Concave Ferrite Core for Wireless Power Transfer (WPT)

Abstract

High-efficiency wireless power transfer (WPT) systems can present a perfect solution for fast-charging autonomous guided vehicles (AGV) to improve working hours in high-tech warehouses. Stationary charging stations reduce separation distance, improving coupling factor and power transfer efficiency. Analysis and design of the WPT system focused on maximum power at the load with a SS compensation circuit to reach high efficiency while applying the theory of power transformers design to maximize the power handleability with the physical dimensions. The proposed concept fits small AGVs. This paper proposes a unique ferrite structure for the transmitter ferromagnetic core. This novel shape introduces horizontal angular misalignment resistance due to the transmitter’s omnidirectional concave disc ferrite core combined with an E-core ferrite at the receiver side. The proposed WPT system can output 200 W at 100 kHz. A realistic 3D model has been designed into a symmetrical equivalent to reducing complexity and computational effort. The visualization of the magnetic flux distribution demonstrated that the proposed design has a better path to flow without concentrating flux in small regions, reducing heating losses.

1. Introduction

Wireless power transfer (WPT) systems have become a convenient technology for many applications as the cordless solution, which prevents trip hazards and increases mobility for various devices, such as vacuum cleaners, electric toothbrushes, smartwatches, and kitchen appliances [1,2,3,4]. For some medical implants, sensors, and pacemakers, WPT offers safe charging without any physical contact or the necessity of going under surgery [5,6,7,8,9]. Other advantages of WPT systems are easy implementation, moderate to low maintenance, and small area required for operation [10]. Consequently, high-power applications, such as electric vehicles (EV) [11,12,13,14], autonomous guided vehicles (AGV) [15,16,17], and unmanned aerial vehicles (UAV) [18,19,20], are charging wirelessly on roads, in shopping centres, high towers, roofs, warehouses, ports, and other harsh environments. For AGV, WPT systems have improved health and safety, as the contact-based charging system can create electrical sparks during physical connection, which might lead to fire hazards in dusty environments. Despite the many benefits of WPT systems, the power transfer efficiency could be better due to the low coupling factor, ferromagnetic core saturation limit, and heating losses. These issues have been the objective of numerous research in recent years. A WPT system is composed of a transmitter and a receiver. The transmitter (Tx) has a DC power source, an inverter, a compensation circuit, and the Tx coils. The RX coil is integrated with a compensation circuit and a rectifier on the receiver side before reaching the AGV battery (load), as illustrated in Figure 1. Compensation circuits are not a requirement for WPT. However, these circuits are an outstanding feature to reach power factor unity, lowering losses, and maximizing power transfer [21,22].

WPT systems are implemented for dynamic [16] or stationary charging [23]. In research such as [24,25], dynamic systems have been discussed and improvements have been made. However, dynamic charging systems are subject to misalignment on the horizontal and vertical plane, whereas the transmitter is inserted into ground tracks that become irregular over time. Additionally, the overall efficiency is low due to the large air gap, which leads to a poor coupling factor [26,27]. WPT dynamic systems, defined as loosely coupled, are implemented for larger separation between transmitter and receiver. As an example, EVs such as automobiles, buses, and trains [24].

On the other hand, stationary WPT systems present better coupling factor and higher power output. Because AGV navigation is precise and robust [28], there is a reduction in misalignments when docking at the charging station, further improving the stationary WPT application. Therefore, the air gap for AGV applications is smaller. Standard system structures are magnetically based on resonant inductive power transfer (IPT) [29], composed of two planar plates and coil windings. Circular and square coils are classified as nonpolarized couplers, whereas the magnetic flux path begins and finishes within the same coil. Because of the symmetric flux distribution, AGV charging stations structure implement these systems. The cost to build such a system is low as making coils in these shapes is simple. The planar ferrite core is also applied to these applications. Although the relevant advantages, the low coupling factor is still present for larger air gaps.

Polarized couplers have been introduced in WPT systems with multiple transmitter and receiver coils [30,31,32]. The DD coil is a well-known topology for a polarized coupler [33,34,35], whereas two square overlap each other, generating magnetic flux at the same time, consequently creating a polarized flux. Polarized WPT systems have demonstrated improvement in misalignment and coupling factor compared to traditional circular coil designs. Other polarized designs are DDQ and BP [27,36,37,38], in which quadrature coils further improve the coupling factor and magnetic flux. However, the manufacturing cost for adding more coils has increased. Furthermore, polarized charging systems are heavier and larger, becoming unsuitable for smaller applications.

In this article, a novel ferrite structure for the transmitter is proposed to improve the coupling factor while maintaining the coupling factor for angular misalignment. The design process did not consider any charging and discharging process or the WPT system integration to the Demand-Side-Management, since this paper only focuses on the magnetic flux density evaluation of the proposed ferrite structure. An omnidirectional WPT system is the initial concept for this structure, however the ferrite core is a double concave design with six coil windings at the inner structure. Despite the novelty of the transmitter, the receiver ferrite is a standard, E-core, whereas the coil windings are concentrated around the inner leg. A mathematical analysis of the WPT system is formed in Section II, leading to maximum power output and system efficiency. Section III details the proposed ferrite core, whereas design considerations related to measurements, electrical specifications, power capability, and magnetic flux density path are defined. The 3D model was developed and simulated over a time-dependent solver with a highly non-linear method. The Finite Element Method (FEM) simulation provided an adequate result that allowed the analysis of the proposed structure in terms of power output and magnetic flux density, described in Section 4. Thus, this paper emphasizes the novel structure concept.

2. Analysis of WPT

Assuming a WPT stationary charging station for AGV where misalignment is negligible, the equivalent circuit diagram is illustrated in Figure 1. In this scenario, the coupling factor is considered low so that only current fundamental components affect the electric circuit. The harmonic component must be analysed in highly coupled systems to achieve high performance [39]. For the resonant frequency, the series–series (SS) compensation is applied, and the frequency is defined as

$$\omega =\frac{1}{\sqrt{C_1{L}_1}}=\frac{1}{\sqrt{C_2{L}_2}}$$

(1)

An alternating power source $V_1$ energizes the transmitter side, composed of resistor $R_1$, capacitor $C_1$ and coil windings $L_1$. These windings are magnetically coupled with the windings at the receiver side $L_2$, where an induced current $I_2$ circulates following the resonant frequency ω. The circuit is composed of a receiver capacitor $C_2$, receiver resistor $R_2$, and load $R_L$ on the secondary side. Finally, the mutual inductance between the transmitter and receiver is represented by M.

Applying Kirchhoff’s Voltage Law (KVL) in Figure 2, the following equations are made:

$$V_1=\left(j\omega L_1+\frac{1}{j\omega C_1}+R_1\right)\ast I_1+{\left(\omega M\right)}^2\ast I_2$$

(2)

$$0=\left(j\omega L_2+\frac{1}{j\omega C_2}+R_2+R_L\right)\ast I_2-{\left(\omega M\right)}^2\ast I_1$$

(3)

Simplifying the impedance from transmitter and receiver to $Z_{Tx}$ and $Z_{Rx}$, the Equations (2) and (3) become

$$V_1=Z_{Tx}\ast I_1+{\left(\omega M\right)}^2\ast I_2$$

(4)

$$0=\left(Z_{Rx}+R_L\right)\ast I_2-{\left(\omega M\right)}^2\ast I_1$$

(5)

Observe that Equations (4) and (5) were formed taking in consideration that the turn ratio between transmitter and receiver is 1:1. Applying the turn ratio in reference to the currents, Equations (6) and (7) into (4) and (5), Equations (8) and (9) are obtained:

$$Turnratio=\frac{N_1}{N_2}=\alpha$$

(6)

$$\frac{I_1}{N_1}=\frac{I_2}{N_2}$$

(7)

$$V_1=\left(\frac{{\alpha Z}_{Tx}+{\left(\omega M\right)}^2}{\alpha }\right)\ast I_1$$

(8)

$$I_2=\frac{{\left(\omega M\right)}^2}{\left(Z_{Rx}+R_L\right)}\ast I_1$$

(9)

Therefore, the transmitter and receiver coil current can be obtained from (8) and (9), given as

$$I_1=\left(\frac{\alpha }{{\alpha Z}_{Tx}+{\left(\omega M\right)}^2}\right)\ast V_1$$

(10)

$$I_2=\frac{{\alpha \left(\omega M\right)}^2}{\left(Z_{Rx}+R_L\right)\ast \left({\alpha Z}_{Tx}+{\left(\omega M\right)}^2\right)}\ast V_1$$

(11)

Considering that the voltage $V_2$ across the load represents the output terminals in Equation (12), the maximum power output is obtained by (13):

$$V_2=R_L\ast I_2$$

(12)

$$P_{omax}=\frac{V_2^2}{R_L}=\frac{{{\alpha }^2\left(\omega M\right)}^4\ast V_1^2}{{\left(\left(Z_{Rx}+R_L\right)\ast \left({\alpha Z}_{Tx}+{\left(\omega M\right)}^2\right)\right)}^2}$$

(13)

Assuming a highly efficient WPT system where losses on the inverter and rectifier are neglected, the power efficiency of the sole wireless transmission component, considering maximum power output, is described in Equation (16) by applying Equations (14) and (15):

$$Z_{in}=\frac{V_1}{I_1}=\left(\frac{{\alpha Z}_{Tx}+{\left(\omega M\right)}^2}{\alpha }\right)$$

(14)

$$P_{in}=\frac{{V_1}^2}{Z_{in}}=\frac{{V_1}^2\ast \alpha }{\left({\alpha Z}_{Tx}+{\left(\omega M\right)}^2\right)}$$

(15)

$$\eta =\frac{P_{omax}}{P_{in}}=\frac{{\alpha \left(\omega M\right)}^4}{{\left(Z_{Rx}+R_L\right)}^2\ast {\left({\alpha Z}_{Tx}+{\left(\omega M\right)}^2\right)}^3}$$

(16)

Analysing Equations (13) and (16), the mutual inductance coupling M strongly affects the power output and system efficiency. Additionally, the turn ratio compensates and balances the low mutual coupling factor whenever the number of turns at the receiver is higher than the transmitter. The impedances at the transmitter and receiver are considered fixed and constant, as well as the load. Therefore, the mutual inductance and the turn ratio are the constraints applied when designing a novel WPT system. Further analysis of the magnetic flux patch is made to understand the WPT system measurements and the ferrite core effects on the close patch and mutual inductance.

Conventional WPT systems with a circular coil on a planar ferrite core have magnetic flux density as illustrated in Figure 3. Considering that the coil windings, copper rectangles, have current field entering the page plane on the left side and leaving on the right, then, the magnetic field direction at the edges is from Tx to Rx, while in the WPT system it is the opposite direction. Since the magnetic flux density from the coils at the centre has the same direction and intensity, the amplitude is stronger in comparison to the edges. When discarding the losses, the centre magnetic flux is equal to ${\varphi }_m$ while the edges have a flux equivalent to ${\varphi }_m/2$, as described in Figure 3.

Furthermore, there is leakage flux at the coil sides, as described by the blue arrows. This leakage flux occurs due to the air low permeability that introduces resistance for the magnetic flux to flow. Ferromagnetic materials, such as ferrite, have higher permeability, creating a low resistance path for the magnetic flux. Consequently, ferrite core increases the power density and handleability of WPT systems. Furthermore, the conventional circular planar WPT middle section have larger area, as well as the ferrite core surface area, providing an enclosure path for the magnetic flux to flow, improving the system coupling and decreasing losses. For larger air gaps, the magnetic flux at the centre loses intensity because of the leakage flux occurrence around the coil windings.

Although the ferromagnetic core improves circular planar WPT systems, these ferrite plates have a lower cross-section area than the coil area. The magnetic flux density is concentrated into the ferrite, leading to losses. Additionally, the magnetic flux leakage illustrated by the blue arrows describes the magnetic flux path in the air gap. The middle section of this WPT system has double magnetic flux than the sides, whereas the air gap is more significant, and there are more losses into the air because the ferromagnetic area is smaller in comparison to the middle section.

Equations (17) and (18) describe the behaviour of the mutual flux over a closed path, considering B the magnetic flux density, A the area where the flux is flowing, and $I_1$ the current at the transmitter. As expected, the amplitude of the magnetic flux affects the mutual inductance and power transfer between WPT pads. Higher mutual inductance is ideal for most systems for high linkage between transmitter and receiver. This coupling factor can be improved when applying multiple coils [40,41,42,43] or implementing ferrite cores to create a better path for the magnetic flux to flow, decreasing the losses due to air gap and leakage flux.

$${\varphi }_m=\oint \overrightarrow{B}\overrightarrow{dA}$$

(17)

$$M=\frac{{\varphi }_m}{I_1}=\frac{\oint \overrightarrow{B}\overrightarrow{dA}}{I_1}$$

(18)

Current WPT market solutions for AGV are the German company, Wiferion, and the Japanese company, DAIHEN. The first has two models, etaLINK3000 and etaLINK12000 [44], capable of delivering 3 kW and 12 kW, respectively, while DAIHEN models D-BROAD CORE and D-BROAD SLIM [45] output 4 kW and 2 kW. Wiferion products have a patent, consisting of wall box, for connection with grid or warehouse power supply, wired to a transmitter pad. The receiver pad is a few centimetres smaller than the transmitter, also wired to a small box that contains the electronic AC/DC converters to charge the AGV’s battery. Wiferion claims that the wires from both transmitter and receiver are flexible enough to integrate this WCS to any AGV design, and this system is compatible with any battery in the market. The company also claims that the etaLINK models has 93% of efficiency. Similar to the etaLINK from Wiferion, the D-BROAD model has the coil pad unit wired to the power unit. However, in these models, the transmitter and receiver pads are equally sized, different from the etaLINK that has a larger transmitter. The DAIHEN offers an optional capacitor unit for compensating and additional efficiency of WCS. These models are designed to fit on middle to large AGVs with wide surface area, as described in

Table 1. Specifications of current AGV WPT market solutions.

Model	Value
etaLINK3000—Power	3 [kW]
etaLINK12000—Power	12 [kW]
D-BROAD CORE—Power	4 [kW]
D-BROAD SLIM—Power	2 [kW]
etaLINK3000—Tx surface area	$62,500[{\mathrm{m}\mathrm{m}}^2$]
etaLINK12000—Tx surface area	$147,000[{\mathrm{m}\mathrm{m}}^2$]
etaLINK3000—Rx surface area	$62,500[{\mathrm{m}\mathrm{m}}^2$]
etaLINK12000—Rx surface area	$118,400[{\mathrm{m}\mathrm{m}}^2$]
D-BROAD CORE—Pad surface area	$114,840[{\mathrm{m}\mathrm{m}}^2$]
D-BROAD SLIM—Pad surface area	$57,000[{\mathrm{m}\mathrm{m}}^2$]

.

3. Structure Design

The proposed WPT transmitter ferrite is a unique double-concave disc where the magnetic flux flows from six circular coils located at the inner part of the concave disc, as illustrated in Figure 4 and Figure 5. The first concave disc has six rods at the bottom, where the transmitter coils are placed. Each inner rod has a thickness equal to α. At the top, the single cylinder rod has a 2α diameter. This top rod acts as a conductor for the magnetic flux, concentrating this flux into one spot and decreasing the leakage flux. The outer concave disc is physically connected to the six rods from the inner concave bottom. The outer disc thickness is equal to Φ so that the flux has an equal cross section area to flow through the disc ends, as shown by the black arrow in Figure 4. Additionally, the disc structure permits the magnetic flux to flow on an Omni dimensional, increasing of power density without overheating specific areas due to high magnetic flux concentration.

Omnidirectional structures have been proposed in the WPT community [46] to solve position issues and increase stations mobility. The magnetic flux density observed on [47] spherical ferrite core was evenly distributed. Meanwhile, the three orthogonal plane core has better efficiency and improved distance for power transmission. Thus, the idea of a concave disc was developed from these results.

In traditional omnidirectional WPT systems, the coils surround the spherical ferrite core from the outer side. The idea of inserting a coil inside the concave structure comes from the satellite concept. The concave design focuses electromagnetic waveforms in one direction but also shields this signal from white noise and interference.

Although, the transmitter is a novel structure, the receiver is composed of an E-core with coil windings in the middle. This E-core tooth’s dimension is α, with the same thickness as the outer concave disc. This configuration allows the receiver to have any angular position on the horizontal plane while aligned at the centre rod maintaining a continuous path for the magnetic flux to flow between Tx and Rx. In addition, the E-core centre concentrates the magnetic flux from the two ends, focusing it and directing it back to the transmitter closing the loop. This proposed structure has been designed to maximize the ferrite cross-section areas, which leads to improvement in magnetic flux density flow and a decrease in leakage losses.

A transmitter six coils design was designed to increase the coupling factor as the magnetic flux at the inner concave disc. As observed in the 3D model Figure 5, these coils are positioned next to each other, and the coil windings do not have a physical connection between the adjacent ones. Thus, the magnetic field in the air gap between each winding is nullified, preventing eddy currents from being generated. In addition, these coils are positioned in series to operate on a higher coupled mode since the mutual inductance is increased by this multiple-coil topology [38]. Furthermore, the magnetic flux density has a lower resistive path to flow because the outer concave disc side is near coil windings.

3.1. Measurement Considerations

This proposed model is designed to fit on a small AGV and be built as a single unit or array. Thus, the size of the transmitter and receiver are defined based on the E-core ferrite commercial measurements. The E-core ferrite chosen was from NEOSID, which has the measurements detailed in

Table 2. Proposed Model Measurements.

Variable	Value
Outer concave disc thickness—α	7.20 [mm]
Circular rods—2α	3.60 [mm]
Outer concave disc diameter	15.00 [mm]
Inner concave disc diameter	10.20 [mm]
E-core width	30.80 [mm]
E-core height	15.20 [mm]
E-core thickness	6.80 [mm]

. Consequently, the concave measurements were defined based on the E-core size. For the outer concave and middle cylinder thickness α, a slightly larger size was set compared to the E-core thickness, so a minor misalignment between the transmitter and receiver can be permitted without compromising the coupling factor.

Figure 6 illustrates how the proposed WPT system can be assembled. The Tx unique structure is developed as an array of multiple units inside the charging station. On the other hand, multiple Rx E-core units cover the AGV side which increase the surface area utilized for wireless charging, resulting in an adaptive WPT system capable of fitting any AGV. Angular misalignment permits that the Rx units to be inserted into the AGV corners and difficult areas whereas a conventional planar pad cannot be installed.

3.2. WPT System Requirements and Power Capability

After defining the proposed model measurements, the power capability is analysed to achieve electrical and power requirements. WPT systems are based on an electric transformer, whereas the primary and secondary are fully decoupled by the air gap. Thus, the power transformer design Equations (19) and (20) can be applied to a WPT system to compare the physical size with the power handleability. The constraints that hold the WPT system of higher power output are the core saturation point $B_{sat}$, window area for coil windings $W_a$, cross-section area $A_c$, frequency $f$, and current density $J$.

$$A_{pphysical}=W_a\ast A_c$$

(19)

$$A_{ppower}=\frac{P_t\ast {10}^4}{K_f{K}_u{B}_{sat}fJ}$$

(20)

The E-core ferrite from NEOSID has a core saturation point $B_{sat}$ of 400 mT. Therefore, the first fixed constraint is defined for both transmitter and receiver cores. Window area and cross-section area are set from the cores dimensions and calculated. The $W_a$ utilization factor is defined to suit the maximum number of coil windings but still considering that there are empty spaces between windings. Furthermore, the $W_a$ in the transmitter is smaller than $W_a$ of E-core, which increases the windings number at the receiver, resulting in a small turn ratio, ideal for the project as demonstrated in Section 2, Equations (13) and (16).

As for the frequency, the IEEE standard C95.1-2005 [48] defines that the operational frequency for a WPT is limited to 100 kHz. Pacemakers are damaged for values higher than this limit, and skin heating occurs.

Considering previous work [49] on a novel WPT system for AGV, whereas the conventional circular coil has been inserted into a set of U-core ring, and the system dimension is more than five times this project proposed dimensions, the output power is also considered five times smaller. Therefore, 200 W is defined for this project. The commercial AGV battery has an input voltage of 48 VDC, and therefore, the voltage wave amplitude at the load should be 68 VAC. Consequently, the current at load is approximately 5.8 A DC. Furthermore, the power source is a sine wave with a waveform coefficient $K_f$ of 4.44.

The specifications described previously are listed in

Table 3. Designs specifications.

Variable	Value
Voltage DC	48 [V]
Power output	200 [W]
Current DC	5.8 [A]
$\mathrm{Waveform}\mathrm{coefficient}\left(K_f\right)$—Sine wave	4.44
$\mathrm{Window}\mathrm{utilization}\mathrm{factor}(K_u$)	0.9
Frequency	100 [kHz]
Ferrite Saturation Point—Bsat	400 [mT]
Cross section area	162.86 [mm2]
Window Area—Tx	21.6 [mm2]
Window Area—Rx	635.35 [mm2]
Turn ratio	14:50

. Applying all variables to Equations (19) and (20), the values found are 124,579.8 mm⁴ and 2813.46 mm⁴, respectively. As a result, the physical dimensions support the designed power constraints. Furthermore, the power specifications could be increased to amplify the magnetic flux at the transmitter coils. Nevertheless, the bottom rod window area available to insert windings has been utilized to the maximum.

3.3. Simulations Decisions

Figure 5 represents the initial concept of the proposed transmitter core. However, the complete 3D model has the E-core and receiver coil above the transmitter structure. The environment described for FEM simulation is magnetic field physics, in which Maxwell’s Equations explain the domain’s behaviour. The air domain of the model has been chosen to be twice the size of the 3D model, as small air domains can lead to miscalculations and errors due inaccuracy of waveforms. Later, coil and ferrite domains were defined as copper and ferrite permeability of 2500 μ. Since the E-core ferrite core was selected from NEOSID, it was considered that the proposed transmitter core was manufactured with the same electromagnetic properties as the E-core. Thus, the initial material description was substitute for the NEOSID ferrite properties with a saturation point of 400 mT, operational frequency up to 120 kHz, and HB curves for temperatures from 50 °C to 120 °C. These domains are included in Ampère’s Law, whereas magnetic field constitutive relation is described as HB curve.

Besides the ferrite cores domains, coils have different definitions and descriptions. All model coils are circular volumes, defined as homogenized multi-turn, whereas the turn ratio is 14 and 50 for the transmitter and receiver coil, respectively. The electric field in the coil’s domain follows the material’s relative permittivity. This copper coil excitation is defined through the electrical circuit which is modelled in another physics environment. Therefore, no values of current excitation are entered in this domain. However, the current direction must be defined in this domain. Because Maxwell’s Law states that the induced current flows in the opposite direction from the source, the transmitter current is set as entering the coil domain plane while the receiver is leaving. Thus, this model has transmitter current initially flowing at clockwise, while the induced current at receiver is anti-clockwise.

After completing the electromagnetic physics, the multiphysics feature is applied to describe the electrical circuit physics whereas power source, resistors, capacitors, and coil windings are defined. In this physics, the schematic illustrated in Figure 2 is built, and the current excitation is calculated. All elements are inserted and connected over the node number position.

Table 4. FEM Model Parameters.

Variable	Value
Input Voltage	68 [V]
Load resistor	10 [Ω]
Inductance—Tx	1.60 [µL]
Inductance—Rx	1.51 [µL]
Capacitance—Tx	10.52 [µF]
Capacitance—Rx	9.95 [µF]
Air gap	2 [mm]

lists all values entered for the FEM model parameters. The voltage source is defined as a sinewave with a frequency of 100 kHz and amplitude of 68 V. Sinewave is more suitable for WPT systems because the amplitude variation over time is slower than square waves, and it operates with the ferrite polarization principals, such as the hysteresis curve. An external inductor represents the coil windings that are connected to the coil domain from the magnetic field physics. Therefore, the electric current that flows through primary and secondary is calculated over the electrical circuit. The coupling of the windings operates as a voltage source within the electrical circuit at the receiver because the current flowing there is induced by the transmitter windings, similar to a power transformer.

Because the proposed model is a complex ferrite structure, and the computational effort to execute a simulation of the complete model is high, in order to lower the simulation time, the 3D model was divided into small segments. This smaller model variable is later multiplied by the length and the area in which the structure was divided.

First, the coil analysis set up the initial values from the coil windings. Then, these values are applied to initialize the time-dependent simulation for the dynamic model simulation. Because of ferromagnetic core hysteresis, simulations start at 0 s and finish after executing 50 period cycles with a time step of 0.01*period cycle. This time step is enough to have a quality resolution in the output waveform.

Even though the model has been broken down to a small symmetrical equivalent, these structure equations are still highly nonlinear. The PARDISO solver has been employed, with a Newton highly nonlinear termination method. This parallel sparce direct solver runs a multithread algorithm that suits multi-core processors and has a feature to store matrix factorizations as block of disk memory. The control method applies a pivot perturbation strategy which observes the pivot magnitude with a constant composed of permutation and diagonal scaling matrices. As a result, desktops with high power CPU and limited RAM memory, such as the one used in this project, are more suitable with the PARDISO solver. An initial course mesh was built for visualization and model preliminary analysis. Later, a finer mesh was applied to increase solution accuracy.

4. Simulations Results

The differences between coarse and finer mesh were minor. In Figure 7, the simplified 3D model illustrates the magnetic flux density when the input current changes directions at the transmitter. Concentrated magnetic flux areas are present around coil windings and at the inner concave disc. The magnetic flux density in these areas can reach up to 0.45 T. The flux distribution is not homogenous since a wide range of the ferrite core has low to no magnetic flux density. These highlighted areas emphasize where eddy currents are located while the concentration within the coils space occurs due skin effect. The skin effect at higher frequencies pushes the electric charges to the wire borders, which leads to high charges near the ferrite domain. This stronger charge concentration at the wires boundaries increases the magnetic field from the windings, which also oscillates at high frequency, resulting in a stronger current induction at the ferrite surface, following Faraday’s Law. Therefore, the high concentration of charged particles, near the windings in this model borders, induced eddy currents in the transmitter core surface.

Although the eddy currents generate losses, it can be observed from Figure 7 that the origin point at the concave disc centre shows no or very low eddy current. Because the coil windings are placed parallel to each other, the magnetic field between adjacent coils is nullified; a similar concept is seen in transformers that have bifilar structures. Observing the flux pattern, most high-density spots (red and orange areas) are located near the outer concave disc and up the inner concave disc.

Some eddy currents losses can be observed in Figure 7. However, the overall magnetic flux is significantly lower in comparison to an ordinary shaped transformer core. While the proposed transmitter core has lower magnetic flux, the receiver traditional E-core has dense flux at the centre and concentrated flux, especially in the 90° angle. The concave disc design generates a larger path for the magnetic flux which creates a homogenous distribution. Additionally, the addition of the rods connecting inner and outer discs reduces the occurrence of 90° corners that concentrate the flux and generate heat losses. Observing Figure 8a, the central top rod has flux amplitude between 4 × 10⁻⁴ T to 5 × 10⁻⁴ T, and as expected, this part has highest flux at the transmitter core as this rod operates as a magnetic conductor between Tx and Rx. The second area that has higher magnetic flux is the junction between coil rod with inner and outer concave disc. The flux amplitude at this area goes from 3 × 10⁻⁴ T to 4 × 10⁻⁴ T. For this corner, the transmitter coil has an uneven window area that leads to a smaller area for windings near the outer concave disc. Consequently, near the smaller window area, the magnetic flux is denser than the larger one. Although changing the corner angle might improve the flux distribution and path, the window area for the receiver coil is reduced which is a physical limitation. Furthermore, this proposed core can handle more magnetic flux without reaching saturation.

Higher magnetic flux concentration is observed at the receiver, where the amplitude is on average 6 × 10⁻⁴ T, Figure 8b. Additionally, the corner between E-core and the coil windings has dense magnetic flux density, highlighted in red, and with an amplitude of approximately 8 × 10⁻⁴ T. This corner shows a higher magnetic flux because of the 90° angle that focuses the path into a smaller area. Although heating losses occur in this corner, these losses are minimized as the flux density amplitude is low, and the time of occurrence is small.

The concave disc has proved that the omnidirectional approach reduces the flux concentration and distributes it over the coils without generating further eddy currents and heating losses. Additionally, the outer concave disc has lower magnetic flux than the inner disc, as the surface area is larger which leads to a longer path for the flux to flow.

As detailed in

Table 5. Results from FEM simulations.

Variable	Value
Input Voltage	1.345 [V]
Output Voltage	0.86 [V]
Input Power	1.77 [W]
Output Power	0.45 [W]
Efficiency	25.42%
Mutual Flux	2.14 × 10−6 [T]
Power Factor	0.53

, the power efficiency is 25.42% with transmitter voltage of 1.345 V AC and receiver voltage of 0.86 V AC. The induced voltage at secondary is 63.94% of the transmitter amplitude. Because there is a swift angle between voltage and current due to low power factor, the final efficiency dropped. Besides the low power factor, the separation between Tx and Rx coils is larger than conventional loosely coupled WPT system. While conventional WPT has an air gap of 2 mm, the distance between Tx coils and the Tx surface is approximately 20 mm, with an air gap of 5 mm between transmitter and receiver. Consequently, the mutual flux is smaller, leading to lower power transfer and low efficiency. However, for this larger separation and the low power factor, the proposed WPT has an outcome with an encouraging result that has high voltage efficiency and better magnetic flux distribution.

Furthermore, a parasite effect is present in the electric circuit that increases the voltage drop around the transmitter coil, reducing the primary voltage and current. Because of the transmitter coils proximity and the angle between each coil, the magnetic field generated from adjacent coils is interfering and reduces the coil’s electromagnetic power. Besides this reduction in the magnetic field, the proximity between copper windings creates two energized and opposite conductors, similar to capacitor places. Thus, the charges stay at the conductors’ edges, introducing other parasite effects to the system.

Comparing the WPT surface power density between market solutions and the proposed model in

Table 6. Surface power density comparison of market solutions and proposed WPT system.

Model	Surface Power Density
etaLINK3000	0.048 [W/mm2]
etaLINK12000	0.08216 [W/mm2]
D-BROAD CORE	0.000035 [W/mm2]
D-BROAD SLIM	0.000035 [W/mm2]
Concave Disc	0.000255 [W/mm2]

, Wiferion solutions present higher power transfer. The larger surface area combined with an inserted capacitor optimizes the etaLINK coupling. However, when comparing D-BROAD solutions with the novel model, denser surface power is applied on concave system. Thus, the overall power transfer is in pair with current solutions. The proposed model delivers power accordingly to the available surface at the AGV application.

5. Conclusions

A novel concave ferrite core structure has been developed and analysed for the WPT AGV system transmitter. Dimensions considerations for this system were made to fit on small AGVs, while transferring high power and improving the magnetic flux density path. The model has an E-core at the receiver side, which allows the secondary to be installed in any horizontal angular direction without compromising power transfer due to angular misalignment.

The proposed model was investigated in a FEM simulation environment, according to AGV electrical and power requirements. Simulation at the time domain was executed by applying a symmetrical 3D model design to decrease computational effort and improve the execution time. Due to the highly nonlinear design and complex structure, the initial simulation applied a coarser mesh to have a finer mesh after the first simulation analysis.

While the proposed model could benefit from certain improvements, the presented simulation results demonstrated that the omnidirectional feature from the concave disc improves the magnetic flux path. While the bifilar 6-coil structure created an interference between adjacent magnetic fields, eddy currents were reduced. Due to these promising results, the novel structure approach has the potential to fulfill the power requirements of AGVs in the near future, replacing wired connections in hazardous environments.