LAPPEENRANTA – LAHTI UNIVERSITY OF TECHONOLOGY LUT School of Energy Systems
Master`s Degree Programme in Electrical Engineering
Emil Iakupov
INCREASING THE BANDWIDTH OF AXIAL AMB FOR THE HIGH-SPEED SYSTEM
Examiners: Docent D.Sc. Rafal Piotr Jastrzebski Docent D.Sc. Lassi Aarniovuori
Supervisors: Docent D.Sc. Rafal Piotr Jastrzebski Docent D.Sc. Lassi Aarniovuori
2 ABSTRACT
LAPPEENRANTA – LAHTI UNIVERSITY OF TECHONOLOGY LUT School of Energy Systems
Master`s Degree Programme in Electrical Engineering
Emil Iakupov
Increasing the bandwidth of axial AMB for the high-speed system
Master’s Thesis
2020
63 pages, 47 figures, 27 tables and 3 appendices Examiners: Docent D.Sc. Rafal Piotr Jastrzebski
Docent D.Sc. Lassi Aarniovuori
Keywords: active magnetic bearings (AMB), axial magnetic bearing, electromagnetic actuator, compressors, high-speed (HS) systems, power bandwidth, optimization, dynamic performance, finite element analysis (FEA)
Active magnetic bearings are increasingly presented as a new alternative to conventional bearings, especially where the latter can no longer be used, as they do not fully meet the newly imposed technical requirements. This is especially true for high-speed equipment, in which the rotating shaft weighing several tons rotates at a high speed. Such applications are for instance gas and steam turbines, compressors, blowers, etc. Besides, at high speeds AMB actuators require sufficient bandwidth needed to achieve desirable performance. In this thesis, the possibilities of force bandwidth improvements are studied that allow to receive the required actuator dynamic performance at high speeds. For these purposes, an analytical model for the axial actuator is created and then it is compared with a finite element model.
ACKNOWLEDGEMENTS
This Master’s thesis was done at School of Energy Systems, Lappeenranta University of Technology.
I would like to express my gratitude to my supervisors Dr Rafal Jastrzebski for his scientific guidance and support that finally allow me to complete the thesis, and to Dr Lassi Aarniovuori for his valuable advices. I also thankful to Dr Katja Hynynen for necessary guidance and willingness to help during all the study at LUT.
I wish to thank the tutor Vladimir Ishin from the Saint Petersburg Electrotechnical university «LETI»
for his help with the organizing of studying process throughout the academic year.
Finally, I am grateful to my family for their omnifarious support.
Emil Iakupov
Lappeenranta University of Technology November, 2020
4 TABLE OF CONTENTS
1. INTRODUCTION ... 9
1.1 Applications of high-speed systems ... 9
1.2 Technical challenges of high-speed systems ... 10
1.3 Summary ... 12
2. MAGNETIC BEARINGS... 14
2.1 Active magnetic bearing main working principles ... 14
2.2 Electromagnets ... 15
2.3 Inductance in coil and current slew rate ... 18
2.4 AMB load capacity and its limitations... 19
3. ANALYTICAL MODEL AND PERFORMANCE IMPROVEMENT ... 21
3.1 Axial bearing parameters ... 21
3.2 Axial actuator analytical model ... 25
3.3 Axial actuator design limitations ... 29
3.4 Geometry optimization ... 32
3.5 Increasing bandwidth by increasing the air gap ... 33
3.6 Increasing bandwidth by changing the permeability ... 36
3.7 Summary... 37
4. FINITE ELEMENT ANALYSIS OF AXIAL AMB ... 38
4.1 The creation of computational models and forces’ calculations ... 38
4.2 Bandwidth calculations for original geometry ... 42
4.3 Bandwidth calculations for optimized geometry ... 51
4.4 Summary... 55
CONCLUSION ... 57
REFERENCES ... 58
APPENDICES ... 61
Appendix 1: Plot of forces’ amplitudes for 8 A (original geometry) ... 61 Appendix 2: Plot of forces’ amplitudes for 8 A (optimized geometry) ... 62 Appendix 3: Plot of forces’ amplitudes for 3 A (original geometry) ... 63
6 NOMENCLATURE
Symbols
a Stator width of C-type actuator mm
Acoil Cole area mm2
Aflux Minimum pole face area for saturation peak force mm2
Ain Inner pole surface area mm2
Aout Outer pole surface area mm2
b Pole width of C-type actuator mm
Bbias Bias flux density T
Bmax Maximum flux density T
Bsat Saturation flux density T
𝑐i Dynamic coefficient of the effective reluctance
d Stator height of C-type actuator mm
d1 Disk thickness total mm
d2 Coil high mm
d3 Back iron thickness mm
din Disc inner diameter mm
dout Disc outer diameter mm
Do Inner pole inner diameter mm
D1 Inner pole outer diameter mm
D2 Outer pole inner diameter mm
D3 Outer pole outer diameter mm
f Frequency Hz
Fmax Maximum load capacity N
𝐹p Perturbation force N
ℎ Flotor length of C-type actuator mm
Hmax Maximum magnetic field strength A/m
ibias Bias current A
ip Perturbation current A
Im Amplitude of current A
ki Current stiffness N/A
kx Position stiffness N/m
lFe Flux path length in iron mm
lc Mean length of a coil turn mm
L Inductance H
n Rated speed of the machine rpm
nmax Maximum speed of the machine rpm
n Rated speed rpm
Pl Joule losses kW
PN Rated power kW
R Resistance Ohm
𝑅m0 Static reluctance 1/H
Sair Air – gap cross sectional area mm2
Sc Winding’s wire cross sectional area mm2
t Time s
umax Maximum voltage V
UN Rated voltage Vrms
Vmax Maximum amplifier output voltage V
Wa Weighting parameter for pole length
Wb Weighting parameter for pole width
Wce Electromagnetic co – energy J
𝑧 Number of cuts of an axial bearing
δ Logarithmic decrement
𝜌1 Allowable disk radius (outer radius limitation) mm
𝜌2 Inner radius limitation mm
𝜌Cu Copper resistivity Ohm·mm2/m
𝜇 Relative permeability
𝜇0 Magnetic permeability of free space
𝜈 Poisson number
𝜎𝑠 Yield strength MPa
𝜎st Stator conductivity S/m
𝜎rt Stator conductivity S/m
𝜗max Maximum circumferential speed of a thrust disk m/s
𝜒 Angle between electromagnetic poles deg
𝜔−3dB Actuator bandwidth, corresponding to –3 dB Hz
𝜔BW Actuator power bandwidth Hz
8
LIST OF ABBREVIATIONS AND SYMBOLS
AC Alternating current
AMB Active Magnetic Bearing
DC Direct Current
FEA Finite Element Analysis
FEM Finite Element Method
ISO International Organization for Standardization
HS High Speed
MMW Multi Mega Watt
USD United States Dollar
1. INTRODUCTION
Rotating machines belonging to the group of heavy engineering equipment, such as compressors, expanders, turbines, pumps and fans, centrifuges and separators are the basis of industrial plants generating heat and electric energy. Higher performance and power, coinciding with a reduced maintain and production costs are the key demands, dictated by the modern market, for the machines’
manufacturers. Besides, the government of many countries impose severe emission norms of and energy efficiency in oil and gas industry. To satisfy these demands, owing the modern developments in semiconductor and power electronic industries, material technologies, the rotating machines design trend is in the direction of increasing of their rotational speeds [1].
1.1 Applications of high-speed systems
During the latest decade high-speed (HS) systems have been finding their application and are becoming more and more widespread in various industrial fields. Their implementation allows to better address the environmental, regulatory and maintenance issues than traditional solutions. HS machines are successfully used in aircraft electrical starter-generator systems and flywheel energy storages, in spindles for machine tools and medical surgery tools, in compressors and turbochargers, blowers, turbo- molecular vacuum pumps, compact electrical power suppliers and micro co-generation gas turbines and so on. For instance, in the recent years, there is an increasing demand in machine tool industry for high-speed spindle applications. At a high speed desirable high mechanical precision can be achieved with reduced misalignment and vibrations, as a result high-quality work and surface manufacturing are provided. The high speed is an essential requirement in the precision grinding of small parts [2].
One of the main areas for high-speed machines is gas compression. This area covers small turbo- molecular pumps up to turbo generators and compressors in Megawatt range [3].
The HS turbo-molecular pumps, where rotational speeds attain 100 000 rpm at low powers, are exploited in semiconductor manufacturing where high rotational speeds for efficient operation and quality improvements are needed [4], [5].
Compressors provide many vital manufacturing processes and are irreplaceable in such key industrial sectors as petrochemical, energy production, etc. They are used for extremely important tasks, from temperature control to gas transportation and mixing. That is why, the efficiency and reliability improvements of these machines are a serious challenge nowadays and many steps are taken in this direction [6]. One of them is the implementation of high-speed machines in compressor systems, which allows to receive higher power density and a smaller footprint of the device [7].
10
Nowadays such large companies as MAN (Germany), General Electric (USA), Siemens (Germany), etc.
produce high-speed compressor systems, utilizing gearboxes as a coupling between a motor and a working machine (compressor), for different industry areas, such as:
Oil and gas industry
Power generation
Petrochemical industry
Energy recovery
To sum up, the implementation of high-speed drives allows to reduce weight, footprint, maintenance cost of the system and improve its efficiency and reliability [8]. However, these benefits come with the cost of a significant effort required for the system development and new challenges for the designers of electrical machines for such applications.
1.2 Technical challenges of high-speed systems
The design of an HS electrical machine is a complex task. Increasing both rotational speed and shaft output power is not easy and require the machines to have enhanced electrical and magnetic characteristics. Additionally, the mechanical parameters such as rotor robustness, cooling, lubrication and damping vibrations must be improved [4], [10].
In case of traditional HS compressor systems produced nowadays, there are some issues, which impose restrictions on their efficiency performance. The compressor systems utilize gearboxes, which leads to inevitable use of oil and a complex oil supply system. Furthermore, the oil unit and gear are significantly bigger that standard electrical motor which is rotating the whole system. It is worth noting that friction losses caused by lubrication oil are a major losses component in the system. As a result, traditional HS compressors have the following drawbacks [11]:
The need for a complex oil supply unit
Decrease of system efficiency because of gearbox and oil supply unit
Requirement for large space
Complicated structure which leads to complex control system
In some areas (such as medical industry) totally oil free operation is required
In addition, there is a trend toward oil-free compressors, which is due to modern requirements for energy efficiency and environmental friendliness. Next, because of high speeds the reliability, lifetime and resistance to the wear of the elements of HS systems become serious concern. This is especially important for units that connect rotating and non-rotating parts of a machine, namely bearings. Today, there
are two main widely used types of bearings applied in industry (Figure 1.1). All of them have their own advantages and drawbacks, which are presented in Table 1.1 [12], [13].
Table 1.1. – Comparison of different bearing types
Type Advantages Drawbacks
Rolling – element bearings
Relatively low cost;
Low friction losses and heating;
Low lubricant consumption;
Reduced maintenance requirements
High sensitivity vibration loads due to the high rigidity of the bearing structure;
Low reliability in high-speed drives due to excessive heating and the danger of destruction of the separator from the action of centrifugal forces;
Relatively large radial dimensions;
Noise and vibration at high speeds.
Sleeve bearings (Fluid bearings)
Low coefficient of friction;
Significant damping ability;
Work quieter and create less vibration than rolling bearing;
Small gap change during high loads;
Simple design compared to rolling bearings.
Energy dissipation in bearings;
Can suddenly wedge or collapse in critical situations;
Allow shaft precession (change in the orientation of the rotational axis of a rotating body) which reduces the service life and worsens the quality of the bearing;
Some types of sleeve bearings are complex and expensive due to the use of pump;
There is a possibility of liquid or gas leaks outside the bearing.
Figure 1.1. Rolling [14] and sleeve (fluid) [15] bearings, respectively.
12
In Table 1.1. it can be seen that rolling bearings are absolutely unsuitable for HS applications. As for the sleeve bearings, they are often used in high load, high-speed or high-precision applications where ordinary rolling bearings would have a short life or produce high noise and vibration. However, this type still has the drawbacks, which do not make it the ideal candidate for use in the HS systems.
Moreover, as it was stated above, the introduction of oil-free systems is also one of the priority directions in the modernization of compressor equipment and fluid bearings that usually utilize oil as a lubrication fluid do not meet these requirements.
1.3 Summary
Taking into account the facts discussed earlier, it can be argued that it makes sense to remove gearboxes from HS compressors. A direct connection of a high-speed drive to the working machine without a mechanical transmission reduces the footprint and possibility of failures, increase the efficiency of the entire system [1].
There are two types of coupling between the motor and the working machine – integrated and standalone.
In former one the working machine is connected directly to the drive, and in latter there is a coupling between the motor shaft and the machine [11].
+ speed control
+ wide operation speed range + better efficiency
+ smaller footprint
Integrated
HS motor Load
+ higher speed + smaller impeller + lighter rotor + reduced footprint Frequency
converter Integrated
HS motor Load
Frequency converter
Coupling Coupling
Motor
Gear
Load Coupling
Coupling Motor
Gear
Load Coupling Frequency
converter
Figure 1.2. Drive solutions for compressor and blower applications. Adapted from [11].
Additionally, as it was stated above the rolling and sleeve bearings are not the best solution for high speed systems. Thus, the equipment which can be utilized instead of the gearbox and traditional bearings shall be used.
The proposed solution is to use in HS motors magnetic suspension system or active magnetic bearings (AMB). The HS systems equipped with magnetic bearings have the following advantages:
The absence of physical contact between the slicing surfaces means that there is no friction, as a result the wear of components is eliminated;
No friction means that there is no need for lubricants, therefore no oil is needed;
Since there is no need for oil, the system can be simplified and its cost reduced;
The ability to use a system in hazardous conditions (in wide range of temperatures, in vacuum, etc.).
These benefits of an HS machine with magnetic bearings make it an ideal candidate for high-speed compressors. However, they can face with forces caused by high rotational speeds and the bearings must have appropriate dynamic capabilities to dampen the response and provide the appropriate performance over the entire speed range (up to the maximum speeds) of the system. Thus, the improvement of the dynamic capabilities of the machine is a major concern.
14 2. MAGNETIC BEARINGS
Active magnetic bearings are becoming more and more widespread today. Currently, they are one of the most innovative developments in the field of engineering industry and find their industrial applications in rotary machines for various purposes. Magnetic bearings are more flexible and wide technical possibilities are opened with them. In this section AMB operation principles and devices are discussed.
2.1 Active magnetic bearing main working principles
The working principle of AMB is based on electromagnetic levitation achieved via magnetic forces. The magnetic field is created by controlled DC-currents currents in the core windings and the bearing itself consists of a stator that has pole windings and rotor (or core) which is mounted on a shaft. A magnetic flux is created by the stator at the air gap, which makes the rotor to levitate without any mechanical contact. To keep the rotor in necessary position a control system is needed. The magnetic field is controlled by power electronics which gets its feedback from position sensors. The sensor system continuously monitors the position of the shaft and sends signals to the automatic control system to return the shaft to the central position by adjusting the positioning magnetic field of the stator. The force of attraction on the desired side of the shaft is made stronger or weaker by adjusting the current in the stator windings of active bearings.
The amplifier supplies current to the control coils of the stator to create the necessary controlling magnetic force. In Figure 3.4 the principle of AMB is presented [3].
Figure 2.1. AMB primary components [3]
The whole bearing system includes three groups of bearings: axial, radial and auxiliary (or touchdown). The last one is used as a backup bearing in case of an abrupt shutdown. Axial and radial magnetic bearings keep the rotor in required position opposing to disturbance forces in axial (e.g gravity) and radial (e.g. unbalance forces) directions respectively. A power amplifier and electromagnets form the electromagnetic actuator.
This element can be considered as a defining element of any AMB. Its purpose is to convert the controller signals into forces applied to the rotor.
2.2 Electromagnets
The electromagnet can be described through a simplified model presented in Figure 2.2. The coil current i creates a magnetic flux Ф, forming a closed magnetic circuit between the actuator and the rotor (or core). The magnetic field H is generated by the current i in coils with N turns according to Ampere’s Law [6]:
∮ 𝐻 ⅆ𝑙 = 𝑁 ⋅ ⅈ (1)
where the left had side represents the actuator magnetomotive force and the right hand side the corresponding current linkage. N is the number of turns and l is the length of the coil.
Figure 2.2. Simplified model of electromagnet, forming magnetic circuit [13].
The length of flux paths in iron, air, and the relative permeability of iron are set as lFe, δ, µFe, respectively, and HFe and Hair are the magnetic field strengths of iron parts and air. Equation (1) may be written as [16]:
𝑙Fe⋅ 𝐻Fe+ 2𝛿 ⋅ 𝐻Fe= 𝑁 ⋅ ⅈ (2) Taking into account that flux density is:
𝐵 = 𝐻𝜇0𝜇Fe (3)
and assuming that flux density is constant in the both media, equation (2) is
16 𝑙Fe 𝐵
𝜇0𝜇Fe+ 2𝛿 𝐵
𝜇0 = 𝑁ⅈ (4)
Taking into account, that the permeability of the ferromagnetic μFe >> 1 the air-gap flux density [14]:
𝐵 = 𝜇0 𝑁ⅈ (𝑙Fe
𝜇Fe+ 2𝛿)
=𝜇0𝑁ⅈ
2𝛿 (5)
Here it is necessary to note, that flux density is limited by the Bsat that is defined by the material properties [6]:
𝐵 ≤ 𝐵sat (6)
To define the magnetic force F of the electromagnet, the equation defining the stored in electromagnet co- energy Wce is used:
𝑊ce = ∫ ∫ 𝐵𝑑𝐻𝑑𝑉
𝐻
𝑉 0
= ∫ ∫ 𝐻𝑑𝐵𝑑𝑉
𝐻
𝑉 0
= 1
2𝜇0∫ 𝐵2𝑑𝑉
𝑉
= 1
2𝜇0𝐵2𝑆air2𝛿 (7) where Sair is the cross-section of the actuator or of the air-gap, V is the volume of the space where magnetic energy is stored – mostly in air.
According to the principle of virtual displacement, force F equals the partial derivative of the field energy Wce with respect to the air gap δ [3]:
𝐹 = 𝜕𝑊ce
𝜕𝑙 =𝐵2𝑆aircos(𝜒)
𝜇0 (8)
Using the equation (5) and assuming the infinity permeability of iron μFe the force can be expressed as:
𝐹 =𝜇0𝑁2ⅈ2𝑆aircos(𝜒)
4𝛿2 , (9)
where 𝜒 – the angle between the poles of electromagnet or the force acting angle (actuator has a round shape) [16].
To keep the rotor in a central position two counteracting magnets are generally applied as shown in Fig. 2.3. Assuming that the number of turns N, nominal air gap δ and the pole face area Sair for both magnets are the same, the net forces acting on the rotor along x and y directions are [6] [16]:
𝐹x = 𝐹1x− 𝐹2x= 𝜇0𝑁2𝑆aircos(χ)
4 ( ⅈ1x2
(𝛿 − 𝑥)2− ⅈ2x2
(𝛿 + 𝑥)2) (10)
𝐹y = 𝐹1y− 𝐹2y = 𝜇0𝑁2𝑆aircos(χ)
4 ( ⅈ1y2
(𝛿 − 𝑦)2 − ⅈ2y2
(𝛿 + 𝑦)2) (11)
x and y are the values of displacement from central position of the rotor.
Figure 2.3. Electromagnets operating in a differential mode. Adapted from [6].
The current in the right magnet can be presented as the sum of the bias current ⅈbias (premagnetization) and the control current ⅈc,
ⅈ1x= ⅈbias+ ⅈcx (12)
and in the left magnet as the difference between ⅈbias and ⅈc:
ⅈ2x = ⅈbias− ⅈcx (13)
The bias current is selected according to the bearing design and desired load capacity. It should be not too low, in order not to limit the linear range of an AMB, and not too high to avoid actuator’s saturation. Usually, the bias current is taken as a fraction of the maximum coil current ⅈmax, between 0.2 and 0.5 [3],[6]. This method linearizes the force-current relationship as [3]:
𝐹x =𝜇0𝑁2𝑆aircos(χ)
4 ((ⅈbias+ ⅈcx)2
(𝛿 − 𝑥)2 −(ⅈbias− ⅈcx)2
(𝛿 + 𝑥)2 ) (14)
This equation for force, linearized about the operating point (x = 0, ic = 0) can be written as follows:
𝐹x = 𝑘iⅈcx+ 𝑘x𝑥 (15)
where current-stiffness factor (or actuator gain) ki and position-stiffness factor kx, can be defined as [16]
𝑘i = 𝜕𝐹
𝜕ⅈc =𝜇0𝑁2𝑆airⅈbiascos(χ)
𝛿2 (16)
18 𝑘x= 𝜕𝐹
𝜕𝑥= 𝜇0𝑁2𝑆airⅈbias2 cos(χ) 𝛿3
(17)
This linear approximation of force works well in practice, but in case of saturation, low bias currents and rotor – stator contact non-linear models should be used [3].
2.3 Inductance in coil and current slew rate
The windings of the electromagnet form RL-circuit, supplied by the voltage u from the power amplifier with the current i running in this circuit [6]:
𝑢 = 𝑅cⅈ + 𝐿ⅆⅈ
ⅆ𝑡 (18)
Taking into account the equation (5), that the inductance L of the coil can be defined as:
𝐿 =𝑁𝛷
ⅈ =𝜇0𝑁2𝑆air
2𝛿 (19)
The value of the coil resistance is relatively low, and it can generally be ignored [6]. By combining these equations the rate of current change (slew rate) can be defined:
𝑑ⅈ 𝑑𝑡= 𝑢
𝐿 = 2𝛿𝑢
𝜇0𝑁2𝑆air (20)
One can see that the current slew rate depends on the input voltage from the amplifier u and Ld. Obviously, the smaller the inductance and the higher the voltage, the faster current grows. Thus, the maximum slew rate:
max (ⅆⅈ
ⅆ𝑡) =𝑢max
𝐿 = 2𝛿𝑢max
𝜇0𝑁2𝑆air (21)
Where umax is the maximum voltage, depending on the amplifier hardware [3]. The expression (21) is identical to the equation for the current limitation of dI/dt proposed by a standard ISO 14839 [17]. The limitation is described in the next section.
The inductance L also depends on the operating point of the magnetic material B-H diagram and the relation between B and H is not linear. Thus, the dynamic inductance Ld can be defined, which corresponds to the gradient of flux-linkage-current "𝜓-i diagram" [3] [16]:
𝐿d =ⅆ𝜓
ⅆⅈ = 𝑁ⅆ𝛷
ⅆⅈ =𝜇0𝑁2𝑆air
2(𝛿 − 𝑥) (22)
2.4 AMB load capacity and its limitations
The load capacity is one of the main parameters of an AMB system since it determines its dynamic performance and finally the performance of the whole system. Poor AMB’s dynamic characteristics make the applications considerably sensible to instabilities caused by the surges of power, high speeds, etc.
The ISO 14839 standard defines the load capacity as a maximum load of a magnetic bearing acting on the rotor at its fixed middle position, and usually it is limited by the magnetic saturation of the ferromagnetic material of stator and rotor, the maximum coil current available from the power amplifier and the maximum voltage of the power amplifier umax (DC link voltage) [17] [18].
Figure 2.4. Load capacity of an AMB. 1 - static load capacity; 2 - peak transient load capacity; 3 - dynamic load capacity; F- force, f – frequency of magnetic force;a - maximum current limit of AMB, b - AMB temperature limit or coil temperature limit, c - voltage limit.
When the frequency f increases, the maximum load capacity decreases because of the influence of the inductance L [17]. The force produced by an AMB actuatoris a function of the current in its coils. The maximum rate of change of the force depends on the maximum slew rate of current (when the rotor is centered (x = 0) [6]:
ⅆ𝐹 ⅆ𝑡 = ⅆ𝐹
ⅆⅈ ⅆⅈ
ⅆ𝑡= 𝑘i 2𝑙0𝑢
𝜇0𝑁2𝑆air (23)
Substituting ki the equation for maximum slew rate of force is received:
max (ⅆ𝐹
ⅆ𝑡) =2𝑢maxⅈbiascos(χ)
𝑙0 (24)
20
As one can see in (9) the static load capacity Fmax is limited by the size and geometry of the bearing.
It can be approximated by assuming that the value of tje control current ic equals the bias current ibias
[6]:
|𝐹max| =|𝜇0𝑁2ⅈbias2 𝑆aircos(χ)|
𝑙02 (25)
For higher frequencies, the force slew rate limits the load capacity of the bearings.
Assuming sinusoidal output force F with maximum magnitude M and frequency ω, it could be written [6]:
|𝑀| = 1
𝜔max (ⅆ𝐹
ⅆ𝑡) (25)
Taking into account the relationships discussed above Figure 2.4 can be redrawn as:
Figure 2.5. Plot of the load capacity for an AMB. The available load capacity of the AMB is given by the intersection of the static load capacity and the force slew rate [6].
The load capacity can also be determined by power bandwidth 𝜔BW, which results from the limited DC link voltage, and the maximum current. It can be presented as related to the rise time of the maximum force 𝐹max (25) and the maximum amplifier performance 𝑃max = 𝑢DCⅈmax, as [16]:
𝜔BW = ln(9)𝑢DC
𝐿ⅈmax =ln(9)𝑃max
𝐿ⅈmax2 =ln(9)𝑃maxcos(χ)
2𝛿𝐹max = 𝑃maxcos(χ)
𝛿𝐹max (26)
The performance of the AMB actuator design can be analyzed using analytical models as well as FEM models.
3. ANALYTICAL MODEL AND PERFORMANCE IMPROVEMENT
The parameter which is used for the evaluation of the dynamic performance (especially for high frequencies) is the power bandwidth 𝜔BW, described earlier. The increase of this parameter allows to achieve higher AMB dynamic performance [16], [19].
The system studied in the work is the multi mega Watt (MMW) AMB-rotor System, which is presented in Figure 3.1.
Figure 3.1. Solid rotor structure with 3 sensor-AMB pairs (A, B, C) [26]
The main aim is to achieve higher dynamic characteristics of the AMB, so that it can provide a stable performance of the machine, at least, at its maximum rotational speed of 20 000 rpm (see Table 3.1).
In order to obtain the required dynamic characteristics, the bandwidth must be higher than the maximum rotational speed of the machine. Thus, its value should be greater than 333 Hz.
Table 3.1. MMW machine parameters
Parameter Value
Rated power PN, [kW] 2000
Rated speed n, [rpm] 12 000 – 15 000 Maximum speed nmax, [rpm] 20 000
Rated voltage UN, [Vrms] 660
Compared to finite element models, which might take hours or days to determine the actuator bandwidth, models based on analytic equations can be easily used to receive the actuator geometry parameters that provide the maximum bandwidth and define how their changes influence it [20].
3.1 Axial bearing parameters
Because of production and material costs, magnetic thrust bearings are non-laminated. As a result, according to the Faraday’s law, eddy currents, induced within iron, generate a magnetic field that opposes the change in a field generated by the varying actuator coil current. This opposing field leads
22
to a reduction in the produced electromagnetic force, and its slower change. Thus, eddy currents have a critical impact on the dynamic capacity and bandwidth that can be attained by the thrust AMB. One possible way to achieve a desirable bandwidth and improved dynamic behavior is the segmentation of the thrust bearing solid stator. That is why the studied axial AMB has slits (Figure 3.2), which prevent eddy currents and therefore the bandwidth and force slew rate increase. However, the force capacity is slightly reduced [21].
In Tables 3.2 and 3.3 the initial parameters of the axial AMB are presented. These values should be optimized to receive the highest possible bandwidth of the actuator.
Figure 3.2. Axial magnetic bearing with slits
Table 3.2. Geometry and electrical parameters of the axial AMB.
Parameter Value
Inner pole inner diameter D0 (radius r0) [mm] 102.5 (51.25) Inner pole outer diameter D1 (radius r1) [mm] 129 (64.5) Outer pole inner diameter D2 (radius r2) [mm] 161 (80.5) Outer pole outer diameter D3 (radius r3) [mm] 179 (89.5)
Yoke thickness d3 [mm] 13.5
Disc thickness total, d1 [mm] 13 Disc outer diameter dout [mm] 180 Disk inner diameter din [mm] 88
Coil height d2 [mm] 14
Nominal air-gap length δ [mm] 0.7
Number of turns N 71
Winding space factor with rectangular wire 0.82
Coil wire [mm×mm] 2.07×1.06
Peak current from amplifier [A] 16 (21?)
Applied DC link voltage [V] 300
For this geometry the operational parameters of the axial AMB are calculated. The inductance is found via equation (19):
𝐿 =𝜇0𝑁2𝑆air
2𝛿 = 0.022𝐻 (27)
where 𝑆air = 4812 mm2 that corresponds to the pole face area.
The winding resistance R:
𝑅 =𝜌Cu𝑙c𝑁
𝑆c (28)
where 𝜌Cu= 0.017 Ω·mm2/m is copper resistivity, lc - mean length of the coil turn, defined as [22]:
𝑙c = 𝜋1(𝐷0+ (𝐷1− 𝐷0) + 𝑑2) = 0.45m (29) and Sc is the winding’s wire cross section:
𝑆𝑐 = 2.07 × 1.06 = 2.194 · 10−6m2 (30) Thus, the winding resistance R equals 0.25 Ω and the resistive voltage drop with the rated 16 A current is just 4 V leaving, in practice, the whole DC-link voltage as the voltage reserve to control the current.
To calculate the saturation force Fsat the saturation current corresponding to Bsat = 1.2 T is defined first through equation (5):
ⅈsat = 2𝛿𝐵sat
𝜇0𝑁 = 18.8A (31)
Thus the saturation force is
𝐹𝑠𝑎𝑡 =𝐵sat2 𝑆air
𝜇0 = 5514N (32)
The maximum force acting in the AMB is calculated as:
𝐹max = 𝐵max2 𝑆air
𝜇0 (33)
where 𝐵maxis defined as:
𝐵max = 𝜇0𝑁ⅈmax
2𝛿 (34)
Thus, the values of 𝐵max and 𝐹max are defined with peak value of current, which is 16 A:
24 𝐵max = 𝜇0𝑁ⅈmax
2𝛿 = 1.0T (35)
𝐹max= 𝐵max2 𝑆air
𝜇0 = 3981N (36)
The current and position stiffnesses are
𝑘i = 𝜇0𝑁2𝑆airⅈbias
𝛿2 = 373.25N/A (37)
𝑘x =𝜇0𝑁2𝑆airⅈbias2
𝛿3 = 3199341.22N/m (38)
where bias current ⅈbias is assumed to be 6 A.
Joule DC losses at maximum drive DC current 16 A:
𝑃l= 𝑅 · 162 = 63.26W (39)
All calculated parameters are shown in Table 3.3.
Table 3.3. Operational parameters of the axial AMB.
Parameter Value
Analytical estimate of the inductance, L [H] 0.02
Resistance, R [Ω] 0.25
Force when one coil active Fsat [N] for isat = 18.8 A at iron
saturation Bsat = 1.2 T 5514
Force Fmax [N] for imax = 16 A 3981
Current stiffness, ki [N/A] 373
Position stiffness, kx [N/m] 3196017
Joule DC losses at maximum driver current (16 A) [W] 63.26
Using the data in the tables the electrical properties of the axial AMB can be defined. The stator and the rotor (disk) of the bearing were made of two different steels – S45C and SUS403, respectively. To calculate the permeabilities of the stator 𝜇stand rotor 𝜇rtthe flux density corresponding to the maximum current (ⅈmax = 16 A) 𝐵max =1T is used. Then the magnetic field strengths are defined via BH-curves of the materials:
𝐻max= 700A
m (40)
Finally, the permeabilities are
𝜇Fest = 𝜇Fefl = 𝜇Fe= 𝐵max
𝜇0· 𝐻max= 1120 (41)
The values of the materials’ properties are listed in Table 3.4.
Table 3.4. AMB materials’ properties
Parameter Value
Stator Conductivity σst [S/m] 4.76·106 Permeability 1120 Rotor (disk) Conductivity σrt [S/m] 1.76·106
Permeability 1120
Since the bandwidth is related to the actuator material and its geometric characteristics, desirable dynamic performance can be achieved by varying these parameters. So, firstly, it is necessary to derive an analytical model of the segmented axial electromagnetic suspension system that predicts the actuator frequency response from the geometry and material properties.
3.2 Axial actuator analytical model
The analytic model for segmented thrust AMB is based on the model for C-type geometries developed by Zhu [23] and Knospe [24]. An axisymmetric thrust bearing, cut like a pie (Figure 3.3), has stator segments that resemble individual C-type actuators that are curved and fit together. It can be seen that the geometry of the cut stator resembles a C-type actuator with some exceptions: the stator is curved, the pole widths and lengths are different for the inner and outer poles, and the flotor (thrust disk) extends beyond the edges of the stator [25]. In this way, the geometric similarities between the single segment and C-type actuator are used to develop the analytical model of the segmented axial AMB.
Figure 3.3. Single segment of a cut thrust AMB. Adapted from [20].
26
The C-type model (Figure 3.4) is based on the principle of effective reluctance. It is divided into four regions, for which so called effective reluctances are determined. These are two regions for each air- gap, one for the flotor and one for the stator. The approximate effective reluctance of each region is defined as:
𝑅m,i(𝑠) = 𝑐i√𝑠 + 𝑅m,i0 (42)
where 𝑅m,i0 is the static reluctance and 𝑐i is the dynamic coefficient of the effective reluctance. The reluctances of each segment form the branches of a parallel magmatic circuit. The total effective reluctance of a C-type actuator Rm(s) is the sum of all its components. However, a segmented flotor could not be implemented in rotating machines, because of the mechanical strength requirements.
Thus, for an actuator with z cuts the total reluctance is defined as [24]:
𝑅m(𝑠) = 𝑐√𝑠 + 𝑅m0 = ((𝑐st+ 2𝑐g
𝑧 + 𝑐fl) √ 𝜎
𝜇Fe𝜇0) · √𝑠 +𝑅m,st0 + 2𝑅m,g0
𝑧 + 𝑅m,fl0 (43)
Figure 3.4. C-type actuator geometry [20].
As the stator and flotor has different conductivities the expression (45) should be rewritten as:
𝑅m= ((𝑐st+ 2𝑐g
𝑧 ) √ 𝜎st
𝜇Fest𝜇0+ 𝑐fl√ 𝜎fl
𝜇Fefl𝜇0) · √𝑠 +𝑅m,st0 + 2𝑅m,g0
𝑧 + 𝑅m,fl0 (44)
The analytical model of the segmented thrust AMB is described through half-order transfer function [24]:
𝐹p(𝑠) =𝛷bias 𝜇0 ( 1
𝐴in+ 1
𝐴out) 𝑁
𝑐√𝑠 + 𝑅m0 ⅈp(𝑠) (45)
𝐹p – perturbation force, 𝛷bias – bias flux, ip(s) is the perturbation current, which is the same as the control current ic that shows deviation of magnet current from bias current value ibias (ip = ic = i – ibias), and Ain and Aout inner and outer pole surface areas, respectively, calculated as:
𝐴in = π(𝑟12− 𝑟02) = 4.818 · 10−3𝑚2 (46) 𝐴out= π(𝑟32− 𝑟22) = 4.807 · 10−3𝑚2 (47)
Figure 3.5. Axisymmetric geometry of thrust disk and stator electromagnet of an axial magnetic bearing. Only one electromagnet of the opposing pair is shown [19].
Due to their circular geometry, the adaptation of a segmented thrust AMB to an analytic model developed for C-type geometries requires a choice of effective pole length (2𝑎̂𝑚) and width (2𝑏̂), since these values differ for the inner and outer poles of the wedge shaped electromagnet (Figure 3.3). These geometric parameters can be calculated through the next equations [20]:
2𝑎̂m =π[(1 − 𝑊a)(𝑟0− 𝑟1) + 𝑊𝑎(𝑟2+ 𝑟3)]
𝑧 (48)
2𝑏̂ = (1 − 𝑊b)(𝑟3− 𝑟2) + 𝑊b(𝑟1− 𝑟0) (49) where Wa = Wb = 0.5 arethe weighting parameters for pole length and pole width, respectively. Since the flotor is not segmented, the expression for effective pole length 2𝑎̂ for the flotor is
28
2𝑎̂ = π[(1 − 𝑊a)(𝑟0− 𝑟1) + 𝑊a(𝑟2+ 𝑟3)] (50) However, for many geometries, a simple average (Wa = 0.5 and Wb = 0.5) is effective [20].
The C – type actuator geometry parameters are presented in Table 3.5.
Table 3.5. C-type actuator geometry parameters
Parameter z h [m] d [m] δ[m] 2𝑎̂𝑚 [m] 2𝑎̂[m] 2𝑏̂ [m]
Value 32 0.038 0.014 0.7 0.014 0.449 0.011
The bandwidth is defined by the frequency at which the output magnitude is attenuated by 3 dB from its DC value (i.e. the ratio of gains is √2/2) [20]. Thus
| 𝑅m0
𝑐√𝑗𝜔 + 𝑅m0| =√2
2 (51)
Taking into account that
√𝑗 =√2
2 + 𝑗√2
2 (52)
it can be written
𝑐2𝜔 + √2𝑐𝑅m02√𝜔 − 𝑅m02 = 0 (53) So, the approximate bandwidth of a segmented axial AMB is given by
𝜔−3dB = (2 − √3) (𝑅m0 𝑐 )
2
(54) where the component values (𝑅m,fl0 , 𝑅m,st0 , 𝑅m,g0 ) of 𝑅0,are calculated as follows:
𝑅m,st = ℎ + 2𝑑 + 𝑑3
𝜇Fe𝜇0 ⋅ 4 ⋅ 𝑎̂m⋅ 𝑏̂= 350.8 · 1031
𝐻 (55)
𝑅m,fl = ℎ
𝜇Fe𝜇0⋅ 2 ⋅ 𝑎̂ ⋅ 𝑑1 = 2.25 · 1031
𝐻 (56)
𝑅m,g = 𝛿𝑎
𝜇0⋅ 4 ⋅ 𝑎̂m⋅ 𝑏̂= 3570 · 1031
𝐻 (57)
𝑅0 = 2𝑅g+ 𝑅st
𝑧 + 𝑅fl = 236.3 · 1031
𝐻 (58)
and c component values (𝑐st,𝑐fl, 𝑐g):
𝑐st = ℎ + 2𝑑
4(𝑎̂m+ 𝑏̂)= 1.317 (59)
𝑐fl = ℎ
4(𝑎̂ + 𝑏̂)= 0.042 (60)
𝑐g = 𝑏̂
6𝑎̂m− 32𝑏̂
π5𝑎̂m2 · tanh(π𝑎̂m
2𝑏̂ ) = 0.069 (61)
𝑐 = (𝑐st+ 2𝑐g
𝑧 ) √ 𝜎st
𝜇Fe𝜇0+ 𝑐fl√ 𝜎rt
𝜇Fe𝜇0 = 4044 (62)
The bandwidth for initial axial AMB parameters is:
𝜔−3dB = 145.6Hz (63)
3.3 Axial actuator design limitations
There are several limitations which must be considered for the axial AMB design, related to geometrical parameters and forces acting in the bearing [19]:
Geometrical limitations:
1. Outer radius r3 – limited by the available space within the housing and hoop (tangential) stress in the disk at maximum speed 𝑛max
2. Inner radius r0 – must be large enough to accomodate the shaft
3. Axial length must not exceed bounds determined from the machine design 4. Geometrical limitation: r0 < r1 < r2 < r3
Force limitations:
5. Peak force – pole face area must be sufficient to provide the specified peak (maximum) force. In addition, the cross-sectional area of all segments, which trough the flux passes, must be bigger than or equal to the pole face area, ensuring that saturation flux density can be achieved at the pole face.
6. Continuous force – cross-sectional area of the coil must be enough to provide the number of ampere-turns necessary to create required continuous force without coil overheating. In addition, the slot area should have sufficient space to accommodate the required number of coil turns (N = 71).
The pole and coil face area are defined, respectively:
30 𝐴flux = 𝐹sat· 𝜇𝑜
𝐵sat2 = 4.812 · 10−3m2 (64)
It should also be considered that the air-gap cross section area Sair should not be lower than 4812 mm2 (coinciding with face area of pole) to provide the maximum force capacity Fmax (see equation (33)).
𝐴coil =1 𝐽
(
(𝐵bias𝐴min𝑅0) + 𝜇0𝑅0𝐹max 2 (𝐴min
𝐴in +𝐴min 𝐴out) 𝐵bias
)
= 157mm2 (65)
where
𝐴min = min(𝐴out, 𝐴in) = 𝐴out = 4.807 · 10−3m2 (66) Bias flux density is calculated as [19]:
𝐵bias = 𝑁ⅈbias
𝐴min𝑅0 = 0.38T (67)
Restraints for the slot area is derived from the following considerations. The total coil area for the winding with 71 turns, consisting of wires with 2.07×1.06 mm is:
𝐴coil = 71 · 2.07 · 1.06 = 155.8mm2 (68) It should also be taken into account that there should be some space between the coils and the walls of the slot. The length of the space is taken equal to 0.75 mm. For the geometry presented below:
Figure 3.6. Geometry and rectangular winding consisting of 71 turns
the constraints for the slot dimensions are:
𝑑2 > 0.75 + 1.06 · 12 = 13.5m (69) 𝑟2− 𝑟1 > 0.75 · 2 + 2.07 · 6 = 13.9m (70) To define the outer radius r3 limitation ρ1 first it is needed to find the maximum circumferential speed of the thrust disk that determined by its material properties, namely the yield strength, which value is 𝜎s = 390 MPa [26]. Thus, the maximum circumferential speed 𝜗max is [3]:
𝜗max= √ 8𝜎s
(3 + 𝜈)𝜌= 271.42𝑚/𝑠 (71)
the yield strength is taken with 40% safety margin and thus equals 234 MPa, 𝜌 is the density and ν = 0.3 is the Poisson number.
Finally, the maximum allowable disk radius or the outer AMB radius can be calculated as:
𝜌1 = 𝑟3max = 𝜗max
𝛺max= 𝜗max 2𝜋𝑛max
60
= 0.13m = 130mm (72)
The inner radius r0 limitation ρ2 is defined by the shaft radius. In the designed AMB it is determined by the inner disk radius:
𝜌2 =𝑑in
2 = 44mm (73)
The constraints presented in form of mathematical expressions are listed in Table 3.6.
Table 3.6. Constraints for axial magnetic actuator optimization problem.
Constraint Mathematical expression Values
Geometry
Outer radius r3 < ρ1 r3 ≤ 130 mm
Inner radius r0 > ρ2 r0 > 44 mm
Axial length d1+2d2+2d3 < ρ3 d1+2d2+2d3 ≤ 100 mm Dimensional ratio r0 < r1 < r2 < r3 r0 < r1 < r2 < r3
Slot dimensions - 𝑑2 ≥ 13.5 mm
- 𝑟2− 𝑟1 ≥ 13.9 mm
Force
Peak force (𝐹𝑠𝑎𝑡) and pole cross-sectional area
𝐴flux< π(𝑟12− 𝑟02) 4812 mm2 < π(𝑟12− 𝑟02) 𝐴flux < 2𝜋𝑟1𝑑1 4812 mm2< 2π𝑟1𝑑1 𝐴flux< π(𝑟32− 𝑟22) 4812 mm2< π(𝑟32− 𝑟22)
𝐴flux < 2π𝑟1𝑑3 4812 mm2< 2π𝑟1𝑑3 Continuous force (𝐹𝑚𝑎𝑥) and
coil cross-sectional area 𝐴coil < (𝑟2−𝑟1)𝑑2 157 mm2< (𝑟2−𝑟1)𝑑2
32 3.4 Geometry optimization
The optimization is implemented in order to maximize the value of the bandwidth 𝜔−3dB described via Equation (54), by varying the set of geometrical parameters of the axial AMB {r0, r1, r2, r3, d1, d2, d3} within the previously specified limits (see Table 3.6). For this purpose the built-in PTC Mathcad optimization function maximize is used. It returns the vector of values, at which the optimized function takes the largest value. The implementation in Mathcad is showed below:
Initial values
BW r0 r1( r2r3d1d2d3)
2 3
Ro
co
2
2
r0 51.25 mm d1 13 mm r1 64.5 mm d2 14 mm r2 80.5 mm d3 13.5 mm r3 89.5 mm
Given
r1 r0 8 mm 2 r1d1 4812 mm 2 r2 r1 13.9 mm 2r1d3 4812 mm 2 r3 r2 6.5 mm
r12 r02
4812 mm 2 r0 44 mm
r32 r22
4812 mm 2 r3 130 mm (r2 r1) d2 157 mm 2r0 r1 r2 r3 d1 d2 d3
Maximize
BW r0 r1r2r3d1d2d3
r0 r1 r2 r3 d1 d2 d3
0.093 0.101 0.115 0.121 0.01 0.014 0.032
m
BW r0 r1( r2r3d1d2d3) 346.91
s
The dimensions describing the initial AMB actuator design and those of the optimized design are presented in Table 3.7.
Table 3.7. Initial values and values after optimization for dimensions and performance of the AMB.
Dimension Initial value Optimized value
r0 [mm] 51 93
r1 [mm] 65 101
r2 [mm] 81 115
r3 [mm] 90 121
d1 [mm] 13 10
d2 [mm] 14 14
d3 [mm] 13.5 32
Performance Initial value Optimized value
𝜔−3dB [Hz] 145.6 346.9
As a result, the bandwidth has doubled. The radial dimensions of the actuator have increased and the stator radius r3 has almost achieved the limit value ρ1 = 130 mm. Width of the poles were reduced while saving the initial peak force. The disk d1 became narrower while the yoke thickness increased significantly (about 2.3 times). The coil slot width (r2 - r1) is largely unchanged, and the coil slot depth d2 did not change, providing the sufficient coil area for required continuous force. However, the bandwidth can be enlarged further by increasing the radial dimension and making the poles’ widths even smaller.
3.5 Increasing bandwidth by increasing the air gap
Another way to enhance the power bandwidth according to the equations (54) – (62) is increasing the length of the air gap. However, it results in a decrease of the maximum load capacity of the bearing.
This can be seen from the equation:
𝐹max = 𝜇0𝑁2ⅈmax2 𝑆air
4𝛿2 (74)
The corresponding graph is shown in Figure 3.7. For initial geometry, the gap length was varied from 0.7 mm to 1.3 mm and corresponding bandwidths were calculated using the formula (52) of the approximate analytic model. They are presented in Table 3.8 and showed graphically in Figure 3.8.
34
Figure 3.7. Load capacity vs air-gap length (the current is constant).
Table 3.8. Bandwidth as a function of axial air-gap (N = 71 and imax = 16 A).
As it can be seen in Table 3.8 the required bandwidth (333 Hz or higher) is reached with the air-gap value of 1.1 mm (marked red) and equal to 335.6 Hz. But at the same time, the force capacity is decreased to 1612 N.
0 500 1000 1500 2000 2500 3000 3500 4000
0.7 0.8 0.9 1 1.1 1.2 1.3
Load capacity, Fmax[N]
Air - gap length, δ[mm]
Air gap [mm] Bandwidth [Hz] Force [N]
0.7 148,1 3981
0.75 167,4 3468
0.8 187,8 3048
0.85 209,5 2700
0.9 232,3 2409
0.95 256,4 2162
1.0 281,6 1951
1.05 308,1 1770
1.1 335,6 1612
1.15 364,3 1475
1.2 394,2 1355
1.25 425,37 1249
1.3 468,51 1154
Figure 3.8. Bandwidth as a function of axial air-gap (N = 71 and ⅈmax = 16 A).
There are two ways that allows to keep the force constantly equals 3981 N. The first one is through increasing the number of turns. From the equation (74) the necessary number of turns is defined, which is 112 turns. However, with the increase number of turns the inductance increases, since there is a quadratic relation between inductance L and number of turns N, according to equation (27). As a result, with N = 112 the inductance L increases up to 0,034 H and that leads to the decrease of the bandwidth according to equation (26), which value will be 296 Hz (by formula (54)).
The second possible way is changing of the rated current ⅈmax to keep the force value. Using the equation (74) it can be defined that ⅈmax = 25 A is necessary. In this case, according to the formula (54) the value of the bandwidth is 323 Hz. But the value of the current is restricted by the maximum amplifier current 21 A, for which 𝐹max = 2777 N and ωBW = 379 Hz.
One can see that both methods leads to the decrease of the bandwidth. However, in case of the current increase the drop in the bandwidth is less then this is for the increase of the number of turns. This also follows from the equation (27) rewritten as:
𝜔BW = ln(9) · 𝑢DC· 2𝛿
𝜇0 · 𝑁2· ⅈmax · 𝑆𝑎𝑖𝑟 (75)
200 300 400 500 600 700 800
0.7 0.8 0.9 1 1.1 1.2 1.3
Bandwidth, ωBW[Hz]
Air - gap length, δ[mm]
36
3.6 Increasing bandwidth by changing the permeability
Further, the influence of relative permeability on the actuator bandwidth is investigated. According to the equation (54) increasing the permeability allows to achieve higher bandwidth values. The permeability values vary from 1000 to 6000. The air-gap value in both cases is nominal 0.7 mm.
Bandwidth values corresponding to the different permeabilities, defined by formula (54), are presented in Table 3.9 and Figure 3.9.
Figure 3.9. Bandwidth as a function of permeability for initial geometry.
Table 3.9. Bandwidth as a function of permeability for initial geometry.
Permeability Bandwidth [Hz]
1000 98.4
1500 147.61
2000 196.81
2500 246.01
3000 295.21
3500 344.41
4000 393.62
4500 442.82
5000 492.02
5500 541.22
6000 590.42
0 100 200 300 400 500 600 700
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Bandwidth, ωBW[Hz]
Permeability, μFe
