Power Magnet Devices: A Multi-Objective Design Approachsudhoff/ECE61014/Lecture 4 - Chapter 5 (V3).pdf5.4 Case Study •Specifications Table 5.4-1 Electromagnet specifications. Symbol

Power Magnet Devices:A Multi-Objective Design Approach

Chapter 5: Introduction to Electromagnet Design

S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

5.1 Common Electromagnet Architectures

2S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

5.2 Analysis of an EI Core Electromagnet

• Configuration



• Electrical analysiscl w wA w d=

( )2 (2 2 )cl w w c c wV d w l w wπ= + +

2

2cl

clpf cl

V NR

k A σ=

cl

viR

=



• Magnetic analysis 2R ( )2 μ ( /( ))

s c eb

b c B b c

w w ww l w l

+ +Φ =Φ

2R ( )2 μ ( /( ))

s c ei

i c B i c

w w ww l w l

+ +Φ =Φ

2R ( )2 μ ( /( ))

h belb

e c B e c

d ww l w l

+Φ =Φ

R ( )μ ( /( ))

s helg

e c B e c

d dw l w l

−Φ =Φ

2R ( )2 μ ( /( ))

h bclb

c c B c c

d ww l w l

+Φ =Φ

R ( )μ ( /( ))

s hclg

c c B c c

d dw l w l

−Φ =Φ



• In expressions for magnetizing reluctances

( / 2) 2 / 3/ 2 / 4 2 / 3

w s w w sh

s w w s

d d d d ddd d d d

− ≥= + ≤



• End airgap permeance2eg egd efoc efic effP P P P P= + + +

0 e cegd

w lPg

μ=

0 1 min( , )lnci s befoc

l w d wPgπμ

π + +=

0 min(2 , )2 1ln4

s scefic

d wlPg

πμπ

+=

0 min( , )1ln i s beeff

w d wwPg

μ ππ

+ += 7S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach


• Center airgap permeance

2 2cg cgd efic cffP P P P= + +

0 c ccgd

w lPg

μ=

0 min( , )1ln i s bccff

w d wwPg

μ ππ

+ +=



• Slot leakage

0 (3 2 )6( )c s w

vsls

l w wPd g

μ −=+

( )0 3 23

c s whsl

s

l d dP

wμ −

=



• End slot leakage



• We obtain( )0 min(2 , )3 2 2 1ln

23e cs w c

hslss

w wd d lPww

μ ππ

− += +

0 min( , )(3 2 ) 2 1ln6

i bs w cvsl

ss

w ww w lPd gd g

πμπ

− += + ++



• Derivation of new portion of Phsl






5.2 Analysis of an EI Core Electromagnet• Front face leakage

2 2eal ealieale

P PR

= +

2 2 2 2

0 2

( ) (2 )( )w b e w b e b e b eeale

c b e

w w w d w w w w w wR

w w wμ+ + + + +

=

2e s ww d d= −

4 3 2 22 1 2 1 2

02 2 3 4 2

1 2 11

16 16 2 4

2 2256 2 2 ln 1c

ealiw w

k k k k kwP kw d k k k

k

μ + + −

= + +

1 2w wk d w= −22 2

/ 2 2w w w

w w w

w d wk

d d w

>= ≤ 15S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach


• Expression for force

2 2 212 2

1 1 11 122 2

eg cg vsle

eg cg

P P Pf N ig gP P g

∂ ∂ ∂+= − Φ − ∂ ∂ ∂

21 P ( )2

D

be b

b B

xf Fx∈

∂=∂



• We have

2eg egd efoc efic effP P P P Px x x x x

∂ ∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂ ∂

02

egd e cP w lg g

μ∂= −

∂

0 min( , )min( , )

efoc c i s b

i s b

P l w d wg g g w d w

μπ

∂ += −∂ + +

02 min(2 , )4 min(2 , )

efic c s s

s s

P l d wg g g d w

μπ

∂= −

∂ +

0 min( , )min( , )

eff e i s b

i s b

P w w d wg g g w d w

μπ

∂ += −∂ + +



• and have

2 2cg cgd efic cffP P P Pg g g g

∂ ∂ ∂ ∂= + +

∂ ∂ ∂ ∂

02

cdg c cP w lg g

μ∂= −

∂

0 min( , )min( , )

cff c i s b

i s b

P w w d wg g g w d w

μπ

∂ += −∂ + +

( )0 2min( , )(3 2 )

min( , )6c i bvsl s w

s s i bs

l w wP w wd g d g w wg d g

μπ

∂ − += − + + +∂ +



• Limits as the gap goes to zero



• Example 5.2A. Let’s take a look at a EI core• Dimensions (in mm):

• Winding– N=40, ac = 11 μm2

37sw = 27.3bw = 28.7ew = 57.4cw =

48.8sd = 30.5iw = 119.8cl = 2.96g =

15.8ww = 44.4wd =



• Predicted resistance: 24.7 mΩ• Measured resistance: 30.5 mΩ



• Force versus current (predicted)



• Force over power versus current (predicted)


5.3 EI Core Electromagnet Design

• Configuration



• Design space

* * * *[ ]Tcr cd c ec ic bc c w dT T w r r r a N N N=θ



• Widths

• Conductors

12e ec cw r w= 1

2i ic cw r w= 12b bc cw r w=

*round ( )c WG ca a=

cc

ar

π=

*round( )N N=26S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach


• Dimensions*round( )w wN N= *round( )d dN N=

2w c b ww r k N= 2w c b dd r k N=

s w ww w c= +

s w dd d c= +

2 2c s e cl w w w= + +

E i s bh w g d w= + + +

2 2E s e cw w w w= + +

2E c wl l w= +27S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

5.3 EI Core Electromagnet Design• Metrics

E E E EV h w l=2

Ecl

vPR

=



• Constraints

1 gte( , )d wc N N N=

cpf

w w

Nakw d

=

2 ,lte( , )pf pf mxc k k=

c

iJa

=

3 mxlte( , J ( ))cdc J T=29S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach


• More constraintsmax( , , )min( , , )

E E EL

E E E

h w lh w l

α =

4 lte( , )L mxc α α=

5 lte( , )E Emxac V V=

6 lte( , )E Emxac P P=

8 gte( , )e Emnrc f f=7c =



• Pseudo-code for fitness function

calculate EV , EP

calculate constraints 1c through 6c

1 2 3 4 5 6SC c c c c c c= + + + + +

6IC =

if ( S IC C< )

11

sC CC

ε − = f

return end construct MEC and calculate 7c and 8c

7 8s sC C c c= + +

2I IC C= +

if ( S IC C< )

11

sC CC

ε − = f

return end

1

1E

E

V

P

=

f

return 31S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

5.4 Case Study

• SpecificationsTable 5.4-1 Electromagnet specifications. Symbol Description Value

v applied voltage 12 V g air gap 1 mm mxaα maximum allowed aspect ratio 3

EmxaV maximum allowed volume 1 L

EmxaP maximum allowed loss 50 W

pf mxak − maximum allowed packing factor 0.7

bk winding build factor 1.05 Emnrf minimum required electromagnetic force 2500 N

wc winding clearance in width 2 mm dc winding clearance in depth 2 mm


5.4 Case Study

• Design spacePar. Description Min. Max. Enc. Gene

crT Core material 1 5 int 1

cdT Conductor material 1 2 int 2

cw Center leg width 2·10-3 10-1 log 3

ecr Twice ew to cw ratio 0.5 1.5 lin 4

icr Twice iw to cw ratio 0.25 1.5 lin 5

bcr Twice bw to cw ratio 0.25 1.5 lin 6 *ca Cross sectional conductor area (m2) 10-9 10-4 log 7 *N Desired number of turns 1 103 log 8 *wN Desired slot width in conductors 1 103 log 9 *dN Desired slot depth in conductors 1 103 log 10


5.4 Case Study


5.4 Case Study


5.4 Case Study


5.4 Case Study


5.4 Case Study


5.4 Case Study


5.4 Case StudyEI Core Data Winding Data Metrics

Material = Hiperco50 wc = 2.02 cm we = 1.03 cm wi = 8.36 mm wb = 9.93 mm ws =2.13 cm ds = 5.57 cm lc = 8.34 cm

Material = Copper AWG = 18 ac = 0.823 mm2 N = 882 Nw = 18 Nd = 50 dw = 5.37 cm ww = 1.93 cm

VE = 764 mL PE = 29.9 W iE = 2.49 A JE = 3.03 A/mm2 RE = 4.82 Ω hE = 7.50 cm wE = 8.34 cm lE = 12.2 cm


5.4 Case Study


5.4 Case Study


Documents

Power Magnet Devices: A Multi-Objective Design Approachsudhoff/ECE61014/Lecture 4 - Chapter 5 (V3).pdf5.4 Case Study •Specifications Table 5.4-1 Electromagnet specifications. Symbol