i UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF ENGINEERING AND PHYSICAL SCIENCES Mechatronics Research Group Doctor of Philosophy A Linear to Rotary Magnetic Gear By Thang Van Lang
Introduction
Motivation and problem statement
Over the years, advancements in technology have enabled the efficient harvesting of energy from renewable resources to combat climate change and global warming In line with the EU Renewable Energy Directive, the UK government has established a renewable energy target to promote sustainable energy practices.
By 2030, the UK aims to meet at least 32% of its total energy demand from renewable sources, primarily through bioenergy, wind (both onshore and offshore), solar, hydropower, and wave and tidal energy In 2019, renewable electricity generation accounted for 37.1% of the total, with wind energy experiencing significant growth—22.9% from offshore and 28.3% from onshore sources—together representing over 50% of renewable electricity In contrast, wave and tidal energy contributed only about 1% to the UK's total electricity generation.
Waves serve as a primary energy source in the marine environment, and energy harvesting systems often employ intermediate mechanisms to convert the low-frequency linear motion of waves into high-frequency rotational motion, thereby minimizing device size and cost The inherent low frequency of waves necessitates larger and more expensive directly coupled linear generators To address this, ball screws have been utilized in wave energy harvesting systems to effectively transform slow linear motion into high-frequency rotary motion Notably, Hendijanizadeh developed an electromagnetic harvester using a ball screw at the University of Southampton.
This PhD research aimed to enhance a hybrid wave energy harvester by integrating a linear generator into Hendijanizadeh’s electromagnetic harvester, resulting in a published conference paper [5] The findings demonstrated a significant increase in generated power while maintaining the original system's parameters Additionally, a case study involving four distinct linear generators illustrated the improved output of harvested power.
The power output of linear generators could increase by 18% to 50%, but this comes with the trade-off of added weight and space, impacting power density calculations While linear generators can be large and costly, the friction between the screw and nut in ball screws leads to non-linear behavior, resulting in high maintenance costs, noise, and vibration In contrast, magnetic gears offer a superior solution for multimedia applications, enhancing efficiency, energy density, maintenance, and service life.
The recent advancement of magnetic gears offers a promising alternative to traditional mechanical gears, primarily due to their physical separation between driving and driven shafts This design eliminates the need for lubrication, reduces maintenance costs, enhances reliability, and provides inherent anti-jamming properties, all while producing minimal noise and vibration However, challenges such as the high cost of permanent magnets and larger size for equivalent torque compared to mechanical gears limit their broader industrial application.
Research on rotary magnetic gears is extensive, but there has been limited exploration of linear to rotary magnetic gears, with only two notable developments: the rack and pinion magnetic gear and the helical magnetic gear, akin to a nut and screw mechanism Recently, Anglada's PhD thesis introduced a foundational concept for a linear to rotary magnetic gear based on a transverse-flux machine, outlining its working principles without delving into theoretical or analytical frameworks This thesis aims to conduct fundamental research on this innovative device to establish the necessary theories and analyses.
Magnetic gears
1.2.1 State of the art of magnetic gears
Gears are essential mechanisms that transfer torque and speed between shafts through meshing teeth or permanent magnets Magnetic gears, such as magnetic spur and planetary gears, share structural similarities with traditional mechanical gears In these systems, magnets on the gearing wheels facilitate torque transfer, creating a fictive torsion spring effect between the wheels This effect can be visualized by fixing one wheel while rotating the other, resulting in a torque application that varies with the angle of displacement of the second wheel.
A magnetic gear has a maximum torque limitation, known as the pull-out torque (T max in Nm), beyond which the gear wheels will slip if the applied torque exceeds this limit To ensure optimal performance, a magnetic gear should always operate below its torque threshold Additionally, the torque spring in a magnetic gear is not constant, as it represents the rate of change in torque relative to the change in angle.
Torque density is a crucial factor in evaluating the performance of magnetic gears, defined as torque per unit rotor volume or torque per total unit volume Accurate determination of rotor volume can be achieved through computer-aided design In magnetic gears, the rotor volume correlates with the maximum transferred torque, which is referred to as the active torque density \( \rho_A \).
To compare the torque density for magnetic gears with mechanical gears, a total torque density can be measured for the magnetic gears, given the total outer gear volume V T
Mechanical gears have a corresponding torque density factor, ρ F , which is calculated from the rated torque, T N
The efficiency of magnetic and mechanical gears is defined as the ratio of the power at the output shaft divided by the power at the input shaft
The torque density of various magnetic gears is compared with those of mechanical gears in Table 1.1 Some magnetic gear that are mimic mechanical gears [10, 11, 12, 13, 16,
18, and 33] have torque density less than 12kNm/m 3 , which is far less than that provided by the mechanical spur gear (from 100 to 200 Nm [10]).
1.2.2 Overview of magnetic gear topologies
The first magnetic gear (MG) was introduced by Armstrong in 1901, resembling a mechanical spur gear but utilizing contactless magnetic interactions instead of tooth meshing In 1941, Faus proposed a magnetic worm gear based on a mechanical design; however, it garnered little interest due to its low torque density, which was only about 5% of that of mechanical spur gears, attributed to the inadequate performance of available permanent magnet materials like SmCo5 The advent of rare earth permanent magnet materials, such as NdFeB, in the 1980s revitalized interest in magnetic gears, paving the way for their potential industrial applications.
The advancement of new permanent magnet (PM) materials has led to increased research funding; however, the topologies of magnetic gears (MGs) remain predominantly based on mechanical gears, often referred to as converted MGs Initial research primarily concentrated on worm and spur type topologies While a higher number of magnetic poles can enhance torque capacity, these MGs face a limitation in torque density due to the single pole-pair interaction between the two shafts Additionally, planetary magnetic gears have also garnered interest in recent studies.
Magnetic gears, particularly those utilizing bevel gear designs, have been developed due to their high gearing ratios and torque density advantages However, these converted magnetic gears exhibit significantly lower torque densities compared to traditional mechanical gears, resulting in limited interest and application in industrial settings.
Table 1: Torque density of different gears [10]
Multi-element MG [16] 24:1 Variable reluctance
Concentric magnetic gear [36] 1.7-100 Variable reluctance
Recent advancements in magnetic gear (MG) technology have led to the development of field-modulated magnetic gears, which utilize ferromagnetic pole-pieces to enhance the magnetic field from both outer and inner rotors This innovative design results in a higher torque density, as all permanent magnets contribute to torque transmission Consequently, field-modulated magnetic gears are increasingly recommended for applications in wind energy and electric vehicles.
Recent studies have explored various topologies of motion generators (MGs), including the transition from linear to linear and rotary to linear motion These MGs present several advantages over traditional mechanical gears, including reduced friction, lower maintenance requirements, and enhanced overload protection Such benefits make this type of MG particularly valuable for applications in wave energy and railway traction.
MGs face challenges such as limited maximum torque capability, higher costs, and increased weight compared to conventional gears While various topologies have been explored, additional research and practical validation are essential to confirm their benefits Most MG topologies exhibit low torque density, ranging from 10-80% of mechanical spur gears, and feature complex structures with numerous assembly parts To enhance the appeal of MGs for practical applications, it is crucial to develop new methodologies and principles that address these limitations.
As mentioned above, this section describes the development of the converted MGs that include operational principle, topologies, and general characteristics
In 1987, Tsurumoto and Kikuchi introduced involuted magnetic gears, as illustrated in Figure 1.1 These gears utilize SmCo5 permanent magnets, achieving a gear ratio of 1:3 and a maximum transmitted torque of 5.5 Nm at the driven magnetic gear.
In 1993, the authors introduced a magnetic worm gear, as shown in figure 1.2 The following year, they advanced this design with a trial construction of a magnetic skew gear However, these magnetic gears typically feature a complex arrangement and exhibit a low torque density, ranging from approximately 5% to 10% compared to mechanical spur gears.
Figure 1 1 Involute magnetic gears [16] Figure 1 2 Magnetic worm gears [17]
Intensive research on converted MGs has concentrated on parallel-axis designs, particularly spur type MGs The first notable spur MG was patented in the United States.
In 1970, Rand proposed a structure that was closely mirrored by Hetzel in 1974 Despite their potential applications, these early suggestions lacked comprehensive analysis to assess their operational capabilities In 1980, Hesmondhalgh et al established a theoretical framework for the operation of these structures and created a demonstrator for experimental purposes Subsequent analytical calculations by Ikuta et al and Yao et al examined both external and internal spur MGs, supported by finite element analysis (FEA) Additionally, Jorgensen et al and Fulani conducted two-dimensional analytical calculations that yielded results consistent with the FEA findings While these analyses contributed to enhancing the characteristics of spur MGs, their applications remain constrained due to inadequate torque density.
Magnetic couplers serve as an effective mechanism for transmitting torque between two coupling halves at the same speed, and they can be categorized into axial and coaxial types Research by Fulani has analyzed the parameters of axial couplers using both Finite Element Analysis (FEA) and torque formula methods, while Yao et al compared these methods for torque calculations, revealing that increasing the number of poles enhances torque through magnetic coupling with minimal air gaps Coaxial couplers have been explored by Wu et al., who utilized 2D and 3D FEA for design optimization to reduce the volume of magnetic materials Additionally, Elies conducted a study in 1999 that analytically computed the radial stiffness of cylindrical-air gap magnetic couplings, demonstrating that this stiffness can become zero with certain axial shifts, independent of angular shifts and the torque transmitted.
Figure 1 3 Radial parallel-axes spur MGs: a) External type b) Internal type c) axial type
Figure 1 4 Magnetic torque couplers: a) Axial coupler; b) Coaxial coupler [28]
These couplers offer superior torque transmission compared to earlier models, making them suitable for specific applications, such as in the chemical industry, where separation between the driving and driven components is essential.
A linear to rotary magnetic gear based on transverse-flux machine
This section introduces the concept of a linear to rotary magnetic gear In a transverse-flux machine, the stator winding generates magnetic flux that influences the rotor magnets via C-cores, which modulate the magnetic flux The rotor rotates when the current in the winding reverses direction.
Figure 1.16 presents a front view of the transverse-flux machine developed at the University of Southampton The stator hub features two rows of C-cores, each equipped with a coil These C-cores, shaped like the letter 'C', are made of magnetic materials and play a crucial role in modulating the magnetic field.
Figure 1 16 A phase of the transverse-flux machine including the rotor and stator
The operation principle relies on the change in reluctance with position due to the orientation of the magnets As illustrated in Figure 1.17, the magnetic field follows distinct paths in two different positions When a current flows out of the paper (indicated by a dot in the figure), the right-hand rule indicates that the magnetic field will be anti-clockwise, generating an electromagnetic force that attempts to move the rotor towards the magnets producing a magnetic field in the same direction.
Figure 1 17 Path of the magnetic field depending on the current [9]
When current flows through the paper, it generates a clockwise magnetic field, causing the electromagnetic force to align oppositely compared to the previous scenario Figure 1.18 depicts the magnetization of a magnet alongside the corresponding current loops The total flux through the equivalent current loop can be directly calculated for both the stator and rotor.
18 by integrating along the magnet thickness the differential flux in through each of the different current where M is the magnetisation of the magnet
Figure 1 18 A rectangular magnet with the equivalent current loops [9]
Figure 1.19 illustrates the magnetic flux within a C-core of a transverse-flux machine, which can generate a net positive torque when supplied with an appropriate multi-phase current waveform The Southampton-built machine features two phases, each with a 90-degree electrical phase difference and 20 C-cores, containing two rows of 40 magnets with alternating polarity Figure 1.20 presents the developed model of the airgap in the transverse-flux machine, showcasing magnets in various positions When the rotor is aligned, the torque is zero due to the balanced forces of the magnets atop the C-cores Conversely, in an unaligned position, the rotor begins to rotate due to the repelling forces from the magnets and the flux from the C-cores, achieving maximum torque at the half-misaligned position.
Figure 1 19 Magnetic flux in a C-core for a single-phase transverse-flux machine
Figure 1 20 Position of the rotor with the C-cores a) aligned, b) unaligned
The electromagnetic interaction in this machine differs significantly from that of a conventional generator, making it challenging to use analytical methods for torque estimation A key parameter in torque calculations is the flux factor, \( K_B \), which indicates the portion of the total magnetic field generated by the windings that contributes to torque production.
In a progress report, Anglada introduced a magnetic gear based on the principles of the original transverse flux machine The design features a rotor, T-cores, and four rows of magnets embedded in two outer corebacks, as illustrated in a schematic cross-section and a 3D view The T-cores play a crucial role in modulating the magnetic field experienced by the rectangular magnets However, the proposed configuration is notably complex.
Manufacturing the T-core will be challenging due to its complexity Additionally, the high reluctance caused by the extended magnetic path, combined with considerable leakage flux, diminishes the efficiency of magnet utilization.
Figure 1 21 A schematic and 3D view of a magnetic gear suggested by Anglada [9]
The transverse-flux machine configuration, illustrated in Figure 1.22, replaces traditional windings with a stack of magnets, where C-cores are transformed into two I-cores, serving as ferromagnetic pole pieces To alter the magnetic flux direction in the I-cores, the magnetic stack is translated, switching the magnets from the North Pole to the South Pole This movement of the magnet stack induces rotor rotation, analogous to changing the current direction in conventional windings Thus, the mechanism of the transverse-flux machine, combined with the magnet stack, effectively converts the translation of magnets into rotor rotation, while the I-cores remain stationary.
Figure 1 22 Transverse-flux machine with replacing the windings by a magnet stack
The transverse-flux machine is characterized by low speed and high torque, attributed to its numerous rotor poles In energy conversion systems, especially wave energy devices, the goal is to transform the low linear movement of waves into high rotary speed for improved efficiency Consequently, minimizing the number of poles on the rotor in the magnetic gear is essential to achieve maximum speed.
The proposed magnetic gear, illustrated in Figure 1.23, comprises three key components: a rotor, translators, and I-cores (ferromagnetic pole-pieces), all separated by an air gap The system features two poles designated as phase 1, with magnets aligned with those on the rotor Additionally, two more stacks are incorporated into the translator, functioning as a second phase, positioned at half a pitch relative to the first phase The translators operate along the Z-axis while the rotor rotates around the same axis.
Figure 1 23 Structure of the magnetic gear a) front view, b) isometric view
Aims and objectives
This study aims to develop a comprehensive theory and analysis for evaluating the performance of a novel magnetic gear It will consider design parameters and utilize 2D and 3D finite element analysis (FEA) to assess the device's torque and force The linear to rotary magnetic gear has diverse applications, including linear actuators for medical and chemical devices, wave harvesting systems, and aerospace technologies A detailed investigation of this magnetic gear is essential to understand its advantages and disadvantages The objectives of this thesis are clearly defined.
1 A proposed magnetic gear is introduced based on the operation principles of a transverse-flux machine In order to understand the working principles of the proposed magnetic gear, 2D and 3D models are developed by using the finite element analysis method to predict the flux density distribution in the air gap and the ferromagnetic pole-pieces Then transmitted torque characteristics are determined
2 Design optimisation studies are undertaken in order to determine the effects of the permanent magnets’ key parameters, the sizes of pole-pieces and the air gap length
23 on the performance of the magnetic gear Consequently, optimal dimensions of a linear to rotary magnetic gear demonstrator are determined
3 Constructing and testing a magnetic gear prototype to validate the analytical and finite element analysis
4 Future work will provide the dynamic behaviour of the proposed magnetic gear and it is examined in terms of oscillation of the rotor Then, the magnetic gear is modelled in a driven train system with an input force on the translator and a load on the magnetic rotor shaft Also, the proposed magnetic gear is considered for constrained applications, such as in wave energy harvesting.
Magnetic Field Theory
Introduction
This chapter reviews fundamental theories and methods for calculating the magnetic field in the airgap of magnetic machines, which is crucial for determining performance characteristics like torque and force Three primary techniques are employed to solve field problems: experimental, analytical, and numerical methods While experimental methods require costly and time-consuming laboratory setups, analytical methods offer precise solutions but can be challenging for complex models In contrast, numerical methods have gained popularity due to advancements in computing power, providing approximate solutions with sufficient accuracy for engineering applications Additionally, numerical methods facilitate swept parametric analysis, allowing for parameter variation and serving as a validation tool for analytical methods, ultimately leading to efficient final design solutions.
Basic magnetic field theory
In this section, basic concepts in magnetic circuit theory will be presented A simple magnetic circuit will provide a fundamental relationship between the key parameters
When an electrical current flows through a conductor, it generates a magnetic field represented by either the magnetic field intensity vector, 𝐻⃗⃗, or the magnetic flux density vector, 𝐵⃗ The relationship between these two vectors is determined by the properties of the conductor material.
The permeability of free space, denoted as \$m\$, is a fundamental constant, while \$\mu_r\$ represents the relative permeability of a material in comparison to free space For instance, the relative permeability \$\mu_r\$ is equal to 1 for air, but it can reach values in the thousands for ferromagnetic materials.
The Biot-Savart law is essential for understanding electromagnetic fields, allowing the calculation of the magnetic field vector \$\mathbf{dB}\$ or the magnetic field intensity vector \$\mathbf{dH}\$ at a specific point defined by the position vector \$\mathbf{r}\$ This calculation is based on the contribution from an elemental current \$i \, dl\$.
The electric field at a specific point can be determined by integrating the right-hand rule, provided the dimensions and shape of the current-carrying conductor are known.
𝑙𝑒𝑛𝑔𝑡ℎ (2.3) where, length indicates that the integration is to be carried out over the length of the conductor
This law states that line integral of the vector 𝐻⃗⃗ along any arbitrary closed path is equal to the current enclosed by the path Mathematically:
Ampere’s circuital law, expressed as \(\oint \mathbf{H} \cdot d\mathbf{l} = I\), relates the magnetic field intensity \(\mathbf{H}\) to the current \(I\) along a closed path This law is often preferred over the Biot-Savart law in magnetic circuit problems For an infinite straight conductor carrying current \(i\), the magnetic field at a distance \(d\) can be calculated using a circular path of radius \(d\), where the field strength remains constant and is directed tangentially Consequently, the equation simplifies to \(H \times 2\pi d\), providing a straightforward method to determine the magnetic field strength.
In deriving the final result, integration is unnecessary, allowing for a swift conclusion It is essential to select an appropriate path by examining the current distribution and justifying that the field's magnitude remains constant along this path before effectively applying the law.
Ampere’s law, as expressed in Eq (2.4), is fundamental to understanding how a magnetic field is generated by an electric current To illustrate this concept, consider the example depicted in Figure 2.1, which features a rectangular core with a winding of N turns of wire encircling one leg of the core.
Figure 2 1 Schematic view of a magnetic circuit [9]
In a ferromagnetic core, the magnetic field generated by the current is entirely contained within the core Consequently, the integration path in Ampere’s law corresponds to the mean path length of the core, denoted as \( l_c \) The current along this integration path, \( I \), is calculated as the product of the number of turns \( N \) and the current \( i \), since the coil intersects the path \( N \) times Therefore, Ampere’s law can be expressed as:
Now, the total flux in a given area is given by
The magnetic flux, denoted as 𝜙, is calculated using the equation 𝜙 = ∮ 𝐵 𝐴 ⃗ 𝑑𝐴 (Wb), where dA represents the differential unit of area When the flux density vector is perpendicular to the area A and remains constant across the entire surface, the equation simplifies significantly.
𝑙 𝑐 , (2.9) where A (m 2 ) is the cross-sectional area of the core
In a simple electric circuit, a voltage source V drives a current I around the circuit through a resistance R The relationship between these quantities is given by Ohm’s law:
By analogy, the corresponding quantity in a magnetic circuit is called the magnetomotive force (ℱ ) which is equal to the effective current flow applied to the core, or
The magnetomotive force (ℱ) is measured in ampere-turns (At) and is defined by the equation ℱ = 𝑁𝑖 In an electric circuit, the applied voltage generates a current (I), while in a magnetic circuit, the applied magnetomotive force induces the flow of magnetic flux The relationship between voltage and current in an electric circuit is described by Ohm's law, paralleling the relationship between magnetomotive force and magnetic flux in a magnetic circuit.
Where ℛ is reluctance of the circuit
The reluctance of a magnetic circuit is the counterpart of electrical resistance, and its units are ampere-turns per weber (At/Wb)
The permeance of a magnetic circuit serves as the magnetic equivalent of conductance in an electric circuit, where it is defined as the reciprocal of reluctance, just as conductance is the reciprocal of resistance.
The relationship between magnetomotive force and flux can thus be expressed as
The resulting flux in the core is given by Eq (2.9)
By comparing Eq (2.11) with Eq (2.14), we can see that the reluctance of the core is
Reluctances in a magnetic circuit obey the same rules as resistances in an electric circuit The equivalent reluctance of several reluctances in series is
We can extend this concept to more complex geometries, like the one illustrated in Figure 2.2, which resembles the previous example but includes an air gap within the core.
Figure 2 2 Schematic view of a magnetic circuit with an airgap [9]
The magnetic circuit in this case has two reluctances, one for the core ℛ 𝑐 , and one for the air gap ℛ 𝑔
(2.20) and the total flux is
Not all the magnetic flux generated by the magnetomotive force (mmf) is confined to the core; some flux lines traverse the air gap, as shown in Figure 2.3 The reluctance of air is greater than that of the core, resulting in a relatively small amount of leakage flux.
Figure 2 3 Magnetic flux in core and fringing effect
The flux lines generated by an exciting current traverse the air gap from the top to the bottom surface of the core, with the upper surface acting as a north pole and the lower surface as a south pole Consequently, the flux lines are not solely vertical or limited to the core face area; some lines of force extend beyond this region.
The fringing effect occurs when magnetic flux follows curved paths outside the core's face area, resulting in a phenomenon known as fringing flux This effect is minimal with small air gaps but becomes significant with larger gaps Essentially, fringing causes the flux density in the air gap to be lower than in the core, as the same amount of flux is distributed over a larger area than the core's cross-sectional area.
Magnetic field analysis
The reluctance network, grounded in an equivalent magnetic circuit, is utilized to address magnetic flux challenges This approach offers a straightforward formulation that, despite yielding approximate results, enhances understanding and provides valuable insights into the system Consequently, numerous researchers have sought to refine this methodology to improve its accuracy for various applications.
Figure 2 4 Magnetic equivalent circuit of an induction motor [61]
Ostovic's research [61-63] focuses on estimating various flux paths using an equivalent magnetic circuit in the airgap, accounting for fringing, inter-polar flux, and leakage This approach is represented as a system of matrix equations, as depicted in Figure 2.4, which illustrates the magnetic equivalent circuit of an induction machine's airgap While the increased complexity of these reluctance networks yields more accurate results, it also complicates physical interpretation, making insights into the system less straightforward.
Amrhein and Krein [64] established a comprehensive framework for 3D geometrical calculations by connecting components of the magnetic circuit through nodes As illustrated in Figure 2.5, an arbitrary 3D node can have six branch reluctances While their approach enhances computational accuracy, the increased number of nodes results in a large matrix that demands significantly more computation compared to traditional analytical methods.
Figure 2 5 A node with the reluctance branches and fluxes [63]
Finite Element Analysis (FEA) is a numerical technique employed to address challenges in engineering and mathematical physics, particularly for complex geometries, loadings, and material properties where analytical solutions are not feasible The general process of finite element analysis is illustrated in Figure 2.6.
The method introduced by Chari and Silvester in the 1970s has become the leading computational approach for electromagnetic field analysis in electric machines Its compatibility with various commercial software, such as Ansys and Opera, as well as open-source options like OpenFOAM and FEM, has facilitated its widespread adoption These software solutions employ advanced algorithms for meshing and solving, ensuring seamless integration with CAD software, which enhances the effectiveness of Finite Element Analysis (FEA) in both industrial and academic settings.
Figure 2 6 The process of finite element analysis [65]
2.3.3 Calculation of the torque with FEA
The magnetic field distribution in the airgap is crucial for calculating key machine characteristics, such as torque, which is a primary objective in the design process Most Finite Element Analysis (FEA) software employs the Maxwell Stress Tensor (MST) for torque or force calculations in electromagnetic analysis Utilizing Kirtley's expression to derive the stress tensor proves to be highly effective.
The traction, denoted as τ, is determined by the cross product of the current vector in 3D and the flux density at a specific point Additionally, another component of the traction arises from the variation of permeability Kirtley provides an empirical expression for the flux density based on this concept.
2(𝐻⃗⃗ ∙ 𝐻⃗⃗ )∇𝜇 (2.22) where 𝐻⃗⃗ is the magnetic field intensity and à is the permeability of the material at each point
As the current density is the curl of the magnetic field intensity so Eq (2.22) can be rewritten as:
2(𝐻⃗⃗ ∙ 𝐻⃗⃗ )∇𝜇 Then the force density can be written as:
If Eq (2.23) applies for k’th component of the force density, is obtained as
(2.24) where 𝛿 𝑖𝑘 is Kroneker’s delta that is 𝛿 𝑖𝑘 = 1 if 𝑖 = 𝑘 and 0 otherwise Instead of working with sums, the tensor T and density can be defined as the divergence of the tensor as
For the three components we can use the vector operator:
This deduction is applicable in the context of low-frequency electromagnetic fields, as analyzed by Kirtley [67] To achieve a comprehensive formulation, it is essential to incorporate the Poynting vector; for a more in-depth analysis, refer to reference [68].
In this section, we focus on the application of our analysis to an electric machine within a Cartesian coordinate system For an arbitrary point represented as \( r = (x, y, z) \) at a specific time \( t \), we define the magnetic field strength, denoted as MST, T, in relation to the coordinates \( x, y, \) and \( z \).
For example, if we want to calculate the force density in the x component:
To obtain the total force for a solid object we can integrate this expression over a volume V:
If S is a closed surface that surrounds V, then Eq (2.28) can apply the divergence theorem as following:
In Eq (2.29), it is crucial to understand that calculating the total force on a body does not require determining the force density throughout the entire volume Instead, it is sufficient to know the magnetic surface tension (MST), or equivalently, the field intensity over a surface that encloses the body.
The method is utilized for a two-dimensional model of a rotating electric machine to calculate the torque produced using the MST The tangential traction, denoted as \$\tau_\theta\$, is derived through a specific calculation process.
If the length of the machine is L and 𝛤 is a closed path in the air-gap, then the total tangential force is the following:
Finite Element Analysis (FEA) is a powerful tool for investigating the electromagnetic behavior of electric machines, significantly enhancing solution accuracy through refined meshing algorithms The integration of FEA with Magnetic Simulation Tools (MST) allows for immediate calculations of torques and forces Currently, FEA dominates the design trends in magnetic machines, following a typical procedure that includes conceptualization, geometry parameterization, field calculation, and optimization This project aims to utilize FEA to explore the proposed magnetic gear discussed in Chapter 1.
Magnetic materials
A magnetic material is defined and characterized by its B-H characteristic, illustrated by a typical hysteresis loop As the magnetizing force increases, a saturation point is reached where additional increases in H do not result in a significant rise in B This saturation flux density is a critical parameter in understanding the material's magnetic properties.
Bs, and the required magnetising force, Hs, to saturate the core are shown with dashed lines
Figure 2 7 Typical B-H or Hysteresis loop of a soft magnetic material [69]
In free space or air, the relationship between magnetic flux density (B) and magnetizing force (H) is linear, with permeability μ0 as the proportionality constant However, this relationship becomes nonlinear, as illustrated by the hysteresis loop The permeability of a magnetic material indicates how easily it can be magnetized, defined as the ratio of flux density (B) to magnetizing force (H) This ratio, or permeability (μ), varies, as depicted in the graph showing the relationship between permeability and flux density The graph also highlights the flux density at which permeability reaches its maximum value.
Figure 2 8 Variation in permeability à with B and H [69]
The area enclosed within the hysteresis loop represents the energy lost in the core material during each cycle, comprising hysteresis loss and eddy current loss Hysteresis loss occurs as the magnetic material undergoes cycling, while eddy current loss arises from induced electrical currents, known as eddy currents, which generate heat in the core High electrical resistance in the core leads to lower eddy currents, making high-resistance materials desirable for minimizing losses Core loss is a critical factor in the design of magnetic components, and it can be managed by choosing appropriate materials and thicknesses Proper material selection and adherence to operational limits are essential to prevent overheating, which can damage wire insulation and potting compounds.
Summary
This chapter reviews fundamental magnetic field theory and various methods for magnetic field analysis While analytical methods offer precise solutions for general applications, they become challenging to apply in complex models In contrast, the Finite Element Analysis (FEA) method is highly accurate and easily integrated into the design process, thanks to modern numerical software capabilities Consequently, this thesis will primarily utilize FEA to explore the proposed magnetic gear introduced in the previous chapter.
Chapter 3 A linear to rotary magnetic gear
This chapter introduces an innovative magnetic gear based on the principles of the transverse-flux machine, detailing practical configurations and utilizing finite element analysis (FEA) to assess its characteristics The structure of the proposed magnetic gear, including the outer core-back and ferromagnetic pole pieces, is thoroughly examined A comparative design analysis is conducted between the proposed magnetic gear and a helical magnetic gear with similar geometric parameters Additionally, a variant of the magnetic gear featuring a different pole arrangement is derived for comparison with an alternative screw magnetic gear The chapter concludes with a summary that emphasizes the key findings.
The concept of magnetic gear is derived from a transverse flux machine, as outlined in the introduction An illustration of an inner rotor version of a two-phase transverse flux machine is depicted in Figure 3.1, where the housing is omitted and not all C-cores are visible The rotor features four rows of heteropolar magnets, with each stator phase coil surrounded by an array of laminated C-cores, where the number of C-cores per phase is half the number of rotor permanent magnet (PM) poles In this machine design, the magnets of the two phases are aligned, while their corresponding C-cores are offset by 90 electrical degrees Consequently, when vector control is applied in the maximum torque per ampere mode, the current in phase two reaches its peak while the current in phase one remains at zero.
Figure 3 1 An illustration of a two-phase outer rotor VRPM machine showing a partial number of C-cores for clarity The stator housing that holds the C-cores is not shown
To enhance the understanding of how a transverse flux machine can be adapted for use in a magnetic gear, a two-pole version is illustrated in Figure 3.2, highlighting its inefficient use of magnetic material This inefficiency can be addressed by increasing the number of poles It's important to note that the current in each phase is shifted by 90 electrical degrees relative to the others The alternating current flowing through the coils produces an alternating magnetic field in the air-gap adjacent to the C-cores, achieving maximum strength when the C-cores are misaligned with the rotor's magnets.
In this innovative design, C-cores are transformed into I-cores, with coils replaced by arrays of hetero-polar magnets, known as translators, positioned at the outer ends of the I-cores As these magnets move axially, they generate alternating flux through the airgaps When the phase 1 magnet array aligns with the I-core and the phase 2 array is spatially shifted by 90 electrical degrees, the resulting flux pattern mimics that of traditional two-phase coils This configuration enables rotor movement through the linear motion of the magnets, effectively creating a linear-to-rotary magnetic gear system.
Figure 3 2 A 2-pole variant of the transverse flux machine
Figure 3 3 A linear-to-rotary magnetic gear derived from the 2-pole transverse flux machine
As well known, the coils in the transverse flux machine, generate a homo-polar flux in all C-cores surrounding them, however, the magnets generate flux in each set of I-cores
From a technical perspective, additional I-cores can be positioned on the opposite side of each phase, utilizing a magnet array shifted by 180 electrical degrees It is important to note that both phases' I-cores are aligned in the same plane and interact with a common rotor, as illustrated in Figure 3.4 This design allows for the replication of the structure along the axial direction, enhancing the torque generated by the rotor However, it is crucial to consider that increasing the number of I-cores leads to a rise in cogging torque and a decrease in stiffness.
Figure 3 4 The final topology of the linear-to-rotary magnetic gear derived from a 2-pole VRPM machine
The interaction between the magnets in Figure 3.5 reveals that the I-cores are not essential for this configuration's operation Removing them can reduce core loss and replace two gaps with one, thereby enhancing flux, force, and torque However, I-cores are necessary when using bar magnets to connect the rectangular geometry of the translators to the cylindrical rotor A cylindrical version of the machine is also possible, although the arc magnets on the translators may incur higher costs.
This configuration enhances the interaction between the magnets on the translators and the rotor, allowing for the removal of the I-cores This improvement increases performance characteristics by replacing two gaps with a single one.
In terms of transmission, either the translator or the rotor can be used as the driving part
This mechanism is versatile, applicable in both motoring and generating functions When utilized as a driving rotor, a motor can be coupled to the rotor's shaft to produce linear movement of the translator Conversely, if the translator serves as a driving port, such as in an oscillating system, its reciprocation induces rotor rotation, which can subsequently power a rotary generator.
Figure 3 5 The cylindrical topology: a) with I-cores b) without I-cores
In a practical design, one component, either the rotor or the translator, must be longer than the other to ensure continuous engagement of their magnets during movement The shorter component is known as the active length, while the longer one is referred to as the stroke length, which will be discussed in the following chapter.
A basic unrolled system featuring two simple rows of magnets on the rotor, as illustrated in Figure 3.6, demonstrates that the gear functions similarly to a discrete helical gear, even in the absence of I-cores and core-backs.
Figure 3 6 Unrolled cylindrical magnetic gear without I-cores
Figure 3 7 A rotary-to-linear magnetic gear with 8 peripheral rotor poles
We can envision a rotor featuring multiple peripheral poles, where the translator magnet arrays create a polygon with sides corresponding to the number of rotor poles As the number of peripheral poles increases, the arrangement of the I-cores and magnets approaches a helical configuration.
The findings presented in [55] indicate that the implementation of 47 enhances pull-out torque and improves the linear relationship between torque and force, thereby minimizing the gear ratio's dependence on the load.
The speed of rotor is associated with the number of poles on the rotor in radial direction
Reducing the number of poles on the rotor leads to an increase in rotor speed, with the optimal configuration being two poles for maximum speed This design features two magnetic gear poles, as illustrated in Figure 3.8.
Air gap length ɡ Active length LT
Figure 3 8 A two-phase magnetic gear Table 2: Parameters description
When the translator shifts by one pole pitch, the rotor also rotates by one pole pitch This indicates that the translator moves two pole pitches for each complete rotation of the rotor Therefore, a narrower pole pitch leads to an increased angular velocity.
48 rotor If speed of the translator moves V meters per second The rotor speed in term of revolution per second would be
Topology modification
This section discusses modified versions of outer core-backs and I-cores (ferromagnetic pole-pieces) Previous analyses indicated that outer core-backs were separated in each magnet row on the translator, facilitating a straightforward fabrication and assembly process Likewise, I-cores are configured as a single piece arranged in four rows, aligning with the magnet poles on the rotor These configurations aim to minimize flux leakage between phases on the translator.
It is assumed that the core-backs are made sufficiently thick so as to avoid saturation However, effect of outer core-back on flux leakage problem influences the characteristics
Figure 3.16 presents two designs for the outer core-backs, featuring both a combined and a separate configuration This analysis employs a magnet arc segment on the translator, leading to the formation of arc-shaped ferromagnetic pole-pieces The objective of this comparison is to identify a suitable topology rather than to achieve an optimal design.
Figure 3 17 A magnetic gear: (a) single core-back, (b) separated core-back, and (c) isometric break out view
Three-dimensional analyses in Ansys were conducted to compare torque and force characteristics, as illustrated in Figure 3.17 using the parameters from Table 3.2 The analysis revealed minor discrepancies in the values of force and torque between the two versions, indicating that both combined and separated core-backs yield reliable results.
Translator Ferromagnetic pole-pieces Rotor
Figure 3 18 Torque and force characteristics of the separated and single outer core-back: a) rotor torque; b) translator force
Rotor torque (mNm) a) Rotor angle (deg)
Translator force (N) b) Rotor angle (deg)
Figure 3 19 Magnetic gears with single ferromagnetic pole-pieces: (a) single core-back and, (b) separated core-back, and (c) isometric break out view of (b)
An alternative topology features a continuous ferromagnetic pole-piece that encircles the rotor In Figure 3.19 (a), both a single core-back and ferromagnetic pole-pieces are illustrated, while Figure 3.19 (b) presents the separate core-back design It is important to note that in Figure 3.19 (a), flux leakage can occur between adjacent magnet rows of the translator due to the flux path (indicated in black) from the magnet rows flowing through the ferromagnetic pole-piece.
In Figure 3.19 (b), flux leakage can happen through the ferromagnetic pole-pieces; however, because of the separated core-back, the flux travels through the adjacent pole instead of returning to the original magnet rows.
Figure 3.20 presents the rotor torque curves for three cases: (1) separated core-back and ferromagnetic pole-pieces; (2) separated core-back and single ferromagnetic pole-pieces; and
The study compares three configurations: single core-back and ferromagnetic pole-pieces Findings indicate that the torques from the latter two configurations decrease by about 8% compared to the first Notably, the starting torque for the first configuration is positive, whereas the other two yield negative values, likely due to flux leakage.
Figure 3.20 illustrates the rotor torque for three distinct scenarios: (1) when both the core-back and ferromagnetic pole-pieces are separated; (2) when the core-back is separated and a single ferromagnetic pole-piece is used; and (3) when both the core-back and ferromagnetic pole-pieces are combined into a single unit.
Seperated both Icores and Coreback Single I-core
Single both I-core and coreback
Comparative design
In configurations where rows of arc segment magnets are used on the translator and the pole pitch on both the translator and rotor are equal, the I-cores are not necessary for operation Removing the I-cores can effectively reduce core losses and replace two gaps with a single gap, leading to an increase in flux, force, and torque.
Figure 3 21 A magnetic gear topology without I-cores: a) front view and b) isometric view
A three-dimensional finite element analysis is performed to compare the performance of the proposed magnetic gear with a helical magnetic gear structure, following parameters similar to those in [55] The isometric view of the magnetic gear, depicted in Figure 3.21, shows a rotor with an outer radius of 50 mm, an active length of 40 mm for the translator, and a magnet thickness of 5 mm on both components The magnets on the translator are arranged in a discretized helix, with each segment angle set at 90 degrees This configuration aligns the magnets on the translator of phase 1 with those on the rotor, while the magnets on the translator of phase 2 are shifted by 90 electrical degrees, effectively moving by half a pole pitch in the axial direction When the inner rotor rotates 360 degrees with the translator stationary, a magneto-static axial force is generated.
Figure 3.22 illustrates the torque and force generated by the magnetic gear due to the rotation of the inner rotor along the z-axis Theoretically, the force and torque values on the translator and rotor components are equal but act in opposite directions Three-dimensional finite element analysis confirms that the obtained force and torque profiles align with these assumptions The graph indicates that the maximum torque from the proposed non-I-core topology is approximately 4.78 Nm.
Torque (Nm) a) Rotor angle (deg.)
Force (kN) b) Rotor angle (deg.)
63 maximum torque of the helical topology [55] reaches 8.95 Nm Similarly, maximum force achieved from two topologies was about 1.43 kN and 2.89 kN, respectively
Figure 3 23 a) Torque and b) Force of each individual phase of the magnetic gear due to rotation of the inner rotor
Torque (Nm) a) Rotor angle (deg.)
Force (kN) b) Rotor angle (deg.)
The helical structure produces torque and force values that are nearly double those of the proposed magnetic gear, as illustrated in Figure 3.23 The graph indicates that the majority of the rated torque is provided by phase 1, while phase 2 remains constant at a low value Additionally, the force profile is primarily generated by phase 1.
2 It can be concluded that all active magnets in the helical topology contribute to force and torque characteristics while the proposed magnetic gear has only half of the active magnets contributing.
Varying pole pitch topology
In 2001, Attalah and Howe introduced a rotary to rotary topology featuring a distinct pole pitch between the inner and outer rotating members, where the number of poles on the ferromagnetic components matches the total poles of both rotors Building on this concept, they developed a linear to linear topology in 2005, maintaining a similar principle regarding pole count Additionally, Wang et al from the same research group created a magnetic screw gear that utilizes helical magnets on both the screw and the nut, based on the innovative idea of a magnetic screw-nut.
In 2016, Kouhshahi et al developed a rotary to linear magnetic gear topology utilizing the principle of varying pole numbers and a helical structure The design features inner rotor magnets and ferromagnetic components arranged in a helical pattern, while the outer rotor magnets are fixed in a cylindrical configuration This innovative approach incorporates ferromagnetic pieces as translators, effectively eliminating non-active magnet material.
This section introduces a magnetic gear topology featuring a varying number of poles The design retains a structure similar to the proposed magnetic gear but incorporates different pole pitch lengths on the rotor, translator, and I-cores As illustrated in Figure 3.24, this topology comprises three main components: an outer translator with \( p_t \) pole pairs, an inner rotor with \( p_i \) pole pairs, and a ferromagnetic piece containing \( n_s \) pole pieces The arrangement of pole pairs is oriented along the axial direction.
65 each component was selected to fit within the axial length L, and to satisfy requirements in
Figure 3.25 depicts the cross-section in the axial direction and principal parameters Thus, the length of pole pairs can be calculated by
𝜆 𝑠 = 𝐿 𝑛⁄ 𝑠 (3.7) where 𝜆 𝑖 , 𝜆 𝑡 , 𝜆 𝑠 are the lengths of the pole pairs of the rotor, translator and ferromagnetic pole pieces respectively, therefore,
Magnets on the rotor and translator can be seen as a descrete helical structure that includes two 180 degree arc magnet segments Eq (3.4) may be presented in the form of wavenumbers, i.e.,
Figure 3 24 A different poles magnetic gear with p i = 6 pole pairs, pt = 7 pole pairs, and n s
Figure 3 25 Cross-section view of the magnetic gear showing pole pitch for rotor, translator and I-cores
Figure 3 26 Flux density lines of a 2D model of the different pole magnetic gear shown in the axial direction w s λ t λ i λ s w t w i
Table 5: Geometric dimensions and material values
Pole pitch w s 10 mm thickness h 6 mm
Magnet remanent flux density B r 1.23 T steel 1008 resistivity 14.2 μΩ-cm
A 2D Finite Element Analysis (FEA) model is developed based on the geometric dimensions and material values provided in Table 5 to illustrate the modulating effect of the ferromagnetic components depicted in Figure 3.26 When the inner rotor completes a 360° rotation while the translator and I-cores remain stationary, it generates an axial force along the z-axis and produces torque, as demonstrated in Figure 3.27.
Figure 3 27 Torque and force on different parts of the varying pole magnetic gear
Summary
A magnetic gear was developed based on the transverse-flux machine concept, focusing on the transmission between linear and rotary motion while determining the gear ratio Three-dimensional finite element analysis (FEA) models were utilized to estimate the magneto-static torque and force, revealing a nonlinear relationship between them Additionally, the cogging torque was found to be low compared to the rated torque.
This chapter introduces a novel magnetic gear developed from the structure of a transverse flux machine The original C-cores used to modulate the flux field were transformed into I-cores, and the winding was substituted with four rows of hetero-polar magnets This design generates flux ripples in the airgaps, which are essential for producing rotor torque Additionally, the rotor's pole count plays a crucial role in determining the gear ratio; specifically, an increased number of rotor poles results in a lower gear ratio.
We can envision a rotor featuring multi-peripheral poles, where the translator magnet arrays create a polygon corresponding to the number of peripheral rotor poles An increase in the number of peripheral poles leads to a closer approximation of a helical arrangement of the I-cores and magnets According to findings in [31], this configuration enhances pull-out torque and improves the linearity between torque and force, thereby reducing the load dependency of the gear ratio.
A comparative design of the proposed magnetic gear utilized arc segment magnets on the translator while eliminating I-cores to reduce core losses By consolidating two gaps into one, there was a notable increase in flux, force, and torque, with geometric dimensions resembling those of helical gear topology Through 3D finite element analysis (FEA), the torque and force of each phase were evaluated, revealing that only half of the active magnets on the translator contributed to these characteristics As a result, the maximum torque and force values of the proposed magnetic gear were approximately half that of the screw magnetic gear topology.