Magnetic Bearing Actuator Group 6

1. List of symbols:

B Flux Density

H Field Intensity

Θ Magneto-motive Force (MMF)

Rc Resistance of the coil

Rext Resistance of the external circuit

R Total resistance

E Electro-motive force (EMF)

I Current

lav Average length of the wire

N Number of turns

Acond Cross-Sectional Area of the conductor

Acoil Cross-Sectional Area of the coil

j Current Density

r Radius to the centre of the coil

ρ Resistivity

ρOT Resistivity at operating temperature

µo Permeability of air

ROT Resistance at operating temperature

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Magnetic Bearing Actuator Group 6

2. Introduction:

This report is to explain the necessary steps that were taken to achieve the task of theoretically building a Magnetic Bearing Actuator. This specific report entails the design details of a radial 8-pole, hetero-polar magnetic bearing actuator. The design had to be within certain specifications had to adhere to. The bearing had to be optimized in accordance to certain design criteria (such as coil area, resultant force on the journal, minimum core volume etc).

There are two parts to the design a magneto-statics component which was used to obtain the load capacity and a thermal component that determines the temperature operating range of the bearing depending on the insulation class given.

The main aim of the design was to make sure that:

- The bearing develops the required load capacity (slightly higher) – result must be confirmed by FE model and relevant calculations.

- The winding temperature was within the acceptable range for the required insulator class.

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Magnetic Bearing Actuator Group 6

3. Theory:

Magnetic Bearings:

Magnetic bearings are used to in lieu of rolling element or fluid film journal bearings in some high performance turbo machinery applications. Specific applications include pumps for hazardous/caustic fluids, precision machining spindles, energy storage flywheels, and high reliability pumps and compressors.

Magnetic bearings yield several advantages. Since there is no mechanical contact in magnetic bearings, mechanical friction losses are eliminated. In addition, reliability can be increased because there is no mechanical wear.

Besides the obvious benefits of eliminating friction, magnetic bearings also allow some perhaps less obvious improvements in performance. Magnetic bearings are generally open loop unstable, which means that active electronic feedback is required for the bearings to operate stably. However, the requirement of feedback control actually brings great flexibility into the dynamic response of the bearings. By changing controller gains or strategies, the bearings can be made to have virtually any desired closed-loop characteristics. For example, flywheel bearings are extremely compliant, so that the flywheel can spin about its inertial axis--the bearings serve only to correct large, low frequency displacements.

Typical Bearing Geometry

Conceptually, the typical magnetic bearing is composed of eight of horseshoe-shaped electromagnets. This configuration is shown in Figure 1. The eight magnets are arranged evenly around a circular piece of iron mounted on the shaft that is to be levitated. Each of the electromagnets can only produce a force that attracts the rotor iron to it, so all eight electromagnets must act in concert to produce a force of arbitrary magnitude and direction on the rotor.

Fig.1: Eight Pole Magnetic Bearing with 4 poles active at any time

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Magnetic Bearing Actuator Group 6

4. Design Process – Electromagnetic (parts a-k)

4.1 Initial implementation of the design:

The design procedure involved several steps:

- Bearing dimension calculations- Coil calculations- Thermal calculations

Bearing Dimension Calculations:

a) Selection of a reasonable flux density:

The example given from the lecture notes was B j of 1.6 – 1.7T. For the design of the model took the average of the example value hence B j=1.65T. This then required steel that will provide the necessary flux density. Through trial and error it was discovered that Steel M-14 would provide the best results for our design.

b) Estimate the flux density in the air gapBg. Assuming 10% leakage:

Bg=0.9B j

∴Bg=0.9 (1.65 )=1.485T

c) From the known load capacity (LC or F) calculate force per/pole F1:

For the design the decision was taken to make three active poles:

Pole Pitch: τ=360p=3608

=450

Hence F=F1+2 F1 cos 45=2.41 F1

∴F=2.41 F1

F1=F2.41

=10002.41

=414.214 N

d) Using the approximate expression for force/pole,

F1=12 μ0

Bg2 Ag

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Magnetic Bearing Actuator Group 6

Calculate the required cross-sectional are of the stator pole Ag , to do this make Ag the subject of the formula:

Hence Ag=2F1μ0

Bg2 =

414.214(4πx 10−7)1.4852

=472.1mm2

e, f) Calculation of the width of the poleW p, journal thickness W jand journal outside diameterD j:

2W j=π16 (Ds+2W j )=

π16

(115 x10−3+2W j)

∴2W j=0.0226+0.393W j

W j=0.0226

(2−0.393)=14.1mm

Therefore the width of pole: W p=2W j=2 (14.1 )=28.2mm

Hence to obtain the journal OD: D j=Ds+2W j

D j=(115 x10−3 )+2 (14.1 )=143.2mm

g) Calculate the axial length of the bearingLb:

Lb=A g

W p= 472.128.2

=16.74mm

h) Estimate the pole (radial) lengthLp:

Used 1.25 as it was the average between the 1 and 1.5.

Lp=(1¿1.5 )W p=1.25 (28.2 )=35.3mm

i) Calculate back iron (radial) width:

W bi=0.5W p

W bi=0.5 (28.2 )=14.1mm

j) Calculate the stator outside diameter OD:

OD=D j+2 (g+Lp+W bi)

OD=143.2+2 (0.3+35.3+14.1 )=242.6mm

k) Calculate the required MMF/pole; assuming (20-25) % leakage and infinite permeability of the steel:

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Magnetic Bearing Actuator Group 6

ϑ=(1.2−1.25 )Bg

μ0(g)

ϑ=1.225 1.4854πx 10−7

(0.3 x10−3 )=434.284 At

l) The area of the coil was assumed to be quite small for the initial calculations and had to be optimized

in the process of achieving the specified load capacity.

m) Calculate number of turns and wire diameter:

To obtain this value required the calculation oflav ,this was done by assuming the shape of the coil to be a trapezium.

The value of Z y is taken as the distance between the centroid (point were the diagonals intersect) and the line DC.

For this modelZ y=5.567, taken from the FE model.

∴lav=2 (W p+Z y )+2 (Lb+Z y )

¿2 (28.2+5.567 )+2 (16.74+5.567 )=112.148mm

Standard copper wire is to be used: resistivity at 20 C is 20 = 0.17241*10-7 m and temperature

coefficient = 0.0039 1/C.

Due to the class H insulation maximum operation temperature was 1800C. Assuming an acceptable temperature range means winding temperature between 65% and 80%.

Therefore class H would be (0.65 to 0.8)*180 = 1170C to 1440C

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Magnetic Bearing Actuator Group 6

To obtain resistivity at maximum operating temperature is as follows:

ρ144=0.17241 x10−7 [1+(144−20 )0.0039 ]=2.55 x 10−8Ωm

Assuming J=4.5 x 106A/m2

∴N= Vj ρ 144 lav

¿ 604.5 x106 (2.55 x10−8 ) (112.148 x10−3 )

=4662.373 turns

Acond=ϑNj

= 434.2844662.373 (4.5 x106 )

=0.02069mm2

Therefore actual Acond taken from the standard metric wire sizes = 0.02270mm2

Coil filling co-efficient k f was not assumed but was calculated and then adjust to produce the best results.

k f=N A cond

Acoil=4 (0.02270)126.27

=0.83

k f ¿Max) = π4 =0.78

The coil filling factor is too high and this was unacceptable (k f >0.78)

n) Calculate resistance and current

At the actual area of conductor = 0.02270mm2 the corresponding nominal resistance at 200C is 0.7596Ω/m.

Therefore at 1440C the nominal resistance is:

RΩm

=0.7596 [1+ (144−20 )0.0039 ]=1.1269 Ωm

The resistance at 1440C is:

R144=R Ωm

∗N∗lav

¿1.1269∗4662.373∗(112.148 x10−3)=589.22Ω

I= VR144

= 60589.22

=0.102 A

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Magnetic Bearing Actuator Group 6

o) Calculation of Actual MMF and MMF density

Actualϑ=N∗I=4662.373∗0.102=475.562 At

ϑ density= Actual ϑA coil

=442.086126.27

=3.766 x 106 Atm2

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Magnetic Bearing Actuator Group 6

Figure 1: Schematic of the initial design implementation

4.2 Final Optimization of Design:

a) Selection of a reasonable flux density:

The example given from the lecture notes was B j of 1.6 – 1.7T. For the design of the model took the average of the example value hence B j=1.65T. This then required steel that will provide the necessary flux density. Through trial and error we discovered that Steel M-14 would provide the best results for our design.

b) Estimate the flux density in the air gapBg. Assuming 10% leakage:

Bg=0.9B j

∴Bg=0.9 (1.65 )=1.485T

c) From the known load capacity (LC or F) calculate force per/pole F1:

For the design the decision was taken to make four active poles:

Pole Pitch: τ=360p=3608

=450

Hence F=2F1 cos22.5+2F1 cos67.5=2.61F1

∴F=2.61 F1

F1=F2.61

=10002.61

=383.14 N

d) Using the approximate expression for force/pole,

F1=12 μ0

Bg2 Ag

Calculate the required cross-sectional are of the stator pole Ag , to do this make Ag the subject of the formula:

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Magnetic Bearing Actuator Group 6

Hence Ag=2F1μ0

Bg2 =

383.14 (4 πx10−7)1.4852

=436.7mm2

e, f) Calculation of the width of the poleW p, journal thickness W jand journal outside diameterD j:

2W j=π16 (Ds+2W j )=

π16

(115 x10−3+2W j)

∴2W j=0.0226+0.393W j

W j=0.0226

(2−0.393)=14.1mm

Therefore the width of pole: W p=2W j=2 (14.1 )=28.2mm

Hence to obtain the journal OD: D j=Ds+2W j

D j=(115 x10−3 )+2 (14.1 )=143.2mm

g) Calculate the axial length of the bearingLb:

Lb=A g

W p=436.728.2

=15.5mm

h) Estimate the pole (radial) lengthLp:

Used 1.25 as it was the average between the 1 and 1.5.

Lp=(1¿1.5 )W p=1.25 (28.2 )=35.3mm

ii) Calculate back iron (radial) width:

W b i=0.5W p

W bi=0.5 (28.2 )=14.1mm

j) Calculate the stator outside diameter OD:

OD=D j+2 (g+Lp+W bi)

OD=143.2+2 (0.3+35.3+14.1 )=242.6mm

k) Calculate the required MMF/pole; assuming (20-25) % leakage and infinite permeability of the steel:

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Magnetic Bearing Actuator Group 6

ϑ=(1.2−1.25 )Bg

μ0(g)

ϑ=1.225 1.4854πx 10−7

(0.3 x10−3 )=434.284 At

Coil design Calculations

l) Calculate the cross-sectional area of the coil AC:

This value was not calculated but was done using trial and error until the maximum or optimal load capacity was achieved.

Acoil=545.73mm2 , this is obtained from the FE model.

m) Calculate number of turns and wire diameter:

To obtain this value required the calculation oflav, this was done by assuming the shape of the coil to be a trapezium.

The value of Z y is taken as the distance between the centroid (point were the diagonals intersect) and the line DC.

For this modelZ y=8.266mm, taken from the FE model.

∴lav=2 (W p+Z y )+2 (Lb+Z y )

¿2 (28.2+8.266 )+2 (15.5+8.266 )=120.64mm

Standard copper wire is to be used: resistivity at 20 C is 20 = 0.17241*10-7 m and temperature

coefficient = 0.0039 1/C.

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Magnetic Bearing Actuator Group 6

Due to the class H insulation maximum operation temperature was 1800C. Assuming an acceptable temperature range means winding temperature between 65% and 80%.

Therefore class H would be (0.65 to 0.8)*180 = 1170C to 1440C

To obtain resistivity at maximum operating temperature is as follows:

ρ144=0.17241 x10−7 [1+(144−20 )0.0039 ]=2.55 x 10−8Ωm

Assuming J=4.5 x 106A/m2

∴N= Vj ρ 144 lav

¿ 604.5 x106 (2.55 x10−8 ) (120.64 x10−3 )

=4334.18 turns

Acond=ϑNj

= 434.2844334.18 (4.5 x106 )

=0.0226mm2

Therefore actual Acond taken from the standard metric wire sizes = 0.02270mm2

Coil filling co-efficient k f was not assumed but was calculated and then adjust to produce the best results.

k f=N Acond

A coil=4334.18(0.02270)

545.73=0.18

It can be seen that the coil filling factor was low k f <0.78

n) Calculate resistance and current

At the actual area of conductor = 0.02270mm2 the corresponding nominal resistance at 200C is 0.7596Ω/m.

Therefore at 1440C the nominal resistance is:

RΩm

=0.7596 [1+ (144−20 )0.0039 ]=1.1269 Ωm

The resistance at 1440C is:

R144=R Ωm

∗N∗lav

¿1.1269∗4334.18∗(120.64 x10−3)=589.22Ω

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Magnetic Bearing Actuator Group 6

I= VR144

= 60589.22

=0.102 A

o) Calculation of Actual MMF and MMF density

Ac tual ϑ=N∗I=4334.18∗0.102=442.086 At

ϑ density= Actual ϑAcoil

= 442.086545.73x 10−6

=0.81 x106 Atm2

The value of mmf density that was used in Quick-Field did not produce the required force and required further optimization.

This was done by recalculating with a thicker wire diameter but keeping the same number of turns.

ChosenAcond=0.05515mm2 and the nominal resistance was 0.3126 Ωm

m) The new k f is:

k f=4334.18(0.05515)

545.73=0.64

This value of k f is higher than the original but is still lower than the expected value

n) RΩm

=0.3126 [1+ (144−20 ) 0.0039 ]=0.464 Ωm

The resistance at 1440C is:

R144=R Ωm

∗N∗lav

¿0.464∗4334.18∗(120.64 x 10−3 )=242.49Ω

I= VR144

= 60242.49

=0.247 A

o) Calculation of Actual MMF and MMF density

Actualϑ=N∗I=4334.18∗0.247=1070.5 At

The actual MMF is higher than the initial MMF but at this value we were able to obtain the correct MMF density to be used in the simulation.

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Magnetic Bearing Actuator Group 6

ϑ density= Actual ϑAcoil

= 1070.5545.73 x 10−6

=1.96 x106 Atm2

Figure 2: Schematic of the Final design implementation

4.3 Thermal design:

Using the maximum allowable temperature for class H insulation of 1800C, ambient temperature of the shaft is 400C and of the air is 200C

The temperature operating range for class H insulation assuming (65 to 80)%of the winding temperature from the maximum 180oC.

RANGE DEGREES KELVIN(+273)0.65*180OC 117OC 390K0.8*180OC 144OC 417K

Copper loss in the winding (coil):

ΔP cu=V 2

R144

R=N lav ρ1441

A cond

¿4334.18 (120.64 x10−3 ) (2.55 x 10−8 )

0.05515 x10−6 =241,79Ω

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Magnetic Bearing Actuator Group 6

Hence

ΔPcu=602

242=14.85W

Volume of the coil:

Vol=A coil lav=N Acond

k flav=(545.73 x10−6 ) (120.64 x10−3 )=65836.867mm3

Volume of Coil = 65836.867mm3, taken from the FE Model

Power density in the coil (in W/m3):

p=ΔP cu

Vol=

V 2 k f

ρ144N2lav2 = 14.85

65836.867 x 10−9=225557.5 W

m3

5. Simulation Results:

5.1 Initial design implementation:

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Magnetic Bearing Actuator Group 6

Figure 3: Showing the initial implementation, where we obtained a less than required flux density in the core (1.45T as compared to 1.6T)

This was the initial simulation of the magnetic bearing actuator design. Please note that the actual load capacity for this model was 767.92N. This was unacceptable as the specified load capacity was given to be 1000N. Further optimization was necessary.

5.2 Final design implementation:

Figure 6: Showing the final implementation of the design, with the correct flux flowing through the journal

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Selecting the x-component and utilizing the equations to follow

Magnetic Bearing Actuator Group 6

Figure 4: Showing the Force calculation interface

When setting the problem properties of the model on Quick-Field, we set the model class Plane-Parallel

valueLz=Lb=15.5mm. By doing this the force obtained in Quick-field already accounted for the length

of the bearing. Thus the resultant force becomes:

Fqf=0.5∗F1

But F=2.61∗F1

F (¿ LC )=2.61∗2∗Fqf=2.61∗2∗194.25=1013.985N

This result is slightly higher then the required load capacity. The error obtained can

% Error =( 1013.985−10001000 )100=1.3985%The load capacity obtained is acceptable as the requirement was to produce the load

capacity given or produce slightly higher.

5.3 Thermal design results:

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Magnetic Bearing Actuator Group 6

Figure 5: Showing the thermal response of the model

6. Summary of Final Design Parameters:

Parameter Unit ValueWinding No of turns - 4334.18

Wire diameter (std) mm 0.05515Average length of turn mm 120.64Operating temp. resistance Ω 242.49Developed MMF A-t 1070.5Coil volume mm3 65836.867Power loss density W/m3 225558

Force per pole (based on FE model) N 383.14Number of poles switched on - 4Axial length of the bearing mm 15.5Winding max. temperature (FE model) 0C 119

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Magnetic Bearing Actuator Group 6

7. Discussion of results:

The simulation design of the magnetic bearing was to achieve the maximum load capacity that was initially given and for the thermal properties of the bearing to in the range of the maximum temperature.

Initial Approach:

Initially it was decided to design the bearing using 3-pole activation. By activating three active poles it produced a high force per pole (414.214N) as a result of the high force, the cross-sectional area of the stator pole was large. Reason being the cross-sectional area of the stator pole is directly proportional to the force per pole obtained. Since the cross-sectional are of the stator pole is high it resulted in the axial length of the bearing to be high.

The actual value of MMF (475.562A-t) calculated, using the number of turns and current which was calculated using the area of conductor and coil. Resulted in a higher value of MMF, although this value was acceptable it produced an error of:

Initial value of MMF = 434.284A-t

%Error = ( 475.562−434.284434.284 )∗100=9.5%19

Magnetic Bearing Actuator Group 6

The MMF density that was produced using the actual MMF and the area of coil was relatively high. However when used in the simulation of the model the MMF density did not produce the expected results such as the force and flux density. The force produced using this design was 767.92 a value well below the expected load capacity of 1000N, an error of:

%Error = ( 1000−767.921000 )∗100=23.2%This was unacceptable, as the requirement for the design was to produce the given load capacity or slightly higher.

The flux density was assumed to 1.65T but in the simulation at some points the flux density was 1.45T. The coil filling co-efficient was 0.83, above the maximum of 0.78.

As a result of the results not meeting expectations, we decided to change the approach used.

Final Approach:

In this approach we decided to use 4-pole activation, although by doing this the value of the force per pole would decrease, directly influencing both the cross-sectional area of the stator pole and the axial length of the bearing (a decrease in both).

This design produced an actual MMF closer to the initial calculations being 442.086A-t; the decrease was a result of using a larger coil that dropped the average length. This decrease the number of turns used. The error between the actual and initial is:

%Error = ( 442.086−434.284434.284 )∗100=1.8%The model produced a smaller MMF density as the area of coil was much larger and the MMF itself was lower. When used in the FE model once again the load capacity was lower and so was the coil filling co-efficient.

On optimizing this model by increasing the area of conductor; the result was a large coil filling co-efficient. This changed caused a decrease in the resistance, producing a higher current. The number of turns stayed the same.

A result of the above change produced an actual MMF considerably larger then the initial calculation, an error of:

Actual MMF= 1070.5A-t

%Error = ( 1070.5−434.284434.284 )∗100=146.5%This is large error; however the MMF density that was calculated using this MMF produced a high value.

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Magnetic Bearing Actuator Group 6

When used in the simulation the MMF density generated through the journal produced the required load capacity although higher, the value is acceptable.

Load capacity achieved = 1013.985N

%Error = ( 1013.985−10001000 )∗100=1.4% a minimal error.

The thermal design used the design that was just discussed. The result of the simulation of thermal design produced a temperature of 392K. The expected range of the winding temperature was 390K to 417K. The model produced a temperature in range of the insulation class H (1800C max).

8. Conclusion:

The aim of the design was to simulate a magnetic bearing actuator using Quick-Field. The design had to adhere to certain constraints whilst some could be optimized.

The results of our design had met the specifications asked such as the achievement of the load capacity and the thermal properties.

With respect to the load capacity it required it to have a minimum volume to maximum force ratio. Although we had not met this requirement to exact levels, we still produce a high load capacity. Another aspect was the high MMF we achieved on the design, this value produced the required results.

The thermal design had utilized the same model used for magneto-statics, this allowed for maximum expected results as the design had already been optimized. The difficulty was achieving the optimal power density that would be used in the simulation. Once we obtained the correct power density and boundary conditions we were able to produce the required temperature of the winding.

In all we had met most of the requirements, errors can be expected. We had worked through most difficulties and produce required expectations.

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Magnetic Bearing Actuator Group 6

9. References:

1. Lecture notes distributed by Professor M. Hippner , based on magneto-statics andmagnetic circuit analysis using Quick Field.

2. Electro-mechanics and Electric Machines , by S.A. Nasar and L.E. Unnewehr.

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Magnetic Bearing Actuator Group 6

10. Appendix:

10.1 Optimization 1:

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Magnetic Bearing Actuator Group 6

The value of Z y is taken as the distance between the centroid (point were the diagonals intersect) and the line DC.

For this modelZ y=5.567, taken from the FE model.

∴ lav=2 (W p+Z y )+2 (Lb+Z y )

¿93.24mm

Standard copper wire is to be used: resistivity at 20 C is 20 = 0.17241*10-7 m and temperature

coefficient = 0.0039 1/C.

Due to the class H insulation maximum operation temperature was 1800C. Assuming an acceptable temperature range means winding temperature between 65% and 80%.

Therefore class H would be (0.65 to 0.8)*180 = 1170C to 1440C

To obtain resistivity at maximum operating temperature is as follows:

ρ144=0.17241 x10−7 [1+(144−20 )0.0039 ]=2.55 x 10−8Ωm

Assuming J=4.5 x 106A/m2

∴N= Vj ρ 144 lav

¿ 604.5 x106 (2.55 x10−8 ) (93.24 x10−3 )

=5607.84 turns

Acond=ϑNj

= 434.2845607.84 (4.5 x106 )

=0.01720mm2

Therefore actual Acond taken from the standard metric wire sizes = 0.01767mm2

Coil filling co-efficient k f was not assumed but was calculated and then adjusted to produce the best results.

k f=N A cond

Acoil=5607.84(0.01767)

365.199=0.27

k f ¿Max) = π4 =0.78

The coil filling factor was too low and unacceptable

n) Calculate resistance and current

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Magnetic Bearing Actuator Group 6

At the actual area of conductor = 0.01767mm2 the corresponding nominal resistance at 200C is 0.9757Ω/m.

Therefore at 1440C the nominal resistance is:

RΩm

=0.9757 [1+(144−20 )0.0039 ]=1.4475 Ωm

The resistance at 1440C is:

R144=R Ωm

∗N∗lav

¿1.4475∗5607.24∗(93.24 x10−3 )=756.80Ω

I= VR144

= 60756.80

=0.0792 A

o) Calculation of Actual MMF and MMF density

Actualϑ=N∗I=5607.24∗0.0792=444.093 At

ϑ density= Actual ϑAcoil

=444.093365.199

=1.216 x106 Atm2

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Magnetic Bearing Actuator Group 6

Figure 6: Showing the Flux line distribution and hence flux density

Figure 7: Showing the schematic where we used a greater concentration of nodes on the core, rather than the air gap in order to increase the accuracy of the force calculated on the journal

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Magnetic Bearing Actuator Group 6

Figure 8: Showing the actual flux density distribution, it can be noted that were not achieving approx. 1.6T in the air gap

F (¿ LC )=2.61∗2∗Fqf=2.61∗2∗144.18=752.619N . The load capacity obtained was unacceptable

as the requirement was to produce the given load capacity of 1000N.

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Selecting the x-component, the required load capacity was not achieved