Open AccessArticle

Model Decomposition-Based Approach to Optimizing the Efficiency of Wireless Power Transfer Inside a Metal Enclosure

Romans Kusnins

Sergejs Tjukovs

Janis Eidaks

Kristaps Gailis

and

Dmitrijs Pikulins

Institute of Photonics, Electronics and Telecommunications, Riga Technical University, 12 Azenes Street, LV-1048 Riga, Latvia

Author to whom correspondence should be addressed.

Appl. Sci. 2024, 14(24), 11733; https://doi.org/10.3390/app142411733

Submission received: 16 October 2024 / Revised: 25 November 2024 / Accepted: 13 December 2024 / Published: 16 December 2024

(This article belongs to the Section Energy Science and Technology)

Download

Browse Figures

Review Reports Versions Notes

Abstract

This paper describes a numerically efficient method for optimizing the high power transfer efficiency (PTE) of a resonant cavity-based Wireless Power Transfer (WPT) system for the wireless charging of smart clothing. The WPT system under study unitizes a carbon steel closet intended to store smart clothing overnight as a resonant cavity. The WPT system is designed to operate at 865.5 MHz; however, the operating frequency can be adjusted over a wide range. The main reason behind choosing a resonant cavity-based WPT system is that it has several advantages over the competitive WPT methods. Specifically, in contrast to its Far-field Power Transfer (FPT) and Inductive Power Transfer (IPT) counterparts, resonant cavity-based WPTs do not exhibit path loss and significant PTE sensitivity to the distance between the Tx and Rx coils and misalignment, respectively. The non-uniformity of the fields within the closet is addressed by using an optimized Yagi-like transmitting antenna with an additional element affecting the waveguide mode phases. The changes in the mode phases increase the volume inside the cavity, where the PTE values are higher than 50% (the high PTE region). In the present study, the model decomposition method is adapted to substantially accelerate the process of finding the optimal WPT system parameters. Additionally, the decomposition method explains the mechanism responsible for extending the high PTE region. The generalized scattering matrices are computed using the full-wave simulator Ansys HFSS for three sub-models. Then, the calculated S matrices are combined to evaluate the system’s PTE. The decomposition method is validated against full-wave simulations of the original WPT system’s model for several different parameter value combinations. The simulated results obtained for a sub-optimal model are experimentally verified by measuring the PTE of a real-life closet-based WPT system. The measured and calculated results are found to be in close agreement with the maximum measured PTE, as high as 60%.

Keywords:

wireless power transfer; cavity-based systems; remote charging

1. Introduction

Over the last several decades, Wireless Sensor Networks (WNSs) have been an active field of research due to the multitude of useful functions they can perform in various areas of human activity, including construction, medicine, biology, health control systems, etc. [1]. Owing to recent advances in microelectronics and materials, a wide variety of miniature and powerful sensors have become available. This enables wearable sensor nodes to efficiently and reliably monitor users’ health state, e.g., smart clothing incorporating a number of dedicated sensors. The smart clothing can perform various functions, including continuous measurement of body temperature and spinal posture, and gait monitoring during exercise to protect the user (a patient) from falling and becoming injured [2,3]. It might be beneficial for persons suffering from various impairments and disabilities as it may reduce the risk of serious injuries, thereby increasing the quality of life [4]. While capable of performing their intended tasks autonomously and establishing communication links with other WSN nodes, the batteries powering the electronic components need periodic recharging or replacement. The recharging necessitates sockets and jack plugs, which affect the integrity of the suit structure and increase the risk of electric shock or fire when used in adverse conditions, such as increased humidity. Untethering these devices would considerably increase the user’s comfort and reduce the risk of injuries due to malfunctioning clothing components [5].

The present study aims to evaluate the feasibility of a simple-to-use, and cost-effective remote powering system for charging a battery incorporated into the smart clothing (smart suit) recently developed by our partners [6]. The most suitable candidate for smart suit charging is resonant cavity-based WPT systems constructed by deploying a metallic chamber enclosing the workspace, which effectively becomes a resonant cavity. Since the EM field is confined to a finite region, no power is wasted due to path loss, resulting in higher PTEs. This allows reliable remote charging of home appliances, toys, gadgets, etc., placed inside the enclosure when their batteries become depleted or during the idle period, for example, overnight, so as not to expose the users to strong EM fields [7].

In fact, this study extends our previous research on the WPT system utilizing a carbon steel closet [8], which was confined to full-wave simulations only. In [8], different antenna (probe) designs, including dipole antennas, reflector-backed dipole antennas, and dipole antennas with both reflectors and an additional conducting strip (Yagi-like antennas), were examined. The Yagi-like antenna was found to extend the region of the closet’s interior, where the PTE is high due to a strong electric field. However, no theoretical analysis was carried out to investigate the mechanisms responsible for such a WPT system’s behavior, and experimental verification was needed. Thus, the primary objective of the present study is to find the cause of the increase in the high PTE region size and validate the simulation results via direct measurements of the received power performed in our faculty’s laboratory.

A standard carbon steel closet was chosen due to its availability in the mass market and relatively low cost. Alternatively, designing an efficient WPT system utilizing such common furniture elements would be attractive as it would obviate the need for constructing tailor-made room-scale enclosures, which necessitate quite a large quantity of copper and lead to less economical cost (it is more cost-effective while being slightly less efficient).

The Inductive Power Transfer (IPT) method is a viable alternative to the cavity-based method [9]. Its impressive power transfer capacity allows quick remote charging. Recent studies have shown that the IPT can ensure maximally achievable power transfer efficiency (PTE) of up to 96.1% at an output power level of 20 kW [10]. The downsides of the IPT are short operation distances and sensitivity to misalignments, which are the main obstacles to achieving sufficiently high PTE in the suit charging application. Namely, the precise coil placement required to avoid misalignments and large separation is challenging because the Rx coil is attached to a soft material of the suit. The antenna-based power transfer systems [11,12] show great flexibility owing to high spatial freedom and larger operating distances, making them advantageous in applications where power receivers are mounted high above the ground, e.g., tree health monitoring, high-rise building structural integrity monitoring, etc. [13]. However, high path loss at high frequencies and large antennas at low ones make them unsuitable for smart suit charging. Though adaptive beamforming can mitigate the high path-loss issue, it requires more complex and sophisticated tracking mechanisms and bulky antenna arrays [14,15].

The conducting walls make cavity-based WPT systems immune to the interference caused by external RF signal sources while causing little interference to electronic devices in the cavity exterior. For this reason, cavity-based WPT systems might be particularly important in applications that benefit from the use of operating frequencies within licensed communication bands. Hence, this WPT technology is expected to be widely applicable in a range of practical scenarios, such as charging closets, toy storage, wireless charging rooms, etc. However, the need for a conducting enclosure confines the use of cavity-based WPT systems to indoor applications. Recently, it was demonstrated that by exciting a cavity with copper walls by electrically small loops ensuring weak coupling with the magnetic field, a PTE of up to 72% can be achieved [16]. Short capacitive dipoles (capacitive probes) also ensure efficient power transfer inside metallic enclosures, as reported in [17]. The coupled mode theory [18] has been shown to provide reasonable PTE estimation accuracy for both the loop- and dipole-based configurations, enabling circuit analysis-based treatment to accelerate the WPT system’s optimization [19].

Operating a cavity in a multi-mode regime yields a higher PTE over larger volumes than in the single-mode regime, as reported in [20], making the PTE less sensitive to receiver position and orientations. In [21], the implantable receiver orientation insensitivity was achieved using a biaxial receive resonator system designed for animal studies involving rodents. A retro-directive beamforming-inspired method was proposed in [22] to improve the PTE achieved by an over-moded cavity.

Cavities operated in a single-mode [23,24] or multi-mode [25] sub-wavelength regime (quasi-magneto-static regime) can also ensure sufficiently high PTE and power levels while meeting safety requirements. This enables the deployment of room-scale enclosed WPT systems for charging various electronic devices. In [26], it is shown that a PTE higher than 37.1% can be achieved throughout a 3 m × 3 m × 2 m enclosure.

Operating a resonant cavity at frequencies far above the dominant one, resulting in over-moding, has also been shown to extend the coverage. This can be achieved by affecting the field distribution within an over-moded cavity by varying the operating frequency (frequency stirring) [27] or altering the cavity interior (mechanical stirring) [28]. The effect of several relatively large scattering objects inside a heavily over-moded cavity was studied in [29], showing that it is possible to ensure a PTE of 30% even when there is no Line-of-Sight between the receiver and transmitter. The inability to establish reliable communication links between the cavity interior and the outside world was addressed [30,31,32] by using meshed walls.

In the present study, the power transfer inside the carbon steel closet is accomplished using a pair of antennas. Figure 1 shows the schematic diagram of the cavity-based (closet-based) WPT system under study. The closet contains the transmitting and receiving antenna, the RF signal generator, and the digital multimeter employed to measure the voltage at an RF-DC converter output. The Tx antenna is driven by a sine wave delivered from the generator’s output to the Tx antenna input by a 50 Ω coaxial cable. The RF signal the receiving antenna receives is converted into a DC voltage using a voltage doubler-based RF-DC converter. A twisted pair of wires connects the RF-DC converter output and the multimeter.

The fields in the closet are excited by a Yagi-like antenna, which, along with the reflector, also has an additional conducting strip (director). The Tx antenna design of this antenna resembles that of the Yagi–Uda antennas widely employed in practice, hence the name. Additionally, the closet-based WPT system under study is optimized to maximally extend the volume inside the closet where a PTE higher than 50% is observed. The optimal WPT system’s model parameters are found using an exhaustive search algorithm applied to a decomposed model, which results in substantial reduction in the computational burden. Specifically, the original closet-based WPT system’s model is decomposed into several sub-models. Two of the three sub-models contain the Rx and the Tx antenna without the director. The director of the transmitting Yagi-like antenna is included in the third sub-model to reduce the number of parameter combinations for which the scattering data are to be calculated.

The operating frequency is 865.5 MHz, which is the central frequency of the sub-GHz ISM band. At this frequency, the closet acts as a resonant cavity in a multimode regime—seven modes can propagate in the closet at the same time. However, only three of them, TE01, TM21, and TE21, can transfer power from the generator to the receiver due to their strong interaction with the dipole antennas.

The decomposition method is validated by comparing the WPT system’s S parameters with those obtained for the original system using Ansys HFSS 2023 R1. In addition, a sub-optimal model exhibiting an even wider higher PTE region than the optimal one is constructed at the expense of a narrow dip in the PTE curve in the middle of the region. Also, the sub-optimal model is experimentally validated by measuring the received power. Two PTE measurement scenarios are examined: with and without an RF-DC converter. The Tx antenna is driven by an RF signal generator during the experiments. Depending on the measurement scenario, the received power is measured using an RF power meter or the voltage at the output of the RF-DC converter connected to the Rx antenna is measured using a digital multimeter. Then, the received power is calculated based on the voltage value and the power conversion efficiency of the RF-DC converter.

To summarize, the primary objectives of the present study are as follows:

Ascertain the mechanisms responsible for the increase in the size of the high PTE region, by performing an in-depth analysis based on the structure decomposition method. Specifically, this question is answered by examining the interaction of sub-models involving a dipole antenna and director with power-carrying waveguide modes (ensuring the energy transfer from the Tx end to the Rx end). As will be shown in the remainder of this paper, the observed elongation of the high PTE region is due to changes in the power-carrying modes’ phase resulting from the modes’ interaction with the director strip, which essentially acts as a primitive phase shifter.
Apply the model decomposition method to the WPT system at hand, validate it, and assess its accuracy by comparing the results obtained using this method with those retrieved by performing the full-wave analysis of the original WPT systems model (without decomposing it) for a large number of different parameter values.
Experimentally verify the calculated results by making measurements at different horizontal and vertical positions of the receiving antenna inside the closet. Two measurement scenarios are examined: with and without the RF-DC converter.
Asses the sensitivity of the cavity-based WPT system’s PTE to variations in the operating frequency by performing frequency sweeps for several Rx antenna horizontal and vertical positions.

This paper is organized as follows: Section 2 outlines a cavity-based WPT system involving two Yagi-like antennas inside a carbon steel closet, presents the Ansys HFSS model of the WPT system constructed to compute the system’s PTE, and provides a detailed description of the decomposition method. Section 3 describes the WPT system’s optimization process and presents the optimized model results. Section 4 details the PTE measurement procedure and compares the simulated and experimentally obtained ones for different positions of the receiving antenna. Section 5 concludes the paper by summarizing the outcomes of this study.

2. Theoretical Analysis of Cavity-Based WPT Systems

The WPT system under study employs two printed dipole antennas placed in a metal closet. The WPT system was designed to wirelessly power sensors incorporated into a smart suit. The receiving antenna employed in the present study is a classical dipole antenna, except that it has an additional conducting strip along the rear edge intended to suppress the propagation of waves in the opposite direction. The transmitting antenna has a Yagi-like design. As demonstrated in our previous study [8], using a Yagi-like antenna with a single director is more advantageous. Specifically, an additional parasitic element (conducting strip) affects the cavity’s field pattern so that high PTE regions may become wider, thereby reducing the PTE’s sensitivity to the receiver position.

The transmitting antenna is attached to the metallic shelf located in the upper part of the closet. It is intended to place an RF generator and control circuitry used during the measurements of the PTE. An adjustable antenna holder was constructed to keep the Rx antenna in a specific position during measurements of the received power. The holder consists of two vertical U-shaped aluminum profiles running parallel to the closet’s side walls and the two horizontal U-shaped PVC profiles to which the antenna is to be attached. The holder’s support comprises several aluminum profiles located on the closet’s bottom side. The holder is designed so that the antenna position can be changed both in the vertical and horizontal plane during the measurements to assess the effect of the receiving antenna’s location on the WPT system’s PTE.

Notably, the distinct advantage of cavity-based WPT systems is that the conducting walls substantially confine the high-intensity fields needed for WPT to the cavity’s interior, thereby eliminating any significant interference from nearby electronic devices. Thus, it is possible to exploit frequency bands outside those intended for unauthorized use (e.g., ISM bands), which increases the WPT system’s adaptability.

The calculation of the WPT system’s PTE and assessment of its behavior are performed using the commercially available full-wave analysis software Ansys HFSS 2023 R1. The full-wave modal analysis of the cavity under study is performed to calculate the S parameters: S₁₁ (reflection coefficient) and S₂₁ (transmission coefficient). Figure 2 illustrates a model of the closet containing the receiving and transmitting antenna constructed to estimate the system’s PTE. Two lumped ports are defined for field excitation and calculation of the S parameters.

Once the scattering parameters are found, the PTE can be calculated as follows:

PTE = {|S_{21}|}^{2} \cdot 100 % .

(1)

The closet is assumed to be empty (filled with air). In the simulation, the losses in the closet walls are considered by imposing impedance boundary conditions on the entire closet surface and the shelf in the upper part of the closet. Note that, in the closet utilized in the experiments, there is a gap between the closet door and shelf edge, which slightly affects the field distribution inside the enclosure. The HFSS model constructed for the full-wave field simulations has the same dimensions and geometry as the carbon closet used the real-life measurements of the PTE. The closet’s dimensions are as follows: width—477 mm, length—290 mm, height—1420 mm.

Both antennas are printed on a rigid FR-4 substrate (standard thickness 1.6 mm) with a dielectric constant and loss tangent of 4.2 and 0.02, respectively. The HFSS model of the receiving antenna is shown in Figure 3, whereas that of the transmitting antenna is illustrated in Figure 4.

The conducting parts of the antennas are modelled as planar objects on which impedance boundary conditions are imposed. While the impedance boundary conditions approximate the relation between the tangential field component, they consider the conduction losses in the conducting antenna parts with reasonable accuracy.

The transmitting antenna’s radiator (the dipole) is excited by a feed line consisting of two copper strips located on opposite sides of the substrate and of identical dimensions. The transverse dimensions of the feed line are such that at the operating frequency of 865.5 MHz, only the TEM wave can propagate, while the higher-order modes decay so quickly that their effect can be completely neglected. The widths of the upper and lower strips constituting the feed line are identical and chosen so that the line’s characteristic impedance is 50 Ω.

2.1. WPT Model Decomposition

To facilitate the analysis of the WPT system at hand, the original system’s model is simplified so that the resulting model is a rectangular metallic chamber containing a pair of antennas. The chamber’s dimensions are identical to those of the closet. In contrast to the original mode, this model ignores the effects of the metallic shelf and the non-uniformity of the door’s surface on the fields. Since one of the three linear closet dimensions (the height) is much larger than the other two (the width and the depth), it is more advantageous from the theoretical analysis point of view to treat the chamber as a rectangular waveguide terminated into conducting plates (chamber’s top and bottom sides) on both sides. The waveguide-based treatment of the fields within the chamber is chosen in favor of the conventional resonant cavity analysis to considerably facilitate this study’s aim of gaining insight into the mechanisms responsible for the high PTE region expansion due to the additional conducting strip in the Tx antenna design. Specifically, the number of waveguide modes needed to approximate the fields adequately is much smaller than the number of resonant cavity modes.

The most computationally expensive step of the WPT system design is the optimization process, which requires calculating the WPT system’s overall power reflection and transmission coefficient data for a large number of parameter values. To circumvent this issue, a simplified system’s model is decomposed into three sub-models. Then, each of these models is treated separately.

The first sub-model contains the dipole antenna in an otherwise empty waveguide section whose length is slightly greater than the length of the dipole antenna substrate. Both ends of the waveguide sections are treated as waveguide ports. Since the number of propagating modes at the operating frequency is seven, at least modal fields corresponding to this set of modes must be used to approximate the fields on the surface of each port. In many cases, this approximation ensures satisfactory result accuracy; however, in some specific cases, the result accuracy may not be satisfactory. The reason this way of approximating the fields sometimes fails is that the evanescent waveguide modes still affect the field pattern at the port location even though they decay exponentially with the distance from a discontinuity (dipole antenna or Yagi-antenna’s director). The rate at which the fields of modes whose cut-off frequency is not much higher than the operating one decay is not sufficiently large to ignore their effect on the port fields. To mitigate this issue, the number of modal fields for approximating the port field is increased to 12. Numerical studies carried out by the authors show that using 12 waveguide modes guarantees excellent agreement between the fast model decomposition-based method and the direct full-wave analysis of the closet-based WPT model at hand.

The WPT system sub-model involving the dipole antenna without the director constructed for the generalized S matrix calculation using full-wave software Ansys HFSS 2023 R1 is depicted in Figure 5. The blue arrow indicates the location of the reference plane defined for the waveguide port at the rear end of the section. The relevant reference plane is positioned at the tip of the arrow, i.e., it is aligned with the rear edge (reflector side) of the dipole antenna.

The second sub-model involves the transmitting antenna director only. The director strip can be separated from the dipole model due to a relatively large distance between them in the optimal antenna configuration, as will be shown next in this paper. The WPT system sub-model involving the Yagi-like transmitting antenna director built for the evaluation of the relevant generalized S matrix using Ansys HFSS 2023 R1 is illustrated in Figure 6.

The last sub-model is the receiving dipole antenna, which is constructively identical to the receiving one with the director removed but, in general, may have different dimensions of its elements. Of course, the proposed model is by no means complete in terms of various subtle features present in the real-life closet serving as the waveguide, such as metallic anchors, a small gap between the metal shelf and the closet door, and non-flatness of the closet door.

Once the generalized matrices describing the behavior of the dipole antenna and its director are calculated for various combinations of their parameters, they are exported to MATLAB files to be used by MATLAB routines developed for computing the overall S parameters of the WPT system by combining the sub-model matrices. The director has only one variable parameter—its length, whereas the dipole antenna has three variable parameters: reflector’s length, dipole’s length, and dipole’s feed line length.

To further facilitate the derivation of the generalized S matrix of sub-model combinations, the generalized matrices can be decomposed into several lower-order sub-matrices relating the mode amplitudes and phases of the incident and reflected waves at a pair of ports. Thus, each submatrix may be interpreted either as a generalized reflection or a transmission matrix. For example, for some arbitrary two-port structure operated in multimode regime, the submatrix matrix

S_{11}

of the structure’s generalized scattering matrix,

S

, relates the amplitudes of the incident and reflected modes at Port 1 and, therefore, may be interpreted as the extension of the reflection coefficient at Port 1. Likewise,

S_{12}

is a submatrix of

S

relating the amplitudes of modes excited at Port 1 to those arriving at Port 2. Hence, this matrix may be regarded as the generalized transmission matrix relating amplitudes of the modes emanating from Port 2 to those reaching Port 2. Submatrix

S_{12}

is, in fact, the generalized transmission matrix from Port 2 to Port 1, and the last submatrix

S_{22}

is a generalization of the reflection coefficient at Port 2, as it relates the incident and reflected mode amplitudes and phases at the port.

Also, for notational convenience and to simplify the treatment of generalized matrices, the modes belonging to several ports may be treated as a single mode group. For example, the TEM mode of the lumped port and the waveguide Port 1 modes can be combined into a single group to find the S matrix for a dipole loaded waveguide section terminated in a conducting plate at the rear end. In such a case, the generalized reflection and transmission matrices will relate the amplitudes of the modes of different groups instead of ports.

Thus, the generalized scattering matrix for a waveguide section containing the Tx dipole antenna, whose rear end is terminated into a conducting metallic plate, can be found as follows:

S^{TX - DP} = S_{11}^{TX - DIP} + S_{12}^{TX - DIP} T^{TX - PT} {(I^{DR} - S^{PLATE} T^{TX - PT} S_{22}^{TX - DIP} T^{TX - PT})}^{- 1} S^{PLATE} T^{TX - PT} S_{21}^{RX - DIP},

(2)

where

S_{11}^{TX - DIP}

and

S_{22}^{TX - DIP}

are the generalized reflection matrices for mode Groups 1 and 2 of the Tx antenna’s dipole sub-model, respectively,

S_{21}^{TX - DIP}

and

S_{12}^{TX - DIP}

are the generalized transmission matrices governing the transmission of the waves from mode Group 1 (the lumped port exciting the TEM wave in the feed line + waveguide Port 1 at the front end of the waveguide section) to mode Group 2 (Port 2 defined at the section’s rear end) of the same sub-model, and vice versa,

T^{TX - PT}

is a diagonal matrix whose order is equal to the number of modes at Port 2 and the n-th diagonal entry of the matrix is given by

{(T^{TX - PT})}_{n, n} = e^{- γ_{n} d_{TX - PLATE}}

d_{TX - PLATE}

and

γ_{n}

are the distance between the rear edge of the Tx dipole antenna reflector and the rear-end of the waveguide and the propagation constant of the n-th waveguide mode, respectively, and

I^{DR}

is the identity matrix of the same order as

T^{TX - PT}

Similarly, one can obtain the generalized scattering matrix for the receiving antenna backed by another metallic plate (the front end of the waveguide in the Ansys model or the closet bottom wall in the real-life closet used in the experiments):

S^{RX - DP} = S_{11}^{RX - DIP} + S_{12}^{RX - DIP} T^{RX - PT} {(I^{DR} - S^{PLATE} T^{RX - PT} S_{22}^{RX - DIP} T^{RX - PT})}^{- 1} S^{PLATE} T^{RX - PT} S_{21}^{RX - DIP},

(3)

where

S_{11}^{RX - DIP}

and

S_{22}^{RX - DIP}

are the generalized reflection matrices for mode Groups 1 and 2 of the Rx dipole’s sub-model, respectively,

S_{21}^{RX - DIP}

and

S_{12}^{RX - DIP}

are the generalized transmission matrices describing the transmission of the waves from Group 1 (lumped port + Port 1 defined at the front end of the waveguide section containing the Rx antenna) to port Group 2 (Port 2 defined at the rear end) of the Rx dipole’s sub-model, and vice versa,

T^{RX - PT}

is a diagonal matrix whose order is equal to the number of waveguide modes at Port 2 and whose n-th diagonal entry of the matrix is given by

{(T^{RX - PT})}_{n, n} = e^{- γ_{n} d_{RX - PLATE}},

and

d_{RX - PLATE}

is the distance between the Rx dipole antenna and the front end of the waveguide.

The matrix describing the reflection of modes from the conducting plate

S^{PLATE}

may be chosen to be equal to the negative of the identity matrix, corresponding to a PEC (perfectly electrically conducting) plate. Some numerical studies conducted by the authors revealed that such an approximation may significantly compromise the amplitude and phase response accuracy. Therefore, in this study, the reflection of the conducting plate is treated as a surface with imposed impedance boundary conditions. Although this model is also approximate, it dramatically reduces calculation errors. The surface impedance is calculated based on the closet walls’ estimated conductivity and relative magnetic permeability. In this case, matrix

S^{PLATE}

is a diagonal matrix whose n-th diagonal entry can be written as

{(S^{PLATE})}_{n, n} = (Z_{S} - Z_{n}) / (Z_{S} + Z_{n}),

where

Z_{S} = (1 + j) \sqrt{μ_{0} μ_{r} f π / σ}

Z_{n}

is the characteristic impedance of the n-th waveguide mode given by

Z_{n} = j ω μ_{0} / γ_{n}

and

Z_{n} = γ_{n} / j ω ε_{0}

for the TE and TM modes, respectively,

f

is the operating frequency, and

μ_{r}

and

σ

are the relative permeability and conductivity of the closet material, respectively.

It should be noted that the same boundary conditions are imposed on the side walls of the waveguide sections during the full-wave analysis. Also, the effect of the side walls of the empty waveguide sections joining different sub-models is incorporated into

γ_{n},

calculated with Ansys HFSS 2023 R1.

Now, having found the generalized S matrix for the Tx dipole antenna considering the presence of the conducting plate and using the S matrix for the Yagi-like antenna director obtained by analyzing the sub-model involving the director, the generalized scattering matrix for a waveguide section containing both the dipole and the director sub-models separated by a distance

d_{dip - dir}

can be found as follows. For convenience, the expression for each of the four submatrices of

S^{YAG}

is deduced separately. The generalized reflection matrix

S_{11}^{YAG}

can be calculated as follows:

S_{11}^{YAG} = S_{11}^{TX - DP} + S_{12}^{TX - DP} T^{DIP - DIR} {(I^{DR} - S_{11}^{DIR} T^{DIP - DIR} S_{22}^{TX - DP} T^{DIP - DIR})}^{- 1} S_{11}^{DIR} T^{DIP - DIR} S_{21}^{TX - DP},

(4)

where

S_{11}^{TX - DP}

and

S_{22}^{TX - DP}

are the generalized reflection matrices for Ports 1 and 2, respectively,

S_{21}^{TX - DP}

and

S_{12}^{TX - DP}

are the generalized transmission matrices for waves traveling from Port 1 (the lumped port) towards Port 2 (the front end of the section with the dipole), and vice versa,

S_{11}^{DIR}

is the reflection matrix for Port 1 of the director sub-model,

T^{DIP - DIR}

is a diagonal matrix whose order is equal to the number of waveguide modes and the n-th diagonal entry is

{(T^{DIP - DIR})}_{n, n} = e^{- γ_{n} d_{dip - dir}},

and

d_{Tx - to - Rx}

is the distance between the dipole end and the director.

The above expression for combining the generalized matrices can be derived either by enforcing the continuity of the mode amplitudes and phases in waveguide cross-sections away from a discontinuity or by utilizing the fact that the multiple reflection method also applies to time-harmonic processes (sine waves). Likewise, one can calculate the other three submatrices of

S^{YAG} :

S_{22}^{YAG} = S_{22}^{DIR} + S_{21}^{DIR} T^{DIP - DIR} {(I^{DR} - S_{22}^{TX - DP} T^{DIP - DIR} S_{11}^{DIR} T^{DIP - DIR})}^{- 1} S_{22}^{TX - DP} T^{DIP - DIR} S_{12}^{DIR},

(5)

S_{21}^{YAG} = S_{21}^{DIR} T^{DIP - DIR} {(I^{DR} - S_{22}^{TX - DP} T^{DIP - DIR} S_{11}^{DIR} T^{DIP - DIR})}^{- 1} S_{21}^{TX - DP},

(6)

S_{12}^{YAG} = S_{12}^{TX - DP} T^{DIP - DIR} {(I^{DR} - S_{11}^{DIR} T^{DIP - DIR} S_{22}^{TX - DP} T^{DIP - DIR})}^{- 1} S_{12}^{DIR},

(7)

where

S_{22}^{DIR}

is the reflection matrix for Port 2 of the director sub-model,

S_{21}^{DIR}

and

S_{12}^{TX - DP}

are the matrices describing mode transmission from Port 1 to Port 2 of the same sub-model, and from Port 2 to Port 1, respectively,

S_{22}^{YAG}

is the reflection coefficient matrix at Port 2 (waveguide port) of the transmitting Yagi-like antenna, and

S_{21}^{YAG}

and

S_{12}^{YAG}

are the transmission coefficient matrices from Port 1 (lumped port) to Port 2 and vice versa, respectively.

The final step in the WPT system scattering parameter derivation process is the identification of the four crucial S parameters. These parameters relate the amplitudes and phases of the TEM waves at the generator and the receiving end of the system. Unlike the Yagi-like antenna, where the generalized scattering matrix was derived from the parts characterizing the antenna, the resulting S data are a set of four scalar quantities in this case. This is because the Tx and Rx ends operate in a deep TEM regime, where the feed line dimensions are significantly smaller than the wavelength at the operating frequency. The contribution of higher-order modes under these conditions is negligible. As a result, the overall S parameters of the WPT system can be expressed in terms of scattering matrices

S^{YAG}

and

S^{RX - DP}

as follows:

S_{11} = S_{11}^{YAG} + S_{12}^{YAG} T^{Tx - Rx} {(I^{DR} - S_{22}^{RX - DP} T^{Tx - Rx} S_{22}^{YAG} T^{Tx - Rx})}^{- 1} S_{22}^{RX - DP} T^{Tx - Rx} S_{21}^{YAG},

(8)

S_{22} = S_{11}^{RX - DP} + S_{12}^{RX - DP} T^{Tx - Rx} {(I^{DR} - S_{22}^{YAG} T^{Tx - Rx} S_{22}^{RX - DP} T^{Tx - Rx})}^{- 1} S_{22}^{YAG} T^{Tx - Rx} S_{21}^{RX - DP},

(9)

S_{21} = S_{12}^{RX - DP} T^{Tx - Rx} {(I^{DR} - S_{22}^{YAG} T^{Tx - Rx} S_{22}^{RX - DP} T^{Tx - Rx})}^{- 1} S_{21}^{YAG},

(10)

S_{12} = S_{12}^{YAG} T^{Tx - Rx} {(I^{DR} - S_{22}^{RX - DP} T^{Tx - Rx} S_{22}^{YAG} T^{Tx - Rx})}^{- 1} S_{21}^{RX - DP},

(11)

where

S_{11}^{RX - DP}

is the reflection coefficient matrix at Port 1 of the receiving dipole antenna,

S_{21}^{RX - DP}

and

S_{12}^{RX - DP}

are the transmission matrices from Port 1 to Port 2 of the dipole, and vice versa, respectively,

S_{22}^{RX - DP}

is the generalized reflection matrix at Port 2,

T^{Tx - Rx}

is a diagonal matrix of order two whose n-th diagonal entry is

{(T^{Tx - Rx})}_{n, n} = e^{- γ_{n} d_{Tx - to - Rx}},

and

d_{Tx - to - Rx}

is the distance between the Tx and Rx antennas.

2.2. Analysis of the WPT System’s Sub-Models

Now, we examine the sub-models containing a dipole antenna to understand better the interaction between different waveguide modes and the antenna itself. For this purpose, the generalized scattering parameters relating the amplitudes and phases of the waveguide modes and the incident TEM mode in the antenna feed line must be calculated and analyzed.

Notably, not all seven propagating modes are responsible for the power transfer between the generator and the receiver (RF-DC) converter. Specifically, the modes have different field patterns in the cross-sectional planes of the waveguide, resulting in different degrees of interaction between them and the antenna dipole. In this interaction, the electric field vector plays a crucial role, as the strength of the current induced on the dipole is solely determined by the electric field strength and direction at the location of the dipole.

To determine which modes will interact strongly with the dipole and, therefore, contribute to the energy transfer, let us first provide the conditions under which a sufficiently strong voltage can be induced across the ends of the dipole arms. The main conditions are as follows:

A mode must have a relatively large electric field component along the dipole for the current to be induced in it;
The electric field intensity must be asymmetric about the waveguide central plane, which is perpendicular to the dipole antenna plane and runs parallel with the waveguide;
The electric field intensity must be large at the dipole location—the dipole must be located within an electric field maximum, or two maxima with opposite field directions, and the same field strength must occur at the dipole arms.

Only modes simultaneously satisfying the above conditions are strongly coupled to the TEM mode from the generator via the feed line. Given the dimensions of the metallic closet and the operating frequency (865.5 MHz) from the seven propagating modes, the above conditions are fulfilled only for three modes: TE₀₁, TM₂₁, and TE₂₁. Specifically, the TE₀₁ mode has only one electric field vector component parallel to the antenna dipole. Its direction is uniform, and the peak electric field is in the middle of the waveguide, where the antenna should be placed to achieve the highest PTE. The other two modes, TM₂₁ and TE₂₁, are degenerate and exhibit a maximum at the central part of the antenna dipole. Similar to the TE₀₁ mode, the electric field intensity of these is asymmetric relative to the antenna symmetry plane. It is worth noting that these modes have non-uniform field distribution along the dipole, leading to weaker coupling between them and the TEM wave.

First, consider the sub-model consisting of the dipole antenna without the director located in an empty section of a rectangular waveguide. The cross-sectional dimensions of the waveguide are identical to those of the closet used in the real-life WPT system. Both ends of the waveguide section are terminated in a perfectly matched load ensured in Ansys HFSS by using two waveguide ports defined at the ends of the waveguide section. Figure 7 shows the amplitudes of the waveguide modes and the reflected feed line’s TEM wave when the sub-model under consideration is excited by an incident TEM wave whose amplitude is unity. Under this assumption, the amplitudes are equal to the absolute values of the corresponding generalized S parameters, which, in turn, may be interpreted as coupling coefficients between the incident TEM wave and the waveguide modes evaluated at different values of the dipole length.

As can be seen in the absence of the conducting plate terminating the rear end of the waveguide (the end of the waveguide closest to the dipole antenna reflector), the antenna-excited waveguide modes can propagate in both directions despite the presence of the reflector intended to suppress the radiation in the undesired direction. Equivalently, the reflector does not ensure total reflection—some portion of the EM energy still reaches the rear end of the waveguide. This means that, in practice, the distance between the top wall of the metallic closet will affect the transmitting dipole antenna’s performance due to wave reflections.

Additionally, Figure 7 reveals that for a sufficiently long reflector, the amplitude of the main (TE₀₁) wave propagating towards the rear end of the waveguide is the smallest among the six waveguide modes displayed. This indicates that the backwards-propagating component of the mode TE₀₁ experiences very strong reflection from the conducting strip intended to suppress energy from flowing in the opposite direction, thus giving rise to a pronounced standing wave pattern due to multiple reflections from the waveguide terminations and the antennas themselves. In brief, in the case of the TE₀₁ mode, the reflector behaves as required, sending the TE₀₁ mode power in the desired direction. As regards the other two modes responsible for the power exchange between the Tx and Rx (ensuring power delivery to the receiver) and having identical phase velocities, the amplitude of the TM₂₁ mode is nearly twice as high as that of its degenerate counterpart TE₂₁, as apparent in Figure 7. Interestingly, the amplitude of the backwards-propagating TM₂₁ mode is just about 30% smaller than that of the forward-propagating TE₂₁ mode. This result reveals that the interaction between the TM₂₁ mode and the reflector is appreciably weaker than in the case of the TE₀₁ mode, meaning that this mode will contribute significantly to the appearance of the PTE minima and maxima along the waveguide. Moreover, the shape of the PTE versus the Tx-to-Rx distance curve is likely to be sensitive to the distance between the dipole antenna end (reflector’s edge) and the top wall of the closet (rear end of the waveguide). Also, it is observed that the TE₂₁ mode demonstrates the weakest coupling to the TEM feedline mode. However, the forward-to-backwards-propagating mode amplitude ratio for the TE₂₁ mode is almost equal to that of TM₂₁.

In view of poor backwards-propagation suppression of the higher order power transferring modes TM₂₁ and TE₂₁ by the reflector, it is of paramount importance to examine the effect of the conducting wall placed at the rear side of the waveguide on the amplitudes of the modes sent towards the receiving antenna. Similar to the previous case, in place of the receiving antenna, perfect impedance matching is ensured for each of the considered modes at the unterminated end of the waveguide.

It should be noted that the reflection coefficient (S₁₁) of the transmitting antenna characterizes the quality of matching between the antenna and the waveguide (WPT system under study). In the case shown in Figure 7, an almost perfect matching is achieved at resonance, which occurs when

L_{dipole} = 62.5 mm

and

L_{feed} = 70 mm .

These antenna parameter values ensure almost zero reflection from the transmitting antenna, implying that all power produced by the generator is converted into the energy stored in the three waveguide modes that are responsible for power transfer to the intended receiver.

Now consider the effect of a conducting plate used to terminate the rear end of the waveguide. In the real-life WPT system employed in the present study, the top of the closet is made of the same metal as the side walls; therefore, the conducting plate terminating the waveguide end, which corresponds to the closet’s top wall, may be treated as such. As follows from the above discussion, higher-order modes are not sufficiently strongly reflected by the reflector, resulting in a variation in the field pattern with the gap size between the Tx antenna reflector and the terminated waveguide end. The coupling coefficients between the forward-propagating modes and the TEM wave and the TEM wave reflection coefficient are presented in Figure 8a. As can be seen, the amplitude of the TE₀₁ mode is larger than without the conducting plate terminating the waveguide. This amplification is due to the superposition of the forward- and backwards-propagating TE₀₁ modes with equal or at least almost equal phases. The phase difference is determined by the feed line length of the dipole antenna, as the backwards-propagating TE₀₁ mode is largely reflected from the reflector. However, this is not the case with the other two modes—as evidenced by Figure 8a, their amplitudes are only slightly affected by the presence of the conducting plate.

The phases of the coupling coefficients (generalized S parameters) and that of the TEM line reflection coefficient plotted as functions of the transmitting antenna dipole length are presented in Figure 8b. The mode phases demonstrate gentle variation with the dipole length at the length values far from the “resonant” one, where the maximum power coupling between the TX antenna and the waveguide at hand is observed. In the vicinity of the resonant value, which in this case is about

L_{dipole} = 62.5 mm,

the phase curve slopes for all modes become noticeable higher. The TEM line reflection coefficient shows a similar behavior, except that it exhibits a jump of about 160° at the resonant value of

L_{dipole}

The perfect matching between the transmitting and the receiving antenna (no reflection back to the generator) could be achieved if the amplitudes of the three power-carrying modes upon the arrival of the receiving antenna were preserved and the relative phases were the negatives of those radiated by the Tx antenna. Namely, the phasors of the modes arriving at the Tx antenna must be the complex conjugates of those emanating from the Rx antenna. This case is the ideal one—a goal to strive for. The main factor limiting the PTE of such an ideal hollow waveguide-based WPT system would be the attenuation experienced by the modes while travelling from one antenna to the other.

The generalized S matrix relating the mode amplitudes and phases at the Yagi-like antenna model’s ports (the faces perpendicular to the wave propagation direction) can be deduced by combining the S matrices of the dipole antenna and the director, respectively. The effect of the empty lossy wall waveguide section joining the sub-models can be evaluated analytically since, for a homogeneously filled rectangular waveguide, there are closed-form relations between the mode amplitudes at the ends of the section. Only the propagation constants for the considered modes need to be found to obtain the relations, which can be done analytically for rectangular waveguides. Alternatively, one can find the mode phase constants and the modal fields analytically by assuming that the waveguide walls are perfectly conducting; then, the effect of the wall losses on the wave propagation expressed in terms of mode attenuation constants may be quantified using either the power loss model [33,34], or the boundary impedance perturbation model [35,36]. The power loss model has been shown to fail to provide adequate accuracy close to the cut-off frequency. The boundary impedance perturbation method, in turn, provides a reliable estimation of the complex mode propagation constant. However, it requires evaluating surface integrals, which may be cumbersome when treating a waveguide with complex cross-section geometry. Therefore, calculating the propagation constants using full-wave simulation tools is more favorable as it gives sufficiently accurate values, provided appropriate boundary conditions are imposed (see Table 1). The Ansys HFSS simulation data presented in Table 1 show that of the three power transferring modes, the TM₂₁ exhibits the highest attenuation, while the TE₀₁ mode has the lowest.

Another model whose behavior is also of theoretical interest is the one involving the dipole antenna augmented with a conducting strip (director), as it might shed some light on the effect of the antenna director on the field distribution within the resonant cavity. The mode coupling coefficients and the TEM wave reflection coefficient are plotted in Figure 9. Note that, like in the first case, there is no termination at both waveguide ends. The figure reveals that the coupling with the forward-propagating mode TM₂₁ is virtually unaffected by the presence of the director.

The same applies to the forward-propagating TE₂₁ mode. Meanwhile, the amplitude of the forward-propagating TE₀₁ mode is about 30% smaller than without the director. In contrast, the amplitudes of all the backwards-propagating modes are considerably larger. The results show that from the three power-transferring waveguide modes, only the TE₀₁ mode exhibits strong interaction with the additional conducting strip; furthermore, the director reflects the TE₀₁ mode and converts a significant portion of its power into the other two modes.

The last case to be examined is a Yagi-like antenna composed of a dipole antenna and a conducting strip (copper strip), referred to throughout this paper as a director, by analogy with the conventional Yagi–Uda antennas. The director is running parallel to the dipole and separated from it by a certain distance denoted by

d_{dip - dir}

. Hence, the conventional dipole antenna may be viewed as a special case of the Yagi-like antenna with a director length equal to 0. Figure 10a shows the most practically significant generalized S parameters (coupling coefficients) calculated for the Yagi-like Tx antenna.

Interestingly, the resonant value of

L_{dipole}

is the same as in the case of a single dipole without a conducting plate at the rear port. Comparing the results with those of Figure 8a, one finds that the director does not affect the amplitudes of the TE modes, whereas that of the TM₂₁ mode slightly increases. As will be shown later, the main benefit of adding a director to the dipole antenna is not the increase in the coupling coefficient. Instead, the director changes the mode phases in a way that cannot be accomplished by changing the position of the dipole antenna, as phase delay resulting from different phase velocities of the modes cannot be compensated for by increasing or decreasing the wave travel path.

Another essential factor characterizing the transmitting antenna behavior is the coupling coefficients’ dependence on the feed line length. The coupling coefficients show how the generator power delivered by the TEM mode of the feed line is distributed among the main three waveguide modes (TE₀₁, TM₂₁, and TE₂₁). The dipole sub-model is contained in an empty waveguide section terminated into a PEC wall at the rear end. The coupling coefficients for the three modes, along with the reflection coefficient of the TEM mode, are plotted against the feed line length in Figure 10b. It is seen that all three coupling coefficients vary slowly with the length of the feed line, the smallest variation being exhibited by the TE₂₁ mode. The other two modes demonstrate comparable variation with

L_{feed} .

The strongest coupling is observed for the TE₀₁ mode, while the TM₂₁ mode has just a slightly smaller amplitude. The weakest coupling to the Tx antenna in terms of power is observed for the TE₂₁ mode, which is entirely in keeping with the above results.

3. WPT Model Decomposition Approach Validation and Optimization

Once the individual S matrices are calculated for all sub-models, they are exported from Ansys HFSS and saved to separate files. Then, a custom MATLAB R2019b script developed by the authors is used to calculate the overall S parameters (the S parameters of the whole WPT system’s model) by combining the pre-calculated individual S matrices. The empty waveguide sections joining the sub-models are analyzed analytically using the waveguide mode propagation constants computed using Ansys HFSS.

To verify the validity of the proposed structure decomposition method, the numerical results calculated by employing the method were obtained for various WPT system configurations with different parameter values. In fact, a variety of models were verified via full-wave simulations. However, due to the page limit, the results of this quite comprehensive comparative study are presented only for two. Figure 11a shows the absolute values of the resonant cavity-based WPT system’s scattering parameters (TEM mode parameters) against frequency, calculated using both the decomposition method and directly (without dividing the WPT system’s model into several sub-models) with Ansys HFSS for two WPT system models with different parameter values. The reflector lengths of the Tx and Rx antennas in both models are identical and equal to 170 mm. Figure 11b shows the corresponding phase responses of the WPT systems (phases of S₁₁ and S₂₁ against frequency). For both examined models, the amplitude and phase responses are in excellent agreement with the full-wave simulation results. The results are almost indistinguishable in graphical accuracy. The parameters of the two models, whose responses are shown in Figure 11a,b, are presented in Table 2.

To maximally mitigate the PTE sensitivity to the Rx antenna position, which is of primary importance in the smart suit charging application, it is highly desirable to maximize the size of the high PTE region—the region in the closet’s interior where high PTE values are attained. To that end, various antenna parameter value combinations should be examined to find the one giving a PTE versus Tx-to-Rx separation distance curve with the widest, flattest, and highest maximum—the high PTE region.

In practice, various optimization algorithms are employed to make the optimal configuration-finding process less computationally expensive. The optimization algorithms reduce CPU time by reducing the number of objective function evaluations and by choosing model parameter value combinations (trial solutions) efficiently based on the objective function values from previous iterations [37]. The local optimization methods are fast but applicable only in the case of a simple objective function landscape, which is not the case in the present study since the objective function must control the shape of the curve of interest. Such complicated optimization tasks typically give objective functions with multiple local minima. Also, constrained optimization requires introducing penalty functions, significantly increasing the floating-point operation count [38].

The global optimization algorithms also require a large number of objective function evaluations to find the global minimum [39,40]. If the number of parameter value combinations examined per algorithm’s iteration is insufficient, there is a high risk of finding a local minimum rather than a global one. Applying a global optimization method to the entire model involving both antennas would lead to a prohibitively large CPU time.

To find the optimal parameters of the WPT system under study for each sub-model, the generalized S matrix is calculated using different combinations of parameters. Once the sub-model S parameters are found (precomputed for different combinations of sub-model parameters), the overall WPT system’s S parameters can be quickly calculated using a simple MATLAB code—the calculation takes just a few seconds on modern computers. The MATLAB routine combines the precomputed S parameters for different sub-model parameter combinations. As the sub-model parameter space dimensions (the number of parameters varied by the optimization algorithm) are smaller than those of the entire model, the number of combinations for which the S parameters need to be calculated with Ansys HFSS 2023 R1 is considerably reduced, leading to a substantial reduction in the CPU time. This is a distinct advantage of the decomposition method.

Also, the decomposition method allows us to divide the WPT system model into smaller sub-models, which allows the S parameters to be computed much faster due to the smaller model size. Thus, using the decomposition method, the exhaustive search-based optimization takes just a few minutes instead of hours or even days. Thus, there is no need for more sophisticated optimization methods.

It is worth noting that only those sub-model’s parameters are varied that significantly affect the S matrix element values. For instance, in the sub-model involving the dipole antenna, only the feed-line length, dipole length, and reflector length are varied to generate different parameter combinations to be examined. The feed-line width is kept fixed at a value that gives 50 Ω characteristic impedance to ensure impedance matching with the generator and the load (an RF-DC converter or RF power meter, depending on the measurement scenario). The width of the dipole is found to have little effect on the curve shape as long as it is not too small or too large. The same applies to the reflector width. Regarding the sub-model containing the Tx antenna director, only the director length is varied, while the width is kept fixed as it does not significantly affect the results.

Additionally, the WPT parameters ensuring the strongest coupling between the waveguide and the antennas, as described in the previous section, are used as an initial guess for the optimization process. These parameters minimize the power reflected back to the generator.

An exhaustive search is performed across all the precomputed sub-model parameter values and all values of the optimization parameters that are not associated with the sub-model S parameters. The parameters are optimized one at a time; only values corresponding to the precomputed combinations are used. Once all parameters have been changed, the process is repeated. The algorithm executes until no further improvements are obtained by varying the parameter values. The parameters varied by the exhaustive search algorithm are as follows:

Tx antenna’s reflector, feed line, and dipole lengths (Tx antenna’s sub-model parameters);
Rx antenna’s reflector, feed line, and dipole lengths (Rx antenna’s sub-model parameters);
Director length (director’s sub-model parameters);
Separation distance between the Tx and Rx antennas;
Distance between the Rx antenna’s dipole and the director.

It was found that for distances between the Tx antenna’s dipole and the director,

d_{dip - dir},

larger than 195 mm, a little change in the PTE curve was observed. The distance between the Tx antenna’s reflector rear edge and the upper closet side was kept fixed during the optimization process; it was found to have little effect on the high PTE region size.

Figure 12a shows the PTE evaluated for the antenna system optimized to achieve the widest high PTE region while ensuring the highest PTE among the models, which provide comparable high PTE region widths.

The optimal set of parameter values was obtained by the exhaustive search over a set of parameters for which the simulation data were acquired. It was observed that the presence of the additional conducting strip in the Yagi-like design appreciably extended the high PTE region compared with the dipole antenna, which was also optimized for the maximum region’s width. Additionally, it was found that reducing the Rx antenna reflector length can achieve a significantly wider high PTE region, albeit at the expense of a sharp dip occurring close to the right edge of the region, as shown in Figure 12b. In both cases, the PTE was close to 70% over a larger part of the high PTE region. The parameters of the optimized Yagi-like Tx and the Rx dipole antennas are summarized in Table 3 and Table 4, respectively. In contrast, those of the sub-optimal model, whose PTE is presented in Figure 12b, are given in Table 5 and Table 6.

It is worth noting that exactly the sub-optimal model was selected for experimental studies aimed at verifying whether the real-life WPT system implementing the proposed concept can ensure high PTE over relatively large volumes inside the closet to render the received power level less sensitive to the Rx antenna position changes resulting from different suit sizes and imprecise placement.

The blue curve shown in Figure 12b is obtained for the WPT system model, where a hypothetical phase shifter is employed in place of the director. The phase shifter’s generalized reflection matrix is assumed to be the zero matrix. In contrast, the transmission matrices are identical (under the symmetry assumption), with all the off-diagonal elements set equal to zero, implying no mode conversion. The diagonal elements of the transmission matrix are set to unity for modes other than those responsible for power transfer. For the TE₀₁, TM₂₁, and TE₂₁ modes, the amplitude of the transmission coefficients (diagonal entries) is equal to unity, while their phases are treated as variables. The optimal phase values resulting in the widest high PTE region were 92.31°, 111.11°, and −99.13° for the modes TE₀₁, TM₂₁, and TE₂₁, respectively.

4. Experimental Verification and Discussion

To experimentally determine the PTE, a pair of semi-rigid coaxial cables was used to deliver power from a signal generator to the transmitting antenna, and the received power was measured with a power meter located outside the closet. The position of the transmitting antenna was kept fixed during the measurements of the PTE. Two PLA fastening hooks were used to attach the antenna to the shelf in the upper part of the closet. The hooks were created with the aid of a 3D printer. To find the optimal configuration of the antenna system resulting in the highest PTE while providing acceptably wide high-electric field intensity regions, an Ansys HFSS model of the closet containing both the antennas and their fixtures was built. The optimal antenna design parameters were found from the simulation data obtained for different combinations of model parameters. Then, the PTE for the designed WPT system was calculated at different positions along the largest dimension of the closet. Figure 13 presents photos of the experimental setup involving a carbon steel closet used as a resonant cavity containing two Yagi-like antennas.

4.1. Measurements of the WPT System’s PTE

Though the power received by the receiving antenna can be routed to the RF power meter via a coaxial cable similar to that used to deliver the generator’s produced power to the TX antenna, its presence in the interior of the metallic enclosure would likely have an adverse effect on the field distribution and therefore would negatively affect the measurement results. To mitigate this issue, an RF-DC converter was designed and fabricated. The converter was equipped with an SMA connector, enabling it to be connected to the Rx antenna via a male–male transition.

The converter’s PCE must be retrieved to extract pure PTE of the cavity-based WPT system free from the effect of the experimental setup elements, such as the coaxial cables utilized in the measurement and the RF-DC converter. To that end, a separate set of measurements was made, where an RF sine wave was directly fed through the input of the RF-DC converter. No cables were used during the measurements to reduce the measurement uncertainty associated with various reflections and imperfections of connecting elements. Throughout the experimental study, two measurement scenarios were adopted. In the first scenario, the RF signal generator frequency was kept fixed at 865.5 MHz, while the output power of the generator was successively set to the following levels: −10, −3, 0, 3, 5, 6, 7, 8, 9, and 10 dBm.

Since for the PTE extraction, one needs the PCE values for the input power levels that, in general, differ from those used in the measurements, a cubing spline interpolation was employed; it was built-in MATLAB function interp1(), with the fourth and fifth arguments assigned ‘pchip’ and ‘extrap’, respectively.

All the measurement results presented in this contribution were made with the RF-DC converter, thereby obviating the need for a coaxial cable. However, the RF-DC converter’s output voltage was measured by a multimeter connected in parallel with a 1 kΩ load resistor. It should be noted that the load resistance was chosen based on a series of measurements involving the RF-DC converter only. The values of the converter’s matching network were chosen so that the input impedance was as close to 50 Ω as possible to ensure adequate impedance matching. The 1 kΩ load resistance was found to ensure the highest RF to DC power conversion efficiency (PCE). The measured PCE as a function of the input RF power level at the operating frequency is shown in Figure 14a. It is seen that the PCE increases monotonically with the input power level reaching about 70% at 12 dBm.

The sensitivity of the RF-DC converter to frequency variations is also of interest for the present study, as it may affect the overall performance of the WPT system when adjusting the operating frequency to improve its PTE slightly. Figure 14b presents the PCE plotted against the frequency from 800 MHz to 960 MHz, measured at different input power levels listed above. The converter’s PCE shows slight variation with the frequency regardless of the input power level, as apparent in Figure 14b.

A few series of received power measurements were made for different positions of the receiving antenna to verify the simulated results. The main objective was not only to find the maximum achievable PTE but also to determine how sensitive the PTE is to small variations in the position of the receiving antenna. The sensitivity is crucial, as the receiving antenna is intended to be incorporated into a smart suit. Hence, it is not easy to ensure precise antenna placement in practice due to the softness of the suit material. The suit fabric is also soft, resulting in unwanted antenna rotations and displacement from the optimal position. The position of the Rx antenna was ensured with a holder constructed from several U-shaped aluminum profiles. The upper horizontal part of the Rx antenna holder was made of a U-shape PVC profile so as not to affect antenna operation.

4.2. Measurement Results

The values of the conductivity (

σ

) and relative magnetic permeability (

μ_{r}

) needed for theoretical analysis were initially chosen based on the data provided in handbooks on metal properties, including the electromagnetic ones and the available information on the metal the closet walls are made of. The initial values of

σ

and

μ_{r}

were assumed equal to

7 \cdot 10^{7} S / m

and 100, respectively. However, these values caused unacceptably large discrepancies between the measured and calculated WPT system’s responses. This problem was overcome by estimating the values of both quantities based on the agreement between the theoretical and experimental curves. Following this approach, the conductivity and permeability were estimated to be about

10^{5} S / m

and 100, respectively. These values were used in all the simulations, the results of which are presented below.

The measured PTE is plotted on the same graph as the calculated one at different vertical positions in the closet for convenience. Six measurement series were made, each corresponding to a different horizontal position (positions along the x- and z-axes) that was kept fixed during the series. The first set of PTE values was obtained at points along the vertical line with

x_{RX} = 0 mm

and

z_{RX} = 0 mm,

where

x_{RX}

and

z_{RX}

denote the x and z coordinates of the central point of the receiving antenna, respectively. For this horizontal position, the centers of both antennas were located on the same imaginary vertical line. For each set of measurements, the PTE was measured at equidistant points with a step size (distance between adjacent points) of 30 mm.

The experimental results obtained for the first set of positions are shown in Figure 15. The very first vertical position corresponds to the case where the front ends of both antennas come in contact with each other. The figure reveals that the shape of the curve obtained experimentally is close to that obtained by employing the simulator. The maximum value of the measured PTE is slightly above 60%, whereas the highest theoretical PTE value is about 70%. This discrepancy arises because the closet wall conductivity and magnetic permeability values used in the simulations differ from the actual ones.

The field within the cavity is, in fact, a superposition of multiple waveguide modes, three of which ensure the energy transfer between the Rx and Tx. Thus, it is natural that such a field distribution is likely to be highly sensitive to changes in the operating frequency, as each of the propagation constants of the modes is frequency-dependent. Furthermore, the coupling between the modes and the feed line TEM wave is also dispersive.

To ascertain the effect of the frequency on the WPT system’s performance as for each set of measurements described above, a frequency sweep over a range from 800 MHz to 960 MHz was performed to experimentally obtain the frequency dependence of the PTE for a relatively wide frequency range. Several measurements were performed at different fixed vertical positions of the Rx antenna. For each horizontal position, one of the frequency sweeps was made for the vertical position of the Rx antenna, giving the highest PTE at 865.5 MHz. The other sweeps were performed for either the worst or locally optimal Rx antenna positions or for an arbitrary position. In the first case, such an optimal position corresponds to the distance between the antennas equal to 26 cm. Figure 16 presents the measured data, with each curve corresponding to one of six different Rx antenna positions.

As can be seen in the immediate vicinity of 865.5 MHz, the Rx position of 26 cm yields the highest PTE. The second optimal position is when

d_{TX - to - RX} = 29 cm

, in which case the PTE is slightly lower than the optimal one. Another local PTE peak is at

d_{TX - to - RX} = 29 cm .

The rest of the positions show unacceptably low PTE in the frequency range between 850 MHz and 870 MHz. Interestingly, at a frequency of about 830 MHz, all the considered Rx positions give quite a high PTE comparable to the optimal PTE at 865.5 MHz; moreover, the most optimal position at 830 MHz is

d_{TX - to - RX} = 38 cm .

The third pronounced PTE peak is seen in the range from 805 MHz to about 830 MHz.

The second set of PTE values is obtained by measuring and calculating the trans-mission coefficient at different

d_{TX - to - RX},

when the receiving antenna holder is shifted by 40 mm towards one the broader closet’s side walls (along the z-axis). Like in the previous case, the PTE is measured at equidistant points with a step size of 30 mm. The relevant results are shown in Figure 17.

The WPT system’s PTE measured at different frequencies when the Rx horizontal position was

x_{RX} = 0 mm

and

z_{RX} = 40 mm

is presented in Figure 18. The highest PTE position in this case was

d_{TX - to - RX} = 29 cm,

like in the first case discussed above. The frequency dependence was obtained only for four Rx antenna positions.

Figure 18 shows that the vertical Rx positions other than the optimal one give considerably lower PTE even at 865.5 MHz. This behavior may be attributed to a small distance between the Rx antenna and the side wall of the enclosure, resulting in weaker TEM wave coupling to the TE₀₁ mode, which has low electric field intensity near the side walls perpendicular to the z-axis. The other Rx antenna positions fail to provide adequate PTE at 865.5 MHz. Only when the Rx antenna is placed 56 cm away from the Rx antenna can a PTE of about 50% be achieved. Another pronounced PTE maximum was achieved for the same Rx position

(d_{TX - to - RX} = 29 cm),

but in a different frequency range, namely 800 MHz–830 MHz, with the system PTE peaking at 810 MHz. Compared to the previous horizontal Rx position, in the present case, there is a noticeable increase in the number of PTE peaks, with the PTE at these peaks being lower than those at 865.5 MHz and 830 MHz.

The third set of measurement data was acquired similarly, but this time, the measurement position (point) was shifted by 80 mm along the z-axis. The calculated and measured results obtained for the 8 cm shift of the receiving antennas are shown in Figure 19. Both the theoretical and experimental curves exhibit two PTE peaks.

The PTE value in the vicinity of the first peak is significantly higher than that of the second one. However, the highest PTE value in this case is lower than in the previous two, which the resonant cavity excitation theory can explain. More specifically, placing the receiving or transmitting probe close to the cavity walls (provided the walls are highly conducting) makes the coupling between the probes weaker due to the lower electric field.

As can be seen, as the distance between the midpoint of the antenna and the midpoint of the top of the closet increases, the received power level decreases for all considered positions of the Rx antenna along the y-axis. Such a result agrees well with the classical theory of excitation of cavity resonators—the closer the excitation source is to the wall of the resonator (in this case, the closet), the weaker the coupling between the resonator and the generator or any other source of electromagnetic fields.

The dependence of the WPT system’s PTE on the frequency measured at different vertical positions of the Rx antenna is exposed in Figure 20.

It is apparent that from the five different antenna separations, the highest PTE is achieved for

d_{TX - to - RX} = 32 cm,

which corresponds to the peak efficiency at 865.5 MHz seen in Figure 20. Nevertheless, there are more optimal PTE values over the 800 to 960 MHz frequency range. Namely, a higher PTE than that at 865.5 MHz can be achieved by slightly adjusting (increasing) the operating frequency since, according to the data of Figure 19, the PTE attains a local peak value at approximately 874 MHz. Furthermore, there is another distinct PTE maximum over a sub-range from 800 to about 840 MHz, with a peak value of about 72% achieved at 812 MHz. This peak value is the highest PTE achievable by the WPT system over the entire frequency range under study.

The fourth set of measured data was acquired for the case where both antennas lie in the same plane, but the receiving antenna is shifted a distance of 40 mm away from the closet door. The relevant measurement data are shown in Figure 21, where the simulated and measured PTE are plotted against the distance between the Tx and Rx antennas.

Although the calculated and measured data are still in good accord, noticeable discrepancies are observed in the Tx-to-Rx separation range from 35 to about 55 cm. Specifically, a shallow, sharp dip is present in the calculated data (solid blue line). At the same time, the measured PTE, both with and without the RF-Dc converter, shows continuous dependence on the distance between the antennas. Furthermore, the measured PTE maximum, occurring at about

d_{TX - to - RX} = 58 cm,

is shifted by 2 cm from the calculated one attained by the WPT system at

d_{TX - to - RX} = 60 cm

The measured PTE against the frequency for this Rx antenna position is presented in Figure 22. It is observed that the PTE reaches the highest value at vertical Rx antenna positions such that the separation distance between the antennas is

d_{TX - to - RX} = 29 cm .

The WPT system’s behavior resembles that of the previous case; namely, there is a more optimal PTE value over the frequency range from 800 MHz to 960 MHz, which can be achieved by slightly changing the operating frequency. The PTE attains a local peak value at approximately 860 MHz. Furthermore, there is another distinct efficiency maximum over a sub-range from 800 MHz to about 840 MHz, with a peak value at 818 MHz.

Another set of measurements was made for the same horizontal location of the transmitting antenna along the z-axis but a different positional along the x-axis (

x_{RX} = 40 mm

). The received power was again measured at a set of points on a vertical line along the largest closet dimension. The measurement and simulation results obtained for this horizontal Rx antenna position are presented in Figure 23. In contrast, Figure 24 presents several curves revealing how the PTE varies with frequency in the 800–960 MHz range when the antenna is shifted 40 mm along both the x- and z-axes. In this case, the measured PTE is seen to reach the highest value at the vertical Rx antenna position corresponding to the Tx-to-Rx antenna separation distance of

d_{TX - to - RX} = 26 cm .

Again, more than one PTE peak is observed over the frequency range under consideration. Additionally, the Rx antenna vertical position of

d_{TX - to - RX} = 26 cm

proves to give a higher PTE of about 78% at 823 MHz and ensures sufficiently high PTE values in a relatively wide band around this frequency.

Interestingly, when the Rx antenna is placed so that

d_{TX - to - RX} = 74 cm,

the PTE of the WPT system in the frequency range about the desired operating frequency of 865.5 MHz, namely, from 850 MHz to 875MHz, is unacceptably low; in comparison, in a narrow frequency range in the vicinity of 900 MHz, the PTE achieved by the WPT system is just slightly lower than in the optimal case at 865.5 MHz. Also, a few less pronounced PTE maxima occur at 915 MHz, 930 MHz, and 940 Mhz for the Rx antenna vertical positions of 56 cm, 68 cm, and 56 cm, respectively. The PTE attains a local peak value at about 860 MHz.

During the last set of measurements, the antenna was placed so that the receiving antenna was shifted 40 mm away from the closet door and 80 mm towards the same side wall as in the previous two measurements. The measured and calculated PTE for this antenna placement is presented in Figure 25. Similar to the case where the receiving antenna was shifted toward a side wall, the agreement between the theoretical and experimental results is very close, except that the experimental PTE is lower than the calculated one. Overall, the PTE curves obtained via measurements with and without the RF-DC converter and the one obtained via direct full-wave analysis (no decomposition method was employed) using Ansys HFSS 2023 R1 are mostly in close agreement, except for the range of Tx-to-Rx distance values from about 0 to 10 cm. In this range, the curve for the PTE measured with the RF-DC converter connected to the receiving antenna differs substantially from the other two curves. The discrepancy may be attributed to variations in the field distribution inside the enclosure caused by wires connecting the multimeter to the RF-DC converter’s output.

Figure 25 evidences that the most optimal Rx antenna vertical position in terms of the measured system’s PTE for this particular horizontal plane location is when the separation distance between the antennas is 26 cm. However, Figure 26 also reveals that this is only the case over the two frequency ranges: 815 MHz to about 835 MHz and 850 MHz to 875 MHz, including the operating frequency of 865.5 MHz. Remarkably, the highest PTE achieved by the WPT system under study is about 75%, which occurs at 822 MHz. However, this frequency is outside the main band, with the desired frequency of 865.5 MHz. Instead, it is in the first of the above-mentioned optimal frequency ranges. The WPT system PTE in the case when the Rx antenna is placed so that

d_{TX - to - RX} = 38 cm

shows a similar behavior, with two peaks occurring over the same two frequency ranges. Nevertheless, the PTE values are lower than in the optimal case

(d_{TX - to - RX} = 26 cm),

except for a relatively narrow frequency range from 875 MHz to 887 MHz, where for

d_{TX - to - RX} = 38 cm

the PTE provided by the WPT system at hand is seen to be approximately twice as high as that for

d_{TX - to - RX} = 26 cm .

While the other two Rx antenna vertical positions indicated in Figure 26 fail to ensure adequate PTE at the desired frequency, they give slightly lower efficiency peaks at higher frequencies.

To summarize, the above data presented in graphical form show that both measured curves are consistent with those calculated, as in the previous measurements. Thus, they confirm the possibility of reasonably accurately estimating the resonant cavity-based WPT system’s behavior in terms of high PTE region location, the PTE versus the Tx-to-Rx distance curve shape, and the maximum achievable PTE. Additionally, the frequency sweep data unambiguously demonstrate that despite the WPT system’s quite high PTE sensitivity to variations in the system’s operating frequency, one can still ensure acceptable PTE over multiple frequency bands, not only around the desired operating frequency.

5. Conclusions

A resonant cavity-based Wireless Power Transfer (WPT) system intended for remote charging of a battery incorporated into the smart suit developed by our partners was thoroughly investigated and optimized. The WPT systems under study utilize a standard carbon steel closet as a metallic enclosure required to confine electromagnetic fields to the cavity interior. The proposed solution is cost-effective, as it does not require an expensive enclosure made of copper; however, it exhibits a lower power transfer efficiency than its copper counterparts with the same dimensions.

Two printed antennas were fabricated and used to ensure power transfer from an RF generator to the receiver (an RF-DC converter) inside the closet. Due to the softness of the suit material, the precise antenna placement was difficult to ensure. This issue can be mitigated by making the WPT system’s efficiency less sensitive to the receiver position within the closet. The results confirm the findings of our previous study, in which only full-wave analysis was carried out. Namely, the optimization results show that extending the design of the Tx antenna by introducing an additional conducting strip parallel to the dipole located a distance away results in wider regions with high electric field intensity. These strong fields, in turn, ensure sufficiently high power transfer efficiency (PTE) of the cavity-based WPT system for reliable charging of a battery incorporated into smart clothing placed inside the closet. Wider high PTE regions inside the closet are of utmost practical importance in smart suit charging, as this mitigates the sensitivity of the system’s PTE and, therefore, the overall WPT system’s efficiency to the Rx antenna position.

The mechanism responsible for the increase in the high PTE region is explained based on the theoretical analysis of the interaction between the sub-models containing a dipole antenna and a conducting strip (director). It was found that the change in the phases of three waveguide modes (TE₀₁, TE₂₁ and TM₂₁) ensuring the power transfer between the Tx and Rx ends may significantly extend the high PTE region.

An exhaustive search-based optimization was performed to find the optimal antenna parameters, giving the widest high PTE region. Since the exhaustive search-based optimization requires exceedingly large amounts of CPU time when applied directly to the original WPT system’s model, the model decomposition method was employed. Specifically, the WPT model was divided into three sub-models. For each sub-model, the individual generalized S parameters were computed using the full-wave simulator Ansys HFSS 2023 R1 for different sub-model parameter combinations and exported into files. Then, the WPT system’s S parameters were computed using a custom MATLAB routine by combining the precomputed sub-model matrices. A substantial reduction in the CPU time was achieved because the sub-model parameter space dimensions are much smaller than that of the entire model, and the S matrix recombination requires just a few seconds per combination.

The closet-based WPT system’s PTE was calculated at different vertical and horizontal positions of the Rx antenna inside the closet to examine the effect of the antenna position on the system’s performance. The simulated results were also experimentally verified. The comparison of calculated and measured PTE shows close agreement between the results. The shape of the theoretical curves obtained for different positions of the receiving antenna mostly agrees with the experimentally obtained ones. The highest measured PTE of the cavity-based WPT system (about 60%) was achieved when the receiving antenna was located halfway between the side walls of the closet. Some discrepancies between the measured and calculated results are attributable to the presence of coaxial cables and wires, which were considered in the simulation.

Author Contributions

Conceptualization, R.K. and D.P.; methodology, R.K.; software, R.K.; validation, R.K., J.E. and S.T.; formal analysis, D.P.; investigation, R.K.; resources, D.P.; data curation, J.E.; writing—original draft preparation, R.K.; writing—review and editing, D.P., S.T. and K.G.; visualization, R.K.; supervision, D.P.; project administration, D.P.; funding acquisition, D.P. All authors have read and agreed to the published version of the manuscript.

Funding

This research is funded by the Latvian Council of Science, project “Smart Materials, Photonics, Technologies and Engineering Ecosystem” No VPP-EM-FOTONIKA-2022/1-0001.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author(s).

Conflicts of Interest

The authors declare no conflicts of interest.

References

Kandris, D.; Nakas, C.; Vomvas, D.; Koulouras, G. Applications of Wireless Sensor Networks: An Up-to-Date Survey. Appl. Syst. Innov. 2020, 3, 14. [Google Scholar] [CrossRef]
Lin, R.; Kim, H.J.; Achavananthadith, S.; Kurt, S.A.; Tan, S.C.C.; Yao, H.; Tee, B.C.K.; Lee, J.K.W.; Ho, J.S. Wireless battery-free body sensor networks using near-field-enabled clothing. Nat. Commun. 2020, 11, 444. [Google Scholar] [CrossRef]
Bootsman, R.; Markopoulos, P.; Qi, Q.; Wang, Q.; Timmermans, A.A. Wearable technology for posture monitoring at the workplace. Int. J. Hum.-Comput. Stud. 2019, 132, 99–111. [Google Scholar] [CrossRef]
Murphy, M.; Bergquist, F.; Hagström, B.; Hernández, N.; Johansson, D.; Ohlsson, F.; Sandsjö, L.; Wipenmyr, J.; Malmgren, K. An upper body garment with integrated sensors for people with neurological disorders—Early development and evaluation. BMC Biomed. Eng. 2019, 1, 3. [Google Scholar] [CrossRef]
Chang, C.W.; Riehl, P.; Lin, J. Alignment-Free Wireless Charging of Smart Garments with Embroidered Coils. Sensors 2021, 21, 7372. [Google Scholar] [CrossRef]
Ancans, A.; Greitans, M.; Cacurs, R.; Banga, B.; Rozentals, A. Wearable Sensor Clothing for Body Movement Measurement during Physical Activities in Healthcare. Sensors 2021, 21, 2068. [Google Scholar] [CrossRef] [PubMed]
Rahimizadeh, S.; Korhummel, S.; Kaslon, B.; Popovic, Z. Scalable adaptive wireless powering of multiple electronic devices in an over-moded cavity. In Proceedings of the 2013 IEEE Wireless Power Transfer (WPT); Perugia, Italy, 15–16 May 2013; IEEE: Piscataway, NJ, USA, 2013; pp. 84–87. [Google Scholar] [CrossRef]
Kusnins, R.; Pikulins, D.; Eidaks, J.; Tjukovs, S.; Aboltins, A. Study on a Metal Closet Based Wireless Power Transfer System for Smart Suit Charging. In Proceedings of the 2023 Workshop on Microwave Theory and Technology in Wireless Communications (MTTW), Riga, Latvia, 4–6 October 2023; IEEE: Piscataway, NJ, USA, 2023; pp. 56–61. [Google Scholar] [CrossRef]
Rehman, M.; Nallagownden, N.; Baharudin, Z. A Review of Wireless Power Transfer System Using Inductive and Resonant Coupling. J. Ind. Technol. 2018, 26, 1–24. [Google Scholar] [CrossRef]
Mohammad, M.; Onar, O.C.; Su, G.J.; Pries, J.; Galigekere, V.P.; Anwar, S.; Asa, E.; Wilkins, J.; Wiles, R.; White, C.P.; et al. Bidirectional LCC–LCC-Compensated 20-kW Wireless Power Transfer System for Medium-Duty Vehicle Charging. IEEE Trans. Transp. Electrif. 2021, 7, 1205–1218. [Google Scholar] [CrossRef]
Van Mulders, J.; Delabie, D.; Lecluyse, C.; Buyle, C.; Callebaut, G.; Van der Perre, L.; De Strycker, L. Wireless Power Transfer: Systems, Circuits, Standards, and Use Cases. Sensors 2022, 22, 5573. [Google Scholar] [CrossRef]
Valenta, C.R.; Durgin, G.D. Harvesting Wireless Power: Survey of Energy-Harvester Conversion Efficiency in Far-Field, Wireless Power Transfer Systems. IEEE Microw. Mag. 2014, 15, 108–120. [Google Scholar] [CrossRef]
Hidalgo-Leon, R.; Urquizo, J.; Silva, C.E.; Silva-Leon, J.; Wu, J.; Singh, P.; Soriano, G. Powering nodes of wireless sensor networks with energy harvesters for intelligent buildings: A review. Energy Rep. 2022, 8, 3809–3826. [Google Scholar] [CrossRef]
Park, H.S.; Hong, S.K. A performance predictor of beamforming versus time-reversal based far-field wireless power transfer from linear array. Sci. Rep. 2021, 11, 22743:1–22743:9. [Google Scholar] [CrossRef]
Lopezf, J.; Tsay, J.; Guzman, B.A.; Mayeda, J.; Lie, D.Y.C. Phased arrays in wireless power transfer. In Proceedings of the 2017 IEEE 60th International Midwest Symposium on Circuits and Systems (MWSCAS), Boston, MA, USA, 6–19 August 2017; IEEE: Piscataway, NJ, USA, 2017; pp. 5–18. [Google Scholar] [CrossRef]
Chabalko, M.J.; Sample, A.P. Resonant cavity mode enabled wireless power transfer. Appl. Phys. Lett. 2014, 105, 243902. [Google Scholar] [CrossRef]
Chabalko, M.J.; Sample, A.P. Electric field coupling to short dipole receivers in cavity mode enabled wireless power transfer. In Proceedings of the 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, Canada, 19–24 July 2015; IEEE: Piscataway, NJ, USA, 2015; pp. 410–411. [Google Scholar] [CrossRef]
Haus, H.A.; Huang, W. Coupled-mode theory. Proc. IEEE 1991, 79, 1505–1518. [Google Scholar] [CrossRef]
Sasatani, T.; Chabalko, M.; Kawahara, Y.; Sample, A. Geometry-Based Circuit Modeling of Quasi-Static Cavity Resonators for Wireless Power Transfer. IEEE Open J. Power Electron. 2022, 3, 382–390. [Google Scholar] [CrossRef]
Chabalko, M.J.; Sample, A.P. Three-Dimensional Charging via Multimode Resonant Cavity Enabled Wireless Power Transfer. IEEE Trans. Power Electron. 2015, 30, 6163–6173. [Google Scholar] [CrossRef]
Mei, H.; Thackston, K.A.; Bercich, R.A.; Jefferys, J.G.R.; Irazoqui, P.P. Cavity Resonator Wireless Power Transfer System for Freely Moving Animal Experiments. IEEE Trans. Biomed. Eng. 2017, 64, 775–785. [Google Scholar] [CrossRef]
Wang, X.; Wang, X.; Lu, M.A. Retro-reflective Scheme for Wireless Power transmission in Fully Enclosed Environments. In Proceedings of the 2019 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, Atlanta, GA, USA, 7–12 July 2019; IEEE: Piscataway, NJ, USA, 2019; pp. 1465–1466. [Google Scholar] [CrossRef]
Chabalko, M.J.; Shahmohammadi, M.; Sample, A.P. Quasistatic Cavity Resonance for Ubiquitous Wireless Power Transfer. PLoS ONE 2017, 12, e0169045. [Google Scholar] [CrossRef] [PubMed]
Sasatani, T.; Yang, C.J.; Chabalko, M.J.; Kawahara, Y.; Sample, A.P. Room-Wide Wireless Charging and Load-Modulation Communication via Quasistatic Cavity Resonance. Proc. ACM Interact. Mob. Wearable Ubiquitous Technol. 2018, 2, 1–23. [Google Scholar] [CrossRef]
Sasatani, T.; Chabalko, M.J.; Kawahara, Y.; Sample, A.P. Multimode Quasistatic Cavity Resonators for Wireless Power Transfer. IEEE Antennas Wirel. Propag. Lett. 2017, 16, 2746–2749. [Google Scholar] [CrossRef]
Sasatani, T.; Sample, A.P.; Kawahara, Y. Room-scale magnetoquasistatic wireless power transfer using a cavity-based multimode resonator. Nat. Electron. 2021, 4, 689–697. [Google Scholar] [CrossRef]
Abdelraheem, A.; Sinanis, M.D.; Peroulis, D. A New Wireless Power Transmission (WPT) System for Powering Wireless Sensor Networks (WSNs) in Cavity-Based Equipment. In Proceedings of the 2019 IEEE 20th Wireless and Microwave Technology Conference (WAMICON), Cocoa Beach, FL, USA, 8–9 April 2019; IEEE: Piscataway, NJ, USA, 2019; pp. 1–5. [Google Scholar] [CrossRef]
Korhummel, S.; Rosen, A.; Popovic, Z. Over-Moded Cavity for Multiple-Electronic-Device Wireless Charging. IEEE Trans. Microw. Theory Tech. 2014, 62, 1074–1079. [Google Scholar] [CrossRef]
Takano, I.; Furusu, D.; Watanabe, Y.; Tamura, M. Study on Cavity Resonator wireless power transfer to sensors in an enclosed space with scatterers. In Proceedings of the 2017 IEEE MTT-S International Conference on Microwaves for Intelligent Mobility (ICMIM), Nagoya, Japan, 19–21 March 2017; IEEE: Piscataway, NJ, USA, 2017; pp. 79–82. [Google Scholar] [CrossRef]
Akai, S.; Saeki, H.; Tamura, M. Power Supply to Multiple Sensors and Leakage Field Analysis Using Cavity Resonance-Enabled Wireless Power Transfer. In Proceedings of the 2022 IEEE/MTT-S International Microwave Symposium, Denver, CO, USA, 19–24 June 2022; IEEE: Piscataway, NJ, USA, 2022; pp. 271–274. [Google Scholar] [CrossRef]
Yue, Z.; Zhang, Q.; Yang, Z.; Bian, R.; Zhao, D.; Wang, B.Z. Wall-Meshed Cavity Resonator-Based Wireless Power Transfer Without Blocking Wireless Communications with Outside World. IEEE Trans. Ind. Electron. 2022, 69, 7481–7490. [Google Scholar] [CrossRef]
Yue, Z.; Zhang, Q.; Zhao, D.; Wang, B.Z. Three-Dimensional Wireless Power Transfer Based on Meshed Cavity Resonator. In Proceedings of the 2020 International Conference on Microwave and Millimeter Wave Technology (ICMMT), Shanghai, China, 17–20 May 2020; IEEE: Piscataway, NJ, USA, 2021; pp. 1–3. [Google Scholar] [CrossRef]
Zhang, K.Q.; Li, D.J. Electromagnetic Theory for Microwaves and Optoelectronics; Springer: New York, NY, USA, 1998. [Google Scholar]
Jackson, J.D. Classical Electrodynamics, 2nd ed.; Wiley: New York, NY, USA, 1975. [Google Scholar]
Hung, C.L.; Yeh, Y.S. The propagation constants of higher order modes in coaxial waveguides with finite conductivity. Int. J. Infrared Millim. Waves 2005, 26, 29–39. [Google Scholar] [CrossRef]
Jiao, C.Q.; Zheng, N.; Luo, J.R. A comparison of the attenuation of high-order mode in coaxial waveguide due to inner and outer conductor losses. J. Infrared Millim. Terahertz Waves 2010, 31, 858–865. [Google Scholar] [CrossRef]
Nocedal, J.; Wright, S.J. Numerical Optimization; Springer: New York, NY, USA, 2006. [Google Scholar] [CrossRef]
Migdalas, A.; Pardalos, P.M.; Värbrand, P. From Local to Global Optimization; Springer: Berlin, Germany, 2001. [Google Scholar]
Zelinka, I.; Snášel, V.; Abraham, A. Handbook of Optimization; Springer: Berlin, Germany, 2013. [Google Scholar]
Simon, D. Evolutionary Optimization Algorithms: Biologically-Inspired and Population-Based Approaches to Computer Intelligence; John Wiley & Sons Inc.: New York, NY, USA, 2013. [Google Scholar]

Figure 1. A schematic diagram of the cavity-based (closet-based) WPT system under study.

Figure 2. The Ansys HFSS model of the steel closet with the Rx antenna holder.

Figure 3. The HFSS model of the receiving antenna: top view (a), bottom view (b).

Figure 4. The HFSS model of the transmitting antenna: top view (a) and bottom view (b).

Figure 5. The Ansys HFSS model constructed to determine the generalized scattering matrix relating the waveguide mode amplitudes and phases at both ends (ports) of a waveguide section containing a dipole antenna model and the amplitude and phase of the dipole feed line’s TEM wave.

Figure 6. The Ansys HFSS model for finding the generalized scattering matrix relating the waveguide mode amplitudes and phases at both ends of a waveguide section containing the Yagi-like antenna’s director (a) and the same model with a port and the relevant mode integration lines highlighted (b).

Figure 7. The coupling coefficients between the TEM line mode and the three power-carrying waveguide modes (TE₀₁, TM₂₁, and TE₂₁) against the transmitting dipole length, calculated at

L_{feed} = 70 mm

and

d_{dip - dir} = 145 mm

without the PEC terminating plate behind the dipole antenna. The waveguide modes propagating in the desired direction are indicated by (+), whereas those propagating in the opposite direction are indicated by (−).

Figure 7. The coupling coefficients between the TEM line mode and the three power-carrying waveguide modes (TE₀₁, TM₂₁, and TE₂₁) against the transmitting dipole length, calculated at

L_{feed} = 70 mm

and

d_{dip - dir} = 145 mm

Figure 8. The coupling coefficients between the TEM line mode and the three power-carrying waveguide modes (TE₀₁, TM₂₁, and TE₂₁) (a) and their phases relative to that of the incident TEM wave against the transmitting dipole length (b), calculated for a waveguide section terminated into a conducting plate at the rear end containing a dipole antenna with

L_{feed} = 70 mm

L_{feed} = 70 mm

Figure 9. The coupling coefficients between the TEM line mode and the three power-carrying waveguide modes (TE₀₁, TM₂₁, and TE₂₁) against the transmitting dipole length, calculated at

L_{feed} = 70 mm

and

d_{dip - dir} = 145 mm

Figure 9. The coupling coefficients between the TEM line mode and the three power-carrying waveguide modes (TE₀₁, TM₂₁, and TE₂₁) against the transmitting dipole length, calculated at

L_{feed} = 70 mm

and

d_{dip - dir} = 145 mm

Figure 10. The coupling coefficients between the TEM mode of the Yagi-like antenna and waveguide modes TE₀₁, TM₂₁, and TE₂₁ against the transmitting dipole length, calculated at

L_{feed} = 70 mm

and

d_{dip - dir} = 145 mm

(a) and transmitting antenna feed line length, calculated at

L_{dipole} = 62 mm

and

d_{dip - dir} = 130 mm

(b) with a conducting plate placed behind the antenna.

Figure 10. The coupling coefficients between the TEM mode of the Yagi-like antenna and waveguide modes TE₀₁, TM₂₁, and TE₂₁ against the transmitting dipole length, calculated at

L_{feed} = 70 mm

and

d_{dip - dir} = 145 mm

(a) and transmitting antenna feed line length, calculated at

L_{dipole} = 62 mm

and

d_{dip - dir} = 130 mm

(b) with a conducting plate placed behind the antenna.

Figure 11. The absolute values of resonant cavity-based WPT system scattering parameters (TEM mode parameters) (a) and their phases (b) against frequency, calculated using the decomposition approach and directly using Ansys HFSS.

Figure 12. The PTE as a function of the separation distance between the Tx and Rx antennas, calculated for the optimal values of the Yagi-like and dipole antenna parameters (see Table 3 and Table 4) (a) and parameter values giving a more extended high PTE region than the optimal one, but at the cost of a sharp dip almost in the middle of the region and WPT model with a hypothetical mode phase shifter optimized to yield the widest high PTE region (b).

Figure 13. Experimental setup involving a carbon steel closet used as a resonant cavity in the WPT system under study: antenna-based WPT system inside the metal closet (a), experimental setup involving the TX and RX antennas, the signal generator operating at 865.5 MHz, and the power meter used to measure the received power (b).

Figure 14. The measured PCE of the BAT6804 Schottky diode-based voltage doubler RF-DC converter as a function of the input RF power level in dBm (a) and as function of the frequency at different fixed-input RF power levels (b).

Figure 15. The calculated and measured PTE against the separation between the receiving and transmitting antennas

x_{RX} = 0 mm

and

z_{RX} = 0 mm

Figure 15. The calculated and measured PTE against the separation between the receiving and transmitting antennas

x_{RX} = 0 mm

and

z_{RX} = 0 mm

Figure 16. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 0 mm

and

z_{RX} = 0 mm

Figure 16. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 0 mm

and

z_{RX} = 0 mm

Figure 17. The calculated and measured PTE against the separation between the receiving and transmitting antennas at

x_{RX} = 0 mm

and

z_{RX} = 40 mm

Figure 17. The calculated and measured PTE against the separation between the receiving and transmitting antennas at

x_{RX} = 0 mm

and

z_{RX} = 40 mm

Figure 18. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 0 mm

and

z_{RX} = 40 mm

Figure 18. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 0 mm

and

z_{RX} = 40 mm

Figure 19. The calculated and measured PTE against the separation between the receiving and transmitting antennas at

x_{RX} = 0 mm

and

z_{RX} = 80 mm

Figure 19. The calculated and measured PTE against the separation between the receiving and transmitting antennas at

x_{RX} = 0 mm

and

z_{RX} = 80 mm

Figure 20. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 0 mm

and

z_{RX} = 80 mm

Figure 20. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 0 mm

and

z_{RX} = 80 mm

Figure 21. The calculated and measured PTE against the separation between the receiving and transmitting antennas at

x_{RX} = 40 mm

and

z_{RX} = 0 mm

Figure 21. The calculated and measured PTE against the separation between the receiving and transmitting antennas at

x_{RX} = 40 mm

and

z_{RX} = 0 mm

Figure 22. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 40 mm

and

z_{RX} = 0 mm

Figure 22. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 40 mm

and

z_{RX} = 0 mm

Figure 23. The calculated and measured PTE against the separation between the receiving and transmitting antennas at

x_{RX} = 40 mm

and

z_{RX} = 40 mm

Figure 23. The calculated and measured PTE against the separation between the receiving and transmitting antennas at

x_{RX} = 40 mm

and

z_{RX} = 40 mm

Figure 24. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 40 mm

and

z_{RX} = 40 mm

Figure 24. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 40 mm

and

z_{RX} = 40 mm

Figure 25. The calculated and measured PTE against the separation between the receiving and transmitting antennas at

x_{RX} = 40 mm

and

z_{RX} = 80 mm

Figure 25. The calculated and measured PTE against the separation between the receiving and transmitting antennas at

x_{RX} = 40 mm

and

z_{RX} = 80 mm

Figure 26. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 40 mm

and

z_{RX} = 80 mm

Figure 26. The measured PTE against the frequency for the Rx antenna located at

x_{RX} = 40 mm

and

z_{RX} = 80 mm

Table 1. Waveguide mode parameters.

Mode	Attenuation Constant [s⁻¹]	Phase Constant [s⁻¹]	Mode	Attenuation Constant [s⁻¹]	Phase Constant [s⁻¹]
TE10	0.0212	16.9227	TE21	0.0747	6.2534
TE01	0.0279	14.5773	TE30	7.7375	0.0968
TM11	0.0567	13.0302	TE02	11.7587	0.0900
TE11	0.0399	13.0133	TE32	13.2780	0.0911
TE20	0.0404	12.5117	TE41	13.3381	0.0327
TM21	0.1477	6.3300	TE22	13.4590	0.0973

Table 2. WPT model parameters used for the validation of the model decomposition method.

Parameter (Case 1)	Value	Parameter (Case 2)	Value
Feed line length (TX)	75 mm	Feed line length (TX)	75 mm
Dipole length (TX)	70 mm	Dipole length (TX)	90 mm
Director length (TX)	160 mm	Director length (TX)	140 mm
Director-to-dipole separation (TX)	195 mm	Director-to-dipole separation (TX)	195 mm
Feed line length (RX)	70 mm	Feed line length (RX)	70 mm
Dipole length (RX)	65 mm	Dipole length (RX)	65 mm
Top-to-TX distance	34 mm	Top-to-TX distance	40 mm

Table 3. Optimal TX antenna parameter values.

Parameter	Value	Parameter	Value
Feed line length	70 mm	Dipole width	5 mm
Feed line width	5 mm	Substrate width	180 mm
Reflector width	20 mm	Substrate length	280 mm
Reflector length	170 mm	Dipole length	63 mm
Director width	5 mm	Director length	160 mm
Director-to-dipole separation	180 mm

Table 4. Optimal RX antenna parameter values.

Parameter	Value	Parameter	Value
Feed line length	70 mm	Dipole width	5 mm
Feed line width	5 mm	Substrate width	180 mm
Reflector width	20 mm	Substrate length	120 mm
Reflector length	160 mm	Dipole length	64 mm

Table 5. TX antenna parameter values.

Parameter	Value	Parameter	Value
Feed line length	75 mm	Dipole width	5 mm
Feed line width	5 mm	Substrate width	180 mm
Reflector width	20 mm	Substrate length	280 mm
Reflector length	170 mm	Dipole length	64.4 mm
Director width	5 mm	Director length	160 mm
Director-to-dipole separation	190 mm

Table 6. RX antenna parameter values.

Parameter	Value	Parameter	Value
Feed line length	85 mm	Dipole width	5 mm
Feed line width	5 mm	Substrate width	180 mm
Reflector width	20 mm	Substrate length	120 mm
Reflector length	160 mm	Dipole length	65 mm

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Kusnins, R.; Tjukovs, S.; Eidaks, J.; Gailis, K.; Pikulins, D. Model Decomposition-Based Approach to Optimizing the Efficiency of Wireless Power Transfer Inside a Metal Enclosure. Appl. Sci. 2024, 14, 11733. https://doi.org/10.3390/app142411733

AMA Style

Kusnins R, Tjukovs S, Eidaks J, Gailis K, Pikulins D. Model Decomposition-Based Approach to Optimizing the Efficiency of Wireless Power Transfer Inside a Metal Enclosure. Applied Sciences. 2024; 14(24):11733. https://doi.org/10.3390/app142411733

Chicago/Turabian Style

Kusnins, Romans, Sergejs Tjukovs, Janis Eidaks, Kristaps Gailis, and Dmitrijs Pikulins. 2024. "Model Decomposition-Based Approach to Optimizing the Efficiency of Wireless Power Transfer Inside a Metal Enclosure" Applied Sciences 14, no. 24: 11733. https://doi.org/10.3390/app142411733

APA Style

Kusnins, R., Tjukovs, S., Eidaks, J., Gailis, K., & Pikulins, D. (2024). Model Decomposition-Based Approach to Optimizing the Efficiency of Wireless Power Transfer Inside a Metal Enclosure. Applied Sciences, 14(24), 11733. https://doi.org/10.3390/app142411733

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Model Decomposition-Based Approach to Optimizing the Efficiency of Wireless Power Transfer Inside a Metal Enclosure

Abstract

1. Introduction

2. Theoretical Analysis of Cavity-Based WPT Systems

2.1. WPT Model Decomposition

2.2. Analysis of the WPT System’s Sub-Models

3. WPT Model Decomposition Approach Validation and Optimization

4. Experimental Verification and Discussion

4.1. Measurements of the WPT System’s PTE

4.2. Measurement Results

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI