CERN/SI/Int. DL/70-10 7.12.1970

BASIC EXPllliSSIONS POH EVALUATING IRON CORE MAGNETS

A POSSIBLE PROCEDURE TO MINIMIZE THEIR COST

G. Brianti and M. Gabriel

l. Description of the magnet 1.1 Dimensions and geometrical parameters 1.2 Other parameters

2. Active power

3. Volume of conductor

4. Volume of yoke 4.1 Volume of yoke for magnets with two return yokes 4.2 Volume of yoke for C magnet with pole 4.3 Volume of yoke for C magnet of window-frame type

5. Stored energy 5.1 Stored energy for magnets with two return yokes 5.2 Stored energy for magnets with one return yoke

5.2 a) Window-frame type 5.2 b) Pole type

6. Possible values for the coefficients c 1 and Ki

7, A possible optimization procedure 7.1 M - cost of power supply and associated equipment 7.2 M

1 -cost of finished coil

7.3 M~- cost of finished yoke 7,4 M

4- cost of a.c. and d.c. power distribution

7.5 M5

- cost of cooling 7.6 M cost of electricity 7.7 cSst normalization 7.8 Example of application

7.8.1 Cost coefficients ?.8.2 Results of computations and a possible choice

8. Conclusions

Acknowledgements

References

-

- l -

It is often useful to estimate the power and the volume

of a magnet prior to its actual detailed design. Expressions

for these quantities as function of the characteristics of the

beam and of the minimum possible number of magnet parameters may be

helpful both for :

a) evaluating the implications and the cost of a beam;

b) directing the actual magnet design to minimize the

investment cost or the investment plus running cost for

a given time.

In what follows, these expressions are given together with

an actual example. A possible procedure for minimizing the overall

cost is indicated, it makes use of both the magnetic length L

(or gap field B) and the current density j as main parameters.

1. Description of the magnet

A magnet giving a uniform field in the be= reg~o::c. is

assumed throughout the report (bending magnec). All the types

of magnet most commonly used are considered, as sho·.m in Fig. :, c..

and b), and expressions for both d.c. a.'ld pulsed :na.gnets are given.

It will be seen that, in most cases, it is possible to wori:

out general expressions which formally are the same for the various

types of magnets. Of course the values of certain coefficients

change from one type of magnet to another.

symbols is used hereafter.

The following list of

l.l Dimensions and geometrical parameters

a) Equivalent magnetic length = L

b) Average length of conductor in beam direction = K1L

c) Length of iron yoke = Ly= c1L

Remark

- 2 -

d) Deflecting angle

e) Maximum required deflecting angle

f) Maximum beam dimension (horizontal)

g) Radial aperture ( equal to t~e

sagitta plus the beam diameter and margin)

h) Radial extension of coil

i) Vertical extension of finished coil

j) Vertical extension of pole

k) Gap height

1) Dimension of return yoke

m) Average length of conduc<:or due to "~~c

n) Radial width of conductor ir-c~uding

insulation

o) Height of conduccor including ir.z'"'lation

p) Effective conductor cross sec<:ion (taking

into account insu~ation and cooling hole)

q) Additional radial aperture for field

inhomogeneity and vac~ur:-,. cha:n.ber

r) Aspect ratio of the coil

= 8

= w a

= w c

= d = K7

h

= e ~ K8

h

= h

= a

= K3Wc

= cw

= ch

= K6CWCh

= K h 9

KlO K~,h

= = 7'-c It must be noted that in principle it is not correct to

assume the lengths under b) and c) above as si~ply

proportional to L. In fact both

L through expressions of the form

is shown in section 6. a) and b).

and c1

depend upon

or c1 = A ± ~' as it

1.2 Other parameters

a) Particle momentum

b) Number of turn

= p

= N

- 3 -

c) Resistivity of conductor material (at working temperature)

d) Nominal field at center of gap (d.c. or

e) Current required for B (d.c. or peale)

f) Effective current density (d. c. or peale)

g) Saturation coefficient (where .!II is the

fraction of the current needed for the

ampere-turns in the iron) (d.c. or peale)

2. Active po~er

= Po

peale) = B

= I

= j

For all types of magnets, the resisoance of the coil is

given by

One can replace most of the !:lagne·o pararne-;;ers of (1) by

beam parameters in ~he following way :

(l)

W = (sagitta + beam diame"Cer) - margin for ::::.eld inhomogenei t.;· a max

+ K9

t

N NI Bh I=~= ~-'o

Incorporating (2), (3) and (4) into (1) and from P = RI2 ,

one has finally the expression for the d.c. or the peale power

( 2)

( 3)

(4)

- 4 -

2p P = __ c e

'h (K + _!!!\ ll + J s l 8/ 0.3 (I. term)

~0

2 Pc r +- jh b

~o m \ _P~ ( + K h s + II. term)

9 / 0. 3L

(III. term)

The expression (5) is valid for all types of magnets, but

of course the coefficients Xi vary within different ranges for

various types.

(see Fig. lb)).

In particular K7

= 1 for window-frame magnets

... v.ne

The I. te= is concerned ·,vi th the power required for the

part of the coil parallel to the be~ and for the part of the end

connections correspondi~~ to the sagitta, the II. ter~ to the par~

of the end connections correspondir~ ~o the bean diameter ar.d to ~te

margin and finally '!:he ::r. ~er:~ -co "''e 'Jar-;; of the e!ld connections

corresponding to t~e coil radial ex~ension.

It is interestiY'_g to no1;e that in case of a long narrow

magnet not satura~ed (s = l), (5) tends to to the very simple for=

p 2 Pc

jh (, -'- ":r.' DA ::::-- ~

~0 '"¥1 3/ 0.3

which can be considered as the minimum power required to deviate by

an angle 9 a beam and momentw" p.

Introducing the numerical values for ~ and for o of o ·c copper at 30°C and usual units for the other parameters one has :

(5)

(6)

- 5 -

P[KWJ = 29.44 {~2] h [mJ s (K1 + :m) J.93

[T.m] +

K + 23.43 ~ h [mJ

6 7

The relation (7) shows that, in order to obtain an

approximate evaluation of the power required to deviate a beam of

diameter bm and momentum p by an angle em' it is necessary to fix

only j and h and have a knowledge of the coefficients K. from ~

previous design and/or from some other considerations (see section 5).

To complete the electrical parameters, the expressions

for the voltage E and the current I are :

E = 2p NjL(K + em~+ 2p Nj(b + c l 8_,; c m

=~ l ......E.L I N h s 0.3L

i-1o

(8)

(9)

Expressions (8) and (9) contain in an important way actual

magnet parameters like N and L and therefore are of less immediate

use than (5).

Introducing the usual numerical values and practical

units, one has :

- 6 -

e + _2!!.\ +

8

I [AJ -- 0.796 ~ h [mJ s o:~L [TJ x 106

In case of a pulsed magnet, the active power P can be

expressed by :

The peak power P and the peak current I are given

respectively by (5) and (9). As to the voltage E, the expression

(10) gives only the peak resistive part. The inductive part will

be treated in section 5.

3. Volume of conductor

For all types of magnets, the volume of conductor is

(10)

(11)

write

one has

- 7 -

By using the relations (2), (3) and (4), we can also

e V = .1... 1;. hs(K + .2!!.\ ..J/JL +

c ~ J . 1 8/ 0.3 0

Introducing the numerical values and practical units,

1 ( 6m\ .J&_ [ J +

1.59 -j""['""mm"'A""'2

]"'" h [m] s K1 + ""8> 0 _3 T.m

+ 1.59 ~~ h [m] J-

mm2

s (b + K h-, [ml ...J?.L [Tl + m 9 . - 0. 3L -

1 ...:l + 1.27 13]l K K h [m]

j2 ...L 6 7 4

mm .J

(12)

- 8 -

4. Volume of yoke

The dimension a of the yoke (see Fig. 1) can be expressed

in terms of W and W by means of a c

a = K fw + w' 4 a c,

(\1 w \if

a = K4 + c a 2lC +1,..

"'

w r\'1

~

and K4 c a = + 2(2K 3 +1)~ - a

{}

for windor1-fre.me magnets with one or two return yokes

for pole magnets with two return yokes

for pole magnets with one return yoke

The different types of ~agnets of Fig. l require slightly

different expressions for the yoke vol~e.

* The (2K8

+1) does not correspond to e. rigorous treatment of the flux

passing through the coil, but it is simple approximation which 'N

improve3 with increasing c. h

4.1 Volume of yoke for magne-cs witb. two return yokes

(14)

(15)

This expression is valid for both the window-frame

magnets (K8 = 0) and for the pole magnets (K8 ~ 0).

- 9 -

4.2 Volume of yoke for C magnet with pole

+C1

LK4(2K

8+l\,/ (wa + We :'\ + 2C LK h W P 4K8+2/ 1 s a

4.3 Volume of yoke for C magnet of window-frame type

By using the usual expressions(2), (3) and (4), it is

possible to work out general expressions containing the beam

parameters.

nowever, such expressions are rather long and are not

given here for simplicity. It is easy in any given case to replace

the magnet parameters by actual values calculated by means of (2),

( 3) and ( 4).

5. Stored energy

In the case of pulsed magnets one has also to evaluate

the stored energy \VB and its consequences, the inductance LB and the

time constant T·

WB = ~ fv BHdv

(16)

(17)

- 10 -

LB 2 WB

= 12 (18)

2 WB ,- = p

Where v is all the volume in whicl1 B and H are not nil. This volume

inclu.des both the air volume of the aperture L'4'1d the yoke itself. The

energy stored in the latter is usually rather small (- 0 for ~ ~ ro)

compared to the one stored in the air, so that a rather rough appro­

ximation is sufficient for the purpose of this report. The following

expressions apply to the various types of magnets.

5.1 Stored energy for magnets with two return yokes

1

2~ 0

K h\ (pe \2 + 9 ) \!}.3)

(19)

where the last term represents the energy stored in the yoke and

for window-frame magnets. L. is the average flux line in the ~

yoke and is given by :

+ 2 w c

where, as usual, K8 = 0 for window-frame magnets.

5.2 Stored energy for magnets with one return yoke

+ _1_ h r;; +K j;\ 2~ L ~m 9'/

0

( .£L_.}2 + \jl.y

(20)

b ';· p 1 t _s_~ __ :z::£~

- 11 -

W = - 1- h ....!!!.fP~2\ + ..1.._ .£ ft +K h\ .£L + e 2 a 02 B 2~ 8 ~ 2~ L ~m 9~ 0.3

0 0

+ 2W c

(23)

(24)

II.B. The choice of j for a pulsed magnet is not only related to cost

considerations (see par. 7), but also to an appropriate value

for the time constant T· In fact j can be high8r, the magnet

itself smaller and T also smaller than for a d.c. magnet.

6· Possible values for the coefficients c1

and Ki

a) c1 , ratio of the core length to the magnetic length, is

obtained with good approximation by means of

magnetic length ~ core length + gap

- 12 -

b) K1

, ratio of the average conductor length in the beam

direction to the magnetic length, is in general rather

c)

d)

close to c1

• From existing designs, one can assume :

K~, ratio of the average conductor lensth duo to IV toW ~ c c

itself, depends on the eoil design .. .For "flat" coils

(generally possible in pole "'"gnets) it tends to be smaller

than for coils ·.vi th 11 s2ddle-s11aped 11 ends (windo·.v-i'rarne r::.e.~:::.e-::=

and pole magnets rri tL t''e coil ·:rinaovr couplet ely filled).

One can assume the followir~ :

K4

, relates

where K3

is

K3

= 2 + 0.2 for "flat" ends

K3

= 3 ~· 0.8 for ''sadale-shaped 11 ends

a to W and \'1 througL the relations (14), a c defined ur:der ,g) below and is nil for ·.vindow-

frame m~g:1e ts.

The conservation of f2~x in th8 magnetic circuit requires

that

K4 0.5 B --

B. l

nagnets with two return yokes

K4 B

- B. l

magnets with one return yoke

where B. is the average field in the mediru1 plane of the l

return yoke(s).

- 13 -

e) K6 , ratio of the effective copper surface to the overall

cross-sectional surface of the coil. It depends on the

voltage to be applied to the magnet and, in existing

CERN designs, varies between 0.5 and 0.7.

It is a rather important coefficient to be watched since

a too great conservatism in the insulation thickness

(small K6~ leads either to a too high current density and

power consumption or to a bulkier magnet, especially in

the case of window-frame magnets.

In view of this it is suggested to adopt

f) K7

, ratio of the total vertical extension of the coil to

the gap, depends on the type of magnet (pole or window­

frame). One has typically :

for window-frame magnets

K7

~ 2K8

for pole magnets in which the coil is split in

two parts (upper m,d lower pancake) and the

end connections are 11 flat 11 •

for pole magnets with "saddle-shaped" ends

g) K8 , ratio of the pole height to the gap, is obviously

nil in case of window-frame magnets.

Instead of fixing a value for K8 , it is more convenient

for the optimizRtion procedure which follows to fix a value

- 14 -

which in turn gives K7

and K8 (through relations such as

K7

= 2K8 and K7

= 2K8 + l of f) above). In this case K7

and

K8

vary with the para;:ceters j, s and B and depend on the

value of K6 .

h) K9

, ratio between the total margj_n included in the horizontal

aperture (for field inhomogeneity and vacuum chamber) to the

gap, depends on the type of magnet, on the required field

accuracy and on the gap icself.

field accuracy of 10-3

One has generally for a

K9 -- 0.5 l.O for window-fra..11e

K9 = 1.0 .:. 1.5 for pole m.agne'ts

The smaller value is valid for relatively 1 h ~ 0 10 ) - -1 ... ~ ..... l l a'<;l ~ /. ::: ~ e::.. • m , ·.vnl e ror ...,.~.3-.J.. ..... er 0 c..p.:;, \n

K9 is more appropriate.

7. A possible optimi~ation procedure

oagne-:;s

large gaps

0.05 ::.) the

The basic specifications required for initiating the

actual design of a magnet are :

a)

b)

c)

d)

Bending power

Maximum bending angle

(irrespective of momentum)

_Vertical gap

Maximum beam dimension

(in the direction JL to B)

BL

h

b m

[T.ml

[radian J

[mJ

[mJ

higher

e)

f)

- 15 -

Field accuracy in the beam

region

Foreseeable utilization time

liB B

0

T [hour J

If free from limitations in cross-section, length, weight,

etc., the magnet designer can then set up an optimization procedure

With the aim of finding a magnet system (magnet itself, power supply

and distribution, and cool:'.ng) of miniuum cost, taking also into

account the running expe:.diture.

The overall cost of a ::-.agne·;; sys"';e::l can be sul>divided

then into equipment cost and running co2: .. 7hese t·.vo costs ~re rather

different in nature sinca tha former is a capital investe::~ent to be

paid in & relatiyely shor·t time (ore, or t·::o years), while the latter

is spread over the entire working perie>d a:· ,,.,e equipment and, at

the Boment of forecast, :nay contair. some uncer-:;ainty with regard to

the actual utilization time T and other factors. The equipment cost

Me can be expressed by

M - M + M + l\ + '! + '·' e 1 2 '3 ·· 4 ·"5 (25)

where Ml = cost of po·uer supp2.y and associated equipr::ent

M2 = cost of finished coil :1ounted on the yoke

113 = cost of finis:1ed yo:ce

M4 = cost of a. c. and d.c .. power distribution

M5 = cost of cooling

The running cost is given essentially by :

M6

= cost of electricity for the power supply,

the a.c. and d.c. distribution and the

cooling.

By restricting ourselves to the case of d.c. magnets, the v~rious

M1

can be expressed as follows.

- 16 -

7 .1 M1 - cost of power supply and associated equipL1ent

If the power 3upply feeds only the magnet i.n question,

the cost can be expressed by :

(26)

where H01 = cost independent froe1 P (in a certain power range).

It includeo the necl1anicc.l structure, t~:.e basic

cubicle equipment, the electronics, t!:e re::J.ote con-

wri"te

troJ s, etc.

m1

= cost/kw

ai) = multiplication fac"';or in units of [~;~-l of tl:.e

magnet power?, to obtain ~he apparent po~er on

the e .• c. side.

If the power supply feeds .. ~ :: . .::.;~e v2 in serie 2, :::.:.e ca...l'l

' ''. +m..<a?

.L '- n

M01 , m1

and ~depend on the type of rec~ification used :s or 12

phe.se), on the precision ·.van ted a.nd en "tile power ra....~.ge. rt depends

in addition on t!-.e distance between power supply and mR.gnet and

(slightly) on 'he ntmber of magnets fed.

7.2

where

hl - cosc of finished coil 2

For sake of simplicity, this cost can be expressed by

= cost/m3 of conductor in finished coil

(27)

(28)

- 17 -

m2 depends on the insulation technique adopted, on the form of the

coil, e.g. with "flat" or "saddle-shaped" ends, and on the number of

coils to be made.

7.3 M3

-cost of finished yoke

As for M2

, this cost can be expressed by

(29)

where : m3

- cost/m3 of iron in finished yoke.

m3

depends on the precision required, on the type of core (solid or

laminated), on the type of steel and on the number of cores to be

made.

7.4 114

- cost of a.c. and d.c. power distribution

This cost 2 ) includes all equipment and installations

related to the a.c. and d.c. distribution, such as cables, substation,

etc. It can be expressed for simplicity by

(30)

where : m4

~ cost/kw

m4

depends on the layout adopted (power supp1ies concentrated in

one location or installed near to the magnets), on the flexibility

required (power reserve in the cabling) and on the current level.

7.5 M5

-cost of cooling

This cost 3) includes the central cooling installations

and the distribution system. Again for simplicity it can be expressed

by :

- 18 -

M5 = m5 P

where : m5

= cost/kw

m5 depends on the type and total capacity of the plant and on the

layout of the experimental area.

7.6 M6 - cost of electricity

( 31)

It includes the cost of electricity for running the power

supply (including a.c. and d.c. distribution systems) and the cooling.

ct can be expressed by :

(32)

where m6 - cost/kwh of the electricity

S = correction factor for the magnet power P 1 which takes

into account the power cost in the rectifier itself

and in the a.c. and d.c. distribution systems, the

power necessary to run the cooling system, and a

posstble reduction factor since the magnet may not

be excited always at full power.

It is important to note that the cost M6

is rather different in nature

from M • While the latter is a firm investment cost to be paid in a e relatively short time (one or two years), the former is spread over

many years (e.g. ten) and it has not the same degree of certainty

(the time T and the S are somewhat uncertain). In order to compare

it toM the minimum that should be done is to capitalize it to the e moment of the investment by applying a suitable interest factor

which could also be included in S.

7.7 Cost normalization

It appears rather difficult to establish a precise set of

values for most of the costs Mi. In fact, besides the implications

- 19 -

of a particular design, one has to take into account the cost

variation from year to year and the fluctuations of the economy

(expansion, recession) which have been recently even more important

than the cost variation itself, calculated according to the usual

formulas. As far as the equipment cost !II is concerned it seems e

reasona.ble to adopt a normalization which gives the correct relations

between the various Mi but leaves undetermined the absolute level

in actual money.

By doing so, it is obvious that the results of the opti­

mization procedure to minimize the equipment cost remain valid until

a modification occurs in the relative cost of the various items.

If one wants however to determine the actual cost in money, the

value of "unity" in the normalization has to be assessed following

the economic conditions, price variation formulas, etc. For the

running cost, since the cost of electricity in a given laboratory

is rather well known over a number of years, its normalization

implies that "unity'' has to be specifically stated. This will

be done as an example in 7.8. Of course the optimization on the basis

of an essentially variable cost, such as M , and of a more stable e cost, such M6 , may require more frequent revisions than for Me alone.

7.8 Example of application

To illustrate a possible application, the following case

is treated numerically for a H magnet :

.E.L = 5.0 Tm e = 0.05 rad h - 0.05 m 0.3 m

b = 0.0(-) m f::,B

-· l0-3 T -- 30.000 hours m B

0

Most of the coefficients K. are chosen according to l

current pratice as follows :

- 20 -

for 11flat 11 coil connections

for "saddle-shaped" coil connections

s and K4

, which depend on Bi' are taken to vary with B following

a parabolic law.

this corresponds to :

B 0.8 for B l T; B. = -· ~

B 1.2 for B 2 T. B. - =

l

For the saturation parameter s, we take

s = 1 for B ~ 1.25 T

s = for 1.25 < B < 2 T

which gives s = 1 for B - 1.25 T

and s - 1.2 forB= 2 T.

According to 6 f) and g), we have taken

which in turn gives

= K7 2

for "flat" connections

- 21 -

__ K7-l K

8 2 for •saddle-shaped'' connections

7.8.1 Cost coefficients

For the equipment cost M , the following normalized e

coefficients are used :

1.35 for n = l

i) MOl -- O.'JO ml - 0.001 C{ =

1.1'7 for n = 4

which gives

= 0.50 + 0.00135 P [kwl for n = l

= 0.125 + 0.00117 P~wJ for n- 4

ii) M2 = 4 V 0 ~3]

iii) M3 - 0.4 V [m~ y

iv) M4 ·- 0.003 p [!r~

v) M5 =o.oo2PH

Por the running cost, two values have been used

-7 = 3 .10 TP !}:wl~

and

(32)

The former corresponds closely to the present price paid

by CERN and ~ = 0.5.

N.B. This set of normalized costs is preliminary, it takes into

- 22 -

account only current techniques used at CERN, and is based on the

result of the most recent tenders, which are all of the medium size

type (total weight s 1000 tons). These tenders show a considerable

increase with respect to large contracts of the years 1967 and 1968,

beyond what can be calculated from price variation formulas. To

translate the above normalized coefficients to real prices on the

basis of the above tenders, "unity" in the normalization oould

be as high as 105 Sw. Fr., for an order of only few hundred tons and

including new tooling.

sity j and

All computations were

the yoke length L (or y

performed taking the current den­

the magnetic field B) as main A parameters. 'l'he former was made to vary between 2. 5 and 10.0 --2 ,

while for the latter values between 2.5 m (=:~ 2 T) and 5 m (=:~ T)

were considered. Figs. 2 through 10 concern a magnet with "flat"

end connections fed by its own power supply (not feeding any other

magnet).

Fig. 2 gives the power P as function of j for various yoke

lengths. It is seen that the major variation occurs with j,

but a certain variation also occurs with J, (or J3) in the y

sense that the smallec't B in the range considered gives the

smallest P.

Fig. 3 gives the copper volume V as function of j for various c

yoke lengths. Same remark as for Fig. 2.

- ~,ig. 4 gives the yoke volume v as function of j for various y

yoke lengths. It is seen that the decrease of vy with j is

much less pronounced than for v c and that there is a value of

L for which V is a partial minimu1u y y

irrespective A

of j. This

is illustrated by Fig. 5, for j = 4.5 --2 • mm

- ]'ig. 6 gives the equipment cost M as function of j and for e

- 23 -

various yoke lengths. It is seen that there is a rather flat

minimum with respect to j for j = 7 -Az, but there is also a

minimum with respect to L for L = ~ (B "'1.25 T). The latter y y minimum is also rather flat, with the minimum cost increasing

by only ~ 3 % going from I· - 4 m (B ":! 1.25 T) to y min L - 3m (B ":! 1,64 T). y

- l•'ig. 7 gives the total cost M (inclusive of the electricity

cons1Unption for 30.000 hours) as function of j for various

yoke lengths Ly. Here the minimum with j is obviously more

pronounced than for the equipment cost alone. The minimum with

respect to L occurs also at L - 4 m (B ~ 1.25 T), but the y y

difference is rather minor in the range 3.5 m < Ly < 5m, namely

for ~ l T < B < ~ 1.4 T.

- Fig. 8 gives the breakdown of the total cost M as function of

j for a yoke length L -- 4m. y

-Fig. 9 gives the minimal total and equipement costs as function

of h for b = 0.08 m kept constant. m

Fig. 10 gives the same costs as ]'ig. 8 for a case in which

both h and b vary. m

Fig. ll gives the total cost M as function of j for variouo

yoke lengths r, , when the magnet is fed in series with three y

other equal magnets. The part M01 of the cost of the power

supply, which is independent of the power, is then shared

between four magnets. The minimal cost (j = 4-.5 A2

and

Ly ~ 4 m) decreases by ~ 22 % with respect to thiFcase of Fig.7.

- Fig. 12 gives the total cost M as function of j for various

yoke lengths L when using a cost per unit of electricity equal y to half that of Fig. 7 (this may be the case for certain Ame-

rican laboratories).

Fig. 13 gives the total cost M as function of j for various

yoke lengths L , when the window of the magnet is completely y

filled with the coil, requiring "saddle-shaped" end connections.

It is seen that the minimal cost is slightly increased compared

- 24 -

to Fig. 7, if one assumes that the cost/m3 of these coils is

the same than :for coils with "flat" end connections. Since

this might not be completely true, it is probably fair to

say that the cost difference between the two cases may be

slightly larger than shown.

The above results suggest the following considerations

and guide-lines.

'Phe equipment

for j ';i 7 A2

, with

the range ~- 9 A2

j in the above r~e

of electricity

cost M (Fig.6) shows a rather flat minimum e

only minor variations for values of j in

. This gives the opportunity of choosing

taking also into account the consumption

= 5 A2

would lead

equipment cost

(Fig. 7). A value j

M8

and total cost M very~lose to their respective

mj.nimao

As to the yoke length

the m~n~mum occurs for L = y

L = 3 m (B ~ 1.64 T) leads y in the equipment ~ost M . A

e total cost is considered.

8. Conclusions

L (or field B), it ic seen that y 4 m (B ~ 1.23 T), but an

to an increase of only few percent

similar argument applies if the

This report should be considered only as an encouragement

to magnet system designers to take into account overall economical

considerations. It does not replace in any way the detail design

of a magnet, nor does it pretend to be accurate enough for certain

coefficients and for the price formulas.

are welcome.

Comments and criticism

In what follows, we attempt nevertheless to outline some

tentative conclusions, based on various expressions throughout the

report and on the example of 7.8, of course only from an economical

view-point.

- 25 -

1'he power of a magnet syetetl is very important, because it

en':iers into the overa]l cost through two major items, tl:e power supply

and it8 auxiliaries, and t!:.e r..:onsumption of electriclty. ':'his is

particularly true in CEH!I because of' ·che relatively higr, cost of

electricity. Obviously, one can give a different weight to the cost

of the equipment, to be paid in a relatively short time, and to the

cost of electricity, which initially has not the same degree of

certainty than the former and is distrib~~ed over a condiderably

longer time.

Fields outside the range 1,2 - 1,7 r appear disadvantageous,

even for the equipment cost alone; this is o:"' course even :lore true

if one includes the cost of electricity.

Certain aspects of the ~agnet desi6n deserve a special

attention, like the copper filling facto!' ;.;6

• I!l :'act the pov:er is

in firct <cpproximation inversely proportioc,al to K6

, all o-:o:Oer parameters

being equal. Altern:ltively, if j is kept co:cstant, a bigger ::agnet

results from a decrease of K6

. Relatively high values of K, are 0

obtained with a high current, a small nll!!lber of t-urns an6. a..11 insulation

thickness not overdesigned.

It is obviously advantageous to L~eed. ;·Jhenever possible sever3.l

magnets in series, since the psrt oi the _o~er supply coa~ i~de~e~de~~

from t!le power is rather important.

Finally, we draw the attention of ~he reader to the fact

that the actual methods of production of a Jagnet system ~ave an

important influence on the various coeffj_cients appearing in the

price formulas. This may be part of a separate study.

Acknowledgements

Vie wish to thank C. Iselin for some computation and many very

helpful discussions, and K. Braun, M. Georgijevic, R. Mosig and

H. Reitz for supplyinc; most of the infornation on the cos~ oo~ power

supplies and associate~ equipment.

- 26 -

References

1) R, Mosig - d.c. and a.c. power of a rectifier set -

Private communication

2) H. Reitz -Private communication

3) K. Braun - Private communication

4) G. Hartwig - Cost calculation and optimization of bending

magnets and quadrupoles -

BSG - Notiz 70/5

IEKP, Karlsruhe

"'

C><J >< " ___ __/ ···~

.c.

C><l C><J ~r. t-

a I We +- - w~ J We .. -~--- - a

··----~ ---

H Magnet with"flat" end connections of coils.

~

f----...!a!!...---4---'W!-'e'----+--·-'!.{a __ ~--+-----"W"'-e ---+--"-a --1

H Magnet with"saddle-shaped" end connections of coils.

FIG. 1a

l I

<1l I I I

··-----·----~--~------~--

C><JI "0 ><: II .c

. ------ ---------·····

I a +- We +- Wa j We +----"

I

: Window- frame magnet (with one or two return yokes).

'-----···--·· ····--·---"""'' -· ----·

C magnet.

FIG. lb

_Gnp !_·-~,_:_\iqht h.::: ~::~Orhtn Hcr1zcrr~ct LJC<1lll ~:Jirnt:T1~;ion brn !:.-= 60rnrn

_____ "F:!nt 11 co it r:n(1 cc;r-;neclions (i'or other co<-:~tficients ( l_ f'i._d_ 1')(!''('1'•1 C'lf'1'" <!po ·S'"Cl_i.('>fl '7 i'l) , •, •-. • . ,.-.! ~ , 1 , ~· .,lJ ~ "'-• <;__ ,_, " , '.- <

110 _;

Ly

2,5 100-

90 3

L

' ··:····l:-····:··~ +--------;__ i

: ... !. I T

·.:.:: ... :; ' '

10 -f--,--lt-:-:---'--------'-' ,-,--c---1 z;5 ' ! t T

J_

6 --;-----___1____;,--' __ L_• _,___:___j __ c..' - -+----:-'----- --{$>-

7 ___ , 8 9 10

l __ [ ______ f c i : _:_~ i

·:: L:: ·;: ·: ! :: · ;-. · t · · · · :. ·: :

--'-cl-~--"-· • :.Lc'

1 ·==· • Lcill_,j_c_i__:.:L..

'

I ., I

. . . . . . I . .

·- I

,. T- ---+-- -·-r j [A!rnrn2]

. T' . ····;·:·· .·:·-~-- . . , i. , , ___ :.:-J.:....::.i: .. : .. J:..:_~~;.I..:... .. ~L~:.:.J ... : . .:.:L.:.-~Ll_ _ _l_ __ ~- LFIG. 2

0.25 -

0.15

0.1

0.05

vc

1 \ .... "I

• I I

\ r

.. '. \.

.) I

' ..... )

Vclurnc q::; funciion of _j unci vcrkY,JS. yoke~ ten!]ths Ly

sarnc· pol arne' t ·::·rs os in ·fig.2

Ly

............ J ' ..... ,

.. :J I

···········- : .... L :.:.i· __ , l- .: I

l___i_.:_L _ _:_;__· _j_ __ Li.. ... L.:__' ___ • __ L...:_i__,_.J ____ -J$>-

tcL2;L1cl ;_,j~JLcl J:j • .c: fl: .31 ~- ;U ••.•• : i Li0

J[Aimm']

. --··· ~---

vv[m3]

I ····.·.[····· .· I . . . . . 1- · 0.9-~ -

i

I I I

I I

i I

: \

\ . \

'· \ : \

'

\

\~'.

\-' \~- \ .\

. \. "\ -\ ' \ '.\.··· \' '·-·

\ \ :\ \

\,, \ '

\ '· \. \.

\ \

. \

'\ ·\·· \

\

.\

'\. \~.

'\

'•

:::.(1 rriC' pc1

Ly

5

i. ' )_:

1 , __ 8_ ~· I I

I o7 I ~ .. ! __ _ I I .. O.G .

L .. ! I ' I 0.5 -I I '. i I·· i: I ! i-' ' \--·.

l I I I ! ' . I . I I · 0.2 I ~--

1 .

t \f/i~·h

Volun .,-_,

Vy[rn~ -for q qivc;-rl _current cJ2n~1ity ]:

... ""· ··. : '_· .. ~:: N· : " ' ...... ' ' -...,

... -- ........... '<...-...:.._:~~~----------·

2.3 -

2.2 -·

2.1 -

1.9 -

I 1.8 I·· I

I

1.7l­l

1.6

1.5 -

1.4 -

1.2

·- \ \

\ \ \

\\ .\ \\ \ ,, .

\ \ "', ',

of j

Ly

- 3

LJ __ • _.]__· _l_ ___ l__:_,.L_ _, ___ , __ L ---'----1----~--'______J_-----j_ ____ j _____ L_L ___ ·_· ----~ 2,5 ~ .. 4 ? f;) 7 ' 8 . ~ . 10

.......... J[Aimm7]

FIG.6

A

l I

2.'/ !~

2.6

v.

2.3 I

I

2.2 1--

1

2.1 ~-

2 -

1.9

1.8

1.7

1.6

\-·~

' \

\

\ \

\

\

\

I .. • ''); ;' '<) ,l !' ....

'•'¢-·---··--·-4. ··-' .. .-·

r· l

i t co•,;t '/i

I ."' ... ,\ ·, ('" ·-1 •. ·.• .. ·,·· ); , .•• ·.· .. • •. • •

• - .:~ ' ................ ·-... • _._._'. ------------------ ............................................... _........................ i

l

~-..-~·· . /

_.,,_ , ..

son ;c: pa

/ ' /

•") ,,:.,

/

' /

)

1.5 I L--_,' ____ .. L ____ __,_., ____ j _____ _, _____ J _____ ;__:,_L __ L_L_ • .. l._c.....L- .i ___ L_.....i....,..-_i _____ ,_--l'!>-

2.5 3 4 6 ) i $ 9 10 •

:FIG.7

. . . -. i . . .

! ... . 1 .. CL.C.C.C.L .... LI .•

I . .... , __ ,_L_ . L..c .• !. ..

1-,\l\•) \ ;, •

1 -

08

0.7

OG -·

05

0.2

0.1

[·,1 i

( 'l (·< ~- i" e\ ,--~ t I. 7 c:~ rJ' Ll' .. ll ',\ 0 l 'J ···~-t ··-~'-· '-.

. \

\ \

\ \ ' \

\

\ \

' '·

)

' -\

·!·· ,...L _ _,__L__,__

25 4 :J.

net ion oi-

3ct rnl} po. ro

. ·l j_ _ _i_. _i_ __ i__• _L___;_'---J_ _ _,__ J __ __, __ j_ _____ ·-~·

5 6 .... 7 6 ... -~ 10

i[A!mrn 2] I_~_::J:.:J: __ [jj:JJ_:LI_:" ••.. IJ••. I_I __ •• ~·,G .• 8 . ' . . ' . i. -l . .

-· :~_..:__:.:..: ____ ~L ..:...·.:~ .• ;_! __ :,_.:._;__:.:J.:...:...:_~ .•. :J.:..__:~.;...:.__:_.:J_:_ __ :._:.~--~--- _____ .:...L ----. --· •

I I I

'1 r I .T I

I I

2 -

. h-~ in i t h

Huritontnl l"rt dirt1t!n:;.ion·b~11 ::::b{)r-;i:n (con~~:.,

for other co::.:fricic:~n c1.nd pa.t-o1-r1{::\ters Se(:> .sect

I

' .. / I

/ -----l-1.

// /

···_z I

I

//

I i'-

1 /

I

~ Mfr) I

·-.. ' -c:: • 1 j:: lLJ J-\/iJ !ill ..

Ly::4._m

... --: --/~

I /

. ·i. t •

..... , .. .... ,._. . ' '

(...;....c_.;___j___. _• ___ j__ ____j ____ j _ _:_ __ • ____ L ____ :_____] __ : __ ! .•. _· _.;[_· . --'. --'-· .-'-... _.-;.!_' .-'---:-5.0. 50 70 1

. 80 9,0 . I . ! . . [ ] · ··· , .·: .

1 1 J ~ rnm

:,· ...... -. ·-·•-- ----- r i .... i,:: .· · :_ ..... · ;1

.....•... ' ... ) I ... ,1··· I

. . . .... ; . I . , .. : . ... _ .!. . . : . - . i . , . · · ·I . . . : . . . t • • :

l. ..• LL._L. ~l"l""L" ___ L.L_L;..J ••. ]FIG .. 9 __ ;__.: .J .. "~J.~_ .. LLL .. L.i .. L

;1t)

'/b)

lZ::' c .. ond pcirornet~:r:;

3.0 -

i i

I

!. 2.0 ... i I I

I /""

. I ·; I

...... J

'/• I

// /

/

/ ./

I

/

/· /•

/ /

··/' .. /

;·· ·.I

I' ··I ' .

· .. ·.·;/·· I

;/ ... .... j ..

i i . . I

l . 7~ ! 1--~-T-- I

··/ /. · .. .

• ... /. /'

/ ... : .. /.

··; ....... , .

' ' ' . ' . . . . ' . • ' ' • t • < '

--/ ;

I

I . '/ .

~ Loll_' _j_ ~ L

I I 1.0 i

......... J.J_j ___ _j ____ __.I_. __ ·· -........Jc-~iO co 70 80

I• .Iii . ,' .so .J 65. ·. : . . I ' i. ,. :

I i' i ' ....• •.,_ ...•.. l.·.· .. •_··.······ · ... · .. ·_·l.·,_· .. ·,_·_·,·.,.·,.·.·.r, . T . IF I' G! 1 o· L .... ; .. .::~:.1/L.,:: ... L _ _ . . ... tL ... cLc .. C: 1. ' - J • _, .

80 ·. ··i . 95 110. I 1

. . . . . . i .

90 125 .

c .·_-,r ··-· ---- '···

j_::.: /)_~_) :\hnrn'·

I. y . I ! . I,.,

·tv~-:!.,.. rn

________ ; __ -----------~---{'~.~

h mm ··

' )lm[IYlm]

A

l

2.2

2.1 -

2 -

1. 9 -

1.8 -

1.7 -

1.6 ·-

1.5 -

1.4

1.3

1.2 -

rr-n<:·~ii

• \ I

_\ \

\

\ \ ..

I I I

\

:' ''\, \

• \,

\

' \

ur;i t· ,)

~ ~-,_

'<·--

., ~

'· ·- '· -..v •• ,_

.J>----~

i '-1 : ... )

•\ ,:. ·'

'----- .... .(~

_c ____ ___L _____ _, _ _j ___ ..J ____ L ____ ;__:__j ___ j ____ '----'-----L-----'-----.J. __ ,__ __ L_ ________ i>.,._

25 3 /, 6 7 . 8 9 10

i[A/rnm]

2.1 -

L9

.. 181

1.5

···\ \

r unii cf

_____ _t _____ .l 2,~) ~ ···-,·-~-- _I_ .•• J. ____ L __ j___ j . . . . 5 ' -6--"---~;-

FIG.12 i ..... ··- - L

~; Ct~:; in fie!.?

: .. ;·

l. " ")

3

i I ' l

2.7

: 21 1- .

r 20r \ \·

\ ' \

.p-

I

\ \

'•

orn cil rn(_.··r! ~·:~ion

i C ,i C' i .. t t :·:.~ __ , ···'\ :;:.c,c tior1 1. C)

FIG.13

co1·1nc:c

h:::: FJO l!''rn rrtrn

. [A 21 j e,/ mrn J

; :,,1'<··~1· :.· .. t-.