April. 1985 OCR Output

current pulse.

the pulse length of the current and the optimum working point during the

are discussed and are used to define the final parameters of the lenses.

diameter of the core and the thickness of the container wall. The results

including the influence of the variation of different parameters like the

understand the lens behaviour, the design of the core has been studied

as the temperatures and thermal and magnetic pressures. In order to

of the current and the magnetic field distribution in the lenses as well

to collect the antiprotons. In this report are described the calculations

focus the protons onto the antiproton production target and another one

As part of the ACOL project, two lithium lenses have to be built, one to

Abstract

A. Ijspeert, P. Sievers

COHPUTATIONS OF THE DESIGN PARAMETERS FOR THE ACOL LITHIUM LENSES

SCAN-0012004

IIlIIIlIIlN|ll\||||I|lNlllllllllllllNIIINIHIIWIIINIIIIIIICERN LIBRARIES, GENEVA CERN! SPS B5- 10 <ABT>

CERN - SPS DIVISION

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

0007éS

Appendix 2. The axial magnetic forces in the current leads OCR Output

Appendix 1. Analogy between the electric and the thermal diffusion

20. References

19. Summary of the results

18. The axial magnetic forces in the current leads

into the cooling medium

17. The effect of the heat transmission from the container wall

The effect theof wall thickness of the container

The effect theof radius

14. The effect of the current amplitude

13. The effect of the length of the current pulse (damped half~sine)

12. The effect of the length of the current pulse (undamped half-sine)

11. The main results for the reference case under discussion

10. The criterion for the optimum performance of the lens

The temperature dependant parameters

The steady state temperatures

The total pressures and the axial forces

The resistive heating, the energy deposition and the thermal pressures

The magnetic pressures

The magnetic field distribution

The current distribution

The method of calculation

Introduction

C O N T E N T S

.-1-_

in the following paragraph. OCR Output

the lenses, a computer program has been developed which is described

In order to determine the design parameters for the construction of

unnecessary heating by excessively long current pulses.

which is retarded by the "skin effect" and secondly to prevent

allow for sufficient radial current penetration towards the centre

Finally, the pulse shape and duration must be optimised, firstly to

in the lithium and the current leads must be carefully studied.

lithium conductor. Moreover, the magnetic pressures and forces induced

and therefore an efficient cooling system must be designed around the

Due to the high currents, the lens is submitted to strong heat pulses

secondary winding.

winding of a current transformer of which the lens forms part of the

generated by a capacitor discharge and transmitted to the primary

the order of 0.5 to 1 HA are necessary. These current pulses are

of 500 to 1000 T/m within lens diameters of 20 to 40 mm, currents of

container of the lithium. To achieve the required magnetic gradients

materials which can be applied for the lens, especially for the

with other elements. This limits strongly the choice of the engineering

account the low melting point of lithium and its chemical reactions

requires, however, careful design of the lens and one has to take into

particles, lithium, the lightest existing metal, is most suitable. It

In order to improve the transparency of these lenses for the traversing

to collect the produced antiprotons respectively.

polarity, be used to focus the protons onto the production target and

inside the rod. Such fields can, with the proper choice of the current

tial magnetic field is created which rises linearly with the radius

current is sent. with a uniform radial current density a circumferen

basically of a cylindrical lithium rod through which a high axial

devices “Lithium 1enses"will be used. Such lenses consist2'3)

target and to collect the antiprotons are envisaged. As focussing

In particular, efficient systems to focus the incoming protons on the

proton per day, among others by means of an improved antiproton source.

the antiproton production by a factor of about ten to some 10anti—12

Recently the ACOL projecthas been approved. It aims at increasingl)

1. INTRODUCTION

- 1 ....

electrical parameters is resumed in Appendix 1. OCR Output

finite element programme DOT". The analogy of thermal and

employed to evaluate the above "skin effect". We made use of the

analogue equations, computer codes solving the latter problem can be

Since also the non—stationary heat transport phenomena are based on

H _ $@.1... _....—c ar - u at where Hw(t,R) - 2wRI(t)BE (t.R) BH (t.R)

one of Haxwells equations yields

I(t) producing the field H (t,R) at the rim of radius R. Using

where the boundary condition results from the imposed current pulse

BtBr? r Brz E.EE + 3 EEE = v Y BEZ ; (Er = E¢ = 0)

this reduces to the diffusion equation

In cylindrical coordinates, however, using the cylindrical symmetry,

V E"‘Yat

distribution of the electric field is governed by the equation

fields can readily be calculated. The time behaviour of the radial

relevant parameters such as temperatures, pressures and magnetic

and thus the current density as a function of time, all other

Once we know the radial distribution of the axial electrical field

becomes one of an axisym etric one—dimensional field.

calculation of a thin disk cut out of this lens and the problem

steel container.The calculation can then be reduced to the

cylinder tightly fitted in an infinitely long thin walled stainless

As a model for the lithium lens we took an infinitely long lithium

2. THE METHOD OF CALGULATIOH

-2..

pulse. OCR Output

currents inside to vanish some time after the stop of the driving

currents near the rim. These then equalise with the still positive

the radius) and finally the decreasing current pulse causes negative

homogeneous distribution produces a magnetic field proportional to

distribution becomes more uniform over the whole cross section (a

pulse are shown in Fig.2; after the initial rise at the rim. the

The current density distributions at different instants during the

dsk =`/;:; = 5.3 mm

depth" in the lithium is for this reference case

propagates from the periphery towards the centre, and the "skin

From the start of the pulse, the current density j (r,t)

subsequent formulae.

given in Table l and the notations are the same as used in the

The parameters of the lithium and the stainless steel container are

s 2.5 mmContainer wall thickness d

= 10 mmLithium radius

: 600 psCurrent pulse duration n/w

Current pulse damping 6 ; 1750 sec ` (23.7 uH. 83 nfl)

s 500 kACurrent pulse amplitude I

following parameters have been assumedreference case" and where the

lithium lens which we will call "thetypical case of the proton

In fact, in the following paragraphs, results are discussed for a

An example of such a pulse is shown in F1g.1.

0 < t < w/w. I(t) = I e s1n wt-6t

damped half—sine of the form

As current pulse from the power supply we suppose an exponentially

3. THE CURRENT DISTRIBUTION

- 3

rather regularly with the current pulse. OCR Output

Fig.4 shows the distribution of magnetic pressures wich rise and fall

They are zero at the rim of the lithium and maximum in its centre.

Apr(r,t)magn = I fr(r,t)dr

rmagnhydraulic pressures ¤p(r,t)are then given by the equation

has been considered to behave like a perfect liquid and the resulting

of the lithium which is a rather soft material. Therefore the latter

For the reference case. these forces are far beyond the yield limit

Z ¢f (r.t) = j (r,t) B (r.t)

volume is given by

radially directed towards the centre. The force f_(r.t) per unit

The lithium conductor is submitted to magnetic forces which are

5. THE MAGNETIC PRESSURES

towards the centre decreases with diminishing local currents.

when this is zero again the remaining magnetic field distribution

Finally the field at the rim decreases with the current pulse, and

field at the rim is lower than at the peak of the current pulse.

this instant the current pulse has already passed its maximum and the

slope where the current distribution is more or less homogeneous. At

field builds up inside the conductor and reaches a nearly linear

only a rising field at the rim. with the penetration of current, the

start of the current pulse there is no field inside the conductor and

The magnetic fields for the reference case are shown in Fig.3. At the

F z- H · B (r,t) - I 3(r.t)r dr

to the current density j_(r,t) by

The radial distribution of the magnetic field Bm(r,t) is related

A. THE MAGNETIC FIELD DISTRIBUTION

- 4

therm OCR Outputthermal pressures Ap(t)which are uniform over its volume.

infinitely stiff, the thermal expansion of the lithium will lead to

If we still consider the lithium as a liquid and its container as

due to ohmic heating during the pulse.

Fig.6 gives the total energy deposition in a disk of unit thickness

increase.

homogeneous heating and ending with a mainly central temperature

close to the periphery (the skin} followed by a more or less

fig.5. At the beginning of the pulse there is a fast temperature rise

The temperature distributions for the reference case are given in

0 YcvAT(r.t) = j i,iE;ii dt

the material with the temperature rise

to conduct the local heat away. we can assume adiabatic heating of

we consider the pulse time to be short compared to the time necessary

The local currents cause local resistive heating of the material. If

6. THE RESISTIVE HEATING, THE ENERGY DEPOSITION AND THE THERMAL PRESSURES

prevent any formation of voids.

following we assume that sufficient preload has been applied to

preload will be lower (see the following sections). Anyhow, in the

or even totally compensated by the thermal expansion and the required

During operation, however, the magnetic compression will be partially

1·rR’ Opg = -i- [I bpr(r,t) 2wr dr]m8x_

pressure during the pulse (averaged over the cross section)

with a static pressure p_ of at least the maximum average magnetic

magnetic compression of the lithium, the latter should be preloaded

In order to prevent the formation of voids inside the lens due to the

- 5

due to the magnetic forces in the radial current leads. OCR Output

Later on we will discuss the additionnal axial force on the end caps

as this culminates the magnetic one.

the magnetic force but will continue like the thermal force as soon

case of zero preload. the total force variation will start off like

is cancelled by the simultaneous reduction of the preload. In the

at the ends due to the magnetic compression of the lithium cylinder

are only given by the thermal axial forces since the average pressure

The total axial forces on the end caps of the container (see Fig.7c)

totalcontainer wall (bpof Fig.7a).

which is the sum of the local magnetic pressure and that from the

Fig.7b shows the pressure variation in the centre of the lithium

and the pressure will therefore be zero during the first 300 us.

zero preload, the contact with the container will be lost temporarily

during the unloading effect of the magnetic pressure. In the case of

there is a sufficiently high preload to avoid the appearance of voids

variation of pressure is the sum of the two on the condition that

the lithium causing an unloading of the container wall. The total

lithium. The magnetic pressure is due to the magnetic compression of

the one caused by the retainment of the thermal expansion of the

at the periphery of the lithium are shown. The thermal pressure is

In Fig.7a the variations of the pressure (deviation from the preload)

7. THE TOTAL PRESSURES AND THE AXIAL FORCES

v [1 + ¤T(r,t)]’21r drnv " {1 + aiu- ¤>1= - 1 A (t) = K i. = K ........iL......e. P th°'m‘ l

reality

The above assumptions will give pressures which are higher than in

- 6

field (short pulse). OCR Output

result will be a compromise between linear field(long pulse) and high

a lower current amplitude to keep the heating acceptable. The optimum

will have to choose a longer pulse for better linearity with possibly

the field is "most" linear. If this linearity is unacceptable, one

working point for the lens the instant during the current pulse where

possible (for reasons of heating of the lens) we have to select as a

constant current. Being obliged to use pulses which are as short as

gradient. In reality this is only perfectly obtained in the case of a

is strictly proportional to the radius and has thus a constant

The working point of the lens is supposed to be optimum when the field

10. THE CRITERION FOR THE OPTIHUM PERFORMANCE OF TR§_L§NS

has been taken into account.

dependence of the relevant electrical as well as thermal parameters

temperatures. In all the calculations reported here, the temperature

distribution of the current as a function of time due to the local

temperature range from zero to l80.5“C (see Table 1) influencing the

especially the specific electrical resistance nearly doubles over the

The material constants of lithium are temperature dependant,

9. THE TEMPERATURE DEPENDANT PARAMETERS

previous section.

operation adds a thermal pressure to the pressures discussed in the

It must be kept in mind that the temperature increase due to steady

steady state distribution.

the temperatures of the next following pulse are superimposed on this

the temperature cycles are stabilised after a number of pulses and

the radial temperature distribution can be seen for the case where

the container has been kept constant and equal to 20°C. In fig.9

of the cylinder. For this calculation the temperature at the rim of

fig.8 shows the progressive build up of the temperature in the centre

finite element programme DOT. Still referring to the reference case,

build up during repetitive pulsing has been calculated with the

The lens is pulsed once every cycle of 2.4 seconds. The temperature

8. THE STEADY STATE TEH ERATURES

...7

lithium. the gradient "centr.grad" in the centre, the linearised OCR Output

tabulated respectively : the actual field "1ith" at the rim of the

anywhere inside the lithium. Under the heading "magnfields" are

The maximum pressure and the maximum temperature are the maxima found

pulse during steady operation.

time intervals in Table 3a for the first pulse and in Table 3b for a

The main results of the reference case have been tabulated at regular

11. THE MAIN RESULTS FOR THE REFERENCE CASE UNDER DISCUSSION

varying the timing between the current pulse and the proton burst.

certain cases a higher anti—proton yield. This could be optimised by

inversely, a field less than proportional to the radius gives in

could be that a field more than proportional to the radius or,

whether this gradient is really optimal for a given situation. It

standard deviation is taken as quality factor. Experience must show

deviation is considered to be the optimum working point and its

given pulse, the instant where the gradient has the smallest standard

the linearised gradient and its associated standard deviation. For a

The performance of the lens at any instant can then be expressed by

Standard deviation —V’()—\[*···]a ¤ 1 B ` n . G(t) ri Q AQ: /Q 2 §L£&l;§i;l;2

error/local field)

standard deviation of the relative local field errors(local field

As the criterion for the linearity of the field we have taken the

tion of G(t} and which are equally distributed over the radius R.

where n represents the number of field samples used for the calcula

lr

E B§r,t)

G(b) =

BIF]B$r,t} 2

instant is calculated by

The magnetic gradient G(t) 0f the best fitting straight line at any

- 8

current pulse. If we change omega in the formula of the damped OCR Output

its transformer, we have to count with the e—power damping of the

In reality, due to the inductances and resistances of the lens and

13. THE EFFECT OF THE LENGTH OF THE CURRENT PULSE (damped half—sine)

latter improves strongly with increasing pulse length.

obtained and its linearity in the form of the standard deviation. The

Figure 10b shows the time at which the optimum linear field is

endcap F and F' increase practically linearly with the pulse length.

energy En, the temperatures T and T' and the axial forces on thelin

field Brises strongly for increasing pulse lenghts and the

at the rim of the lithium lens is rather constant, the optimum linear

maxto 575 ps long. Over this range of pulse lengths, the field B

gives results equivalent to the reference case is of the order of 550

for the reference case under steady operation. The half sine which

The results are shown in Fig.10a in percentages of the values found

the reference case (312.3 kA).

half—sine pulses with amplitudes equal to the real peak amplitude of

length can best be seen by suppressing the damping term taking

the damping term is time dependant. Therefore the effect of the pulse

will change but also the resulting peak amplitude as the e-power of

in the formula of the pulse. In doing so, not only the pulse length

To vary the length w/w of the current pulse, one has to vary omega

l2. THE EFFECT OF THE LENGTH OF THE CURRENT PULSE (undamped half—sine)

currents.

is still a non negligible magnetic field left caused by the eddy

One can see that at the end of the current pulse, at 600 us, there

as the working point of the lens and the results have been underlined.

particular instant where the field linearity is optimum is considered

deviation shows a rather pronounced minimum at 360 us. This

field distribution from the linear field distribution. This standard

straight line, and the standard deviation "deviation" of the actual

field "lin" at the rim of the lithium as found with the best fitting

- g

where R' is the lithium radius of the scaled case. OCR Output

(R'/RPI(R'} = IO E; e"6t(R/R°)2sin —..EE "

field at the rim

lithium and the peak amplitude has been adapted to keep the same

scaled to the related change in resistance and inductivity of the

keeping the container thickness the same. The current pulse has been

We consider here an increase in radius of the lithium cylinder

15. THE EFFECT OF THE RADIUS

follows a power of 2.0. the temperature T a power of 1.3.

proportional to the current to the power 2.1, the maximum force F

than proportional; the deposited energy En rises fastest and is

less proportional to the current, all the other phenomena rise faster

maxthe field at the periphery of the lithium Bwhich rise more or

neffect shown in Figure 12. Apart from the optimum field Bliand

Changing the current amplitude I_ at a fixed pulse length has the

lh. THE EFFECT OF THE CURRENT AHPLITUDE

operation than for the first single pulse.

temperatures T and resulting forces F are much higher for the steady

at very short pulse lengths and the standard deviation is worse. The

linHowever, due to the skin effect, the optimum field Bis smaller

as the current pulse is less reduced by the exponential damping term.

follows the maximum current and is much higher at short pulse lengthsmax

500 and 700 us. The field Bat the periphery of the lithium

forces. Therefore it is advantageous to choose a pulse length between

the optimum field decreases more quickly than the temperatures and

a lower maximum temperature T. At longer pulse lengths (beyond 600 us)

an important gain in linearity (smaller standard deviation) and also

the energy deposition En. But with increasing pulse length there is

linlengths, the optimum field Bdoes not change very much nor does

drastically as shown in Fig.11a and llb. Over a large range of pulse

different real amplitudes and the calculation results change

current pulse, we will not only get different pulse lengths but also

- 10

container wall cause magnetic forces in axially opposite directions. OCR Output

components of the incoming and outgoing currents through the

and thus much larger than the actual lithium rod, the radial

of the lens container. Since the diameter of this container is 90 mm

The external current connections are clamped onto the circumference

18. THE AXIAL MAGNETIC FORCES IN THE CURRENT LEADS

the standard deviation stay rather constant.

linproportional to k whereas the energy En, the optimum field Band

the temperatures T and especially the forces F inrease inversely

temperature as well as on the other variables. Figure 15 shows that

different heat transmission coefficients k to judge its effect on the

cooling media like water or air, calculations have been made for

outside of the container. To evaluate the efficiences of different

To dispose of the heat in the lens a cooling medium flows along the

COOLING HEDIUM

17. THE EFFECT OF THE HEAT TRANSMISSION FROM THE CONTAINER HALL INTO THE

thickness.

temperature T and especially the force F increase with rising wall

thickness, the energy En does not change very much, whereas thenoptimum field Blidrops inversely proportional to the wall

temperature in the lithium. Figure lé illustrates the effect. The

conductivity of the stainless steel causes a higher steady state

the useful field in the lithium. In addition, the poor thermal

conduct part of the current and its thickness therefore influences

The wall of the container is made of stainless steel which also will

16. THE EFFECT OF THE WALL THICKNESS OF THE CONTAINER

temperature that increases with increasing radius.

rature for one single pulse hardly changes but it is the steady state

rise with powers of 2.8, 2.l and 0.3 respectively. The maximum tempe

pulse being scaled. The force F, energy En,and temperature T however

nBliand its standard deviation do not change very much, the current

The results are displayed in Fig.l3 and show that the optimum field

- 11

can easily be estimated with the formula of section 18. OCR Output

the radial current leads around the core have to be considered. They

only the axial forces from the lithium cylinder but also those from

pessimistic as discussed above. For the design of the lens body, not

higher steady state temperatures which may, however, be somewhat

the huge increase in axial force for the larger lens caused by the

magnetic pressure cancels at the periphery. Moreover, the table shows

the calculations have shown (section 7) that the effect of the

rim occur at the end of the pulse and are due to the thermal stress;

The maximum axial forces on the end caps and radial pressures at the

the container wall and the cooling medium.

current pulse and different heat transmission coefficients k between

The calculations were made for different amplitudes of the input

of 2éOO us, and the damping was scaled accordingly (see section 15).

length was scaled up with the square of the radius to a pulse length

For the antiproton lens with a lithium radius of 20 mm, the pulse

pulse form (600 us long) was used for the proton lens (radius 10 mm).

A summary of the different results is shown in Table 2. The standard

19. SUMMARY OF THE RESULTS

current pulse.

the actual lithium rod. These forces are highest at the peak of the

where R_ is the outside radius of the lens body and R the radius of

_ Fu)- M {n.¤<R>+4}H1‘gt> jg i

under somewhat simplified assumpti0ns(see Appendix 2) and are given

These axial forces F(t) on the current leads have been estimated

- ...

heat transfer problems. 1976. University of California, Berkeley. OCR OutputR. M. Polivka and E. L. wilson. Finite element analysis of non linear

IEEE trans. on Nuclear Science,Vol NS—30, No 4, August 1983.lithium lens and transformer system. 1983 Accelerator Conference,G. Dugan et. al. Mechanical and electrical design of the fermilab

190, (1981), 9.

of high energy particle beams. Nuclear instruments and MethodsB. F. Baganov et. al. A lithium lens for axially symmetric focussing

sntiproton accumulator (ACOL). CERN B3—lO, Oct.l983.

E. J. Wilson (Editor). Design study of an antiproton collector for the

20. REFERENCES

_ 13

NY m2 OCR Output . (r-) PC ,2. . . . 1 s . . . A Magnetic diffusivity ·· (··) Thermal diffusivity r

m'°CElectr. conductivity Y (3E) | Spec. heat/vol. pc (";—")

Magn. permeability u (¥§) ] Therm. resistivity g($§)

(”C)Electric field E (V/m) | Temperature

Units

(stored energy per unit length)

(A) (i)I(t) : [ YE_(t,r)ds Q(t) = I pcT(t,r)ds

Where Where

() ar ` U at ' zwn at m __ Br — X q(t°R) _ 2wRX Bt (m )g;; 1 ag(t; jgg __ 6Hg§t,R) __ L BIH;] y__{__Igf

Boundary condition at rim Heat source at rim

X at; ( ). __V Ez — vT at yim ( ) 2 __E$.2I V TBE z

Diffusion equation Diffusion equation

Euuations

Electrical Thermal equivalent

Analogy between the electric and the thermal diffusion

APPENDIX l

E I I ,:2; B¢(r.z) = F I Jz(r.z)rdr = 2'R2 r OCR Outputg(z)

g(z) S r S R is due to the axial currenf contained inside the boundary $(2):The circumferential magnetic field B in the region -20 S z S 20 and

where I is the total current.

. JY(r’z) = 1R? rdzI gg Q dgg Q z z

Conservation of the current requires:

g(—zO) = O , g(+zO) = R

The reflecting surface g(z) must satisfy:

at which time a practically uniform current density should be achieved.

for the maximum force values which occur during the peak of the current,current density j, is uniform inside the radius R. This is justifiablearbitrary surface g(z) into the axial direction and that the axialThen we assume for simplicity that the radial current is deflected on an

O S r S R and -20 S z S zorange

We calculate first the forces at the ends of the lithium conductor in the

‘i•° *•

r.q tz)

j1("»'i)

i.(n¤)

container can be estimated from the following equivalent flow pattern:The magnetic forces due to the radial current flow through the lens

The axial magnetic forces in the current leads.

APPENDIX 2

2 Aw R OCR Output- LV A F" ln

R

or

-z R

F_ = I I 2wr f_(r,z)dr dzZ0 Ro

For the region with R S r S RO and -20 S z S zu the force F2 is given by

l 16w_ LL. F—

This integral can readily be evaluated and it results:

—z g(z)

F1 = IU I 2vr f_(r,z)dr dzz R

Integrating fz(r,z) over the considered volume gives the total force F1

2w’R‘ r2 dz

uI’ g°(z) dg(z)

fz<r.z> = jr(r.z> Bw<r.z>

Therefore the volume force f,(r,z) in that region becomes:

- 2

Fig.1. The current pulse. OCR Output

TIME GF CQLCLJLQTIGN {MICFW8.-0. BG. 1EC. 2uC. BEC. HOO. MBU. 5GU. EQU. ?2L.ETC

-1GUQCC.1 l I

-50000.%

SUDUG.

ii§ODOU.

Q1j5¤0c=0.

FEQOOOU.

ESDUDG.

SDUUUD.

35000C.

OCR OutputOCR OutputOCR OutputOCR OutputQGOOUD.

The driving current pulse is shown in fig.1. OCR Output

of 80 microseconds each.

The curves are marked with the number of time increments

$5.2. The current density distribution at vcrious instants.

Fi {MN}0.01 1.25 2.E.T 3.75 5.GC- 6.25 7.Ei· 8.75 1G.OO 11.252.5Q

I , I ` Bt { 4

`HUL.

-2"V

F~——1o \ \2uG .

\14C·G.

BDC.

BUG.

1Gn

12OG.

WOO.

The driving current pulse is shown in fig.1. OCR Output

of 80 microseconds each.

The curves are marked with the number of time incrementsFig.3. The field distribution at vgrious instants.

R [MW¤.cc· 1.25 2.55 5.75 5.uc 5.25 2.5c a.25 mn: 11.252.5c

§' I

The driving current pulse is shown in fig.1. OCR Output

of 80 microseconds each.

The curves sre marked with the number of time increments

Fig.4. The distribution of the magnetic pressure at various instants.

H {MW0.00 :.25 2.55 2.70 5.00 5.25 v.$0 0.vs IO 00 11.25E.€;

—1E.I . r

:==1o=9¤——— _.s

15.

ED.

25.

3D.

35.

The driving current pulse is shown in fig.1. OCR Output

as s function of time.Fig.6. The absorbed energy iu s lmnxthick disk

•¤<> .z¤¤ .>¤¤ .~»¤¤ .s¤¤ .s¤• aw

”W-·—¢§m)

xm.) 3 I MOCR Outputlc 4 Ektrqej

The driving current pulse is shown in fig.1. OCR Output

of BO microseconds each.

The curves are marked with the number of time increments

under stead? cvclin

Fig.9. Ihe temperature distribution at various instanta during the_pgg;;

FR (MM}0.uc 1.2s 2.50 a.vs $.00 s.2s msc s.vs www :1.22;;.sn

i:i\L

IC.

2G.

20·

35.

109

LIU.

H5.

SD.

OCR OutputOCR OutputS5.

(The current pulse is a ha1f—¤i¤e). OCR Output

tiBB P0int as functions of the pulse length 1/;.;.

Fig.10b. The optimum stondard deviation of the linear field and its

*°° .1¤¤ .3¤¤ .·+¤<> .¤¤¤ .6¤¤

0ph·v•-o-·~ stand awk v\tv{qEiq-1.

:0

to Ek! Pub! Lt.1··»•JUntime oi- epkimon sire.: Naxos; v¢.laA·{·•¢

noo

1¤¤

Percuni

OCR OutputSon

due to the damping term). OCR Outputsection 3; the amplitude is decreasing with the pulse length(The current pulse is the dumped half-sine described in

Fig.l1a. the calculated variables as functions of the pulse length 1/u.

\¤¤ ,2-¤¤ ___ ,30q__ 4*100 *500 ,600 ,700 ,8¤¤;·|s¢r.

Putbt

__-, First Pulse

Stccxdnj operatic-na

"•··• `P•

80

"·..T1·-. ` ¤,_._ "I

·I•;\.,0

Stcadss ¤?¢v•J:€¤··v•._'°°Put}! Ltmgkh {Db/1 gc.

R!¢¢'¢'Y\C{ (.G$¢ s l°°7•

itc

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