1
Università di Pisa
Laurea Magistrale in Ingegneria Elettrica
Modelling, Analysis, Sizing and
Control of a Radial Active
Magnetic Bearing
Relatore
Prof. Paolo Bolognesi Correlatore
Dott. Luca Papini
Candidato Tiziano D’Aversa
Anno Accademico 2018/2019
2
Index
i Abstract 3 1. Introduction 5 2. Model Description 18 2.1 Mechanical Model 18 2.2 Electromagnetic Model 20 3. Analytical Model 35 3.1 Harmonic Analysis 48 4. FE Model Validation 53 5. Model Generalization and Limits 62 5.1 Model Limits 68 6. Tetrapolar case 697. Control Design 77
7.1 Actuator Dynamic Modelling 77 7.2 Mechanical System Modelling 80 7.3 Converter Modelling 82 7.4 Control Modelling 84 7.5 Current Controller 85 7.6 Optimal Currents Calculation 86 7.7 Kinematic Controller 90 7.7.1 Position Controller 91 7.7.2 Speed Controller 91 7.8 Controller testing 92 8. Conclusions 96 9. Bibliography 98
3
Abstract
Bearings are a critical component in applications where high rotational speed is required. Furthermore, high-end applications demand the capability of operation in harsh and/or oil-free environments with high efficiency, high reliability, reduced maintainance, controllability of the rotor position, and reduced vibrations.
Among all the technologies available, magnetic bearings are the best matchers for most of the above demanding requirements. In fact, they operate by exploiting mainly the normal components of the electromagnetic forces operating across the airgap of electro-magneto-mechanical devices, rather than the tangential components that are at the basis of electric machines. Such forces are transmitted without requiring any physical contact between the stator and moving parts, thus permitting a lubricant-free operation with minimal noise, wearing and maintenance requirements possibly joined to very high reliability and low losses. Moreover, when windings are –also- used to produce the magnetic fields (active magnetic bearings technology - AMB), the plus of controllability is achieved permitting to implement sophisticated functions such as active damping of oscillating forces and vibrations.
In fact, AMBs have become increasingly popular in high speed applications thanks to their capability to actively control the rotor position without the necessity of any contact between the parts. However, thi requires to design an appropriate control and supply platform including suited electronic modules and effective control algorithms which require a good knowledge of the electromagnetic behavior of the structure.
This work deals with an in-depth analysis of a simple poly-phase inner-rotor AMB system mainly focusing on the 3-phase layout and considering the electromagnetic, mechanical and control aspects. The device is analysed first using the
“mid-4
complexity” generalised model for electro-magneto-mechanical devices, adopting a Lagrangian approach based on cylindrical coordinates.
The machine characteristics are then deduced from a magnetic analysis carried out using different levels of model complexity, which are implemented as numerical algorithms in the Matlab simulation environment. This permits to achieve a fast yet accurate evaluation of the machine characteristics, including the dependency of the winding inductances with respect to the rotor displacement that plays a key role in the development of the normal forces permittig AMBs to operate. The accuracy of the above model is then assessed by comparing ist numerical results with those obtained from a 2D FE electromagnetic model of the same device developed for validation. An extended application of the same model to the analysis of a 4-phase AMB layout is also investigated to demonstrate the flexibility of the approach used.
A criterion permitting to determine the phase currents minimizing the Joule losses in the windings while generating the desired radial force is then presented and applied to the control of the proposed AMB adopting a classical dual position-speed nested loop approach focused on the radial position of the rotor. Finally, the dynamics of the whole system is modeled and analysed numerically in the Simulink simulation environment, permnitting to get ready the parameters of the control system referring to a specific case study. The results obtained by such analysis confirm the validity of the model and of the minimal Joule losses criterion to select the most appropriate current reference values permitting to properly govern the electromechanical dynamis of the AMB system proposed.
5
1. Introduction
Nowadays, rotating machines are widely used in many industrial applications. The specifications of the application lead to different engineering solution for what concerns the technology used to guarantee and allow the mechanical motion between the machine components. Considering a generic electro-magneto-mechanical device, it consists of a mover which is in relative motion with respect to the static structure of the device. For each specific device, the mechanical structure must be designed in order to guarantee that the required motion take place safely. Bearings are commonly used in this context to constrain a sub-set of degrees of freedom (DoF) yet allowing others which are relevant for the application in hand. There are different technologies that have been developed to perform the actions characteristic of bearings which can be firstly classified according to the nature of the forces that are generated.
The first general classification is based on the characteristic of the force transmission between the mover and the static component, leading to two main bearings type:
- Contact bearings - No-contact bearings
Both types of bearings have their own application areas. In contact bearings, the forces are transmitted through point of contact between the two structures. Among the contact bearings, is possible to consider the rolling, journal, and flexural bearings.
6
Rolling bearings are widely used in many engineering fields thanks to their simple structure and low cost. The mover and the static part are separated by a series of components which allow their relative motion. They present some advantages as low starting and running friction, low heat generation and reliability. On the other hand, they are very noisy and running friction dramatically grown up at high speeds; however, they have no good resistance to shock loads.
Journal bearing (or sleeve bearings) are mostly used in application where the load is light, and the motion is continuous. The shaft rotates in the bearing with a layer of lubricant separating the two parts. Lubrication is required to prevents metal-to-metal contact and extend the life of the component while reducing the friction losses. Their use is limited to low-load and low-surface speed applications. Although their limitations, they are largely used in industrial for their low-economic impact.
One of the main drawbacks of contact bearings are the friction losses. In order to reduce the losses and increase the controllability of the motion, forces of different nature can be exploited. Hydro-static, hydro-dynamic, sound, and electromagnetic forces are just examples of an alternative to the mechanical contact forces that can be used.
In this overview, it is possible collocate magnetic, externally pressurized, and hydrodynamic fluid-film bearings in no-contact bearings class [1]. The technology of interest in this thesis is related with the magnetic bearings where electromagnetic forces are used to levitate the mover and control the motion without any physical contact between the device components. Despite this technology is known for many years, their use in industrial applications has increased only in the last decades. This has been possible thanks to the reduction of the electronic components cost, the development of new sophisticated and accurate control strategies, and an increased number of applications that can take great advantage of this solution. Nowadays, magnetic bearings can be found in many applications as:
7
- Vacuum, cleanroom, low temperature systems and, more generally, in critical environments;
- Artificial heart pumps and molecular pumps;
- Ultra-high speed and ultra-precise numerical control machine tools; - High speed flywheels;
These, and other applications, are possible thanks to the peculiar characteristic of magnetic bearings. The absence of lubrication system and the fact that they do not suffer any mechanical wear are advantages and important characteristic for niche applications like the ones above. Furthermore, they could be used to compensate vibrations and reduce the noise level [2].
Generally, magnetic bearings can be broken down into three different categories: - AMB: Active magnetic bearings
- PMB: Passive magnetic bearings - HMB: Hybrid magnetic bearings
However, they require power electronics and sophisticated control platform in order to stabilise the system.
Passive magnetic bearings use permanent magnet or induced currents effects [3] in order to maintain the mover in its centred position. They are working thanks to attractive and repulsive forces between magnets.
8
In the figure below [3] it is possible to see some configurations of PMBs.
The main advantage of passive magnetic bearings is that they counterpart external forces acting on the shaft without requiring any control. Furthermore, they have got some other advantages like a positive stiffness [4] and, for their nature, they do not need control. For these reasons they are simpler and cheaper than AMBs and HMBs. On the other hand, they present a very important disadvantage which is identified in the lack of stiffness control. With this in mind, according to Earnshaw’s theorem, they cannot be used alone for stable levitation, which instead can be achieved using AMBs with suitable control system. Differently, with respect to the PMB, AMBs generates actively controlled electromagnetic forces to achieve a stable levitation. This last point introduce higher cost, attributed to control system, power electronics, and therefore overall energy consumption.
Finally, HMB are a combination of both the above technologies: some DoF of the mover are controlled by means of the AMB while other are passively managed with PMB [5].
Each AMBs is composed of four main parts:
9
1- Actuator. Component that output an electromagnetic wrench according to the
mechanical position and electrical currents supplied.
2- Sensors. They detect rotor position and send the signal to the control system.
Normally, they are inductive, eddy current, optical and capacitive displacement sensors arranged on five axes: four radial and one axial axis.
3- Power electronics. Is the interface between the controllers and the actuator.
Enables the supply of the desired current/voltage to the actuator based on the output of the control system.
4- Control strategy. It supplies, through dedicated algorithms, the necessary
signals to the power electronic modules in order to achieve the desired target. The control is usually developed on DSP (Digital Signal Processor) or FPGA platforms.
For what concerns AMBs manufacturing, they look like a common electrical machine. In fact, they consist in a stator and rotor iron core. It is very important that the soft magnetic materials are selected with relative magnetic permeability and saturation level to avoid non-linear operative conditions. In this way, it is possible to neglect iron losses during machine operation and minimize the non-linearities in the behaviour while increasing the force-to-weight ratio. The core is commonly composed by thin electrically insulated sheets of soft magnetic materials stacked together in the axial direction. The sheets shape presents slots and teeth necessary to create electromagnets
10
by means of wrapping copper strand around the stator teeth. Widely used is the “tooth wound” technology for its simple manufacturing and allows to achieve high fill factor but other configurations are feasible. The electromagnets are the components through which is possible to generate flux density in the mechanical clearance between stator and mover. The forces are generated thought the variation of the electromagnetic co-energy of the device which is actively controlled by means of the current flowing in the electromagnet.
Another possible classification of active magnetic bearings is based on degree of freedom available:
1- Axial magnetic bearings called “thrust magnetic bearings”. They allow only one degree of freedom [6];
2- Radial magnetic bearings. They allow two degree of freedom [7]; 3- Radial-axial bearings. They allow three degree of freedom [8];
Among the above topologies of AMBs, radial magnetic bearings (RMBs) are widely used in many industrial applications when rotation of the mover is required.
In order to obtain an electro-magneto-mechanical device capable of controlling each one of the DOF, there are different topological structures which results feasible. Design refinement strongly impact on their performances. In fact, during the design stage, the electrical designer must choose the main features like as the path of flux and the number of independence magnetic loops, to achieve the highest efficiency and the maximum force-to-weight ratio.
Regarding the flux path selection, there are two main possible configurations, namely heteropolar and homopolar.
11
In homopolar bearings, the structure is designed in a way that the main magnetic flux paths are closing in the axial direction as is shown in the figure (4).
This structure operates with reduced iron losses and allows the use of no-laminated rotor. Moreover, eddy-currents have a symmetrical effect with overall null effect. On the other hand, it presents some disadvantages e.g. large-sized and higher weigh compared to heteropolar bearings.
The other configuration possible is heteropolar bearings. In this case, main path of magnetic flux is radial as represented in the figure (5).
Figure 4 Homopolar AMB
12
This configuration is more compact and feature a better exploitation of the soft ferromagnetic core. However, the eddy currents effect does not result negligible therefore requiring a laminated soft magnetic cores structure. In fact, there are major iron losses and eddy-currents effect produce no symmetrical distortion of magnetic flux with consequent noise for the control.
Once it has been chosen the path for the magnetic flux, it is necessary to examine the stator structure, focusing on the number of poles.
The simplest structure available in literature use four electromagnets arranged to create a 4-pole field. In figure (6) are reported examples of AMB structure and its equivalent magnet circuit.
This configuration allows the control of shaft displacement in two different directions.
13
Another widespread configuration of radial magnetic bearings features 8-pole field distribution. Using this kind of configuration, it is possible to separate four different magnetic loops to increase the decoupling effects on the force control.
Carry on with the review of stator configurations, is possible to find 3-pole radial magnetic bearings.
The electromagnets can be independently controlled in order to increase the flexibility in the control of different DOF. The last structure presented, equipped with three power amplifiers, features some advantages with respect to 4-pole and 8-pole structure. These advantages could be resume as [9]:
Figure 7 eight-pole RMB
14
- Lower cost
- Reduced iron losses - Better heat dissipation - Lower power consumption
In the design of AMB, the air gap length is a critical parameter, generally included between 0.3 and 0.5 mm. Rotor and stator core materials must be accurately selected as their properties strongly affect the performances.
The analysis and design of AMB can be performed using different analytical and/or numerical techniques.
First kind of numerical-analytical approach use equivalent magnetic circuit. Neglected the magnetic flux leakage, the airgap reluctance model is the only one considered if the magneto-motive force in the iron core results negligible (high relative magnetic permeability materials). These steps are fundamentals to gain a simplified model otherwise not being suitable for analytical consideration. A general schematic of the equivalent magnetic circuit is presented in the figure below:
Where the symbols represent:
- GA1, GA2, GB1, GB2, GC1, GC2 radial air-gap permeances
- Фm excitation flux
15
- Fm magneto-motive force
The forces in x-axis and y-axis are obtained through the solution of the magnetic circuit as presented below: 𝐹𝑥 = 3 2𝑘𝑥𝑦𝑥 + √ 3 2𝑘𝑖𝑟𝑖𝑥 𝐹𝑦 =3 2𝑘𝑥𝑦𝑦 + √ 3 2𝑘𝑖𝑟𝑖𝑦 𝑘𝑖𝑟 =𝜇0𝑁𝑟𝐹𝑚𝑆𝑟 2𝛿2 𝑘𝑥𝑦= 𝜇0𝐹𝑚 2𝑆 𝑡 2𝛿2
Where each symbol means:
- ix and iy current components in x-axis and y-axis after Clark transformation
- kxy radial force-displacement
- kir radial force-current coefficient
- µ0 magnetic permeability in vacuum (4𝜋 ∙ 10−7 𝐻
𝑚)
- Sr radial magnetic pole face area
- 𝛿 uniform air-gap length (rotor centred case) - Nr Number of turns
Another approach to the problem is based on punctual Maxwell equations and Maxwell stress tensor method. This method is based on the fine knowledge of the flux density distribution in the air-gap. Because of this, it is very important to accurately compute the field generated by the electromagnets and permanent magnet (in case of PMBs or HMBs). In order to achieve that it is possible to use a finite element method
16
or advanced analytical techniques. Known the flux density distributions it is possible to evaluate force per unit area acting on the rotor which is expressed as:
𝑑𝐹(𝜗) =𝐵 2(𝜗, 𝑡)𝑑𝑆 2𝜇0 = 𝐵2(𝜗, 𝑡) 2𝜇0 𝑙𝑟𝑑𝜗 Where:
- l is the equivalent rotor length - r is the rotor radius
- ϑ is the dimensional mechanical angle
The forces in the Cartesian coordinate system can be obtained as before: 𝐹𝑥 =3 2𝑘𝑥𝑦𝑥 + √ 3 2𝑘𝑖𝑟𝑖𝑥 𝐹𝑦 =3 2𝑘𝑥𝑦𝑦 + √ 3 2𝑘𝑖𝑟𝑖𝑦
But, this time, are the formulation of kxy and kir are changed and reported below.
𝑘𝑖𝑟 = 3𝜋𝑙𝑟𝐻𝑚ℎ𝑚𝜇0𝑁𝑟 8𝛿2 𝑘𝑥𝑦 =𝜋𝑙𝑟𝐻𝑚 2ℎ 𝑚2𝜇0 4𝛿3
- Hm is the magnetic field intensity evaluated at the working point of PM;
- hm is the PM length evaluated in its magnetization direction.
If it is necessary to achieve a very accurate model, the best approach is to use finite elements method. This one uses Maxwell’s equations in punctual form therefore leading to a very accurate solution. However, it presents some disadvantages and,
17
between these, the more important are very long computational time and lack of generalisation of the results. In fact, with FEM analysis it is possible to calculate only one machine structure for only one working point.
Another approach uses a changing of co-energy [10]. Co-energy is introduced to aid analysis, but it has no a physical reality and could be evaluated thanks to integration of the flux linkage:
𝐶𝑒 = ∫ 𝜑𝑑𝑖 𝑖𝑜
0
Where:
- φ is the flux linkage - 𝐶𝑒 is the co-energy - i0 is the current
The force in x direction can be therefore computed as: 𝐹 =𝜕𝐶𝑒
𝜕𝑥
The approach used in this thesis is based on a general theory of machines [11] applied at magnetic bearings. This last one, that it will be widely discussed in next chapter, permits an accurate, yet fast and simplified analytical analysis which lead to the understanding of force production principle in AMB.
18
2. Model Description
In this second chapter it will be detailed the general modelling technique used to analyse electro-magneto-mechanical devices. Subsequently, in next chapter, it will be applied to a three-phase radial AMB used as a case study.
This general approach is intended to describe the behaviour of electro-magneto-mechanical devices. In these devices there are electro-magneto-mechanical phenomena caused only by magnetic interactions having neglected electrostatic ones (Coulomb forces).
The model targets to provide an accurate and yet general methodology to analyse a wide range of electro-magneto-mechanical devices without the use of finite element methods who need a very long computational time.
Electro-magneto-mechanical actuator are characterised by three main dynamics that can be identified in the device: electromagnetic dynamic, mechanical dynamic and thermal dynamic. Considering thermally stable operative conditions, it is possible to focus on the study electromagnetic e mechanical dynamics.
2.1 Mechanical Model
The components in which the EMM consist in can be considered as a group of material points. The mechanical dynamic aims to describe the time evolution of their relative position and relative movement. The vector 𝑦̅ is defined as the position of the Np
material points in which the device consists in: 𝑦̅(𝑡) = [
⋮ 𝑦𝑘(𝑡)
⋮
] , 𝑥̅ ∈ 𝑅 3𝑁𝑝𝑥1
That said, it is clear that considering all material points of the machine might not result in the most efficient representation. To achieve a synthetic yet powerful description of the mechanical dynamic, it is possible to introduce generalized coordinates or Lagrangian coordinates. They fully describe the mechanical problem and their
19
cardinality NM is the same of the DoF of the system and is usually 𝑁𝑀 ≪ 𝑁𝑃. Although
cardinality is fixed, Lagrangian coordinates types it could be different. Calling 𝑥̅ the Lagrangian coordinates vector:
𝑥̅(𝑡) = [ ⋮ 𝑥𝑘(𝑡)
⋮
] , 𝑥̅ ∈ 𝑅𝑁𝑀
There exist and invertible relation between the position of the material points and the Lagrangian coordinates considered which can be expressed as
𝑦̅(𝑡) = 𝑌̅(𝑥̅(𝑡))
The time derivative of the set of material point can be therefore expressed as 𝑑𝑦̅(𝑡) 𝑑𝑡 = 𝜕𝑌̅(𝑥̅) 𝜕𝑥̅𝑇 | 𝑑𝑥̅(𝑡) 𝑑𝑡 𝑥̅(𝑡)
Continuing with the second order time derivative we obtain 𝑑2𝑦̅(𝑡) 𝑑𝑡2 = 𝑑 𝑑𝑡( 𝜕𝑌̅(𝑥̅) 𝜕𝑥̅𝑇 | 𝑥̅(𝑡) )𝑑𝑥̅(𝑡) 𝑑𝑡 + 𝜕𝑌̅(𝑥̅) 𝜕𝑥̅𝑇 | 𝑑2𝑥̅(𝑡) 𝑑𝑡2 𝑥̅(𝑡)
Thanks to the above relations, the generalized form of the Newton’s law can be easily computed ∑ ℱ̅𝑘(𝑡) 𝑘 = 𝔐𝑑 2𝑦̅(𝑡) 𝑑𝑡2 = 𝔐 [ 𝑑 𝑑𝑡( 𝜕𝑌̅(𝑥̅) 𝜕𝑥̅𝑇 | 𝑥̅(𝑡) )𝑑𝑥̅(𝑡) 𝑑𝑡 + 𝜕𝑌̅(𝑥̅) 𝜕𝑥̅𝑇 | 𝑑2𝑥̅(𝑡) 𝑑𝑡2 𝑥̅(𝑡) ] The left-side multiplication of the above relation for the transpose of the generalised jacobian matrix (𝜕𝑦̅(𝑥̅)
𝜕𝑥̅𝑇 |
𝑥̅) 𝑇
leads to the generalised form of the Newton’s law of motion which results expressed as
20 𝕿(𝑥̅(𝑡))𝑑 2𝑥̅(𝑡) 𝑑𝑡2 = ∑ 𝑊̅𝑘(𝑡) 𝑘 − 𝑊̅𝐶(𝑥̅(𝑡))
where the generalized inertia matrix, the complementary wrench and the wrenches correspondent to any other force acting on the system are expressed respectively as below. 𝕿(𝑥̅(𝑡)) = [𝜕𝑌̅(𝑥̅) 𝜕𝑥̅𝑇 | 𝑥̅(𝑡) ] 𝑇 𝔐 𝜕𝑌̅(𝑥̅) 𝜕𝑥̅𝑇 | 𝑥̅(𝑡) 𝑊̅𝐶(𝑥̅) = (𝜕𝑌̅(𝑥̅) 𝜕𝑥̅𝑇 | 𝑥̅(𝑡) ) 𝑇 𝔐 𝑑 𝑑𝑡([ 𝜕𝑌̅(𝑥̅) 𝜕𝑥̅𝑇 | 𝑥̅(𝑡) ])𝑑𝑥̅(𝑡) 𝑑𝑡 𝑊̅𝑘(𝑡) = [𝜕𝑌̅(𝑥̅) 𝜕𝑥̅𝑇 | 𝑥̅(𝑡) ] 𝑇 ℱ̅𝑘(𝑡)
The wrench is the generalized form of the well-known concept of force. The above relations allow to describe the mechanical dynamics of the system through a sub-set of coordinates with respect to the overall material point of the system.
Wrench express all kind of interactions. Among all the generalised wrench, the electromagnetic one is of high importance when describing the behaviour of electro-magneto-mechanical system. The electromagnetic wrench is expressed as a function of both Lagrangian coordinates and electromagnetic state variables 𝑊̅𝐸(𝑖̅, 𝑥̅). Its explicit form is derived in the electromagnetic system section. The power related with the wrench components can be synthetically expressed as:
𝑃𝑘(𝑡) = 𝑊̅𝑘 𝑇
(𝑡)𝑑𝑥̅(𝑡) 𝑑𝑡
2.2 Electromagnetic Model
In order to describe the mid-complexity model [11] it is necessary to begin with the hypothesis. First one, it is that the device evolves in low frequency. This approximation allows us to describe the machine with concentrated parameters.
21
Another hypothesis is that conductor elements are characterized by a threadlike shape: one dimension (longitudinal dimension) is very longer that other two transversal dimensions. Moreover, their path composes a geometrical loop (no short circuit). Thanks to these assumptions it is possible to use their border to construct the surface to which the flux linkage can be computed.
Each one of the abovementioned geometrical loops is defined as “phase”. Generally, inside an electro-magnetic-mechanical device there are NE phases. They can be
connected only externally of device having that any internal connection is not permitted.
Each phase it will be described by means of a set of three components: current, voltage and flux. Those components, according to user convention, are assumed positive like as represented in the figure (10).
Since, there are NE phases it is possible to collect all current, voltage and flux in three
vectors: 𝑖̅ = [ ⋮ 𝑖𝑘 ⋮ ] ; 𝑣̅ = [ ⋮ 𝑣𝑘 ⋮ ] ; 𝜑̅ = [ ⋮ 𝜑𝑘 ⋮ ] ; 𝑖̅, 𝑣̅, 𝜑̅ ∈ 𝑅𝑁𝐸𝑥1
where 𝑖̅, 𝑣̅ and 𝜑̅, in general, are function of the time. It is assumed that the electrical currents are flowing only in the conductive components. Under this assumptions, all secondary effects like as eddy current in ferromagnetic cores and them contribute on field map characterization will be neglect.
I
𝑉
𝐴𝐵
𝛹
A
B
22
Soft ferromagnetic materials are supposed to be isotropic, without hysteresis and saturation. Furthermore, their B-H characteristic is supposed to be linear.
In order to resolve map field problem and characterise the behaviour of the device in terms of electromagnetic performances, it is necessary to introduce a set of independence variables, namely currents as electrical state variables. The electromagnetic and mechanical state function are therefore introduced as below:
𝜑̅ (𝑖̅, 𝑥̅) = 𝜑̅𝐶 0(𝑥̅) + 𝜑̅𝐶 𝐼(𝑖̅, 𝑥̅) ∶ 𝑅𝑁𝐸,𝑁𝑀 𝐶 → 𝑅𝑁𝐸 𝑊̅𝐸( 𝐶 𝑖̅, 𝑥̅) = 𝑊̅ 𝐸0 𝑐 (𝑥̅) + 𝑊̅ 𝐸𝐼 𝑐 (𝑖̅, 𝑥̅) ∶ 𝑅𝑁𝐸,𝑁𝑀 → 𝑅𝑁𝑀
Where the first one represents the linkage fluxes while the second one corresponds to the electromagnetic wrench. Letter “c” is used to clarify that the electromagnetic analysis has been developed using current as state variable. Applying the pseudo-linear decomposition, the two components in which those consist in can be separated and are introduced as:
- 𝐶𝜑̅0(𝑥̅) and 𝑊𝑐̅𝐸0(𝑥̅) are flux and wrench components evaluated in open circuit, i.e. without current flowing; the only field source possible are, if present, hard magnetic materials;
- 𝐶𝜑̅𝐼(𝑖̅, 𝑥̅) and 𝑊𝑐̅𝐸𝐼(𝑖̅, 𝑥̅) are, respectively, incremental component of flux linkage and electromagnetic wrench.
According to the pseudo-linear decomposition of the flux linkage function, is possible to introduce the concept of incremental inductance matrix:
𝜑̅ 𝐼(𝑖̅, 𝑥̅)
𝐶 = 𝑳
𝐼(𝑖̅, 𝑥̅) 𝑖̅ 𝐶
However, is interesting to note that the flux linkage function can be differentiated with respect to the electromagnetic state variables leading to the definition of the differential inductance matrix
23
𝑳𝐷(𝑖̅, 𝑥̅) =𝜕 𝑳𝐼(𝑖̅, 𝑥̅)
𝐶
𝜕𝑖̅𝑇 𝐶
They are related by following relationship: 𝑳𝐷(𝑖̅, 𝑥̅) 𝐶 = 𝑳 𝐼(𝑖̅, 𝑥̅) 𝐶 + 𝑳 𝐶 𝐶 (𝑖̅, 𝑥̅)
In this equation compare 𝐶𝑳𝐶 that is defined as the matrix of the complementary inductances and it is expressed as:
𝑳𝐶 𝐶 (𝑖̅, 𝑥̅) = −𝜕 𝑳𝐼(𝑖̅, 𝑥̅) 𝐶 𝜕𝑖̅𝑇 𝑫[𝑖̅] 𝑫[𝑖̅] = [ [𝑖̅] 0 0 0 ⋱ 0 0 0 [𝑖̅] ]
Under the assumption of linear behaviour of the soft ferromagnetic materials we have that the matrix 𝐶𝑳𝐶(𝑖̅, 𝑥̅) = 𝟎, therefore leading to the identity between the
incremental and differential matrix of the inductances. 𝑳𝐷(𝑖̅, 𝑥̅)
𝐶 = 𝑳 𝐼(𝑖̅, 𝑥̅) 𝐶
On the other hand, is also interesting to note that the flux linkage function can be differentiated with respect to the Lagrangian variables leading to the definition of the matrix of the motional coefficient
𝑴(𝑖̅, 𝑥̅) = 𝜕 𝜑̅(𝑖̅, 𝑥̅) 𝐶 𝜕𝑥̅𝑇 , 𝑴 𝐶 : 𝑅𝑁𝐸,𝑁𝑀 → 𝑅𝑁𝐸𝑥𝑁𝑀 𝐶 𝑴𝐼(𝑖̅, 𝑥̅) 𝐶 = 𝜕 𝑳𝐼(𝑖̅, 𝑥̅) 𝐶 𝜕𝑥̅𝑇 𝑫[𝑖̅]
24
Following the same steps but applied to the electromagnetic wrench state function, it is possible to obtain: 𝑊̅𝐸𝐼(𝑖̅, 𝑥̅) = 𝑲𝐶 𝐼(𝑖̅, 𝑥̅) 𝑖̅, 𝑲 𝐼: 𝑅𝑁𝐸,𝑁𝑀 → 𝑅𝑁𝑀𝑥𝑁𝐸 𝑲 𝐶 𝐷(𝑖̅, 𝑥̅) = 𝜕 𝑊𝐶̅𝐸(𝑖̅, 𝑥̅) 𝜕𝑖̅𝑇 𝑲𝐶(𝑖̅, 𝑥̅) = 𝜕 𝑲𝐼(𝑖̅, 𝑥̅) 𝐶 𝜕𝑖̅𝑇 𝐶
In equation above we have that𝐶𝑲𝐼, 𝐶𝑲𝑫 and 𝐶𝑲𝐶 represent the matrix of the incremental, differential and complementary wrench coefficient, respectively. They are obtained thanks to decomposition respect at electrical state variable. Instead, using decomposition respect mechanical variable it is possible to achieve:
𝑺 𝐶 (𝑖̅, 𝑥̅) = 𝜕 𝑊𝐶̅𝐸(𝑖̅, 𝑥̅) 𝜕𝑥̅𝑇 𝑺𝐼(𝑖̅, 𝑥̅) 𝐶 = 𝜕 𝑲𝐶 𝐼(𝑖̅, 𝑥̅) 𝜕𝑥̅𝑇 𝑫[𝑖̅]
Where 𝐶𝑺 is defined as stiffness matrix.
If there are ferromagnetic linear material it will be possible express incremental wrench component as:
𝑊̅𝐸𝐼(𝑖̅, 𝑥̅) 𝐶 =1 2𝑫 𝑇[𝑖̅] 𝜕𝑳 𝑇(𝑥̅) 𝜕𝑥̅ 𝑖̅
Introduced those arrays it is possible to approach map field problem. Firstly, with the view to simplify problem, it will be decupled amplitude and direction field problems. Firstly, it will be discussed direction of flux density and inductive lines.
25
In electrical machines there are different materials with different electrical and magnetic properties. Neglecting the magnetic saturation, we focus on the interface between materials featuring different magnetic permeability. Considering the fundamental magnetic laws, namely Gauss’s law and Ampere’s law, is possible to write ∇ ⃗⃗ ∙ 𝐵⃗ = 0, ∯ 𝐵⃗ ∙ 𝑛̂ 𝑑𝑆 = 0 Σ ∇ ⃗⃗ × 𝐻⃗⃗ = 𝐽 , ∮ 𝐻⃗⃗ ∙ 𝑢̂ 𝑑𝑙 = ∬ 𝐽 ∙ 𝑛̂ 𝑑𝑆 Σ Γ
Those equations represent stationary case. On the left it is illustrated their differential form and, on the right, their integral form.
In order to apply Gauss’s law, it will be considered an infinitesimal cylinder as represented in blue in figure (11). Instead, the red plane is tangential plane to the surface on the point under investigation which sits on the boundary between two materials with different magnetic permeability.
Evaluating the flux on all surfaces it is possible to see that:
26
𝐵1𝑛 = 𝐵2𝑛
In other words, normal flux density component it will be conserved. This first result has been obtained through the Gauss’s law; using Ampere’s law twice, before among 𝑝̂ then among 𝑞̂, path as illustrated in the schematics of figure (12), it possible to evaluate second boundary condition.
Carrying out a contour integral it may possible to see that tangential field component it will be conserved if there are not any surface current in the considered section:
𝐻⃗⃗ 1𝑡 = 𝐻⃗⃗ 2𝑡
This conditions it will be useful to determinate field lines direction because in the device there are materials with considerable permeability differences. Moreover, supposing isotropic materials and no affect to residual flux density , the link between 𝐵⃗ and 𝐻⃗⃗ it will be determinate to constitutive equations:
𝐵⃗ 1 = 𝜇1𝐻⃗⃗ 1 𝐵⃗ 2 = 𝜇2𝐻⃗⃗ 2
27
Called tan 𝛼1 and tan 𝛼2 the ratio between the absolute value of the tangential and the normal field strength components in the material characterised by 𝜇1 and 𝜇2 magnetic permeability, respectively, is possible to write the above relations
tan 𝛼1 = ‖𝐻⃗⃗ 1𝑡‖
‖𝐻⃗⃗ 1𝑛‖, tan 𝛼2 = ‖𝐻⃗⃗ 2𝑡‖ ‖𝐻⃗⃗ 2𝑛‖
The field lines deflection in their passage form the one media to the other is governed by
tan 𝛼1 𝜇1 =
tan 𝛼2 𝜇2
Particularly, if 𝜇2 is bigger then 𝜇1 (ideally infinite) the value of tan 𝛼2 collapse to 𝜋
2.
Thus, field lines which exits and enter in materials with infinite relative permeability are normal at the interface surface. Knowing the interface surface envelope therefore imply to have known the initial direction of the field lines in the media with low relative permeability.
As so far demonstrated, it is very useful to resolve a field map problem, to know materials features and geometries.
The electrical machines analysed in this treatise are characterized by following features:
1- “Drum-type layout with coaxial rotor and stator members separated by layer of non-ferromagnetic and low-permeability material (e.g. air);
2- Stator and rotor magnetic cores featuring equal length. Last one is much larger than the external transversal size and an axially extruded geometry with grossly cylindrical envelope profiles of the surface facing the main airgap and belt-shaped yokes encircling the active parts of the machine respectively on the external and internal sides;
28
3- Straight slots in the magnetic cores, when present, with rather small openings along the sides facing the main air-gap;
4- Salient poles with grossly cylindrical main head surfaces, when present; 5- Phase windings composed of thin wire, each one featuring a geometrically
closed shape with straight active sides laid parallel to the machine axis either inside the slots or around the salient poles or in the air-gap attached to the surface of a core;
6- Permanent magnets, when present, located for instance only on the external side of the rotor core and featuring a rather small thickness and a planar grossly radial magnetization;
7- Suitably narrow main air-gap interposed between the permanent magnets layer and the internal stator surface” [11].
Under the above hypothesis, an intrinsic 3-dimensional problem can be studied considering only the 2-dimensional field map in the central cross section of the structure. A possible machine cross-section to which the model is applied is represented in figure (13):
29
The behaviour of the machine can be fully characterised by two main typologies of physical fluxes:
- Principal flux {Φ𝑝}: it contributes to the power conversion.
- Secondary flux o dispersed flux {Φ𝑠}: their contributes, referred to the wrench production, it may be neglect.
The model is developed considering only principal fluxes. Neglecting magneto-motive-force drop in the ferromagnetic materials, the energy density in concentrated in the air-gap region which is the only one considered in the analytical model.
The overall dimension of the air-gap thickness is relatively small with respect the other geometrical dimension. For this reason, rotor and stator boundaries create a magnetic equipotential surface. In addition, the punctual air-gap flux density in the air-gap can be approximated with its average value equally distributed along the air gap thickness. In order to develop the analysis, is necessary to define a reference system. According to the problem it is possible to choose between a stator reference and rotor reference. The choice is arbitrary and depends only on the simplification that they lead to the mathematical treatment.
Defined the reference system, is possible to identify every tangential point with a geometrical a variable 𝜆 that belongs to the range 𝜆 𝜖 [0,1). As shown in the figure it is assumed that 𝜆 increases in anticlockwise direction.
30
In order to evaluate the field map contribution of each phase it is important to indicate how many groups of active conductors, where they are located, and the direction of the current flowing in it.
In general, there are nc groups of active conductors. Arranging all information in a
compact form, it will be introduced an “allocation winding function array” 𝑵𝐶 𝜖 𝑍𝑁𝐸×𝑛𝑐. Each array element identifies how many active conductors for each
phase there are in a group and the current direction indicated by its sign. Reference convention used is Nhk > 0 when the current flows inside the section plane when seen
from the side of machine arbitrarily chosen as the front.
In this way it was determined both number and direction of each active conductor for phase. With the aim of characterize their position it will be introduce the vector 𝜆̅𝑐(𝑥̅):
𝜆̅𝐶(𝑥̅) = | ⋮ 𝜆𝑐ℎ(𝑥̅)
⋮
| , 𝜆̅𝐶: 𝑅𝑛𝑀 → 𝑅𝑛𝐶
Building a path by combining all points in the gap where average field and real field are the same leads to a curve Γ𝐺 in the section plane.
Furthermore, it is possible to introduce a relationship between a 𝑑𝜆 variation and dl i.e. an infinitesimal displacement on the Γ𝐺 path. Those concepts are represented in figure (15). With this in mind it is possible to describe the electromagnetic problem by means of analytical equations applying Gauss’s and Ampere’s laws.
Calling magneto motive drop as ℱ𝐺 and neglecting the part related with the stator and
rotor iron cores, we have that
ℱ𝐺(𝜆, 𝑖̅, 𝑥̅) = ∫ 𝐻⃗⃗ 𝐺∙ 𝑑𝑙
31
Where the integration path is represented in figure (15).
Applying Ampere’s law as in figure (15), it is possible to write equations: ℱ𝑅𝑆 (𝜆, 𝑖̅, 𝑥̅) − ℱ𝑅𝑆 (0, 𝑖̅, 𝑥̅) = ℱ𝐼 (𝜆, 𝑖̅, 𝑥̅)
Above equation considers all contributes that, as it knew, is null. For this reason, it will be introduced a 𝛿(𝜆, 𝜆𝐶ℎ(𝑥̅)) function with the scope of identify only the active
conductors included in a generic Ampere loop which start from 𝜆 = 0 and arrive to 𝜆 = 𝜆𝐶ℎ(𝑥̅). This function is defined as:
𝛿(𝜆, 𝜆𝐶ℎ(𝑥̅)) = {0 𝑖𝑓 𝜆 𝜖 (0, 𝜆⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ) 𝐶ℎ(𝑥̅) 1 𝑖𝑓 𝜆 𝜖 (𝑦, 𝜆⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (𝑥̅))𝐶ℎ It is possible now to explicit the ℱ𝐼 equation as:
ℱ𝐼 (𝜆, 𝑖̅, 𝑥̅) = ∑ 𝑖𝑘∑ 𝑁𝐶𝑘ℎ 𝑛𝑐 ℎ=1 𝑁𝐸 𝑘=1 𝛿(𝜆, 𝜆𝐶ℎ(𝑥̅))
32
Where, calling 𝑁𝑘as winding k-phase function: 𝑁𝑘(𝜆, 𝑥̅) = ∑ 𝑁𝐶𝑘ℎ 𝑛𝑐 ℎ=1 𝛿(𝜆, 𝜆𝐶ℎ(𝑥̅)) ⇒ ℱ𝐼 (𝜆, 𝑖̅, 𝑥̅) = ∑ 𝑁𝑘(𝜆, 𝑥̅)𝑖𝑘 𝑁𝐸 𝑘=1 ℱ𝐼 (𝜆, 𝑖̅, 𝑥̅) = 𝑁̅(𝜆, 𝑥̅)𝑇𝑖̅ 𝑁̅(𝜆, 𝑥̅) = 𝑵𝐶𝛿̅(𝜆, 𝜆̅𝐶(𝑥̅))
Using these equations and constitutive equations it is possible to define 𝜀𝐸 equivalent
gap thickness in case without PM is:
𝜀𝐸(𝜆, 𝑥̅) = 𝜀𝐺(𝜆, 𝑥̅)
Where 𝜀𝐺 and is the airgap thickness. Defining 𝜇𝐺 as the airgap magnetic permeability, is possible to continue with analysis and compute 𝐻𝐺:
𝐻𝐺(𝜆, 𝑖̅, 𝑥̅) = ℱ𝐸 (𝜆, 𝑖̅, 𝑥̅) 𝜀𝐸(𝜆, 𝑥̅)
As before, it is possible to define an equivalent and total air-gap magneto-motive force as
ℱ𝐸(𝜆, 𝑖̅, 𝑥̅) = ℱ𝑇 (𝜆, 𝑖̅, 𝑥̅) + ℱ𝑅𝑆 (0, 𝑖̅, 𝑥̅) ℱ𝑇(𝜆, 𝑖̅, 𝑥̅) = 𝑁̅𝑇(𝜆, 𝑥̅)𝑖̅
Defining 𝜌 as factor form:
𝜌(𝜆, 𝑥̅) = 𝜏𝐺(𝜆, 𝑥̅) 𝜀𝐸(𝜆, 𝑥̅)
33
It can be introduced an equivalent permeability as: 𝜇𝐸(𝜆, 𝑥̅) = 𝜇𝐺(𝜆, 𝑥̅) 𝜌(𝜆, 𝑥̅)
The solution of the magnetic problem consists in the calculation of ℱ𝑅𝑆 (0, 𝑖̅, 𝑥̅) which can be achieved applying the Gauss Law to the entire device gap surface.
∮ 𝐵𝐺(𝜆, 𝑖̅, 𝑥̅) 𝜏𝐺(𝜆, 𝑥̅)𝑑𝜆 = 0 The total magneto-motive force can be therefore written as:
ℱ𝐸(𝜆, 𝑖̅, 𝑥̅) = ℱ𝑇(𝜆, 𝑖̅, 𝑥̅) −∫ 𝜇𝐸(𝜆, 𝑥̅) ℱ𝑇(𝜆, 𝑖̅, 𝑥̅)𝑑𝜆 1 0 ∫ 𝜇𝐸(𝜆′, 𝑥̅) 1 0 𝑑𝜆′
The above relation lead to the definition of a “descriptive anisotropy function” 𝜐: 𝜐(𝜆, 𝑥̅) ≜ 𝜇𝐸(𝜆, 𝑥̅)
∫ 𝜇𝐸(𝜆′, 𝑥̅)𝑑𝜆′ 1
0
Whit all elements introduced it is possible to define the equivalent winding function: 𝑁̅𝐸(𝜆, 𝑥̅) = 𝑁̅(𝜆, 𝑥̅) − ∫ 𝜐(𝜆, 𝑥̅) 𝑁̅(𝜆, 𝑥̅) 𝑑
1
0
𝜆
Finally, combining all the functions introduced, the inductance matrix can be calculated according to the following relationship:
𝑳𝑃(𝑥̅) = 𝑙 ∫ 𝜇𝐸(𝜆, 𝑥̅) 𝑁̅𝐸(𝜆, 𝑥̅) 𝑁̅𝐸𝑇(𝜆, 𝑥̅)𝑑𝜆 1
0 𝐶
34
The computation of the principal inductance matrix is a fundamental step in the analysis of electro-magneto-mechanical devices as it directly relates the electromagnetic and mechanical model through the wrench developed.
35
3. Analytical Model
In this third chapter the modelling approach previously discussed is applied to study a three-phase radial active magnetic bearing. The numerical analysis is performed through implementation on MatLab environment. The target of this analysis is to determinate an expression of the wrench in order to compute the forces required to compensate unplanned and gravity forces during normal operation of the machine. Since there are two degree of freedoms, the mechanical Lagrangian coordinates vector features cardinality in the amount of two:
𝑥̅(𝑡) = [𝜌(𝑡) 𝜗(𝑡)]
Where 𝜌 is the radial displacement and 𝜗 is the tangential displacement as portrayed in figure:
36
In the figure above, 𝜆 is indicated as the variable which is used to map the tangential points along the circumferential development of the structure profile considering the stator reference frame system.
For what concerns the electromagnetic state variables, the current vector is defined and presented as below considering 𝑁𝐸 = 3
𝑖̅ = [ 𝑖1 𝑖2 𝑖3
]
Starting from the quantities which do not depends on the Lagrangian coordinates, is important to define the winding descriptive functions. In this case-study it will be considered a three-phase winding function constituted by three steps shifted by 120 electrical degrees with amplitude ND:
𝑁1(𝜆) = 𝑁𝐷(𝑢(𝜆) − 𝑢 (𝜆 −1 3)) 𝑁2(𝜆) = 𝑁𝐷(𝑢 (𝜆 −1 3) − 𝑢 (𝜆 − 2 3)) 𝑁2(𝜆) = 𝑁𝐷(𝑢 (𝜆 −2 3) − 𝑢(𝜆 − 1)) These winding functions are represented in following image:
37
According with the general model discussed, it possible evaluate the equivalent winding function as:
𝑁̅𝐸(𝜆, 𝜌, 𝜗) = 𝑁̅(𝜆) − ∫ 𝜐(𝜆, 𝜌, 𝜗) 𝑁̅(𝜆) 𝑑1
0
𝜆 So, i.e. equivalent winding function for phase one is computed as:
𝑁𝐸1(𝜆, 𝜌, 𝜗) = 𝑁𝐷(𝑢(𝜆) − 𝑢 (𝜆 − 1 3)) − ∫ 𝜇𝐸(𝜆, 𝜌, 𝜗) 𝑁𝐷(𝑢(𝜆) − 𝑢 (𝜆 −13)) 1 0 ∫ 𝜇𝐸(𝜆, 𝜌, 𝜗)𝑑𝜆 1 0
38
Stator radius and rotor radius must be introduced to define the geometry of the problem. In particular, for centred-case, the rotor radius is constant and change whenever there is a rotor displacement. In figures (18) are represented stator, rotor and gap radius having supposed the rotor geometry under sinusoidal approximation:
The function that describe the rotor radius is determined for all displacement position, and it allows to compute 𝜏𝐺(𝜆, 𝜌, 𝜗) defined as:
𝜏𝐺(𝜆, 𝜌, 𝜗) = 2𝜋 𝑟𝑔(𝜆, 𝜌, 𝜗) Where 𝑟𝑔(𝜆, 𝜌, 𝜗)is gap-radius.
Figure 18 Stator, Rotor and Gap radius
39
As a first approximation, it is supposed a sinusoidal variation of both air-gap radius and equipotential gap surface. This leads to an approximated expression for the equivalent permeability function 𝜇𝐸 as:
𝜇𝐸(𝜆, 𝜌, 𝜗) = 𝜇𝐺𝜏𝐺(𝜆, 𝜌, 𝜗)
𝜀𝐸(𝜆, 𝜌, 𝜗) ≅ 𝜇𝐸0(𝜌, 𝜗) + 𝜇𝐸𝐼(𝜌, 𝜗)cos (2𝜋𝜆 − 𝛾𝐸𝐼(𝜌, 𝜗)) Furthermore, according to the geometry and electromagnetic field distribution, is possible to simplify the above relation having that the amplitude of the quantities is only affected by the radial displacement while the 𝜗 Lagrangian coordinate only impact on the phase of the Fourier expansion.
𝜇𝐸(𝜆, 𝜌, 𝜗) ≅ 𝜇𝐸0(𝜌) + 𝜇𝐸𝐼(𝜌)cos (2𝜋𝜆 − 𝛾𝐸𝐼(𝜗))
Moreover, as a fair approximation we can consider 𝛾𝐸𝐼(𝜗) ≅ 𝜗. Evaluating the permeability in both centred case and eccentric case, it is possible to see highlight the difference as shown in the following pictures:
40
Figure 21 Equivalent permeability
Substituting the Fourier approximated form of the equivalent permeability in the equivalent winding function we have:
𝑁𝐸1(𝜆, 𝜌, 𝜗) = 𝑁𝐷(𝑢(𝜆) − 𝑢 (𝜆 −1 3)) − 1 3𝑁𝐷+ √3𝜇𝐸𝐼(𝜌) 2𝜋𝜇𝐸0(𝜌)cos (𝜗 − 𝜋 3) With the same passages it is possible determinate the equivalent winding function for the other phases:
𝑁𝐸2(𝜆, 𝜌, 𝜗) = 𝑁𝐷(𝑢 (𝜆 −1 3) − 𝑢 (𝜆 − 2 3)) − 1 3𝑁𝐷+ √3𝜇𝐸𝐼(𝜌) 2𝜋𝜇𝐸0(𝜌)cos (𝜗) 𝑁𝐸3(𝜆, 𝜌, 𝜗) = 𝑁𝐷(𝑢 (𝜆 −2 3) − 𝑢(𝜆 − 1)) − 1 3𝑁𝐷+ √3𝜇𝐸𝐼(𝜌) 2𝜋𝜇𝐸0(𝜌)cos (𝜗 − 5𝜋 3 ) In the figures below are represented the equivalent winding function in centred case:
41
Considering rotor displacement in the radial direction of 22% of the air-gap in centred case and a tangential displacement of 118 mechanical degrees, the results are presented in figure (23).
Figure 22 Equivalent winding function centred case
42
Comparing the two cases it is possible to see that equivalent winding function is not constant but changes with respect to the rotor displacement.
With the target of electromagnetic wrench computation in mind, it is important to evaluate the inductance matrix so that is possible to estimate its derivative along the two Lagrangian coordinates. Recalling the inductance equations yet introduced in the chapter before:
𝑳𝑃(𝜌, 𝜗) = 𝑙 ∫ 𝜇𝐸(𝜆, 𝜌, 𝜗)𝑁̅𝐸(𝜆, 𝜌, 𝜗)𝑁̅𝐸𝑇(𝜆, 𝜌, 𝜗)𝑑𝜆 1
0 𝐶
Substituting each parameter, it is possible to evaluate a generic self-inductance as: 𝐿𝑗,𝑗(𝜌, 𝜗) = 𝑙 (4𝜇𝐸0 27 𝑁𝐷 2+2√3 9𝜋 𝑁𝐷(𝑁𝐷 − 𝜇𝐸𝐼(𝜌)) cos (𝜗 − 𝜋 3− 2𝜋 3 𝑗) + 1 𝜋2 𝜇𝐸𝐼(𝜌) 𝜇𝐸0(𝜌)(𝜇𝐸𝐼(𝜌) − 𝑁𝐷)𝑐𝑜𝑠2(𝜗 − 𝜋 3− 2𝜋 3 𝑗) +3√3 8𝜋3 𝜇𝐸𝐼2 (𝜌) 𝜇𝐸0(𝜌)𝑐𝑜𝑠3(𝜗 − 𝜋 3− 2𝜋 3 𝑗)) Where “j” is the phase index that span in the range 𝑗 ∈ {1,2,3}.
In the below figures are represented the self-inductances where the 𝜌 -axis is portrayed the radial displacement expressed as percent of gap thickness in the centred case:
43
For what concerns the mutual inductances, they are expressible, in general form, through the above relations:
𝐿ℎ,𝑘 = 𝑙𝑁𝐷2(1 3𝑁𝐷 2𝜇 𝐸0(𝜌) − 2 9𝑁𝐷𝜇𝐸0(𝜌) + √3 2𝜋((𝑁𝐷− 1 𝜇𝐸0(𝜌)− 1 3) 𝜇𝐸𝐼(𝜌) − + 𝑁𝐷 3 ) cos (𝜗 − 𝜋 3− 2𝜋 3 𝑖) + √3 2𝜋( 2 3𝜇𝐸𝐼(𝜌) − 1 3𝑁𝐷) cos (𝜗 − 𝜋 3− 2𝜋 3 𝑗) + + ( 9 4𝜋2 𝜇𝐸𝐼2 (𝜌) 𝜇𝐸0(𝜌)− √3 2𝜋 𝜇𝐸𝐼(𝜌) 𝜇𝐸0(𝜌)) cos (𝜗 − 𝜋 3− 2𝜋 3 𝑖) cos (𝜗 − 𝜋 3− 2𝜋 3 𝑗)) Figure 24 Self-inductances
44
Where h,k are generically phases and i,j are support coefficient. The relationships between h,i and k,j are: h= 1,2,3 → i = 0,1,2 and k= 1,2,3 → j = 0,1,2.
In the figures above are represented only L12, L13 and L23 elements because, taking
advantage of the symmetric nature of the inductance matrix, the full matrix can be populated following the rules below:
𝐿21 = 𝐿12, 𝐿13 = 𝐿31, 𝐿32 = 𝐿23
45
Once the self and mutual inductances are evaluated, it is possible to determine their derivative with respect to the Lagrangian coordinates.
𝑳𝜌′(𝜌, 𝜗) = 𝜕𝑳(𝜌, 𝜗)
𝜕𝜌 𝑳𝜗′ (𝜌, 𝜗) =𝜕𝑳(𝜌, 𝜗)
𝜕𝜗
In following figures are illustrated some components of the derivative of the inductance matrix with respect to the generalized coordinate 𝜌:
46
The above three figures show the self-inductance derivative while the mutual inductance derivative are presented below.
Analogously, it is possible to represent the components of the derivative of the inductance matrix along the 𝜗 direction:
Figure 27 Mutual-inductances derivatives respect to ρ
47
The above results show that the variation of the inductance matrix can be modelled using a mid-complexity model of a generalised electro-magneto-mechanical device modified to account for the rotor eccentricity. The variation of the inductance shows the capability of the device to generate the forces required for maintain the rotor in centred position which is the topic of the last chapter of this work. In particular, variation of the inductance follows specific path and trend which are investigated in more detail in the following section.
48
3.1 Harmonic Analysis
The AMBs are design and controlled with the aim of counteract external forces which act as disturbance to the system. To this end, it may be useful to determine which harmonic of the matrix of the incremental wrench coefficients provides the major contribution at this scope. Furthermore, this enables to draw considerations that direct towards the decupling of the system in a set of independent state variables.
To achieve this target, it is necessary to perform the harmonic analysis by means of the Fourier transformation. The derivative of the inductance matrix can be therefore expressed in terms of Fourier series as
𝑳 𝜌′ (𝜌, 𝜗) = 𝑳 𝜌′(𝜌) + ∑ 𝑳𝑘 𝜌′(𝜌) cos(𝑘𝜗 + 𝝍(𝜌)𝑘 ) ∞ 𝑘=1 0 𝑡 𝑳 𝜗′ (𝜌, 𝜗) = 𝑳𝜗′ (𝜌) + ∑ 𝑳 𝜗 ′ (𝜌) cos(𝑘𝜗 + 𝝍(𝜌)𝑘 ) 𝑘 ∞ 𝑘=1 0 𝑡
Is interesting to note that the harmonic analysis is performed considering the Lagrangian coordinate which account for the tangential rotor displacement as the variable. Therefore, the resulting Fourier coefficients are dependent on the radial Lagrangian coordinate.
In the following graphs, the average value and the harmonic content of the Fourier coefficients is presented as a function of the rotor radial displacement. The aim of this analysis is the evaluation of the contribution of each element to the total electromagnetic wrench.
49
The pictures above represented the self-inductance L11 element as the other matrix
elements follows a similar trend.
50
It is possible to see that the major contribution is given by both the average value and the first harmonic component. That said, all the other harmonics could be neglected to achieve a quick but yet accurate model of the AMB behaviour.
Using same line of thinking, the fast Fourier transform is applied to the derivative of the inductance matrix along 𝜗 direction.
Figure 32 Average value and Harmonics
51
The first picture represents the average value of the derivative of the self-inductance along 𝜗 direction with respect to the radial displacement. In the second picture, instead, the harmonics spectrum is presented against the rotor radial displacement.
52
Is possible to highlight that the contribution of the average value to the tangential component of the electromagnetic wrench is negligible. Furthermore, the higher harmonics are also featuring a magnitude considerably lower than the one presented in the case relative to the derivative of the inductance matrix with respect to 𝜌.
This allow to neglect the contribution of the derivative of the inductance matrix with respect to the tangential coordinate in the overall electromagnetic wrench generated. In the following section, FE analysis is performed to confirm this result.
53
4. FE Model Validation
The analytical model is validated by means of comparing the results with respect to FE ones. The more accurate field distribution and force calculation of the FE allows to understand the goodness of the analytical calculations performed in the previous chapter. On the other hand, FE analysis required long computational time and good hardware support.
In this chapter it is described how the geometry is created and the type of simulation that have been performed are described.
The geometry is created taking advantage of the possibility to control the FE software (MagNet) through MatLab. A MatLab script is created to construct the geometry of the problem which can be easily parametrised to investigate variation in the structure. Particular attention has been focused on the creation of the airgap and the sections of virtual air required for the accurate calculation of the electromagnetic forces.
In the table below there are listed the main parameters of the AMB geometry:
Object
Size
Internal Stator Diameter 𝜙 50 𝑚𝑚 External Rotor Diameter 𝜙 49.2 𝑚𝑚
Internal Rotor Diameter 𝜙 25 𝑚𝑚 External Stator Diameter 𝜙 100 𝑚𝑚
Slot depth 15 𝑚𝑚 Internal Shaft Diameter 𝜙 17 𝑚𝑚 Gap Thickness 0.04𝑚𝑚 Structure Depth 50 𝑚𝑚
54
In following figure, it is represented a schematic configuration of the AMB in centred case:
The first case-study analysed is considering linear ferromagnetic characteristic for the iron core materials. More specifically, in table (1) there are listed the magnetic properties of the materials implemented in the FE environment:
Object
Size
Stator Core 𝜇𝑟 104 Rotor Core 𝜇𝑟 104 Airgap 𝜇𝑟 1 Shaft 𝜇𝑟 1 Conductors 𝜎 5.77∙ 107 𝑆 𝑚 Table 1A sensitivity analysis on the mesh has been performed in order to determine the optimum mesh size. To define the most appropriate element size, different static problems have been solved and the optimum value has been found as a good compromise between accuracy of the results and computational time. According to the difference between the torque/force computed on the stator and the rotor component, the optimum value can be selected.
55
Problem Maximum Element
Size
Problem
Maximum Element
Size
1
10
6
0.005
2
1
7
0.003
3
0.1
8
0.002
4
0.01
9
0.001
5
0.008
10
0.0008
Table 2In following analysis, the mesh shown in figure (38) (2D on the left and 3D on the right) has been used.
Figure 37 Mesh sensitivity analysis
56
Next step in the analysis is the evaluation of the inductances for the purpose of comparing the results with the one obtained by means of the analytical model. The flux linkage with each phase is evaluated and the phase inductance is therefore computed (considering linear magnetic material) according to the relation below
L𝑖,𝑗 = Ψ𝑖
𝑖𝑗 𝑖 = 1, . . , 𝑁𝐸 𝑗 = 1, . . , 𝑁𝐸
In this way, the self and mutual inductances are computed. The steps that have been used to calculate the inductances are:
- Impose unitary current in a phase;
- Determinate the flux linkage with all the phase of the device; - Divide each flux linkage with respect to the current imposed
The above algorithm has been performed for each radial and tangential displacement.
Currents 3 Radial Displacement 10 Tangential Displacement 120 Mapping Points 360 Total simulations 3600 Table 3
The data elaboration is performed in the Matlab environment. By way of example, the FE results are shown only one self-inductance because more importance is given to the comparison with respect to the analytical model results.
57
The error percentage is evaluated through:
𝐿𝑖,𝑗 (𝜌, 𝜗) = 𝐿𝑖,𝑗(𝜌, 𝜗) − 𝐿𝑖,𝑗(𝜌, 𝜗) 𝐹 𝐴 𝐿𝑖,𝑗 𝐴 (𝜌, 𝜗) 𝐸 ∙ 100
In the following picture are presented the contour maps of the percentage error between FE and analytical model.
Figure 39 FEM inductance
58
To give an order of magnitude to the error, it is chosen the to show the self and mutual inductances that features the maximum error:
On the left hand of the previous picture is represented the contour maps and, on the right, the 3D error maps where on the x and y axes are indicated the displacement considered while on z-axis is reported the percentage error.
It is also possible to compare the flux density waveforms computed with the analytical/numerical model and sampled from the FEM solved model. According to the analytical model, the flux density distribution in the main airgap can be evaluated as:
59
𝐵𝐺(𝜆, 𝑖̅, 𝑥̅) =𝜇𝐺𝑁̅𝐸
𝑇(𝜆, 𝑖̅, 𝑥̅)
𝜀𝐸(𝜆, 𝑥̅) 𝑖̅
Firstly, the results concerning the variation of the rotor radial position are discussed.
A good match between the FE and analytical results of the air-gap flux density distribution is presented in figure (42).
Analogously, is possible to compare the analytical flux density estimated by means of the analytical model with respect to the FEM analysis calculations considering
Figure 42 Flux density with different radial position
60
different tangential position of the rotor. The case-study analysed is considering a
radial displacement equal to 13% of the airgap the thickness in the centred case and with different tangential positions as indicated in the labels of the figure below. As before, it is possible to identify one case in order to perform a zoom to highlight similarities and difference as in the figure below.
Figure 44 Flux density with different tangential position
61
Although the error is low, small model modification can drastically increase the accuracy of the prediction. To achieve significant improvements, the slot opening can be modelled as a step variation in the stator inner bore radius. In this way it is possible to obtain the flux density evaluated through the analytical and FEM model as presented in figure (46). This figure represents the flux density in the case of radial displacement of 35% and tangential displacement of 60°.
62
5. Model Generalization and
Limits
The model developed in the previous chapters allows to obtain very accurate prediction as demonstrate in the FE comparison chapter. However, it is possible to generalise the analytical/numerical calculation to further improve the prediction. Furthermore, re-thinking the process of evaluation of the expected flux lines distribution in the main airgap allows to achieve a more stable and general approach to the problem, also in case of critical airgap thickness to airgap radius ratios.
According to the interface conditions between media with different relative permeability, a quasi-exact computation of the field lines path and length can be performed. Considering the principle of minimum reluctance path for the field lines distribution, eccentric rotor conditions leads to a non-radial flux lines distribution. A possible approximation can be done assuming that their path is an arc of a circle.
63
The problem of the field distribution prediction can be solved determining the circumference which belongs to the line “t” such that the field line comes out orthogonal form stator surface and enters orthogonal in rotor surface. In order to fulfil these conditions, it is necessary to obtain the centre C and the radius 𝑃𝐶̅̅̅̅ of the circle as a function of 𝜆. The coordinates of the point S and arc length are computed by means of the calculation of the line integral on circumference Σ evaluated between P and S.
𝐶 = (𝑥1, −𝑥𝑠𝑥1 𝑦𝑠 +𝑦𝑠 2+ 𝑥 𝑠2 𝑦𝑠 ) 𝑟𝐶 = √(𝑥1− 𝑥𝑠)2+ (−𝑥𝑠𝑥1 𝑦𝑠 +𝑦𝑠 2 + 𝑥 𝑠2 𝑦𝑠 − 𝑦𝑠) 2
In those relationships the unknown parameter is 𝑥1 that can be computed by solving the following system of algebraic equations:
{ 𝛼 ∶ 𝑦 = − (𝑥𝑎1− 𝜌 cos 𝜗 𝑦𝑎1− 𝜌 sin 𝜗) 𝑥 + 𝑦𝑎1+ ( 𝑥𝑎1− 𝜌 cos 𝜗 𝑦𝑎1− 𝜌 sin 𝜗) 𝑥𝑎1 𝛽 ∶ 𝑦 = − (𝑥𝑎2− 𝜌 cos 𝜗 𝑦𝑎2 − 𝜌 sin 𝜗) 𝑥 + 𝑦𝑎2+ ( 𝑥𝑎2− 𝜌 cos 𝜗 𝑦𝑎2 − 𝜌 sin 𝜗) 𝑥𝑎2 𝑡: 𝑦 = −𝑥𝑠 𝑦𝑠 𝑥 +𝑦𝑠 2+ 𝑥 𝑠2 𝑦𝑠
Where 𝛼 and 𝛽 are respectively the line orthogonal to the line passing for O’S and O’D.
{
𝑥𝑎 1,2=−𝑘𝜌 cos 𝜗 + 𝑐
2𝜌 cos 𝜗 ± √(𝑘𝜌 cos 𝜗 − 𝑐2𝜌 cos 𝜗)2− (4𝜌2𝑐𝑜𝑠2𝜗 + 𝑐2)(𝑘2− 𝑐2𝜌2(𝑐𝑜𝑠2𝜗 − 2𝑠𝑖𝑛2𝜗) + 𝑐2𝑟 𝑟3) 4𝜌2𝑐𝑜𝑠2𝜗 + 𝑐2 𝑦𝑎1,2= 𝜌 sin 𝜗 + √𝜌2𝑠𝑖𝑛2𝜗 − 𝑥𝑎1,22 + 2𝑥𝑎1,2𝜌 cos 𝜗 − 𝜌2𝑐𝑜𝑠2𝜗 − 𝜌2𝑠𝑖𝑛2𝜗 + 𝑟𝑟2 𝑐 = 2𝜌 sin 𝜗 +2(𝑥𝑠𝑥1− 𝑟𝑠 2) 𝑟𝑠2− 𝑥𝑠2 𝑘 = 𝜌2(𝑠𝑖𝑛2𝜗 − 𝑐𝑜𝑠2𝜗) + 𝑟 𝑟2+ 2𝜌 sin 𝜗 𝑟𝑠2− 𝑥𝑠3 (𝑥𝑠𝑥1− 𝑟𝑠2) + 2𝑥1𝑥𝑠− 𝑥𝑠2− 𝑥𝑠4− 2𝑥𝑠3𝑥1+ 2𝑥12𝑥𝑠2− 2𝑥𝑠𝑥1𝑟𝑠2+ 𝑟𝑠4 𝑟𝑠2− 𝑥𝑠2
64
Implementing the above calculation in the MatLab environment allows a fine calculation of the field lines path as shown in figure (48). It is possible to note that the deviation with respect to the radial path is clear.
The sinusoidal approximation introduced in chapter 3 is not the exact solution of the problem. The computation of the field lines as presented above results in a quite complicated mathematical problem. To find a good compromise between accuracy of the results, fast implementation and reduced calculations, further considerations can be done looking at figure (49).
