Field Model for the Multipoles Factory

FQWG, 17/3/2004 S.Amet, L.Deniau, M.Haverkamp, L.Larsson, T.Pieloni,

S.Sanfilippo, M. Schneider, R. Wolf, G.Ambrosio(+), P.Bauer(+)

and many other contributors from AT-MTMpresented by L.Bottura

(+)Fermilab, TD

Salvific Magnetic Reference

magneticreferencedatabase

machinetopologydatabase

field modelB[t,I,dI/dt,T,I(-t)]

LHC operating conditionst, I, dI/dt, T, I(-t)

predicted field strength (Bm) andharmonics (cn) by octant

reference magnets

… or usine-a-gaz wasting resources ?

(LE, private communication, Chamonix XIII, 2004)

a key to the successful ramp management at the LHC…

ora pro nobis

Overview

A description of the field dynamics in the LHC MB’s and MQ’s: the Field Model: general decomposition in error components static errors (geometric, persistent, saturation) decay and snap-back

Error sources extrapolation errors magnet life-long instabilities modelling errors measurement errors

Expected results uncertainty on settings at injection and flat-top uncertainty on ramp correction (work in progress)

The field model

complex harmonic coefficient Cn in MB’s and MQ’s

depends on time t current I ram-rate dI/dt temperature T powering history I(-t)

simple fits based on physical models or empiric relations (tested against measurements)

€

Cn=Cn t,I,dIdt,T,I(−t)

⎛ ⎝ ⎜ ⎞

⎠ ⎟

Components in the field model

general decomposition in error sources geometric DC magnetization from persistent currents iron saturation decay at injection snap-back at acceleration coil deformation at high field coupling currents residual magnetization

smaller valuessmaller variability

smaller uncertainty

higher valueshigher variability

higher uncertainty

Geometric multipoles

important at all field levels absolute field is linear in current, normalised field is constant

measured in warm conditions (can be extrapolated from industry data)

IBmgeommγ=€

Tgeom=γm

€

cngeom=γn

€

Tcold=fTTwarm+ΔT

€

cncold=fcncn

warm+Δcn

Persistent currents

mostly important at low field (but present throughout) proportional to the magnetization M proportional to Jc

assume that the Jc(B) scaling is maintained (geometry and B distribution effects are condensed in fitting exponents and )

€

M∝JcD

€

Jc∝1B

BBc

⎛ ⎝ ⎜ ⎞

⎠ ⎟α1−B

Bc

⎛ ⎝ ⎜ ⎞

⎠ ⎟β

€

BmMDC=μm

IinjI

IIinj

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

α Ic−IIc−Iinj

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

β

€

TMDC=μmIinjI2

IIinj

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

α Ic−IIc−Iinj

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

β

€

cnMDC=μn IIinj

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

2−α Ic−IIc−Iinj

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

β

add T dependence

Iron saturation

important at high field only associated with details of iron geometry (shape of inner contour,

slits, holes, …) no “theoretical” expression available, apart for the general shape

of the saturation curve (sigmoid) take a convenient fit to the experimental data

€

Bmsat=σmIaσ fσ I,I1σ,ΔI1σ( )+1−aσ( )fσ I,I2σ,ΔI2σ( )[ ]

€

fσ I,Iσ,ΔIσ( )=1πarctanI-Iσ

ΔIσ ⎛ ⎝ ⎜ ⎞

⎠ ⎟+π2

⎡ ⎣ ⎢ ⎤

⎦ ⎥

€

Tsaturation=σm aσ fσ I,I1σ,ΔI1σ( )+1−aσ( )fσ I,I2σ,ΔI2σ( )[ ]

€

cnsaturation=σn aσfσ I,I1σ,ΔI1σ( )+1−aσ( )fσ I,I2σ,ΔI2σ( )[ ]

Decay

appears during constant current excitation associated with current redistribution in the superconducting

cables result of a complex interaction:

current redistribution local field magnetization bore field

assume that the dynamics follows that of current diffusion

€

Bmdecay=Δm aΔ1−e−

t−tinjτ1

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟+1−aΔ( )1−e−

t−tinjτ2

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥

€

Tdecay=ΔmI aΔ1−e−

t−tinjτ1

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟+1−aΔ( )1−e−

t−tinjτ2

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥

€

cndecay=Δn aΔ1−e−

t−tinjτ1

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟+1−aΔ( )1−e−

t−tinjτ2

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥

Powering history effects

average effect of powering history has an uncertainty due to limited sampling (2 % of production ?)

2 magnets3 magnets

Powering history dependence

main parameters: flat-top current flat-top duration waiting time before injection (injection duration)

tFT

tinjection

tpreparation

IFT

I

t

€

Δn=ΔnstdIFT

Istd

⎛ ⎝ ⎜ ⎞

⎠ ⎟A−Be−tFTτ

A−Be−tstdτ

⎛

⎝ ⎜ ⎜ ⎜

⎞

⎠ ⎟ ⎟ ⎟C+De−

tpreparationτ

C+De−tstdτ

⎛

⎝ ⎜ ⎜ ⎜

⎞

⎠ ⎟ ⎟ ⎟

Snap-back

first few tens of mT in the acceleration ramp, after injection pendant to decay: magnetization changes are swept away by

background field result of a complex interaction:

current ramp background field magnetization bore field

magnet family invariant found by serendipity

€

Bmsnap−back=Δb1Ie−

I t()−IinjectionΔI

€

Tsnap−back=Δb1e−I t()−Iinjection

ΔI

€

cnsnap−back=Δcndecaye−I t()−Iinjection

ΔI

Look at the data the right way…

fit of the b3 hysteresis baseline

hysteresis baseline subtracted

b3 snap-back singled out

exponential fit

Same magnet, different cycles

Δb3 and ΔI change for different cycles…

… and they correlate !

An invariant for snap-back !?!

great ! the correlation plot holds for many magnets of the same family

Coil deformation

coil deforms under Lorentz loads at high field the displacement of the strands is proportional to the

electromagnetic force

this effect is small (order of 0.1 … 0.2 units of b3)€

Bmdef=δm

IInom

⎛ ⎝ ⎜ ⎞

⎠ ⎟2

€

Tdef=δmI

Inom2

€

cndef=δnI

Inom

⎛ ⎝ ⎜ ⎞

⎠ ⎟2

Coupling currents

important during the ramp eddy currents flow resistively among superconducting filaments

in strands and superconducting strands in cables these currents couple the superconducting filaments and strands contribution is proportional to dB/dt and constant in B (neglect

magnetoresistivity effects)

this effect is small (order of 0.1… 0.2 units of b3)

€

BmAC=θm

110

dIdt

€

TMAC=θmIinj10I

dIdt

€

cnAC=θnIinj10I

dIdt

Residual magnetization

important at very low field (e.g. warm measurements) iron (or other magnetic parts) maintain a magnetization after

powering at high field

very small values, broadly unknown origin, useful to adapt fits (especially for the transfer function)

mresidualmBρ=€

Tresidual=ρmI

€

cnresidual=ρnI

W/C extrapolation errors

30 % cold measured (realistic ?)

uncertainty estimated on a sector (50 magnets)

take best result (lowest uncertainty) for the estimate, from W/C extrapolation

Magnet stability at long term

coil geometry changes during the magnet life settling and ratcheting of

the composite formed by cables, wedges and insulations

geometric multipoles change

systematic effect observed only on allowed multipoles

Modelling errors

deviation of local fit from average for magnets completely measured the fit residual can be

decreased at will for magnets not completely measured the model may be not

appropriate/sufficient/adapted

uncertainty on decay and SB produced by LHC powering cycle different from the one measured and

modelled 6 TeV vs. 7 TeV expected and unexpected waiting times temperature changes …

Modelling of Decay scaling

Flat Top Duration InfluenceMBSMS5V1

-0.182704

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 500 1000 1500 2000 2500

Flat Top Duration (s)

b3 - Snapback (units)

Measurement

Neuron

Analytic

Flat Top Duration t @ 11750 A

Injection

Error Plot

-30%

-20%

-10%

0%

10%

0 500 1000 1500 2000

Pre Cycle Duration (s)

Relative Error (%)

Analytic

Neuron

Analytical model accurate to ≈20 %

Neural network accurate to ≈ 5 %

the model of the average has an uncertainty

Uncertainty from model

assume model is accurate to 20 % of effect

add uncertainty on average due to limited sample so far 2 % of the population

has been characterised (partially)

assume 20 magnets till the end of the production

NOTE: this is obviously a good reason to have extra magnets on the benches at LHC start-up and after

Uncertainty from measurements

assume coil radius is known to within 50 m sensitivity to harmonic of

order n scales as the radius to the n-th power

error on the harmonics n is proportional to n Radius

add uncertainty from measurement r.m.s.

Uncertainty on settings

injection uncertainty on b1 is the same at injection and flat-top

uncertainty on a1 does not contain the effect of changes of magnet roll

uncertainty on b3 and b5 at injection significantly larger than at flat-top

Summary of estimates

A comment: seems pretty damn good to me, there must be something wrong…

Work in progress (by KW-14)

verify model vs. measured magnets (L. Deniau, V. Granata, N. Sammut)

hardware concept for reference magnets (M. Buzio, A. Masi)

data fusion concept (L. Deniau) experience at HERA, Tevatron, RHIC (L. Bottura) plan and cost estimate (L. Bottura) scope of the review, panel, participants, to be

discussed at the next FQWG