Summary

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].

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]:

𝑙_Fe⋅ 𝐻_Fe+ 2𝛿 ⋅ 𝐻_Fe= 𝑁 ⋅ ⅈ (2) Taking into account that flux density is:

𝐵 = 𝐻𝜇₀𝜇_Fe (3)

and assuming that flux density is constant in the both media, equation (2) is

16 𝑙_Fe 𝐵

𝜇₀𝜇_Fe+ 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 (𝑙_Fe

𝜇_Fe+ 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 ≤ 𝐵_sat (6)

To define the magnetic force F of the electromagnet, the equation defining the stored in electromagnet co-energy Wce is used: 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 = 𝜕𝑊_ce

𝜕𝑙 =𝐵²𝑆_aircos⁡(𝜒)

𝜇₀ (8)

Using the equation (5) and assuming the infinity permeability of iron μFe the force can be expressed as:

𝐹 =𝜇₀𝑁²ⅈ²𝑆_air⁡cos⁡(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]:

𝐹_x = 𝐹_1x− 𝐹_2x= 𝜇₀𝑁²𝑆_air⁡cos⁡(χ)

4 ( ⅈ_1x²

(𝛿 − 𝑥)²− ⅈ_2x²

(𝛿 + 𝑥)²) (10)

𝐹_y = 𝐹_1y− 𝐹_2y = 𝜇₀𝑁²𝑆_air⁡cos⁡(χ)

4 ( ⅈ_1y²

(𝛿 − 𝑦)² − ⅈ_2y²

(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 ⅈ_bias (premagnetization) and the control current ⅈ_c,

ⅈ_1x= ⅈ_bias+ ⅈ_cx (12)

and in the left magnet as the difference between ⅈ_bias and ⅈ_c:

ⅈ_2x = ⅈ_bias− ⅈ_cx (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 ⅈ_max, between 0.2 and 0.5 [3],[6]. This method linearizes the force-current relationship as [3]:

𝐹_x =𝜇₀𝑁²𝑆_air⁡cos⁡(χ)

4 ((ⅈ_bias+ ⅈ_cx)²

(𝛿 − 𝑥)² −(ⅈ_bias− ⅈ_cx)²

(𝛿 + 𝑥)² ) (14)

This equation for force, linearized about the operating point (x = 0, ic = 0) can be written as follows:

𝐹_x = 𝑘_iⅈ_cx+ 𝑘_x𝑥 (15)

where current-stiffness factor (or actuator gain) ki and position-stiffness factor kx, can be defined as [16]

𝑘_i = 𝜕𝐹

𝜕ⅈ_c =𝜇₀𝑁²𝑆_airⅈ_bias⁡cos⁡(χ)

𝛿² (16)

18 𝑘_x= 𝜕𝐹

𝜕𝑥= 𝜇₀𝑁²𝑆_airⅈ_bias² ⁡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 = 𝑅_cⅈ + 𝐿ⅆⅈ

ⅆ𝑡 (18)

Taking into account the equation (5), that the inductance L of the coil can be defined as:

𝐿 =𝑁𝛷

ⅈ =𝜇₀𝑁²𝑆_air

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

𝜇₀𝑁²𝑆_air (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: 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]:

𝐿_d =ⅆ𝜓

ⅆⅈ = 𝑁ⅆ𝛷

ⅆⅈ =𝜇₀𝑁²𝑆_air

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= 𝑘_i 2𝑙₀𝑢

𝜇₀𝑁²𝑆_air (23)

Substituting ki the equation for maximum slew rate of force is received:

max (ⅆ𝐹

ⅆ𝑡) =2𝑢_maxⅈ_bias⁡cos⁡(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]:

|𝐹_max| =|𝜇₀𝑁²ⅈ_bias² 𝑆_air⁡cos⁡(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 𝜔_BW, 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 𝐹_max (25) and the maximum amplifier performance 𝑃_max = 𝑢_DCⅈ_max, as [16]:

𝜔_BW = ln(9)𝑢_DC

𝐿ⅈ_max =ln(9)𝑃_max

𝐿ⅈ_max² =ln(9)𝑃_max⁡cos⁡(χ)

2𝛿𝐹_max = 𝑃_max⁡cos⁡(χ)

𝛿𝐹_max (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 𝜔_BW, 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 =𝜇₀𝑁²𝑆_air

2𝛿 = 0.022⁡𝐻 (27)

where 𝑆_air = 4812 mm² that corresponds to the pole face area.

The winding resistance R:

𝑅 =𝜌_Cu𝑙_c𝑁

𝑆_c (28)

where 𝜌_Cu= 0.017 Ω·mm²/m is copper resistivity, lc - mean length of the coil turn, defined as [22]:

𝑙_c = 𝜋₁(𝐷₀+ (𝐷₁− 𝐷₀) + 𝑑₂) = 0.45⁡m (29) and Sc is the winding’s wire cross section:

𝑆_𝑐 = 2.07 × 1.06 = 2.194 · 10⁻⁶⁡m² (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):

ⅈ_sat = 2𝛿𝐵_sat

𝜇₀𝑁 = 18.8⁡A (31)

Thus the saturation force is

𝐹_𝑠𝑎𝑡 =𝐵_sat² 𝑆_air

𝜇₀ = 5514⁡N (32)

The maximum force acting in the AMB is calculated as:

𝐹_max = 𝐵_max² 𝑆_air

𝜇₀ (33)

where 𝐵_maxis defined as:

𝐵_max = 𝜇₀𝑁ⅈ_max

2𝛿 (34)

Thus, the values of 𝐵_max and 𝐹_max are defined with peak value of current, which is 16 A:

24 𝐵_max = 𝜇₀𝑁ⅈ_max

2𝛿 = 1.0⁡T (35)

𝐹_max= 𝐵_max² 𝑆_air

𝜇₀ = 3981⁡N (36)

The current and position stiffnesses are

𝑘_i = 𝜇₀𝑁²𝑆_airⅈ_bias

𝛿² = 373.25⁡N/A (37)

𝑘_x =𝜇₀𝑁²𝑆_airⅈ_bias²

𝛿³ = 3199341.22⁡N/m (38)

where bias current ⅈ_bias is assumed to be 6 A.

Joule DC losses at maximum drive DC current 16 A:

𝑃_l= 𝑅 · 16² = 63.26⁡W (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 𝜇_st⁡and rotor 𝜇_rt⁡the flux density corresponding to the maximum current (ⅈ_max = 16 A) 𝐵_max =1T is used. Then the magnetic field strengths are defined via BH-curves of the materials:

𝐻_max= 700⁡A

m (40)

Finally, the permeabilities are

𝜇_Fest = 𝜇_Fefl = 𝜇_Fe= 𝐵_max

𝜇₀· 𝐻_max= 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:

𝑅_m,i(𝑠) = 𝑐_i√𝑠 + 𝑅_m,i⁰ (42)

where 𝑅_m,i⁰ is the static reluctance and 𝑐_i 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]:

𝑅_m(𝑠) = 𝑐√𝑠 + 𝑅m0 = ((𝑐_st+ 2𝑐_g

𝑧 + 𝑐_fl) √ 𝜎

𝜇_Fe𝜇₀) · √𝑠 +𝑅_m,st⁰ + 2𝑅_m,g⁰

𝑧 + 𝑅_m,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:

𝑅_m= ((𝑐_st+ 2𝑐_g

𝑧 ) √ 𝜎_st

𝜇_Fest𝜇₀+ 𝑐_fl√ 𝜎_fl

𝜇_Fefl𝜇₀) · √𝑠 +𝑅_m,st⁰ + 2𝑅_m,g⁰

𝑧 + 𝑅_m,fl⁰ (44)

The analytical model of the segmented thrust AMB is described through half-order transfer function [24]:

𝐹_p(𝑠) =𝛷_bias 𝜇₀ ( 1

𝐴_in+ 1

𝐴_out) 𝑁

𝑐√𝑠 + 𝑅m0 ⅈ_p(𝑠) (45)

𝐹_p – perturbation force, 𝛷_bias – 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:

𝐴_in = π(𝑟₁²− 𝑟₀²) = 4.818 · 10⁻³⁡𝑚² (46) 𝐴_out= π(𝑟₃²− 𝑟₂²) = 4.807 · 10⁻³⁡𝑚² (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 (2𝑎̂_𝑚) 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]:

2𝑎̂_m =π[(1 − 𝑊_a)(𝑟₀− 𝑟₁) + 𝑊_𝑎(𝑟₂+ 𝑟₃)]

𝑧 (48)

2𝑏̂ = (1 − 𝑊_b)(𝑟₃− 𝑟₂) + 𝑊_b(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 − 𝑊_a)(𝑟₀− 𝑟₁) + 𝑊_a(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] 2𝑎̂_𝑚 [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 So, the approximate bandwidth of a segmented axial AMB is given by

𝜔_−3dB = (2 − √3) (𝑅_m⁰

𝑐_st = ℎ + 2𝑑

The bandwidth for initial axial AMB parameters is:

The bandwidth for initial axial AMB parameters is:

In document Increasing the bandwidth of axial AMB for the high-speed system (sivua 12-0)