16},{64}}\right)$ transmitters were regarded as an M-element antenna array, and one receiver was used as the testing antenna.
The MPTEs versus $d$ are plotted in Figure 9, and the line of ${1}{/}{{d}^{2}}$ is also plotted as a reference. The factor of ${1}{/}{{d}^{2}}$ is the power varying versus the distance $d$ when the receiver is located in the far-field region of the transmitting array antenna, according to the Friis transmission formula. From Figure 9, the MPTE of the ${1}\times{1}$ SISO is much larger than those of the ${4}\times{1}$ and ${16}\times{1}$ MISO-WPTs when the distance $d$ is less than about ${0}{.}{3}{\lambda},$ indicating that one transmitter is the most efficient way to deliver the power. However, the situation is reversed when the distance $d$ is over ${0}{.}{3}{\lambda},$ where the MPTE becomes higher when the transmitting antenna number is larger, indicating that when the receiver is far from the transmitters, the more transmitters that are used, the higher the MPTE achieved. Moreover, when the distance $d$ is over ${0}{.}{3}{\lambda},$ the MPTEs of the three types of WPT systems decreased nearly ${1}{/}{{d}^{2}},$ varying according to the Friis transmission formula.
Figure 9. The MPTE versus $\mathbf{d}$ when using M-RQ (solid lines) and M-RQ with a 50-Ω load (dashed lines).
In the near-field coupling, from Figure 9, the MTPE is dominantly affected by the distance between the receiver and closest transmitting element. When ${d}\leq{0}{.}{3}{\lambda},$ among the three types of structures, the receiving element of the ${1}\times{1}$ SISO is the closest to the transmitting element, and high transmission efficiency is obtained. However, even if ${d} = {0},$ in the ${4}\times{1}$and ${16}\times{1}$ MISO-WPT cases, the distance between the receiver and its closest transmitting element exceeds ${0}{.}{45}{\lambda}{;}$ therefore, when ${d}\leq{0}{.}{3}{\lambda},$ the ${4}\times{1}$ and ${16}\times{1}$MISO cases obtain lower transfer efficiency than that of SISO. However, when ${d}{>}{0}{.}{3}{\lambda},$ the far-field coupling is dominant, and the array with a large number of elements has a high transmission efficiency. That is the reason why there is a cross point at 0.3m in Figure 9. To further confirm this point, the MPTEs of the ${5}\times{1}$ and ${17}\times{1}$ MISO-WPT are also calculated and plotted in Figure 10, where the ${5}\times{1}$ MISO-WPT is the ${4}\times{1}$ MISO-WPT with an additional half-wavelength dipole located at the center of the transmitting array, while the ${17}\times{1}$ MISO-WPT is the ${16}\times{1}$ MISO-WPT with an additional half-wavelength dipole located at the center of the transmitting array; this means that the transmitting antenna in the SISO-WPT case is kept in the ${5}\times{1}$ and ${17}\times{1}$ MISO-WPT cases. From Figure 10, the MPTE of the ${5}\times{1}$ MISO-WPT is larger than that of SISO when ${d}\leq{0}{.}{3}{\lambda}$ and becomes the same as that of the ${4}\times{1}$ MISO-WPT when ${d}{>}{\lambda}$. The same performance is also observed in the MPTEs of the ${17}\times{1}$ and ${16}\times{1}$ MISO-WPT cases. Due to the center element at the transmitting array structure, the MPTEs of the ${5}\times{1}$ and ${17}\times{1}$ MISO-WPT cases are always larger than those of the SISO-WPT at any distance $d,$ without the cross point in the ${4}\times{1}$ and ${16}\times{1}$ MISO-WPT cases.
Figure 10. The MPTEs of 4 × 1, 5 × 1, 16 × 1, 17 × 1 MISO cases and a 1 × 1 SISO case.
In [17], using the MTPE performance versus the distance, instead of the traditional ${2{D}^{2}}{/}{\lambda}$ boundary criterion, a new criterion for the boundary of the near-field region and far-field region was proposed. In the traditional ${2{D}^{2}}{/}{\lambda}$ boundary criterion, $D$ represents the maximum dimension of the antenna, which means that the region boundary criterion depends on the antenna’s physical size, with no relation to the antenna excitation conditions. From Figure 9, the distance when the MPTE behaves close to the line of ${1}{/}{{d}^{2}}$ when using our proposed approach is different from that when using a 50-$\Omega$ load impedance, demonstrating that the near-field and far-field region boundary also depends on the excitation conditions, as emphasized in [17]. Moreover, from the calculation of the MPTE performance versus the distance, M-RQ can be applied to obtain the MPTE when the receiver is moved from the near-field to the far-field seamlessly. In Figure 9, the MPTEs calculated using Wen’s approach, which assumes that the receiver is loaded by 50 $\Omega$, are also plotted in dashed lines. The MPTEs are quite lower than the MPTEs from our M-RQ, showing that the MPTEs obtained by (3) and (8) are the maximum, which is achieved only when all receiving ports are connected to the optimal impedance, not the referenced impedance ${Z}_{0}.$
Because the optimal excitations obtained by our M-RQ for MISO systems make the transmitting array antenna send the maximum power to the target receiving antenna, our proposed MPTE approach has another potential application to array antenna beamforming if the location of the target antenna is in the far-field region of the transmitting array antenna, as described in [17]. If the location of the target antenna or receiving antenna is changed, the S-matrix will change, and the optimal excitations for each element antenna in the transmitting array will be recalculated and adjusted, while the system still transmits the maximum power to the moved target antenna. In the future, more such applications will be investigated.
Finally, it should be noted that the MPTE obtained from (8) is the maximum for the decided transmitter and receiver structure by applying the optimally excited voltages at the transmitters’ ports and loaded by the optimal impedance at the receivers’ ports. For the same transfer distance, there is an optimal transmitter and receiver structure to achieve the largest MPTE, which is quite a complex problem and will also be our next research topic.
This article reviewed the universal approach of M-RQ to calculate the PTE, MPTE, and condition for achieving the MPTE of an arbitrary MIMO-WPT system. The MPTE is achieved by calculating the eigenvalue of the formulated generalized Rayleigh quotient problem, which is based on an S-matrix. Numerical examples of SISO-WPT and MISO-WPT systems demonstrated the effectiveness and universality of M-RQ. Our proposed method’s features and capabilities are as follows:
Special thanks to graduate students Riri Niizeki, Takumi Aoki, and Hiroshi Satake for their great efforts to promote this research when they studied in my laboratory.
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Digital Object Identifier 10.1109/MMM.2022.3233509