lecture 8: a design example - an electromagnetsudhoff/ee630/lecture08.pdf · a design example - an...
TRANSCRIPT
![Page 1: Lecture 8: A Design Example - An Electromagnetsudhoff/ee630/Lecture08.pdf · A Design Example - An Electromagnet. 2 Objective Use a genetic algorithm to solve an engineering design](https://reader030.vdocuments.mx/reader030/viewer/2022040116/5e92b40a169a0301c703092f/html5/thumbnails/1.jpg)
Lecture 8:A Design Example -An Electromagnet
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2
Objective
Use a genetic algorithm to solve an engineering design example
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3
The ProblemDesign an electromagnet to lift a mass.
f
bv
br
Battery
i
M
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4
SpecificationsMass
10 kG10 cm by 10 cm areaMagnetic material should not overlap massMust lift from distance of 5 mm Gravity is on surface of Earth (G=9.8 m/s2)
Batteryvb = 12 Vrb = 0.5 ΩCurrent should not exceed 6 A
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5
Specifications (Continued)Winding
Should not exceed packing factor of 0.7May use Aluminum (Al) or Copper (Cu)Aluminum
Density (ρw) = 2701 Kg/m3
Conductivity (σw) = 35.4 1 1/(μmΩ)Current Density (jmax) = 6.14 MA/m2
CopperDensity (ρw) = 8900 Kg/m3
Conductivity (σw) = 58.0 1/(μmΩ)Current Density (jmax) = 7.62 MA/m2
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6
Specifications (Continued)
Magnetic MaterialMicrosil
Bsat = 1.4 T, μ = 15000 μ0 , ρm = 7064 Kg/m3
Superperm 80Bsat = 0.7 T, μ = 6000 μ0 , ρm = 8069 Kg/m3
Superperm 49Bsat = 1.2 T, μ = 3500 μ0 , ρm = 7892 Kg/m3
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7
Specifications (Continued Again)
Size (Mass)Total mass of electromagnet should be as small as possible
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8
Electromagnet Dimensions
g
sw ewew
ww
sdwd
Depth intopage = d
iw
bw
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9
Design Variables
WireMaterial Type mw
1=Cu, 2=A)Conductor Cross Section acNumber of Turns N
Magnetic MaterialMaterial Type mm
1=Microsil, 2=Superperm 80, 3=Superperm 49)
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10
Design Variables (Continued)
CoreSlot width wsWire window width wwSlot depth dsWire window depth dwEnd width weI-core width wiBack width wbDepth d
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11
Parameter Vector
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
drrwrdrwmNam
wb
wi
e
dw
s
ww
s
m
w
w
θ
swww wrw =
sdww drd =
ewii wrw =
ewbb wrw =
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12
Data Vector
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=Ψ
maxf
max
max
max
sat
w
m
w
plJiB
gGM
,
σρρ
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13
Electrical Analysis
Goal
Matlab m-file: Electrical_Analysis.m
),( Ψθelec
f
Fij
p=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
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14
Electrical AnalysisPacking factor
Winding volume (conductor)
Winding length
ww
cf dw
Nap =
( )( )wwbwwfw wdwddwpv 222 π++=
c
ww a
vl =
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15
Electrical Analysis (Cont.)
Winding resistance
Current
Current Density
wc
ww a
lrσ
=
wb
b
rrvi+
=
caij =
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16
Magnetic Analysis
Goal
Matlab m-file: Magnetic_Analysis.m
),,(
5,4
3,2
2,1
1,0
Ψθ iF
fBBBB
mag
e
=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
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17
Magnetic Equivalent Circuit
g
sw ewew
sdwd
1N
2N
3N
4N5N
6N
7N
ww
iw
bw0N
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18
Magnetic Equivalent Circuit (Cont)
hlP
Ni1N
2N
3N
4N5N
6N
7N
2,1P
3,2P
4,3P
5,4P
6,5P
7,6P
vlP
0N
eP
1,0P
7,0P
3,6P
1,0Φ
2,1Φ
3,2Φ
5,4Φ
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19
Review: Magnetic Equivalent Circuits
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20
Magnetic Material Permeances
wb
e
dwdw
PP+
==μ2
7,02,1
2/7,63,2ws
e
dddwPP
−==
μ
es
i
wwdw
P+
=μ
5,4
es
b
wwdw
P+
=μ
1,0
g
sw ewew
sdwd
1N
2N
3N
4N5N
6N
7N
ww
iw
bw0N
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21
Air Gap Permeances
g
sw ewew
sdwd
1N
2N
3N
4N5N
6N
7N
ww
iw
bw0N
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−+
−=
s
ews
s
ws
wwdd
wddd
Pπ
πμμ
1ln)(2)( 00
3,6
444 3444 2144444 344444 21
44 344 21321
facesbackfront
ie
faceinner
ss
faceouter
i
facemain
e
gww
gdwd
gwd
gdw
PP
/
00
006,54,3
1ln2
4),min(
1ln2
1ln
⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛+
+⎟⎟⎠
⎞⎜⎜⎝
⎛++==
ππ
μππμ
ππ
μμ
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22
Hor. And Vert. Leakage Permeances
g
sw ewew
sdwd
1N
2N
3N
4N5N
6N
7N
ww
iw
bw0N
gddwP
s
wvl +
= 0
61 μ
⎟⎟⎠
⎞⎜⎜⎝
⎛++=
s
ew
s
whl w
wdw
ddP ππ
μμ 1ln3
231 00
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23
End Leakage Permeance
g
sw ewew
sdwd
1N
2N
3N
4N5N
6N
7N
ww
iw
bw0N
wst ddw −=
bt
btbtww
bt
ebte
wwwwwwdw
wwwdwwP
+++
++
+=
)2(1)2(0
1μ
ww wdk 21 −=
⎪⎩
⎪⎨⎧
≤
>=
www
www
wddwdw
k2
2
222
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛++−
+++
=
1
241
231
22
21
321
42
220
2 221ln168
2412
8)2(
kkkkk
kkkkk
dwwd
Pww
ee
μ
21 eee PPP +=
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24
Nodal Equations
hlP
Ni1N
2N
3N
4N5N
6N
7N
2,1P
3,2P
4,3P
5,4P
6,5P
7,6P
vlP
0N
eP
1,0P
7,0P
3,6P
1,0Φ
2,1Φ
3,2Φ
5,4Φ
( ) ( ) 02,12111,01 =−++− PFFPFPNiF e
( ) ( ) ( ) 03,232722,112 =−+−+− PFFPFFPFF hl
( ) ( ) ( ) 03,2234,3433,663 =−+−+− PFFPFFPFF
( ) ( ) 05,4544,334 =−+− PFFPFF
( ) ( ) 06,5655,445 =−+− PFFPFF
( ) ( ) ( ) 03,6366,5567,676 =−+−+− PFFPFFPFF
( ) ( ) 07,667277,07 =−+−+ PFFPFFPF hl
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25
Nodal Equations (Cont)
columnstPNiPF
11
1,0−=
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
++−−−++−−
−+−−+−
−−++−−−++−
−++
=
7,67,07,6
7,63,66,57,66,53,6
6,56,55,45,4
5,45,44,34,3
3,64,33,24,33,63,2
3,23,22,12,1
2,12,11,0
0000000
00000000000
00000000
PPPPPPPPPPP
PPPPPPPP
PPPPPPPPPPPP
PPPP
P
hlhl
hlhl
e
[ ] TFFFFFFFF =7654321
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26
Fluxes
hlP
Ni1N
2N
3N
4N5N
6N
7N
2,1P
3,2P
4,3P
5,4P
6,5P
7,6P
vlP
0N
eP
1,0P
7,0P
3,6P
1,0Φ
2,1Φ
3,2Φ
5,4Φ
( )
7,077,0
7,6767,6
6,5656,5
5,4545,4
4,3434,3
3,232
2,1212,1
1,011,0
)()(
)()()()(
PFPFFPFFPFFPFFPFFPFFPFNi
=Φ
−=Φ
−=Φ
−=Φ
−=Φ
−=Φ
−=Φ
−=Φ
2,3
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27
Flux Densities
dwB
dwB
dwB
dwB
i
e
e
b
5,45,4
3,23,2
2,12,1
1,01,0
Φ=
Φ=
Φ=
Φ=
g
sw ewew
sdwd
1N
2N
3N
4N5N
6N
7N
ww
iw
bw0N
![Page 28: Lecture 8: A Design Example - An Electromagnetsudhoff/ee630/Lecture08.pdf · A Design Example - An Electromagnet. 2 Objective Use a genetic algorithm to solve an engineering design](https://reader030.vdocuments.mx/reader030/viewer/2022040116/5e92b40a169a0301c703092f/html5/thumbnails/28.jpg)
28
Force
( )( )
( )( ) ( )i
ie
ss
ss
i
ie
wggwdw
dwggdwd
wggdw
gdw
gP
gP
πμ
πμ
πμμ
+−
+−
+−−=
∂
∂=
∂
∂ 0,002
06,54,3 2,min44
min2
( )20
61
gddw
gP
s
wvl
+−=
∂∂ μ
Using co-energy techniques and assuming magnetic linearity it can be shown
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29
Force (Cont.)
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂
∂
∂
∂−
∂
∂−
∂
∂∂
∂
∂
∂−
∂
∂−
∂
∂
=∂∂
0000000
00000
00000
00000
00000
00000000000000
6,56,5
6,56,5
4,34,3
4,34,3
gP
gP
gP
gP
gP
gP
gP
gP
gP
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30
Force (Continued)
[ ] [ ]g
PNP
gPPPN
gL vl
∂∂
+∂∂
=∂∂ −− 2
column1st1
row1st 12
012
2
21 i
gLfe ∂
∂−=
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31
Setting Up Constraints
Less than function
⎪⎪⎩
⎪⎪⎨
⎧
>
Δ−
+
≤
=Δ maxmax
max
max xx
xxx
xx
xxxltne1
11
),,(
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32
Setting Up Constraints
Greater than function
⎪⎪⎩
⎪⎪⎨
⎧
<
Δ−
+
≥
=Δ minmin
min
min xx
xxx
xx
xxxgtne1
11
),,(
![Page 33: Lecture 8: A Design Example - An Electromagnetsudhoff/ee630/Lecture08.pdf · A Design Example - An Electromagnet. 2 Objective Use a genetic algorithm to solve an engineering design](https://reader030.vdocuments.mx/reader030/viewer/2022040116/5e92b40a169a0301c703092f/html5/thumbnails/33.jpg)
33
Geometrical Constraints
Recall, magnetic material should not overlap mass
)1.0,,2( max1 llwwltnec maxes +=
)1.0,,(2 maxmax lldltnec =
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34
Electrical Constraints
Packing Factor
Current Density
Current
)1.0,,( ,,3 maxfmaxff pppltnec =
)1.0,,(4 maxmax Jjjltnec =
)1.0,,(5 maxmax iiiltnec =
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35
Flux Density Constraints
Back of U-Core
Back of Legs
Front of Legs
I-Core
)01.0,,( 1,06 satsat BBBltnec =
)01.0,,( 2,17 satsat BBBltnec =
)01.0,,( 3,28 satsat BBBltnec =
)01.0,,( 5,49 satsat BBBltnec =
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36
Force Constraints
We must lift mass (and I-core)
GwwdwMf mesireq ))2(( ρ++=
)01.0,,(10 reqreqe fffgtnec =
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Performance Metrics
Total mass
( )( )( ) wwwbwwf
mibeses
wdwddwp
dwwwwdwdm
ρπ
ρ2
1
22
))(2(2
++
++++=
g
sw ewew
ww
sdwd
Depth intopage = d
iw
bw
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Fitness Function
Combined constraint
Fitness Function
),,min( 1021 cccc L=
⎪⎪⎩
⎪⎪⎨
⎧
<−
=+=
∑=
110
11
10
1
1
cc
cm
ci
εf
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Code Organization
Design CodesElectromagnet_Design.m
GOSETElectromagnet_Fitness.m
Electrical_Analysis.mMagnetic_Analysis.m
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Code Organization (Cont)
Supporting RoutinesElectromagnet_Drawing1Electromagnet_Drawing2Electromagnet_Design_ReviewElectromagnet_Design_MultirunElectromagnet_Design_Multrun_Review
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Electromagnet_Design.m (1/3)% Electromagnet Design%% Written by:% Ricky Chan for S.D. Sudhoff% School of Electrical and Computer Engineering% 1285 Electrical Engineering Building% West Lafayette, IN 47907-1285% E-mail: [email protected]
% Set Up PopulationGAP = gapdefault;GAP.fp_ngen = 2500;GAP.fp_ipop = 500;GAP.fp_npop = 500;GAP.mc_alg = 4.0;GAP.dt_alg = 3;
% Set Up MigrationsGAP.mg_nreg=5; % number of regionsGAP.mg_tmig=100; % mean time between migrationsGAP.mg_pmig=0.05; % probability of a individual migrating
% Unitsmm=1.0e-3;cm=1.0e-2;
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Electromagnet_Design.m (2/3)% Problem Requirement Data Psi.M = 10; % Mass of object (Kg)Psi.G = 9.8; % Gravity (m/s2)Psi.g = 0.5*cm; % Gap Width (m)Psi.rhow = [ 8900; 2701]; % Wire - Kg/m3 Psi.sigmaw = [ 58.0; 35.4]*1e6; % Wire - A/OhmPsi.jmaxw = [ 7.62; 6.14]*1e6; % Wire - A/m2Psi.descw = ['Copper '; 'Aluminum']; % Wire - DescriptionPsi.bmaxm = [ 1.4; 0.7; 1.2]; % Steel - Bsat, TPsi.rhom = [7064.1; 8069.4; 7892.1]; % Steel - Density Kg/m3Psi.myum = [ 15000; 6000; 3500]; % Steel - Perm., (relative)Psi.descm = ['Microsil '; ...
'Superperm 80'; ...'Superperm 49']; % Steel - Description
Psi.imax = 6; % Maximum current, APsi.lmax = 10e-2; % Maximum lengthPsi.pfmax = 0.7; % Maximum packing factorPsi.vb = 12; % Battery VoltagePsi.rb = 0.5; % Battery Resistance
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Electromagnet_Design.m (3/3)% Genetic Mapping% mw ac N mm ws rww ds rdw we rwi rwb dGAP.gd_min = [ 1 1e-8 10 1 1*cm 0.1 1*mm 0.1 1*mm 0.2 0.2 1*mm];GAP.gd_max = [ 2 1e-4 1e3 3 10*cm 1.0 20*cm 1.0 5*cm 2.0 2.0 10*cm];GAP.gd_type = [ 1 3 3 1 3 3 3 3 3 3 3 3 ];GAP.gd_cid = [ 1 1 1 1 1 1 1 1 1 1 1 1 ];
% Solve Problem[fP,GAS]= gaoptimize(@Electromagnet_Fitness,GAP,Psi,[],[],[]);
% Get Final Answer and Plotfinal_parameters = GAS.bestgenes(:,end);f = Electromagnet_Fitness(final_parameters,Psi,23);
% Save runs=input('Type 1 to Save Run ');if (s==1)
save samplerunend
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Electromagnet_Fitness.m (1/7)function fitness = Electromagnet_Fitness(param,Psi,figNum)
% ELECTROMAGNET_FITNESS %% fitness = Electromagnet_Fitness(param,Psi,x)%% Inputs:% param = design parameters% Psi = data vectors% x = optional input to print out extra information%% Outputs:% fitness = fitness value%% Written by:% Ricky Chan for S.D. Sudhoff% School of Electrical and Computer Engineering% 1285 Electrical Engineering Building% West Lafayette, IN 47907-1285% E-mail: [email protected]
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Electromagnet_Fitness.m (2/7)% Map to nicer variable namesmw = param(1);ac = param(2);N = round(param(3));mm = param(4);ws = param(5);rww = param(6);ds = param(7);rdw = param(8);we = param(9);rwi = param(10);rwb = param(11);d = param(12);
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Electromagnet_Fitness.m (3/7)% Computing the width of winding (ww), depth of winding (dw), width of the% I core (wi), and the width of the back of the U core (wb)ww = rww*ws;dw = rdw*ds;wi = rwi*we;wb = rwb*we;
% Electrical Circuit Analysis[pf, j, i] = Electrical_Analysis(param,Psi);
% Magnetic Analysis[B01, B12, B23, B45, fe] = Magnetic_Analysis(param,Psi,i);
% Constraintslmax = Psi.lmax;pfmax = Psi.pfmax;jmaxw = Psi.jmaxw(mw);imax = Psi.imax;bmax = Psi.bmaxm(mm);rhom = Psi.rhom(mm);rhow = Psi.rhow(mw);
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Electromagnet_Fitness.m (4/7)c1 = ltne(ws+2*we,lmax,0.1*lmax);c2 = ltne(d,lmax,0.1*lmax);c3 = ltne(pf,pfmax,0.1*pfmax);c4 = ltne(j,jmaxw,0.1*jmaxw);c5 = ltne(i,imax,0.1*imax);c6 = ltne(B01,bmax,0.01*bmax);c7 = ltne(B12,bmax,0.01*bmax);c8 = ltne(B23,bmax,0.01*bmax);c9 = ltne(B45,bmax,0.01*bmax);f_req = (Psi.M + wi*d*(ws+2*we)*rhom)*Psi.G;c10 = gtne(fe,f_req,0.01*f_req);
% find the aggregrate constraintc = [c1 c2 c3 c4 c5 c6 c7 c8 c9 c10];cmin = min(c);
% design objectivem1 = (2*ds*we*d + (ws+2*we)*(wb+wi)*d)*rhom + ...
pf*(ww*dw*(2*d+2*wb)+pi*dw^2*ww)*rhow;
% compute the fitness if (cmin == 1)
fitness = 1/(m1+1e-6);else
fitness = sum(c)-length(c);end
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Electromagnet_Fitness.m (5/7)% if more than 3 arguments, plot the result% and write a reportif nargin >= 3
format short g
disp('Parameters'); disp(['Parameter 1: Wire type ',Psi.descw(mw,:)]);disp(['Parameter 2: Wire cross section is ', num2str(ac), ' m^2']);disp(['Parameter 3: Number of turns is ', num2str(N)]);disp(['Parameter 4: Magnetic material is ', Psi.descm(mm,:)]);disp(['Parameter 5: Slot width is ',num2str(ws*100),' cm']);disp(['Parameter 6: Winding width is ',num2str(rww*100),'% slot width']);disp(['Parameter 7: Slot depth is ',num2str(ds*100),' cm']);disp(['Parameter 8: Winding depth is ',num2str(rdw*100),'% slot depth']);disp(['Parameter 9: End width is ',num2str(we*100),' cm']);disp(['Parameter 10: I width is ',num2str(rwi*100),'% end width']);disp(['Parameter 11: Base width is ',num2str(rwb*100),'% end width']);disp(['Parameter 12: Depth is ',num2str(d*100),' cm']);disp(' ');
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disp('Constraints');disp(['Overall width is ',num2str(100*(ws+2*we)/lmax),'% allowed']);disp(['Overall depth is ',num2str(100*d/lmax),'% allowed']);disp(['Packing factors is ',num2str(100*pf/pfmax),'% allowed']);disp(['Current density is ',num2str(100*j/jmaxw),'% allowed']);disp(['Current is ',num2str(100*i/imax),'% allowed']);disp(['Back core flux density is ',num2str(100*B01/bmax),'% allowed']);disp(['Back end core flux density is ',num2str(100*B12/bmax),'% allowed']);disp(['Front end core flux density is ',num2str(100*B23/bmax),'% allowed']);disp(['I core flux density is ',num2str(100*B45/bmax),'% allowed']);disp(['Force required is ',num2str(f_req),' N']);disp(['Force produced is ',num2str(100*fe/f_req), '% required']);disp(' ');disp('Metrics');disp(['Mass is ',num2str(m1),' Kg']);
figure(figNum);clf;Electromagnet_Drawing1(param,Psi);figure(figNum+1)Electromagnet_Drawing2(param,Psi);
end
Electromagnet_Fitness.m (6/7)
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Electromagnet_Fitness.m (7/7)function p = ltne(x,xmax,delta)
if x <= xmaxp = 1;
elsep = 1./(1+abs((xmax-x)./delta));
end
function p = gtne(x,xmin,delta)
if x >= xminp = 1;
elsep = 1./(1+abs((xmin-x)./delta));
end
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Sample Evolution
1 2 3 4 5 6 7 8 9 10 11 120
0.5
1
Parameter Number
Nor
mal
ized
Val
ue
500 1000 1500 2000 2500-6
-4
-2
0
2
f:B(b
) Md(
g) M
n(r)
Generation
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Evolution – Best Fit Individual
(Matlab Demo)
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ParametersParameter 1: Wire type Copper Parameter 2: Wire cross section is 7.8519e-007 m^2Parameter 3: Number of turns is 585Parameter 4: Magnetic material is MicrosilParameter 5: Slot width is 7.1906 cmParameter 6: Winding width is 97.7656% slot widthParameter 7: Slot depth is 0.96031 cmParameter 8: Winding depth is 97.8117% slot depthParameter 9: End width is 1.2104 cmParameter 10: I width is 49.2545% end widthParameter 11: Base width is 52.2008% end widthParameter 12: Depth is 3.7947 cm
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ConstraintsOverall width is 96.1141% allowedOverall depth is 37.9474% allowedPacking factors is 99.3753% allowedCurrent density is 99.4713% allowedCurrent is 99.1917% allowedBack core flux density is 99.442% allowedBack end core flux density is 50.9116% allowedFront end core flux density is 49.215% allowedI core flux density is 99.6887% allowedForce required is 99.5053 NForce produced is 100.3524% required
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Metrics
Mass is 0.86127 Kg
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-0.1 -0.05 0 0.05 0.1
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-0.1 -0.05 0 0.05 0.1
-0.15
-0.1
-0.05
0
0.05
0.1
0.15Profile View
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3-D View
-0.02
0
0.02
-0.050
0.05
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
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Design RepeatabilityRepeat design 25 times:
0100.00100.00mn
3.48106.1693.95N
1.46102.0595.26aw
0100.00100.00mw
Std/MnMax/MnMin/MnParameter
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Design Repeatability
1.46101.9496.46rdw
4.07105.8692.18ds
0.57100.7297.93rww
5.06110.0188.44ws
Std/MnMax/MnMin/MnParameter
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Design Repeatability
5.93109.8288.55we
1.09102.0097.55Fitness
9.02118.9183.18d
5.17108.7889.55rwb
5.06109.2690.00rwi
Std/MnMax/MnMin/MnParameter