Estimation of the Influence of the Coil Resistance on the Power and Efficiency of the WPT System

Abstract

This paper presents the results of an analysis of a low-power Wireless Power Transfer (WPT) system. The system consists of periodically distributed planar spiral coils that form the transmitting and receiving planes. An analytical and numerical analysis of the WPT system, over the frequency range from 100 to 1000 kHz, was carried out. A simpler and faster solution is the proposed use of an equivalent circuit represented by a single WPT cell. The influence of coil resistance changes on the power and efficiency of the WPT system was studied. This was obtained by changing the diameter of the wire from which the coils were wound. In addition, the size of the coil, the number of turns, and the distance between the two planes have changed. After a detailed analysis, the results showed that the highest efficiency values were obtained for a wire diameter of 200 μm, which means the lowest coil resistance. However, the lowest efficiency values were obtained for the smallest wire diameter, i.e., 100 µm, which means the highest coil resistance. In this case, the efficiency decreased by more than 40%. Based on the calculation results, it was also shown that it was better to accept the skin effect (efficiency decreased below 7%) than to reduce the wire diameter to eliminate it.

1. Introduction

Over the past two decades, mobile and wireless communication networks have experienced tremendous growth, now with billions of connected devices. The standardization and deployment of wide area and local area networks is growing rapidly and further development is expected. Wireless communication has become an integral part of everyday life [1,2,3,4]. Today, every household has a variety of devices that require charging. Smartphones, Bluetooth headphones, watches and portable media players are devices that have a wired power adapter in the set. Each of these devices needs a cable with the appropriate connector and the associated power supply. Some devices require you to insert the USB plug in the correct way so that it fits into the socket. Over time, the charging socket wears out and no longer guarantees a secure connection. The result is loose contact and costly repairs. A much better solution would be if no cables or plug connectors were needed for charging. This is why manufacturers have developed wireless charging (WPT), also known as inductive charging (IPT). Wireless charging means non-contact energy transfer via electromagnetic induction over short distances. The principle of operation is based on the inductive coupling of the two coils. One of the coils acts as a transmitter and the other as a receiver [5]. The physical interpretation is related to electromagnetic induction. A time-varying current (alternating current, AC) flowing through the primary coil generates a time-varying magnetic field. The magnetic field of the primary coil partially penetrates the adjacent secondary coil. The magnetic field of the primary coil generates a voltage induced on the receiver coil and, after connecting the load, a time-varying current.

Wireless power transfer (WPT) systems are used in many fields, such as electric vehicle charging [6,7,8,9,10,11], implantable medical devices [12,13,14], and consumer electronics [5,15]. WPT is also used in robotic systems [16], communication sets [17,18], and in the Internet of Things (IoT) [16,17]. WPT has also found its application in biology and medicine. In publication [19], the authors presented an extendable platform for transmitting power to a moving object receiving power from the array. The future application of the described transmitter may be the study of the neurobehavior of free-moving animals and research of the brain–machine interface in medicine. One recent solution is to use textile-based flexible coil engineering design techniques to charge wireless electronics. The authors [20] presented a multi-modal portable respiratory rate monitoring device for childhood pneumonia detection. The authors proposed an automated novel respiratory rate monitoring device consisting of a textile glove and dry electrodes. This glove and system can take advantage of the child’s relaxed posture when resting on the caregiver’s lap. This portable system is non-invasive and made with inexpensive instrumentation integrated into a non-standard textile glove.

Conductive fibers/yarns and inks are an essential part of the next generation of wearable electronics, enabling electronic functions to be seamlessly integrated into textiles through various techniques and processes [21]. The authors [22] made a kind of fabric of conductive and elastomeric fibers to be worn directly on various parts of the body to detect stress. A textile induction coil was developed by researchers for use in electrocardiogram monitoring [23]. To take advantage of wireless power transmission, the coil was constructed of a low-resistivity conductive material, such as copper, which minimized coil losses. Conductive textile materials have a high resistance compared to commonly used metals, which can significantly affect the performance of wireless power transmission systems [24]. For this reason, commercially available coils are made of copper wires. However, the biggest problem with this type of coil is the skin effect, which reduces the efficiency of a WPT system and can even cause reverse current in some cases. Litz wire is used to eliminate this problem. Currently, the commercially available coils used as receivers for wireless charging systems are made by winding one or two layers of the Litz wire. The coils are stiff and not suitable to be embedded in a garment for a wirelessly charging device worn next to the skin for monitoring the physiological properties. Due to the excessive thickness of the wire and the external dimensions of coils made of Litz wire, such a solution is not suitable for planar coils, which are proposed in the article, or coils used in textile electronics. Litz wire was used to reduce the skin effect. This solution is highly recommended at higher frequencies and larger coil sizes, for example, when charging electric cars. However, with small coils, e.g., 10 mm, the use of Litz wire is physically impossible. For this reason, with small coils, it is necessary to study the influence of the coil resistance on the parameters of the system and to search for optimal solutions [25]. The coils made of Litz wire have a wire diameter of not less than 1 mm, which means that with small coils, whose dimensions are in millimeters, it is impossible to wind more turns and thus achieve higher efficiency values.

WPT has also found its application in intelligent textiles. In the article [26,27], the authors analysed the use of a textile-based coil taking into account the QI standard. WPC (Wireless Power Consortium) is a non-profit organization founded in 2008, connecting over 650 member companies from around the world. Together with its members, the WPC develops manufacturer-independent standards for wireless power. The QI standard not only precisely defines the functional aspects of inductive charging, but also issues related to the safety and reliable detection of foreign bodies. The authors [26] studied three different techniques and processes for the development of textile conductive coils, including embroidery, laser cutting and screen-printing, which can be embedded in a garment layer. Within each method, different conductive materials were tried. Limitations on the dimensions of the coils imposed by the manufacturing process of each method have been identified. They have a direct effect on the resistance of the coil and thus its performance. Although the laser-cut coil has a relatively lower resistance, it has difficulty keeping the entire coil within the specified dimension. On the other hand, the silk-screen coil has very high resistance, but the problem is applying a conductive layer and conducting continuously. The embroidered coil was the best of the three solutions in terms of maintaining the specified dimensions and the spacing between turns and the performance quality of the system with this coil. The authors suggested that such a solution is a way to charge wearable sensors in applications, such as health monitoring or protection.

In paper [27], the authors studied three different methods of producing textile rolls. Five different coil materials were used. The manufacturing process resulted in a total of 26 different coils that differed in shape and size, number of turns and starting material. Coils with a large number of turns and a small distance between turns always gave better results. However, textile coils that used copper wire to wrap the yarn showed eddy currents that negatively affected the magnetic field. Another problem in the manufacture of Shieldex yarns is the lack of insulation, which causes short circuits. The authors studied the electrical characteristics of sewn, embroidered, laser cut, and printed induction coils for wireless power transmission. They used a reference coil design of the Qi standard called WE-WPCC TRASM Qi-A5 (Würth Elektronik GmbH & Co. KG, Waldenburg, Deutschland) [26]. Embroidered coils had diameters of 0.04, 0.05, and 0.06 m. One of the coils was made of silver-coated yarn. A coil was laser cut. They showed that this type of coil was not appropriate for a WPT system. The reason was the high resistance of the coil, which resulted in the low efficiency of the system, not exceeding 40%. Laser cutting was also a problem, as it required two cuts, making it impossible to maintain accuracy in the diameter of the turns and the distance between them. This technique made the thickness of the conductive material different and resulted in a lower current density in some parts of the turns. The use of Litz wire is not possible with small coils used for low power charging, such as in smart textiles, small household appliances (e.g., toothbrushes, battery chargers), or portable lighting. In these cases, the size of the coils prevents the use of Litz wire due to the excessively large diameter of the wire. For this reason, the article also discusses a compromise between reducing the wire diameter and avoiding the skin effect or using a larger wire diameter with the skin effect.

In WPT technology, metamaterials have also been used to better control the magnetic field and the possibility of shaping it [28,29]. Thanks to the advantages of metamaterials, it is possible to shape coils in such a way that it is possible to achieve the assumed parameters of the system. In publications [30,31], various coil shapes and the influence of their geometrical and electrical parameters on, e.g., system efficiency, are analysed. These factors have a huge influence on the quality of transmission and thus on the efficiency of the system, which determines the functionality of the devices. The authors of [32] analysed the relationship between coil efficiency and three geometrical parameters of the coupling coils and the operating frequency. For this reason, the solution space of the design parameters was reduced from four dimensions (coil geometry and frequency) to three dimensions (coil geometry), and a particle swarm optimization algorithm was used to solve it. A modified AC resistance evaluation method was also shown in which the conductive resistance is calculated from the frequency-dependent formula. The results of the experiments showed that the proposed method is precise for close wound coils.

In the field of high power transmission, such as the WPT EV system, power transmission efficiency is delayed due to wired charging and losses caused by substrate shielding materials. For this reason, in article [33], the conduction resistance of Litz wire coils without substrate was analysed. The induction resistance of the coil with the substrate materials was also modeled. In this article, the authors analysed a single coil and the influence of substrate stratification on system efficiency. The efficient operation of the WPT system is a major design challenge in WPT applications. Reference [34] presents a method of designing a high-efficiency WPT system. Restrictions on the receiver coil size are included. The authors mainly analysed the variability of the dimensions of the transmitting coil. While most WPT systems are designed for mobile applications, the authors of [35] focused on reducing coil resistance, which translates into higher efficiency with the lowest possible weight. The results showed that a more conductive material increased weight and efficiency.

In article [36], in order to reduce the influence of the internal resistance of the source and improve the efficiency, a source coil was added to the two-coil structure, creating a three-coil structure. In most articles [36,37], the authors deal with coil-to-coil systems that are used to charge electric vehicles and other larger devices. This article introduces another solution that allows you to charge multiple devices or one. This is possible thanks to the use of flat planar coils forming the transmitting and receiving planes. The topology of the system and the phenomena occurring between the coils also affect the parameters of the WPT system. Typical topologies are series and parallel combinations. In [37], a mixed series-parallel topology was presented, which causes the transmission distance to be longer than for the serial topology. Efficiency is higher than in parallel topology. The authors presented a series-parallel-series topology that allows power shift in case of significant misalignment. The analysis was performed only for the serial configuration. Based on the available literature, it can be concluded that the parallel-series system with planar coils has not yet been fully analysed.

This article presents a WPT system with periodically arranged flat circular coils. The influence of variable model geometry (e.g., coil size, number of turns, and distance between planes) on efficiency and power was analysed. Additionally, by changing the diameter of the wire from which the coils were made, the influence of the coil resistance on the efficiency of the system was analysed. A wider frequency range used to charge mobile devices <100–1000 kHz> was analysed. Exemplary results with skin effect are presented. An analytical method was used for the research, which allowed for quick determination of the basic parameters of the system (e.g., power, efficiency). Using the finite element method (FEM), it is shown how to solve a system of planar coils as a single segment. This allows for easier modeling, limiting unknowns and faster solutions than in the solutions presented by other authors, where modeling of the entire system is required. Thanks to the applied parameterization, it is easy to change the parameters and values of the model in order to achieve new results.

There are four sections in this article. Section 2 shows the proposed WPT system model, as well as the numerical and analytical methods used in the analysis. Section 3 presents the discussion and results of the analysis. The characteristics show the influence of wire diameter on transmitter and receiver power and efficiency. Section 4 presents the most important conclusions from the conducted analysis.

2. Wireless Power Transfer System

2.1. Analyzed WPT System Models

The article presents a WPT system containing the transmitting and receiving planes. Each plane consists of periodically arranged flat circular coils (Figure 1a). Between these planes, there is a wireless transmission of energy, by means of inductive coupling, between the energy source and the receiver. Among the typical WPT devices consisting of several coils, systems with many inductive elements may also be considered. A pair of transmitting and receiving circular inductors, at the distance (h) with the same radius (r) and number of turns (n), is the basic part of the WPT cell with outer dimensions d × d (Figure 1b). The windings are wound around a dielectric carcass with an additional compensation capacitor. The periodic distribution of the WPT cells leads to transmitting and receiving surfaces where energy transmission takes place. The transmitting surface consists of TR coils connected in parallel to the sinusoidal voltage source (RMS value U_nd), while the receiver coils are connected to individual loads (Z_l).

The proposed configuration increases the density of the transmitted power in the area between the transmitting and receiving surfaces. Furthermore, the energy supply conditions can be adjusted. It is possible, for example, to transmit power simultaneously to many independent devices, where each WPT cell is directly connected to an individual energy storage. Another possibility is connecting each receiver coil in parallel to one common energy receiver. It is possible to connect the coils in a series and an intermediate series-parallel configuration.

The parameter values of the analyzed models are presented in

Table 1. Parameter values of the WPT system models.

r (mm)	n	h (mm)
10	15	h = r/2 = 5	h = r = 10
	25
	35
	45
25	50	h = r/2 = 12.5	h = r = 25
	60
	70

. Calculations were made for the small and large coils and at two distances, h = r/2 and h = r (where r is the radius of the coil). The distance h, taken into account in the analysis, was selected based on previously made tests. In the literature, when analysing WPT systems, the authors considered distances up to h = 2r. The tests of the proposed WPT system showed that the analysis of the system at the distance h = 2r gives an efficiency below 5%, which practically means no wireless transmission. The choice of two distances, h = r/2 and h = r, was aimed at finding a relationship between, among others, efficiency and distance.

In the multi-variant analysis, the radius of the coils (r) and the number of turns (n) were also changed depending on the size of the coil. The powers of the transmitter and receiver, as well as the efficiency of the system, were compared. Calculations were made over the frequency range of 100 kHz to 1000 kHz.

The analysis of the influence of coil resistance (R_co) on the power and efficiency of the WPT system was carried out, and it was determined by changing the diameter of the wire (200 μm, 150 μm, 100 μm), from which the coils were wound. The parameter values used in the calculations are presented in

Table 2. Parameter values used in the analysis.

Parameter	Symbol	Value
wire diameter	wc	200; 150; 100 µm
wire insulation thickness	wi	5; 1; 1 µm
wire conductivity	σw	5.6 107 S/m
voltage source	Und	1 V
load impedance	Zl	50 Ω

.

In the article are presented two ways of analysis: numerical (Section 2.2) and analytical (Section 2.3). Both methods are described below and allowed results to be obtained quickly.

2.2. Numerical Solution

Numerical methods can be fully used to model physical phenomena [38,39,40,41,42,43]. Their main advantages are:

• The ability to reproduce complex geometries of systems;
• Low application costs, avoiding prototype construction and measurements;
• Execution speed understood as the time of mathematical calculations;
• The ability to perform calculations using typical computers.

Despite the numerous advantages, one should also remember the limitations and disadvantages of numerical methods, e.g.,

• The need to know the programming tools used in modeling the problem;
• Limitations in the construction of models and the need to introduce simplifications;
• Errors resulting from both the method used and the construction of the numerical model (rounding errors or area discretization errors).

The Finite Element Method (FEM) is one of the numerical methods for solving differential equations. These equations, most often, describe phenomena and processes known in nature, physics and technology. The use of FEM in solving technical problems results from the possibility of obtaining the result of an equation that cannot be solved analytically or that is too complex and time-consuming. The numerical approach proposed in the WPT model based on the Finite Element Method (FEM) [41,42] was used to determine the transmitter and receiver powers, as well as the system efficiency. This method allows for performing a multi-variant analysis. The selected method has the ability to refine the mesh in selected areas with discretization of the analyzed area. Discretization is very important because it affects the computation time and accuracy of the results. Differential equations describing physical phenomena occurring during energy transfer are used in the solution. Simulation of physical phenomena using FEM consists of solving a system of algebraic equations. The number of unknowns depends on the number of mesh nodes and the number of unknowns in that node. If we increase the number of nodes, we will gain accuracy in the results, but at the same time, the computation time will increase. Therefore, the key is to find a balance between these elements. Very often, this is done at the testing stage and using practice in this type of numerical calculation. In order to obtain correct results, it is necessary to create large models, the construction of which requires a large amount of RAM and a long calculation time.

With the possibilities offered by the FEM method, it is possible to use periodic boundary conditions that allow the analysis of a single WPT cell (Figure 2a) instead of modeling the whole WPT system. Thanks to this approach, it is possible to reduce the number of unknowns (NDOF), which allows you to obtain results in a short time. This solution is only possible for periodic planar coils. Periodic boundary conditions were imposed on the sidewalls of the cuboidal WPT cell. On the other hand, on the planes parallel to the X-axis, an additional layer was modeled: the Perfectly Match Layer (PML). The basic idea of the PML method is to surround the computational domain by a layer of finite thickness with a specially designed model medium that absorbs all waves propagating from the interior of the computational domain. The coils are made of a very thin wire (w_d) with insulation of thickness (w_i). The compensating capacitor can be modeled as an element with a lumped capacitance C_cp. Additionally, it is possible to omit a carcass if it is made of dielectric and non-magnetic material (μ = μ₀). A voltage source with RMS value U_nd and frequency f is connected to each coil, and the current I_nd flows through the transmitter. The receiving coil connected to the linear load Z_l conducts the induced current I_od.

The Comsol Multiphysics 4.3b package was used to prepare a numerical model that presented the proposed solution and to perform computer simulations. Comsol can model wave problems in the frequency domain using the additional Radio Frequency module. The analysis was performed in the frequency domain, taking Magnetic fields physics in conjugation with the fragmentary Equivalent circuit. The coils were modeled using the built-in current sheet approximation of planar inductors (Multi-turn coil boundary condition), while the lumped voltage source (U_nd), capacitors (C_cp), and load (Z_l) were connected to the coils by internal coupling with a fragmentary Equivalent circuit. In the Comsol package in the Global Definitions section, the values of the model parameters related to the copper wire from which the coil was made were entered. In the next step, geometry was created in the Geometry section. The Magnetic Fields section describes the boundary conditions: Magnetic Insulation, which has been assigned to the sidewalls of the WPT cell cube. In this section, the Lumped Port was also used, which was included in the Multi-Turn Coil for both coils (transmitting and receiving). In the Electric Circuit section, the source type was specified: AC—Source. An AC/DC module was used. The port was positioned along the direction of the current flow. This section was about connecting circuit elements, i.e., resistor, coils, source, capacitors, load. In the next Mesh section, the mesh was selected. The elements were Free Tetrahedral and Free Triangular (Figure 2b,c). The mesh was very dense on the surfaces of the coils. The grid was selected so that there were no fewer than 20 elements per wavelength. In the Study section, the frequency was defined as 1000 kHz. A solver was also chosen: Direct MUMPS, which requires a large amount of RAM, but results were obtained in less time than iterative solvers. In sections, i.e., Results, Tables, Comsol shows the results of the simulation.

The FEM algorithm calculates the distribution of the electromagnetic field (EM) at any node. The induced voltages and currents were then calculated. Next, power, energy, efficiency, etc. were calculated. The simulation uses the magnetic vector potential (A) and the Helmholtz equation:

$$A=[A_x A_y A_z],$$

(1)

$$\nabla \times \left(\frac{1}{{\mu }_0^{}}\nabla \times A\right)-\mathrm{j}2\pi f{\sigma }_{w}A=J,$$

(2)

where J—is an external current density resulting from U_nd. Equations (1) and (2) are solved using the Direct solver, which gives results in less time than iterative solvers. The results are presented in Section 3.

Differential approximations occurred in the adopted grid construction [38,39,40,41,42]. This means that the resultant distribution of the numerical calculation error is not uniform. Due to the order of differential approximations of the EM field, the PML conditions have a central (second order) approximation. The implementation of these conditions, however, is associated with an increase in the analysis area (an additional layer), which results in an increase in the size of the model, expressed in the number of degrees of freedom (NDOF). Two factors were taken into account when choosing the methods for solving Equation (2).

• The possibility of determining the distribution of the EM field for the constructed models is understood as the convergence of the procedure of iterative calculation of the solution to the equation.
• Requirements related to the numerical implementation of the algorithm. The calculation time and the size of the processed data were taken into account.

The consideration of the first factor results primarily from the magnetic vector potential property. According to the theory, the solution to a problem consisting of NDOF unknowns is obtained after NDOF iterations at most (assuming calculations using exact arithmetic).

In the adopted numerical model, all factors were taken into account in order to obtain the correct results. It has been verified that increasing the NDOF by increasing the density of the mesh while discretizing the model gives practically the same results, but the computation time is significantly longer. If the results were the same as the initially assumed grid, there was no need to increase the NDOF. The developed models contained about 460,000 nodes, i.e., the number of degrees of freedom.

2.3. Analitycal Solution

The article analyses the influence of the coil resistance on the parameters of the WPT system, especially on efficiency. In the previous section, the proposed solution for wireless energy transmission using a system of planar coils was solved using a numerical method. As you can see, there are many steps in numerical modeling (discretization, boundary conditions, solver selection) that can be difficult. Incorrect selection results in incorrect results or even no solution.

In order to verify the correct selection of numerical parameters and the obtained results, this section presents an analytical solution. In this approach, it was also possible to simplify the model to one pair of transceivers representing the WPT cell (A_x,y). Physical phenomena occurring between adjacent coils or coils in the opposite plane are included in the formulas in the form of mutual inductances (M_tr) and (M_pe). Figure 3 presents an analytical model.

The mutual inductances of the coil (A_x,y) and the coils (A_x+m,y+n) affect the internal inductance (L_co) of the coil (A_x,y):

$$L_{co}=L_{sf}+{\displaystyle \sum _m{\displaystyle \sum _n\left(M_{x+m,y+n}\right)}},$$

(3)

where: L_co—effective inductance in (H); M_x+m,y+n—mutual inductance between coils adjacent to each other in the horizontal plane in (H).

The self-inductance (L_sf) was calculated from Equation (4) [43,44,45], where: d_w—mean diameter, and c—fill factor:

$$L_{sf}=\frac{{\mu }_0{k}_1{d}_w{n}^2}{2}\left[\ln \left(\frac{k_2}{c}\right)+k_{3}c+k_4{c}^2\right],$$

(4)

$$d_w=\frac{2r+2\left[r-n\left(w_d+w_i\right)\right]}{2},$$

(5)

$$c=\frac{2r-2\left[r-n\left(w_d+w_i\right)\right]}{2r+2\left[r-n\left(w_d+w_i\right)\right]}.$$

(6)

Coefficients k₁, k₂, k_3, and k₄ in Equation (4) depend on the shape of the coil. For circular coils, the coefficient values are: k₁ = 1, k₂ = 2.5, k₃ = 0, and k₄ = 0.2 [44,45]. The effective inductance of the coil is presented by Equation (7):

$$L_{co}=L_{sf}+M_{pe},$$

(7)

where: M_pe—a sum of mutual inductances in the periodic grid because it reduces the inductance of a coil, it is with a minus. If the load Z_l = ∞ and there are no capacitors in series with transmitting coils, a sum of mutual inductances takes the form:

$$M_{pe}=\frac{\left(\frac{{\underset{\_}{U}}_{nd}}{{\underset{\_}{I}}_{nd,\infty }}-R_{co}\right)}{\mathrm{j}2\pi f}-L_{sf},$$

(8)

where: I_nd,∞—source current in [A]. The mutual inductance M_tr between the coils (Tx, Rx) were calculated from [43]:

$$M_{tr}=\frac{\left(\frac{{\underset{\_}{U}}_{od,\infty }}{{\underset{\_}{I}}_{nd,\infty }}\right)}{\mathrm{j}2\pi f},$$

(9)

where: U_od,∞—voltage induced in the receiving coil in [V]. The compensating capacity can be presented by:

$$C_{cp}\left(f\right)=\frac{1}{4{\pi }^2{f}^2{L}_{co}}=\frac{1}{4{\pi }^2{f}^2\left(L_{sf}+M_{pe}\right)}.$$

(10)

For wire width w_d + w_i, the total length was obtained by:

$$l_{co}=2\pi n\left[r-\frac{\left(n-1\right)\left(w_d+w_i\right)}{2}\right].$$

(11)

Taking into account that the transmitter and receiver coils are identical, the resistances of the inductors are R_t = R_r = R_co. The equation for coil resistance without skin effect is:

$$R_{co}=\frac{l_{co}}{{\sigma }_w\pi \frac{w_d{}^2}{4}},$$

(12)

and for resistance with skin effect:

$$R_{co\_ac}=\frac{l_{co}}{{\sigma }_w{a}_{ef}}=\frac{2\pi n\left[r-\frac{\left(n-1\right)\left(w_d+w_i\right)}{2}\right]}{{\sigma }_w{a}_{ef}}.$$

(13)

An effective cross-section of the wire (a_ef) [m²] is presented by:

$$a_{ef}=\pi \left(w_d{\delta }_e-{\delta }_e{}^2\right).$$

(14)

An effective skin depth (δ_e) [m] is presented by Equation (15), while δ is a skin depth and is presented by Equation (16):

$${\delta }_e=\delta \left(1-\exp \left(\frac{-w_d}{2\delta }\right)\right),$$

(15)

$$\delta =\sqrt{\frac{1}{\pi f{\sigma }_w{\mu }_0}}.$$

(16)

A comparison of the power of the receiver (P_od) and the power of the transmitter (P_nd) gives the WPT system efficiency value:

$$P_{od}=R_o{I}_{od}^2,$$

(17)

$$P_{nd}=U_{nd}{I}_{nd},$$

(18)

$$\eta =\frac{P_{od}}{P_{nd}}100\%$$

(19)

The results are presented in Section 3.

3. Analysis of the Results Obtained by Both Methods

The parameters of the WPT system models are presented in Table 1. The parameter values used in the analysis are presented in Table 2. Both tables are presented in Section 2.1. The influence of the coil resistance (R_co) on the power and efficiency of the WPT system was studied, and it was determined by changing the diameter of the wire (200 μm, 200 μm se, 150 μm, 100 μm), from which the coils were wound. Calculations for a wire diameter of 200 μm, taking into account the skin effect, are marked as 200 μm se.

Based on the calculations carried out over the frequency range from 100 kHz to 1000 kHz, characteristics (P_nd) power, (η), and (P_od) power are presented in the diagrams in Section 3.1 for the small coil and in Section 3.2 for the large one. A discussion of the results is presented in Section 3.3. The numerical analysis results are presented as solid lines (marked as (N) in the legend) and the analytical analysis results are presented by dots (marked as (A) in the legend). There were compared results obtained by both approaches and the differences did not exceed 1.1%.

3.1. A Small Coil (r = 10 mm)

In this section, the calculation results of the analyzed WPT system, obtained by both approaches, for the small coil are compared.

3.1.1. Results at Distance h = r/2 = 5 mm

Transmitter power, efficiency and receiver power diagrams are presented in Figure 4, Figure 5, Figure 6 and Figure 7.

The transmitter power (P_nd) decreased with increasing frequency, regardless of the number of turns n (Figure 4a, Figure 5a, Figure 6a and Figure 7a). The transmitter power was almost the same value for a wire diameter of 200 µm with and without the skin effect over the whole frequency range (Figure 4a, Figure 5a, Figure 6a and Figure 7a). The skin effect can be explained by the so-called magnetic diffusion defined by Maxwell’s equations [38,43]. The current in the wire creates a magnetic field around it. Because it is a time-varying field, it penetrates its interior as an electromagnetic wave and is attenuated due to the loss of the medium. Currents are induced on the surface of the conductor, which adds to the source current, while inside the conductor, the source current is reduced by them. If the thickness of the wire is large and the frequency is high, the electromagnetic wave then has a short length, the currents coming from the magnetic field may be higher than the source current and then the current inside the wire starts to flow in the opposite direction, which is very unfavorable.

Because of the skin effect, the equivalent resistance of the conductor in which the alternating current flows changes (effective resistance AC). This is due to the change in the actual cross-section of the conductor through which the current flows. DC wire resistance is determined from the geometric dimensions by multiplying the specific resistivity of the material by the length of the wire and dividing by the cross-sectional area. This means that reducing the cross-section increases the resistance. This is the case when a skin effect occurs. Although the physical cross-section of the wire does not change, the current flows only through part of it (mainly at the surface of the wire), and this part affects the resistance value. AC resistance is most often measured by the technical method as the ratio of voltage to current in a conductor.

For the smallest wire diameter (100 µm), the transmitter power decreases slightly over the whole frequency range, which results from the high resistance of the coil. For the remaining wire diameters, the transmitter power decreased the most in the lower frequency range, while at f = 1000 kHz, these powers were comparable. The highest receiver power (P_od) and efficiency were obtained for a wire diameter of 200 µm (the lowest coil resistance) (Figure 4b,c, Figure 5b,c, Figure 6b,c and Figure 7b,c). The values of efficiency increased as the number of turns increased, regardless of the wire diameter. The highest efficiency occurred for n = 45 (Figure 7b) and amounted to almost 88%. The efficiency decreased as the wire diameter decreased due to the increase in coil resistance (Figure 4b, Figure 5b, Figure 6b and Figure 7b). The smallest decrease occurred for the model with the skin effect (200 µm se). In this case, the efficiency decrease did not exceed 7% for n = 15 (Figure 4b). This decrease was smaller with the increase in the number of turns. The lowest efficiency of the WPT system occurred for the model with the smallest wire diameter (100 µm), which resulted from the highest coil resistance. In this case, the efficiency decreased even by almost 36% for n = 25 (Figure 5b). The receiver power increased as the frequency increased and then began to decrease after reaching the maximum value (Figure 4c, Figure 5c, Figure 6c and Figure 7c). The highest value of the receiver power occurred for n = 15 (Figure 4c) for a wire diameter of 200 µm. For the smallest wire diameter (100 µm), the receiver power changed the least over the whole frequency range due to the high coil resistance. The receiver power decreased as the wire diameter decreased due to the increase in coil resistance. The smallest decrease in the receiver power occurred for the model with the skin effect (200 µm se). This decrease was smaller with the increase in the number of turns. The largest decrease in the receiver power occurred for the model with a wire diameter of 100 µm.

3.1.2. Results at h = r = 10 mm

Transmitter power, efficiency and receiver power diagrams are presented in Figure 8, Figure 9, Figure 10 and Figure 11.

The transmitter power (P_nd) decreased with increasing frequency and regardless of the number of turns for a wire diameter of 200 µm, with and without the skin effect (Figure 8a, Figure 9a, Figure 10a and Figure 11a). For the small wire diameters (100 µm and 150 µm), the transmitter power decreased very slightly over the whole frequency range, which resulted from the high coil resistance. When the distance between coils h was doubled, it caused a decrease in the efficiency and maximum receiver power (P_od), regardless of the analyzed cases (Figure 8b,c, Figure 9b,c, Figure 10b,c and Figure 11b,c). The receiver power and efficiency increased as the frequency increased. The highest receiver power and efficiency were for a wire diameter of 200 µm (the lowest coil resistance). The lowest receiver power and efficiency of the WPT system occurred for the model with the smallest wire diameter (100 µm), which resulted from the highest coil resistance. For n = 45, the efficiency decrease was about 23% (Figure 11b). The smallest decrease of (P_od) and (η) occurred for the model with the skin effect (200 µm se). This decrease was larger with the increase in the number of turns and was about 5% for n = 35 and 45 (Figure 10b and Figure 11b). The values of the efficiency and receiver power increased as the number of turns increased, regardless of the wire diameter. The shape of the characteristics was preserved regardless of the number of turns. The highest efficiency occurred for n = 45 (Figure 11b) and amounted to almost 30%.

3.2. A Large Coil (r = 25 mm)

In this section, the calculation results of the analyzed WPT system, obtained by both approaches, for the large coil were compared.

3.2.1. Results at Distance h = r/2 = 12.5 mm

Transmitter power, efficiency and receiver power diagrams are presented in Figure 12, Figure 13 and Figure 14.

The transmitter power (P_nd) decreased with increasing frequency, regardless of the number of turns and wire diameters (Figure 12a, Figure 13a and Figure 14a). The transmitter power had almost the same value for a wire diameter of 200 µm, with and without the skin effect, over the whole frequency range. For wire diameters of 150 µm and 100 µm, the transmitter power decreased the most in the lower frequency range. At frequencies above about 300 kHz, the transmitter power was comparable, regardless of the wire diameter. The highest receiver power (P_od) and efficiency (η) were obtained for a wire diameter of 200 µm (the lowest coil resistance) (Figure 12b,c, Figure 13b,c and Figure 14b,c). The values of the efficiency increased with increasing frequency but decreased as the number of turns increased, regardless of the wire diameter. The highest efficiency occurred for n = 50 (Figure 12b) and amounted to above 92% for a wire diameter of 200 µm. The smallest decrease occurred for the model with the skin effect (200 µm se). In this case, the efficiency decrease was about 2% (Figure 12b). The lowest efficiency of the WPT system occurred for the model with the smallest wire diameter (100 µm), which resulted from the highest coil resistance. In this case, the efficiency decreased even by almost 24% for n = 70 (Figure 14b). In the lower frequency range, the receiver power decreased as the wire diameter decreased due to the increase in coil resistance (Figure 12c, Figure 13c and Figure 14c). At frequencies above about 500 kHz, the receiver power (P_od) was comparable, regardless of the wire diameter. Then, the receiver power slightly decreased with increasing frequency. The highest receiver power occurred for n = 50 (Figure 12c) for a wire diameter of 200 µm. For the smallest wire diameter (100 µm), the receiver power decreased almost seven times. The shape of the characteristics was preserved regardless of the number of turns.

3.2.2. Results at h = r = 25 mm

Transmitter power, efficiency and receiver power diagrams are presented in Figure 15, Figure 16 and Figure 17.

The transmitter power (P_nd) decreased with increasing frequency, regardless of the number of turns (Figure 15a, Figure 16a and Figure 17a). The transmitter power had almost the same value for a wire diameter of 200 µm with and without the skin effect over the whole frequency range. For the smallest wire diameter (100 µm), the transmitter power decreased slightly over the whole frequency range, which resulted from the high coil resistance value. For the remaining wire diameters, the transmitter power decreased the most in the lower frequency range, i.e., below about 400 kHz, while at f = 1000 kHz, these powers were comparable. When the distance between coils h was doubled, it caused a decrease in efficiency, regardless of the analyzed cases. The highest receiver power (P_od) and efficiency (η) were obtained for a wire diameter of 200 µm (the lowest coil resistance) (Figure 15b,c, Figure 16b,c and Figure 17b,c). The values of efficiency increased as the number of turns increased, regardless of the wire diameter. The highest efficiency occurred for n = 70 (Figure 17b) and amounted to almost 84%. The efficiency decreased as the wire diameter decreased due to the increase in coil resistance (Figure 15b, Figure 16b and Figure 17b). The smallest decrease occurred for the model with the skin effect (200 µm se). In this case, the efficiency decrease was about 5% for n = 50 (Figure 15b). This decrease was smaller with the increase in the number of turns. The lowest efficiency of the WPT system occurred for the model with the smallest wire diameter (100 µm), which resulted from the highest coil resistance. In this case, the efficiency decreased by almost 42% for n = 50 (Figure 15b). This decrease was smaller with the increase in the number of turns. The receiver power increased as the frequency increased and then began to decrease after reaching the maximum value (Figure 15c, Figure 16c and Figure 17c). The highest receiver power occurred for n = 50 (Figure 15c) for a wire diameter of 200 µm. For the smallest wire diameter (100 µm), the receiver power changed the least over the whole frequency range due to the high coil resistance. The receiver power decreased as the wire diameter decreased due to the increase in coil resistance. The smallest decrease in the receiver power occurred for the model with the skin effect (200 µm se). This decrease was smaller with the increase in the number of turns. The largest decrease in the receiver power occurred for the model with a wire diameter of 100 µm. The shape of the characteristics was preserved regardless of the number of turns.

3.3. Discussion

In this section, the discussion about the averaged results obtained by analytical and numerical methods is presented. Calculations were made over the frequency range from 100 kHz to 1000 kHz for the small and large coils at distances h = 0.5r and h = r. Because the difference in the results obtained by both methods did not exceed 1.1%, it can be assumed that the assumptions made by both methods were correct.

The calculated coil resistance for different wire diameters is presented in

Table 3. Calculated coil resistance for different wire diameters.

	n	Coil Resistance (Ω)
		Wire Diameter (μm)
		200	200 se	150	100
	15	0.46	0.60	0.85	1.99
r = 10 mm	25	0.67	0.87	1.30	3.40
	35	0.81	1.06	1.65	4.14
	45	0.88	1.15	1.91	5.00
r = 25 mm0	50	3.57	4.63	6.76	16.09
	60	4.06	5.27	7.83	18.88
	70	4.48	5.82	8.80	21.52

.

The coil resistance increased with the number of turns and the size of the coil. Reducing the diameter of the wire also increased the coil resistance, regardless of the number of turns and the size of the coil. The lowest coil resistance (R_co) was for a wire diameter of 200 µm and the highest for 100 µm, regardless of the number of turns and the size of the coil. The greatest difference (nearly five times) was for a large coil and the number of turns n = 70. Considering the skin effect (200 µm se), coil resistance was calculated at f = 1000 kHz, and it increased with the number of turns, as well as with coil size. In this case, the increase in coil resistance was greater the higher the frequency.

The power transfer efficiency (η), obtained at the frequency of 1000 kHz for different wire diameters, is presented in

Table 4. Calculated efficiency for different wire diameters at distance h = 0.5r and f = 1000 kHz.

n	η (%) at 1000 kHz
	Wire Diameter (μm)
	200	200 se	150	100
15	53.89	47.23	38.27	20.24
r = 10 mm      25	80.31	75.71	67.36	44.43
(h = 5 mm)     35	86.62	83.17	75.60	53.17
45	87.49	84.22	75.75	53.25
r = 25 mm      50	92.39	90.29	86.31	71.57
(h = 12.5 mm)      60	91.86	89.65	85.27	69.78
70	91.31	88.97	84.13	67.73

and

Table 5. Calculated efficiency for different wire diameters at distance h = r and f = 1000 kHz.

n	η (%) at 1000 kHz
	Wire Diameter (μm)
	200	200 se	150	100
15	5.73	4.45	3.13	1.30
r = 10 mm      25	18.71	14.95	10.42	4.30
(h = 10 mm)     35	27.78	22.69	15.51	6.25
45	29.63	24.29	16.15	6.36
r = 25 mm      50	78.13	72.76	63.33	36.32
(h = 25 mm)     60	82.04	77.37	68.48	41.38
70	83.77	79.47	70.73	43.64

.

Analysing the calculation results, it can be seen that the efficiency of the WPT system increased with the increase in the number of turns and the size of the coil. Reducing the wire diameter increased coil resistance, which caused a significant decrease in the efficiency of the WPT system. The greatest decrease in efficiency (above 40%) occurred for a large coil and a wire diameter of 100 µm at h = r. The smallest decrease in efficiency occurred for the model with the skin effect (200 µm se) and did not exceed 7%. In the model with a large coil and a small distance between coils h = r/2 (Table 4), a decrease in the efficiency of the WPT system was observed, regardless of the wire diameter. The reason the value decreased was that as the number of turns increased, thus the coil resistance increases.

A decrease in the efficiency of the WPT system for other wire diameters in relation to the wire diameter of 200 μm, at the frequency of 1000 kHz, is presented in

Table 6. Decrease in efficiency of the WPT system for other wire diameters in relation to a wire diameter of 200 μm, at distance h = 0.5r and f = 1000 kHz.

n	∆η (%) at 1000 kHz
	Wire Diameter (μm)
	200 se	150	100
15	6.6	15.6	33.7
r = 10 mm       25	4.6	13.0	35.9
(h = 5 mm)         35	3.5	11.0	33.5
45	3.3	11.8	34.2
r = 25 mm       50	2.1	6.1	20.8
(h = 12.5 mm)       60	2.2	6.6	22.1
70	2.3	7.2	23.6

and

Table 7. Decrease in efficiency of the WPT system for other wire diameters in relation to a wire diameter of 200 μm, at distance h = r and f = 1000 kHz.

n	∆η (%) at 1000 kHz
	Wire Diameter (μm)
	200 se	150	100
15	1.3	2.6	4.4
r = 10 mm        25	3.8	8.3	14.4
(h = 10 mm)       35	5.1	12.3	21.5
45	5.3	13.5	23.3
r = 25 mm        50	5.4	14.8	41.8
(h = 25 mm)       60	4.7	13.6	40.7
70	4.3	13.0	40.1

.

Analysing the calculation results, it can be seen that the decrease in the efficiency of the WPT system for other wire diameters in relation to the wire diameter of 200 μm increased with the increase in the number of turns for the large coil and at h = r/2, and for the small coil and at h = r. However, this decrease decreased with the increase in the number of turns for the large coil and at h = r, and for the small coil and at h = r/2. Reducing the diameter of the wire (i.e., increasing coil resistance) caused a significant decrease in the efficiency of the system. The smallest decrease in efficiency occurred for the model with the skin effect (200 µm se) and the greatest for the model with the smallest wire diameter (100 µm). For a large coil, the greatest decrease in efficiency (above 40%) occurred for a wire diameter of 100 µm and at h = r. However, for a small coil, the greatest decrease in efficiency (above 30%) also occurred for a wire diameter of 100 µm and at h = r/2. In the case of the model with the skin effect (200 µm se), the decrease in the efficiency of the WPT system was greater the higher the frequency, which was due to the increase in coil resistance.

Because of the skin effect, the equivalent resistance of the conductor in which the alternating current flows changed (effective resistance AC). This was due to the change in the actual cross-section of the conductor through which the current flows. This means that reducing the cross-section increases the resistance. Although the physical cross-section of the wire does not change, the current flows only through part of it (mainly at the surface of the wire), and this part affects the resistance value. For conductors with a radius greater than the skin depth, the effective AC resistance can be determined in a simplified way, assuming that the current density at the surface is constant and that it flows only to the skin depth. Then, the useful cross-section is the cross-section of the ring with the inner radius reduced in relation to the outer one by the skin depth. The cross-section can also be approximated by multiplying the skin depth by the conductor circumference.

As a result of the skin effect, the internal inductance of the conductor in which alternating current flows changes. Inductance, as a measure of the magnetic energy stored in a given system, changes slightly and results from the shielding of the magnetic field inside the cable. A change (decrease) in self-inductance due to the skin effect can be observed, for example, for coaxial (cylindrical) conductors.

The skin effect does not affect the inductance of components made of conductors carrying alternating currents. If, for example, a cylindrical (solenoid) coil is made of the wire, the change in the current distribution inside the wire caused by the skin effect will not change the distribution of the magnetic field in the coil, and thus its inductance, but will affect its quality factor, i.e., the ratio of inductive reactance to coil resistance.

Based on a multi-variant analysis, the main trends are noticed.

• Regardless of the coil size, with a greater number of turns, the efficiency of the WPT system increased only slightly. For this reason, when designing coils, it is worth considering whether it is necessary to increase the number of turns further.
• As the wire diameter increased, the efficiency of the WPT system increased because the resistance of the coil decreases.
• Only for the small coil it was noticed that with the increase in the number of turns, the difference between the efficiency of the WPT system made of a wire with a diameter of 200 µm and the system with the skin effect (200 µm se) was reduced. This relationship was only for a small distance between the transmitting and receiving planes, h = r/2. Whereas for h = r, the opposite is true.

4. Conclusions

In this paper, an analytical and numerical analysis of the WPT system over the frequency range from 100 to 1000 kHz was presented. A simpler and faster solution was the proposed use of an equivalent circuit represented by a single WPT cell. The influence of the coil resistance changes on the efficiency of the WPT system was studied. The change in coil resistance was obtained by changing the diameter of the wire from which the coils were wound. In addition, the size of the coil, the number of turns, and the distance between the transmitting and receiving planes have changed. With a detailed analysis, the results showed that the highest efficiency was obtained for a wire diameter of 200 μm (which means the lowest coil resistance). The lowest efficiency was obtained for the smallest wire diameter, i.e., 100 µm (which means the highest coil resistance). By increasing the diameter of the wire, i.e., reducing the coil resistance, a higher efficiency of the WPT system was obtained. Based on the calculation results, the influence of the skin effect on the efficiency of the system was also determined, which caused a relatively small decrease (from about 1% to less than 7%). To eliminate the skin effect, the diameter of the wire should be reduced, which, however, causes a very large decrease in efficiency, even by more than 40% for a large coil and h = r, because coil resistance increases very strongly. As the calculation results showed, it is better to accept the skin effect than to reduce the wire diameter to eliminate it. The author also did calculations for a wire diameter of 50 µm, which eliminates the skin effect even at the frequency of 1000 kHz because this wire diameter is smaller than the skin depth, which in this case is 67 µm. Unfortunately, the obtained efficiency results were unacceptable, as they did not exceed 13% for a small coil and 30% for a large one.

Notable findings:

• The larger the wire diameter from which the coil is made, the higher the efficiency of the WPT system.
• As the number of turns increases, the efficiency of the WPT system increases.
• Using a coil with a smaller radius, we obtained lower efficiency than with a coil with a larger radius.
• A larger wire diameter, even with the skin effect, resulted in higher efficiency than using smaller wire diameters.
• The proposed coils were made on a flexible substrate, so they could also be used in intelligent textiles.