some considerations and back-of-the- envelope computations on the multipole correctors

Some considerationsand

back-of-the-envelope computations on the

multipole correctors

Giovanni Volpini, CERN, 7 March 2013

Goal: to perform a basic magnet design, without FEM computations.

Assumptions made- 2D- iron with ∞ permeability- all orders (n)

Input data- Bore diameter ( 2 r0) [mm]- Int strenght at reference radius = 50 mm [T·m]- Pole field at aperture radius (r0) [T]- Pole azimuth width fraction [-] - Operating current [A]- SC wire diameter [mm] - Winding radial thickness [mm]- …some other stuff…

Basic magnet design


The Ampère’s law on the circuit OPRSQO gives the ampere·turns required to achieve a given B0 on the iron pole tips (points P and Q).

The flux in the iron depends on the extension of the pole, which in the figure goes from P to W. The integral can be solved exactly.

The minimum iron width (from T to U) must cope with this flux without saturating.

r0

O

P

Q

R

S

Magnet geometry & field

r1

T UW

The iron maximum radius can be found summing r1 + WT (coil heigth) + TU (iron width)

A larger pole gives better field quality but will saturate at W, and will increase the overall iron size, because of the larger flux.


Note: the 4-pole is used as an example. Formulae hold for any n≥2

2 1 1 2

2

1

1

2

r0

r1θ1O

P Q

R

S

The iron pole in the first quadrant extends from Q to R.

Energy in the first quadrant has been integrated in the area OPQRSO

The contribution from the outer regions is large:

If we consider an angle ROQ of we have (all quadrants) B0 = |B(r0)|

𝐸=𝐵0

2

2𝜇0

𝜋𝑟 02

𝑛

𝐸=𝐵0

2

2𝜇0

4𝑟02

𝑛½π-2θ1

Energy stored in the magnet

The contribution from the outer regions is large: if we restrict to the circle of radius r0 we have, for any order n


This large contribution to energy from then region outside the free bore is not surprising, since the field grows with increasing radius.

An explicit form for the energy exists for any value of the angle ROQ, which goes like 1/n.

If we introduce a form factor f(θ1) the energy can be expressed as

𝐸= 𝑓 (𝜃1)𝐵0

2

2𝜇0

𝑟 02

𝑛

Energy stored in the magnet

- in the case of a circle of radius r0, f = π, - for the whole area shown before, assuming reasonable values for θ1 f ~ 4-5

MISSING (so far)- the energy in the coils;- an attempt to estimate the energy in the iron, assuming µ<∞


SC wire

NbTi is the reference solution;

MgB2 solution is being considered in parallel for its potential in terms of temperature margin. Contacts (not contracts…) with Columbus Superconductors are in progress. The main issue is the minimum bending radius, which in the products manufactured so far is 80 mm or larger. We are planning measurements to verify whether bending radii suitable for our designs can be reached.


SC wireNbTi Jc 9,000 A/mm2 @ 2T 1.9 K ( Ic = 609 A )

this corresponds to 2,700 A/mm2 @ 5T 4.2 K d = 0.5 mmα = 1.9any other requirement?

2 4 6 8 1 0F i e l d T

2 0 0

4 0 0

6 0 0

8 0 0

C r i t i c a l C u r r e n t A

What is the definition of «% of the load line?»


Int s

tren

gth

@ 4

0 m

m

Nam

e

Orie

ntati

on

Ord

er

Aper

ture

Int s

tren

ght a

t rad

ius

= 50

mm

Pole

fiel

d at

ape

rtur

e ra

dius

(r0)

Iper

grad

ient

Int i

perg

radi

ent

Mag

netic

leng

th

Pole

azi

mut

hal w

idth

fr

actio

n

θ1 r1 Peak

Pol

e fie

ld

[Tm] [-] [mm] [Tm] [T] [T/m^(n-1)] [T/m^(n-2)] [m] [-] [rad] [mm] [T]0,055 Ciemat-hexa 3 140 0,085938 1,25 255,1 34,4 0,135 50,0% 0,26 78,57 1,570,035 Ciemat-octo 4 140 0,068359 1,40 4081,6 546,9 0,134 50,0% 0,20 76,34 1,82

Ciemat-2-quad 2 150 0,997 1,85 24,7 19,9 0,808 60,0% 0,31 97,83 2,20Ciemat-2-hexa 3 150 0,060 0,99 176,0 24,0 0,136 50,0% 0,26 84,18 1,25Ciemat-2-octo 4 150 0,040 0,97 2299,3 320,0 0,139 50,0% 0,20 81,79 1,26Ciemat-2-deca 5 150 0,040 1,52 48039,5 6400,0 0,133 50,0% 0,16 80,38 2,01Ciemat-2-dodeca-n 6 150 0,119 1,57 661596,7 380800,0 0,576 50,0% 0,13 79,46 2,10Ciemat-2-dodeca-s 6 150 0,020 1,05 442469,1 64000,0 0,145 50,0% 0,13 79,46 1,40

Nam

e

Orie

ntati

on

Ord

er

Pole

azi

mut

hal l

engt

h

No

of la

min

ation

s

Ampe

re-t

urns

Ope

ratin

g cu

rren

t

n of

turn

s

Wire

dia

met

er

Wire

"M

IITs"

Win

ding

cro

ss se

ction

(1

pole

)

R2 Win

ding

radi

al h

eigh

t

Win

ding

thic

knes

s

Out

er ra

dius

[-] [mm] [-] [A·turns] [A] [-] [mm] [A²·s] [mm²] [mm] [mm] [mm] [mm]Ciemat-hexa 3 46,67 33,7 23210,1 100 232,1 0,5 2284,0 72,4 78,57 15 4,82 116,9Ciemat-octo 4 35,00 33,5 19496,5 100 195,0 0,5 2284,0 60,8 76,34 15 4,05 108,8Ciemat-2-quad 2 103,23 202,1 55206,9 103 536,0 0,7 8774,2 327,5 97,83 20 16,38 169,4Ciemat-2-hexa 3 50,00 34,1 19695,4 105 187,6 0,5 2284,0 58,5 84,18 15 3,90 124,2Ciemat-2-octo 4 37,50 34,8 14473,2 105 137,8 0,5 2284,0 43,0 81,79 15 2,86 115,5Ciemat-2-deca 5 30,00 33,3 18143,7 131 138,5 0,5 2284,0 43,2 80,38 15 2,88 110,4Ciemat-2-dodeca-n 6 25,00 143,9 15617,1 135 115,7 0,5 2284,0 36,1 79,46 15 2,40 107,0Ciemat-2-dodeca-s 6 25,00 36,2 10444,5 84,8 123,2 0,5 2284,0 38,4 79,46 15 2,56 107,0


Nam

e

Orie

ntati

on

Ord

er

SC w

ire

leng

th o

f wire

war

med

at

300

K

Mag

net W

indi

ng

Resi

stan

ce @

RT

[-] [m³] [kg] [m] [Ω]Ciemat-hexa 3 0,004 29,2 579,5 7,2 81,1Ciemat-octo 4 0,003 23,0 589,5 6,8 82,5Ciemat-2-quad 2 0,059 461,6 4161,5 225,4 297,1Ciemat-2-hexa 3 0,004 33,0 483,7 5,3 67,7Ciemat-2-octo 4 0,003 26,6 436,9 3,9 61,1Ciemat-2-deca 5 0,003 21,6 499,6 7,3 69,9Ciemat-2-dodeca-n 6 0,011 82,8 1707,0 28,0 238,9Ciemat-2-dodeca-s 6 0,003 20,8 543,6 3,1 76,1

Iron

Nam

e

Orie

ntati

on

Ord

er

Bore

form

fact

or

Stor

ed e

nerg

y

TOTA

L

Tota

l Sto

red

ener

gy/u

nit l

engt

h

Indu

ctan

ce

Inud

ctan

ce/u

nity

leng

th

Dum

p re

sist

or

Peak

vol

tage

at

mag

net's

end

s

Free

dis

char

ge ti

me

cons

tant

"MIIT

s"

[-] [-] [J] [J] [J/m] [H] [H/m] [Ω] [V] [s] [A²·s]Ciemat-hexa 3 4,00 948,0 948,0 7035,2 0,190 1,407 0,500 50,0 0,379 1896,0Ciemat-octo 4 4,00 886,8 886,8 6618,7 0,177 1,324 0,300 30,0 0,591 2956,0Ciemat-2-quad 2 5,51 29523,7 29523,7 36522,1 5,566 6,885 2,000 206,0 2,783 14761,8Ciemat-2-hexa 3 4,00 690,8 690,8 5065,9 0,125 0,919 0,300 31,5 0,418 2302,7Ciemat-2-octo 4 4,00 507,6 507,6 3647,4 0,092 0,662 0,300 31,5 0,307 1692,1Ciemat-2-deca 5 4,00 954,6 954,6 7165,1 0,111 0,835 0,300 39,3 0,371 3181,9Ciemat-2-dodeca-n 6 4,00 3666,5 3666,5 6370,2 0,402 0,699 0,333 45,0 1,208 11010,6Ciemat-2-dodeca-s 6 4,00 412,1 412,1 2849,3 0,115 0,792 0,300 25,4 0,382 1373,7


saturationLASA CIEMAT LASA CIEMAT CIEMAT

Ciemat-2-quad 535,99 738,00 0,73 5,57 6,13 0,91 0,81Ciemat-2-hexa 187,58 228,00 0,82 0,13 0,17 0,75 0,96Ciemat-2-octo 137,84 165,00 0,84 0,09 0,09 0,99 0,98Ciemat-2-deca 138,50 198,00 0,70 0,11 0,16 0,68 0,81Ciemat-2-dodeca-n 115,68 185,00 0,63 0,40 0,52 0,78 0,81Ciemat-2-dodeca-s 123,17 165,00 0,75 0,11 0,11 1,03 0,96

Inductanceno turns

Comparison


a few issues

-Operating current: pro’s and con’s-Field quality requirements, especially for the 4-pole-Pole extension-Maximum voltage permissible (up to 50 V the design is «low voltage», if we exceed this value we can go, to, e.g. 300 V?)- Protection: must be confirmed it is OK in some cases


the end


some considerations and back-of-the- envelope computations on the multipole correctors

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