Mini-Coil Design

Michael von Ortenberg

Fukui University, Okayama University,Humboldt University at Berlin

presented at

Mini Coil Workshop

IMR, Tohoku University, November 24,2006

PARAMETERS of the Mini-Coils constructed:

Coil A B C D E Bore (mm) 3/2.6 3/2.6 3/2.6 3/2.6 3/2.6 Wire (mm) 0.5 Cu/Ag 0.5 Cu/Ag 0.5 Cu/Ag 0.5 Cu/Ag 2x0.5

Cu/Ag Number of layers

12 16 14 12 12

Total height (mm)

20 20 20 15 20

R300 (Ohm) With steel flanges

1.017 1.76 1.304 0.747 0.285

R300 (Ohm) Without steel flanges

- - - 0.728 0.285

R77 (Ohm) with steel flanges

0.257 0.371 - - -

L300 (µH) without steel flanges

- - - 138,7 -

L300 (µH) with steel flanges

292 758 549.7 182.45 86

B/Ucoil (T/V)

2.98/100 1.30/102 2.82/101 3.64/101 3.20/103

Bmax (T)/Ucoil 47.78/1997 41.06/1997 46.29/1997 51.85/1806 51.72/1805 τ/2 =(tBmax-tB0) (msec) at 77 K

0.74 1.03 0.886 0.564 0.414

Operation crowbar crowbar crowbar crowbar crowbar

Schematic of Construction:

nn n+1n+1

BBnn

BBn+1n+1

aann

aan+1n+1=a=ann+d+d

aan+2n+2=a=an+1n+1+d+d

BBnn+B+Bn+1n+1

22

BBn+1n+1==BBnn--ΔΔBn(aBn(ann))

aann+0.5d+0.5d

σσnn=(B=(Bnn+B+Bn+1n+1)*I*(a)*I*(ann+0.5d)+0.5d)22 πd2

4

Schematic of Calculation:

Farad U1 5000:= Volt 90 kJC2 0.96 10 3−⋅:= Farad U2 2000:= Volt 4 kJ

Current density in wire at maximum current:

jI0

0.25 π⋅ d2⋅

:= j 9.065= [kA/mm^2] in wire

jeI0

d2:= je 7.12= :[kA/mm^2] effective homogeneous current density

Fieldfunctions in the ho mogeneous coil:Field [T] in the coil axis at height z [mm] above the center:a [mm] inner radius of the one-layer coil, la [mm] total height of the coil, Icurrent in kA

B z a, la, I,( )1.26 I⋅2 d⋅

0.5zla

−

0.5zla

−⎛⎜⎝

⎞⎟⎠

2 ala

⎛⎜⎝

⎞⎟⎠

2+

0.5zla

+

0.5zla

+⎛⎜⎝

⎞⎟⎠

2 ala

⎛⎜⎝

⎞⎟⎠

2+

+

⎡⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎦

⋅:= : for effective homogneous currentdensity I0/(d^2)

Field [T] at position y [mm] for z=0 inside or outside of the coil:

Math-CAD Program:

Program SpulOkayama-Sendai.mcd for the computation of MINI COILS inOKAYAMA for Cu/Ag-wire with breaking tension 0.78 G-PaVersion 26.10.06 based on SpulNeu.mcd of 7.11.02

i 1−:=

Parameters:a1 1.5:= [mm] inner radius of the coil

la 20:= [mm] total height of coil

I0 1.78:= [kA] current in the wireBm 48:= [T] maximum field of coild 0.5:= [mm] diameter of wireρd 9.96:= [gr/cm-3] density CuNL 12:= : number of windings

Capacitor banks #1 and #2:

C1 7.2 10 3−⋅:=

BE y a, la, I,( )1.26 I⋅2 π⋅ d⋅

0

π

φ

1ya

⎛⎜⎝

⎞⎟⎠

cos φ( )⋅−

1ya

⎛⎜⎝

⎞⎟⎠

2+ 2

ya

⎛⎜⎝

⎞⎟⎠

⋅ cos φ( )⋅−⎡⎢⎣

⎤⎥⎦

0.25yla

⎛⎜⎝

⎞⎟⎠

2+ 1

yla

⎛⎜⎝

⎞⎟⎠

⋅ cos φ( )⋅−ala

⎛⎜⎝

⎞⎟⎠

2+⋅

⌠⎮⎮⎮⎮⎮⎮⌡

d⋅:=

Remark: BE(y) has a pronounced step at the position y=a and reduces to 0 forlarger y.

Fieldstrength BF(0,y,z) inside and outside the coil:y-component:

BFy y z, a, la, I,( )1.26− I⋅

2 π⋅ d⋅

0

π

φcos φ( )

la2 a⋅

za

+⎛⎜⎝

⎞⎟⎠

21+ 2

ya

⋅ cos φ( )⋅−ya

⎛⎜⎝

⎞⎟⎠

2+

⎡⎢⎣

⎤⎥⎦

0.5

cos φ( )−

la2 a⋅

za

−⎛⎜⎝

⎞⎟⎠

21+ 2

ya

⋅ cos φ( )⋅−ya

⎛⎜⎝

⎞⎟⎠

2+

⎡⎢⎣

⎤⎥⎦

0.5+

...⌠⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⌡

d⋅:=

z-component:

BFz y z, a, la, I,( )1.26− I⋅

2 π⋅ d⋅

0

π

φ

ya

cos φ( )⋅ 1−⎛⎜⎝

⎞⎟⎠

la2 a⋅

za

−⎛⎜⎝

⎞⎟⎠

⋅

1 2ya

⋅ cos φ( )⋅−ya

⎛⎜⎝

⎞⎟⎠

2+

⎡⎢⎣

⎤⎥⎦

la2 a⋅

za

−⎛⎜⎝

⎞⎟⎠

21+ 2

ya

⋅ cos φ( )⋅−ya

⎛⎜⎝

⎞⎟⎠

2+

⎡⎢⎣

⎤⎥⎦

0.5

⋅

ya

cos φ( )⋅ 1−⎛⎜⎝

⎞⎟⎠

la2 a⋅

za

+⎛⎜⎝

⎞⎟⎠

⋅

1 2ya

⋅ cos φ( )⋅−ya

⎛⎜⎝

⎞⎟⎠

2+

⎡⎢⎣

⎤⎥⎦

la2 a⋅

za

+⎛⎜⎝

⎞⎟⎠

21+ 2

ya

⋅ cos φ( )⋅−ya

⎛⎜⎝

⎞⎟⎠

2+

⎡⎢⎣

⎤⎥⎦

0.5

⋅

+

...

⌠⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⌡

d⋅:=

Computation of the tension in element (d x d) in homogeneous coil enclosingwire of diameter d:

a: inner radius of the winding considered in [mm]d: wire diameter in [mm]B: Magnetfield produced by the winding abover the considered winding in [T], j: Current density within the considered winding in [kA/mm^2]

Bb1 Bm:= a1 1.5:=

n 2 NL 1+..:=

an an 1− d+:=

Bbn Bbn 1− B 0 an 1−, la, I0,( )−:=

σn 1−Bbn 1− Bbn+( )

2I0⋅

an 1−d2

+⎛⎜⎝

⎞⎟⎠

d2( )⋅ 10 3−⋅:= : G-Pa in square element (d x d) in contrast to wire

cross section πd^2/4

σ

0

0.57

0.663

0.724

0.756

0.758

0.732

0.679

0.599

0.493

0.363

0.21

0.033

0.033−

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=Bb

0

48

43.564

39.166

34.814

30.517

26.284

22.119

18.028

14.016

10.086

6.24

2.479

1.196−

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=a

0

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

σ; strain in wire of winding in G-Pa

a: inner winding radius in mm Bb: magnetic field inside winding in T

NL 12=Total number of windings:

Hence within +/- 2 mm around center field inhomgeneity very small!BB 2( )BB 0( )

0.991=

10 8 6 4 2 0 2 4 6 8 1020

30

40

50

60

BB z( )

z

BB z( )

1

NL 1+

n

B z an 0.5 d⋅+, la, I0,( )∑=

:=

Calculation of the z-dependence of the field in the center axis:

: [mm]Dtotal 15=Dtotal 2 aNL d+( )⋅:=Total diameter of coil:

σNL 0.033=σNL 1+BbNL 1+

2I0⋅

aNL 1+d2

+⎛⎜⎝

⎞⎟⎠

d2( )⋅ 10 3−⋅:=

’·‚³

1

NL

liter

aliterd2

+⎛⎜⎝

⎞⎟⎠

2⋅ π⋅lad

⋅∑=

:= ’·‚³ 1.357 104×= : mm total wire length

n 1 NL 1+..:=

0 2 4 6 8 10 120

50

100

150

Bbn

σn 100⋅

78

n

del 0.25001:= : [mm]

n 0 50..:=

m 0 50..:=

Myn m,

1

NL

liter

BFy del n⋅ del m⋅, aliter, la, I0,( )∑=

:=

Mzn m,

1

NL

liter

BFz del n⋅ del m⋅, aliter, la, I0,( )∑=

:=

Mn m, Myn m, i Mzn m,⋅+:=

Radial Component

My

Axial Component

Mz

M

Computation of the compressive force in k-Newton/meter=N/mm on the wire at the coil edge(radial unit: 0.25 mm):

liter 1 50..:= faradliter Myliter 40, I0⋅:=

faradliter

liter

Computation of the axial force acting on the different layers:m 1 20..:= r 6 8, 28..:=

—Ím1

12

r

My4 2 r⋅+ m 2⋅, I0⋅ ar⋅ 2⋅ π⋅∑=

:= : force produced by layer inposition z=0.25*m on thetotal of coil

—Ím =

: Newton

—Ím

m

Summation of all forces within one coil-half side:

sum

1

20

n

—Ín∑=

:= sum = :Newton , total force of one coil-half

MLm n, Lteil m n,( ):=

n 1 NL..:=m 1 NL..:=

:HenryInduk =Induk

1

NL

i 1

NL

j

Lteil i j,( )∑=

∑=

:=

Lteil 12 12,( ) =Lteil 1 1,( ) =NL =Lteil i j,( ) L ai 0.8⋅ aj 0.8⋅, la, d,( ):=

Computation of the inductance of the magnetic coil in consideration

Henry4 π⋅ 10 7−⋅ 106⋅ 25⋅ π⋅10 3−

1000⋅ =

classical result for "long" coil of one layerand 1000 windings

HenryL 5 5, 1000, 1,( ) =test result of present calculation:

L a b, L, d,( )2 π⋅ 10 10−⋅ a⋅ b⋅

d2

0

2 π⋅

cos φ( )−

L−

2

L

2

zln

L2

z−⎛⎜⎝

⎞⎟⎠

2 L2

z−⎛⎜⎝

⎞⎟⎠

2a2

+ b2+ 2 a⋅ b⋅ cos φ( )⋅+ 10 5−++

a2 b2+ 2 a⋅ b⋅ cos φ( )⋅+ 10 5−+

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

ln

L2

z+⎛⎜⎝

⎞⎟⎠

2 L2

z+⎛⎜⎝

⎞⎟⎠

2a2

+ b2+ 2 a⋅ b⋅ cos φ( )⋅+ 10 5−++

a2 b2+ 2 a⋅ b⋅ cos φ( )⋅+ 10 5−+

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

+

...

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⌠⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⌡

d

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅

⌠⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⎮⌡

⋅:=

Number of windings per layer: la/d

d =la =

Computation of the inductance of the coil constructed by several layers ofwindings:Length measurements in [mm], Induktivität in [Henry].Height of the coil la [mm]Radii of windings a1, a2 to be considered for cross inductanceDiameter of wire d [mm]

Calculation of temperature increase in coil after shot considering the "Action Integral" for ahalf-sinus current pulse of length τ and maximal current density j:0.5*τ*j^2=Integral from starting temperature Tex before the shot to final temperature Tf afterpulse with length τ of integrand [ρd*c(T)/ρCu]ρd: density of Cu (not temperature dependent)c(T) specific heat as function of temperature T

[mm] skin depth T=4.2 KδHe =δHe δRT 0.04⋅:=

[mm] skin depth at T=77 Kδ77 =δ77 δRT 0.13⋅:=

[kV] Voltage for maximum fieldUmax =Umax Rblind I0⋅:=

[Ohm] ratio of U/IRblind =RblindLgescoul

:=

[mm] Skin depth at T=273 KδRT =δRT 0.2 τ⋅ρ

μ0 π⋅⋅:=Skin depth in wire::

V*sec/A*cmμ0 4 π⋅ 10 9−⋅:=

[msec] pulse lengthτ =τ π 1000⋅ Lges coul⋅⋅:=

Lges Induk:=Ohm*cm at T=273 Kρ 1.95 10 6−⋅:=

[Farad] capacity of the bankcoul =coul C2:=

Fixing of the system parameters:

ML =

q12 0.42132:=

Tq11 98.169:=q11 0.33307:=Tq10 84.921:=q10 0.24208:=Tq9 73.68:=q9 0.16937:=

Tq8 66.449:=q8 0.12628:=Tq7 57.528:=q7 0.08050:=Tq6 49.032:=q6 0.04516:=

Tq5 43.162:=q5 0.02744:=Tq4 36.80:=q4 0.01380:=Tq3 30.972:=q3 0.00638:=

Tq2 23.278:=q2 0.00157:=Tq1 14.558:=

T0 4 5, 300..:=

[Ohm*cm]ρCu T0( ) interp vsq Tq, q, T0,( ) 10 6−⋅:=vsq cspline Tq q,( ):=

Tq20 297.855:=q20 1.7055:=Tq19 250.187:=q19 1.3865:=Tq18 208.061:=q18 1.1017:=

Tq17 188.174:=q17 0.9687:=Tq16 175.55:=q16 0.8784:=Tq15 152.720:=q15 0.7191:=

Tq14 133.033:=q14 0.58023:=Tq13 117.178:=q13 0.46746:=Tq12 110.620:=

r5 0.0063:=r6 0.0074:=r7 0.0097:=r8 0.014:=r9 0.0195:=

tesla10 60.0:=tesla11 80.0:=tesla12 100.0:=tesla13 150.0:=tesla14 200.0:=tesla15 250.0:=tesla16 273.0:=

r10 0.065:=r11 0.15:=r12 0.23:=r13 0.46:=r14 0.68:=r15 0.92:=r16 1.0:=

Resistance values:

Temperature dependence of the specific resistance of Cu:Data after HENNING (Fritz Herlach)

( )ρCu(T) specific resistance as function of temperature T

q1 0.00016:=Tq0 4:=q0 0.0005:=

Data after LANDOLT-BÖRNSTEIN:

[Ohm*cm]ρcu T0( ) interp vs tesla, r, T0,( ) ρ⋅:=vs cspline tesla r,( ):=

tesla0 4.0:=tesla1 6.0:=tesla2 8.0:=

r0 0.004:=r1 0.0044:=r2 0.0048:=

tesla3 10.0:=tesla4 15.0:=tesla5 20.0:=tesla6 25.0:=tesla7 30.0:=tesla8 35.0:=tesla9 40.0:=

r3 0.005:=r4 0.0057:=

0

ρcu T0( )

ρCu T0( )

0 300T0

: Ohm*cm

Specific Heat after Debye:

Specific heat after Debye:c T0( )T0343

⎛⎜⎝

⎞⎟⎠

3

0

343

T0x

x4 ex⋅

ex 1−( )2

⌠⎮⎮⎮⎮⌡

d

⎡⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎦

⋅ 1.2017⋅:= c 4.2( ) = c 77( ) =

T0 5 6, 250..:=

c T0( )

ρcu T0( ) 105⋅

ρCu T0( ) 105⋅

c T0( )5

ρcu T0( ) 108⋅⋅

c T0( )5

ρCu T0( ) 108⋅⋅

0 250T0

Tex 4.2:= [K] starting temperature of the coil before shot

j = :maximum current dernsity

w T0( )

Tex

T0

teslac tesla( )

ρCu tesla( )

⌠⎮⎮⌡

d:= f0.5 τ⋅ j2⋅ 1.6501⋅

ρd106⋅:= :Factor 1.6501 resulting from magneto

resistance of the form: (1+0.00766*B(t))[after Fritz HERLACH]

f =

T0 10 40, 500..:=

w T0( )

0 500T0

[K] final temperature of thecoil after shot

Tf =Tf rootw T0( )

f1− T0,

⎛⎜⎝

⎞⎟⎠

:=T0 100:=

w T0( )

0 500T0

T0 10 40, 500..:=

f =

:Factor 1.6501 resulting frommagneto resistance of the form:(1+0.00766*B(t))

f0.5 τ⋅ j2⋅ 1.6501⋅

ρd106⋅:=w T0( )

Tex

T0

teslac tesla( )

ρCu tesla( )

⌠⎮⎮⌡

d:=

:maximum current densityj =

[K] starting temperature of the coil before shotTex 77:=

[K] final temperature of the coilafter shot

Tf =Tf rootw T0( )

f1− T0,

⎛⎜⎝

⎞⎟⎠

:=T0 7:=

φd