8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 16

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 26

Fig 1 Sextupole in a normal position [6]

coordinates are

r2part 2V m

partr2 + r

partV mpartr

+ part 2V m

partθ2 = 0

V m(r0 θ) = f (θ) =infinsumn=0

(αn cos (nθ) + β n sin (nθ)) (2)

The Fourier coefficients of boundary condition are

α0 = 2

π

int 2π0

f (ϕ) dϕ

αn = 1

π

int 2π0

f (ϕ)cos(nϕ) dϕ

β n = 1π

int 2π

0

f (ϕ)sin(nϕ) dϕ

(3)

By separation of variables [7] is obtained

V m(r θ) = α0 +

infinsumn=0

rn (αn cos (nθ) + β n sin (nθ)) (4)

and based on B = minusnablaV m are computed the radial and

azimuthal components of the magnetic field

Br(r θ) =infinsumn=1

1048616 r

r0

1048617nminus1

(Bn sin (nθ) + An cos (nθ))

Bθ(r θ) =infinsumn=1

1048616 rr0

1048617nminus1

(Bn cos (nθ) minus An sin (nθ))

(5)

where

An = minusnαn Bn = minusnβ n and B = ∥B∥ =radic

B2r + B2

θ

Let consider now an ideal sextupol in skew position having

as boundary condition Br(r0 θ) = B0 sin(3θ) with B0 =011T

In Fig 2 3 4 are depicted spatial distribution of Bn(r ϕ)

Bt(r ϕ) |B| (r ϕ) for this ideal device inside its aperture

III ANALYSIS OF EXPERIMENTAL DATA

A Experimental data

The electromagnet depicted in Fig 1 rotated in skew posi-

tion was experimentally characterized by HP method Table I

presents the values of radial magnetic flux density measured

Fig 2 Radial component of the magnetic flux density

Fig 3 Azimuthal component of the magnetic flux density

Fig 4 The module of the magnetic flux density

along horisontal Ox axes for different excitation currents

between I = 50A and I = 580A The measured data are

plotted in Fig 5 6 7

Let consider the spatial variation of radial magnetic field

for a reference current I 0 = 300A

Br = [000 00034 00181 00426 00770 01216]

xprime

= [0 02000 04000 06000 08000 10000]

(6)

Where xprime = rr0 with r0 = 005m and Br are the average

values for symmetric positive and negative coordinates Since

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 36

TABLE IEXPERIMENTAL DATA OF Br[T] [ 8]

x[m] I 1 = 50A I 2 = 100A I 3 = 200A I 4 = 300A I 5 = 400A I 6 = 500A I 7 = 580A-005 0020242 0039452 0078291 0117118 0155458 0191329 0213700

-004 0012976 0024993 0049257 0073475 0097344 0119578 0133432

-003 0007351 0013833 0026891 0039899 0052654 0064424 0071698

-002 0003409 0006005 0011156 0016300 0021297 0025759 0028406

-001 0001107 0001415 0002014 0002574 0003061 0003299 0003318000 0000457 0000109 0000648 0001424 0002211 0003188 0003892

001 0001410 0002012 0003157 0004308 0005432 0006240 0006703

002 0004000 0007174 0013530 0019865 0026131 0031800 0035338

003 0008261 0015618 0030443 0045250 0059941 0073532 0082093

004 0014156 0027311 0053920 0080519 0106886 0131478 0147069

005 0021755 0042434 0084248 0126067 0167577 0206448 0230987

Fig 5 Radial magnetic flux density vs current

Fig 6 Radial magnetic flux density vs radius

the first value is perturbed by an obvious error it was replaced

with Br = 0 the correct value due to the symmetry

B Polynomial approximation

The spatial variation of magnetic field Brminusexp for the

reference current (6) was fitted with a 10th degree even

Fig 7 Radial magnetic flux density vs radius and current

polynomial function

Brminusmod(xprime) = c2xprime2 + c4xprime4 + c6xprime6 + c8xprime8 + c10xprime10 (7)

Using CFTOOL module of MATLAB the coefficients pre-

sented in Table IV were obtained for several polynomial

degrees

A graphical approximation of 10th degree polynomial ap-

proximation compared with experimental data is given in Fig

8 Relative standard deviation estimated with

ε =maxk=111(Brminusexpk minus Brminusmodk)

maxk=111

(Brminusexpk) (8)

is 221

C Linear model

Figure 5 suggests that the magnetic core is not strongly

saturated If the magnetic field has a linear dependence versus

excitation current we can consider

Br = Bref

I

I 0 (9)

where Bref is the field corresponding to the reference current

I 0 It will be computed starting from the matrix in Table I

with n = 11 rows (corresponding to n measuring positions)

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 46

TABLE IITHE REFERENCE FIELD COEFFICIENT AND ITS STANDARD DEVIATION FOR SEVERAL POSITION AND CURRENTS

Bref (i j)[T]x[m] I II III IV V VI VII Bref std -005 01 215 0118 4 0 1174 011 71 01166 01 148 0110 5 0 1166 28 882

-004 00 779 0075 0 0 0739 007 35 00730 00 717 0069 0 0 0734 23 549

-003 00 441 0041 5 0 0403 003 99 00395 00 387 0037 1 0 0402 19 108

-002 00 205 0018 0 0 0167 001 63 00160 00 155 0014 7 0 0168 16 246-001 00 066 0004 2 0 0030 000 26 00023 00 020 0001 7 0 0032 14 414

000 00000 00000 00000 00000 00000 00000 00000 00000 00000

001 00085 00060 00047 00043 00041 00037 00035 00050 14566

002 00240 00215 00203 00199 00196 00191 00183 00204 15962

003 00496 00469 00457 00452 00450 00441 00425 00456 19052

004 00849 00819 00809 00805 00802 00789 00761 00805 23282

005 01305 01273 01264 01261 01257 01239 01195 01256 28915

Fig 8 Polynomial approximation versus measured values

and m = 7 columns (corresponding to m excitation currents)

which contain the n times m measured data with

Bref ij = Brij

I 0I j

(10)

The average value of these reference coefficients for each

measuring position is presented in the Table II (where is

included also its standard deviation)

If the current is considered smaller or equal to I 0 the

results are presented the Table III For currents lower than

the reference current the error of linear model is acceptablebeing under 11 So we can conclude that the reference

current do not saturates the magnetic core

TABLE IIITHE REFERENCE FIELD COEFFICIENT AND ITS STANDARD DEVIATION FOR

CURRENTS LOWER THAN I 0

Bref (i j)[T]

I II III IV Bref std

-005 01215 01184 01174 01171 01186 10153

-004 00779 00750 00739 00735 00750 10141

-003 00441 00415 00403 00399 00415 09693

-002 00205 00180 00167 00163 00179 09579

-001 00066 00042 00030 00026 00041 09357

000 00000 00000 00000 00000 00000 000000

001 00085 00060 00047 00043 00059 09583

002 00240 00215 00203 00199 00214 09532

003 00496 00469 00457 00452 00468 09978

004 00849 00819 00809 00805 00821 10287005 01305 01273 01264 01261 01276 10487

The errors under 3 in the Table II validate the linear model

(10) also for higher currents Therefore the radial flux density

of the characterized electromagnet along horizontal radius

within its aperture has following variation

Br(r I ) = f 1048616 r

r01048617 middot I

I 0 =983131

c21048616 r

r010486172

+ c41048616 r

r010486174

+

+c6

1048616 r

r0

10486176

+ c8

1048616 r

r0

10486178

+ c10

1048616 r

r0

104861710983133

I

I 0

(11)

TABLE IVFITTING WITH EVEN POLYNOMIALS

Degree II IV VI VIII X

c2 01208 01158 01093 00976 006939

c4 mdash 0005962 002744 01062 04759

c6 mdash mdash -001515 -01611 -1586

c8 mdash mdash mdash 00789 213

c10 mdash mdash mdash mdash -09678

suma 01208 01218 01216 01216 01215

sse 5048e minus 6 2037e minus 6 1141e minus 6 5333e minus 6 1642e minus 6

RMSE 0001005 00007136 00006167 00005164 4052 middot

10minus9

Fig 9 Even polynomials

D Least-squares method

The experimental data sets preprocessed by averaging val-

ues measured at symmetric positions were fitted with even

polynomials by using the least squares method The most

reliable model is that having 6th degree At higher grades

the coefficients do not decrease monotonically with the order

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 56

probably due to measurement error and numerical instability

results can be seen in the Table IV and in Fig 9 Similar

results can be seen in the Table V and in Fig 10 where the

results are obtained by using measurements in all 11 positive

and negative positions without previous averaging

Fig 10 Fitting of the 11 experimental values

TABLE VFITTING THE 11 EXPERIMENTAL VALUES WITH EVEN POLYNOMIALS

Degree II IV VI VIII X

c2 01208 01158 01093 00976 006939

c4 mdash 0005962 002744 01062 04759

c6 mdash mdash -001515 -01611 -1586

c8 mdash mdash mdash 00789 213

c10 mdash mdash mdash mdash -09678

suma 01208 01218 01216 01216 01215sse 9712e minus 8 911e minus 5 8931e minus 5 8809e minus 5 8703e minus 5

RMSE 0003116 0003182 0003341 0003547 0003808

IV THE FIELD DISTRIBUTION IN THE APERTURE BASED

ON PERIPHERAL MEASUREMENTS

Fig 11 The 30 degree slice extracted from circular aperture

A slice of 30 degree is extracted from circular aperture

disk (Fig 11) The boundaries of this sub-domain D are

bull C 1 on symmetry axe of the pole

bull C 2 on the polar inter-axes

bull C 3 on the aperture border with radius r0 = 50mm

Magnetic field satisfies the following boundary conditions

bull (c1) Bn = 0 on C 1 because the magnetic field lines are

tangential along this radial direction

bull (c2) H t = 0 on C 2 because the magnetic field lines are

perpendicular to this radial direction

bull (c3) Bn = Br known - in this procedure the radial flux

density is measured on C 3 - the aperture periphery

If we will consider that the angle ϕ originates from C 1 thenfrom (5) and (c1) follows that all coefficients Bn are zero

From (c2) and (5) we have

An cos983080nπ

6

983081 = 0 rArr

rArr (2k + 1)π

2 =

nπ

6 rArr n = 3(2k + 1)

(12)

For any value of the index n which is not a multiple of

three times odd numbers An = 0 Therefore the magnetic

scalar potential in aperture has only cos harmonics with orders

3 9 15 and the radial component has polynomial variation

vs radius of orders 2 8 14

Br(xprime) =sum852008

Anxprimenminus1 cos(nϕ)852009

n = 3 9 15 (13)

For ϕ = 0 it results following variation on C 1

Br(xprime) =sum

(Anxprimenminus1) =

= A3xprime2 + A9xprime8 + middot middot middot = c2xprime2 + c8xprime8 + middot middot middot(14)

The presence of non-zero terms c4xprime4 c6xprime6 and c10xprime10 in (7)

has three possible explanations

bull structural asymmetries

bull measurement errors

bull computational errors

V THE FIELD DISTRIBUTION IN THE APERTURE BASED ON

RADIAL MEASUREMENTS

If these errors would be zero then the coefficients c2 c8 c14extracted from the measurement determining the Fourier co-

efficients of harmonic solution A3 = c2 A9 = c8 A15 = c14

With them using (5) can be determining the variation of the

flux density vector field throughout the aperture Magnetic

field distribution in the electromagnetrsquos aperture can be deter-

mined with (5) where coefficients An Bn can be computed

by using the model (11) extracted from experimental data

It should be determined the coefficients c2 c8 and c14 from

the experimental data looking for the the best approximationof a polynomial with these coefficients (ascending order

2 8 14)

Unfortunately the conditioning number of high polynomial

interpolation is extremely high According Gautschrsquos theorem

in our case this number has the approximate value κ = (1 +radic 2)n+1 = 16238 for n = 10 [9]

This means that a disturbance of 1 in experimental data

is amplified ten thousand times in the computed polynomial

approximation This behavior makes totally irrelevant the

numerical result

In this approach it was actually solved the Laplace equation

with Dirichlet conditions on C 1 and C 2 Moreover there are

not boundary conditions on C 3 but instead an additional Neu-

mann condition was imposed on C 1 Although this problem

has an unique solution it is not well formulated problem

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 66

because it is numerically unstable [10] Moreover this problem

is very similar with Hadamardrsquos famous counterexample [11]

Unlike this problem if Dirichlet or Neumann the boundary

conditions are imposed in each point of the domainrsquos border

D including on C 3 as in section V then the problem is well-

formulated in Hadamard sense Consequently its conditioning

number is not greater than 1 [10]

V I CONCLUSIONS

The field problem based on peripheral measurements is well

conditioned because small changes in boundary data are not

amplified and the corresponding variations of the field in any

point within the aperture D has a relative error not greater

than the relative measurement error

It should be noted that from practical point of view the

effort to make measurements is the same With the same

number of measurements the error in determining the fieldinside aperture is much higher when measurement are used

along radius than in the case of using circular measurements

According to (14) the best polynomial approximation with

three terms is

Br(xprime) = c2xprime2 + c8xprime8 + c14xprime14 (15)

Their coefficients are represented in Table VI and its graphic

is depicted in Figure 12 However as it was proven this is not

a reliable result because its conditioning number is 551614

TABLE VIFITTING WITH THE POLYNOMIAL Br(xprime) = c2x

prime2 + c8xprime8 + c14x

prime14

c2 01161

c8 002028

c14 -001479

sum 012159

sse 1910 middot 10minus6

RMSE 00007978

Fig 12 Fitting of experimental values with the polynomial Br(xprime) =c2x

prime2 + c8xprime8 + c14x

prime14

REFERENCES

[1] S Russenschuck Field Computation for Accelerator Magnets Analyti-cal and Numerical Methods for Electromagnetic Design and Optimiza-tion John Wiley and Sons May 11 2010

[2] L Walckiers ldquoMagnetic measurement with coils and wiresrdquo CERN-2010-004 pp 357ndash385 March 2011 httparxivorgabs11043784v1

[3] A Temnykh Y Levashovb and Z Wolf ldquoA study of undulator magnetscharacterization using the vibrating wire techniquerdquo Stanfort - LCLS-TN-10-2

[4] A K Jain Measurements of Field Quality Using HarmonicCoils httpwwwbnlgovmagnetsstaffguptascmag-courseuspas01 AJ01HarmonicCoil_Slidespdf

[5] D Dan and D Ioan ldquoEroarea de metoda icircn determinarea experimen-tal analitica a cacircmpului unui electromanget pentru acceleratoarele departiculerdquo SNET12 Bucharest 2012 httpsnetelthpubrosnet2012

[6] ICPE-CA ldquoElectromagneti si surse de alimentare pentru proiectul fair- poveste de succes pentru icpe - cardquo 2012

[7] I G Shabac Matematici speciale 2 Bucuresti Romacircnia EdituraDidactica si Pedagogica 1965

[8] ICPE-CA Institutul National de Cercetare Proiectare pentru Ingineriaelectric˘ a httpwwwicpe-carorohttpwwwicpe-caroro

[9] K A Gallivan ldquoPolynomial interpolationrdquo 2013[10] S Zaglmayr High Order Finite Element Methods for Electromagnetic

Field Computation Thesis - Linz Univ 2006[11] R Corless Elementary Partial Differential Equations - Hadamardrsquos

example httpwwwapmathsuwoca 1998

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 26

Fig 1 Sextupole in a normal position [6]

coordinates are

r2part 2V m

partr2 + r

partV mpartr

+ part 2V m

partθ2 = 0

V m(r0 θ) = f (θ) =infinsumn=0

(αn cos (nθ) + β n sin (nθ)) (2)

The Fourier coefficients of boundary condition are

α0 = 2

π

int 2π0

f (ϕ) dϕ

αn = 1

π

int 2π0

f (ϕ)cos(nϕ) dϕ

β n = 1π

int 2π

0

f (ϕ)sin(nϕ) dϕ

(3)

By separation of variables [7] is obtained

V m(r θ) = α0 +

infinsumn=0

rn (αn cos (nθ) + β n sin (nθ)) (4)

and based on B = minusnablaV m are computed the radial and

azimuthal components of the magnetic field

Br(r θ) =infinsumn=1

1048616 r

r0

1048617nminus1

(Bn sin (nθ) + An cos (nθ))

Bθ(r θ) =infinsumn=1

1048616 rr0

1048617nminus1

(Bn cos (nθ) minus An sin (nθ))

(5)

where

An = minusnαn Bn = minusnβ n and B = ∥B∥ =radic

B2r + B2

θ

Let consider now an ideal sextupol in skew position having

as boundary condition Br(r0 θ) = B0 sin(3θ) with B0 =011T

In Fig 2 3 4 are depicted spatial distribution of Bn(r ϕ)

Bt(r ϕ) |B| (r ϕ) for this ideal device inside its aperture

III ANALYSIS OF EXPERIMENTAL DATA

A Experimental data

The electromagnet depicted in Fig 1 rotated in skew posi-

tion was experimentally characterized by HP method Table I

presents the values of radial magnetic flux density measured

Fig 2 Radial component of the magnetic flux density

Fig 3 Azimuthal component of the magnetic flux density

Fig 4 The module of the magnetic flux density

along horisontal Ox axes for different excitation currents

between I = 50A and I = 580A The measured data are

plotted in Fig 5 6 7

Let consider the spatial variation of radial magnetic field

for a reference current I 0 = 300A

Br = [000 00034 00181 00426 00770 01216]

xprime

= [0 02000 04000 06000 08000 10000]

(6)

Where xprime = rr0 with r0 = 005m and Br are the average

values for symmetric positive and negative coordinates Since

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 36

TABLE IEXPERIMENTAL DATA OF Br[T] [ 8]

x[m] I 1 = 50A I 2 = 100A I 3 = 200A I 4 = 300A I 5 = 400A I 6 = 500A I 7 = 580A-005 0020242 0039452 0078291 0117118 0155458 0191329 0213700

-004 0012976 0024993 0049257 0073475 0097344 0119578 0133432

-003 0007351 0013833 0026891 0039899 0052654 0064424 0071698

-002 0003409 0006005 0011156 0016300 0021297 0025759 0028406

-001 0001107 0001415 0002014 0002574 0003061 0003299 0003318000 0000457 0000109 0000648 0001424 0002211 0003188 0003892

001 0001410 0002012 0003157 0004308 0005432 0006240 0006703

002 0004000 0007174 0013530 0019865 0026131 0031800 0035338

003 0008261 0015618 0030443 0045250 0059941 0073532 0082093

004 0014156 0027311 0053920 0080519 0106886 0131478 0147069

005 0021755 0042434 0084248 0126067 0167577 0206448 0230987

Fig 5 Radial magnetic flux density vs current

Fig 6 Radial magnetic flux density vs radius

the first value is perturbed by an obvious error it was replaced

with Br = 0 the correct value due to the symmetry

B Polynomial approximation

The spatial variation of magnetic field Brminusexp for the

reference current (6) was fitted with a 10th degree even

Fig 7 Radial magnetic flux density vs radius and current

polynomial function

Brminusmod(xprime) = c2xprime2 + c4xprime4 + c6xprime6 + c8xprime8 + c10xprime10 (7)

Using CFTOOL module of MATLAB the coefficients pre-

sented in Table IV were obtained for several polynomial

degrees

A graphical approximation of 10th degree polynomial ap-

proximation compared with experimental data is given in Fig

8 Relative standard deviation estimated with

ε =maxk=111(Brminusexpk minus Brminusmodk)

maxk=111

(Brminusexpk) (8)

is 221

C Linear model

Figure 5 suggests that the magnetic core is not strongly

saturated If the magnetic field has a linear dependence versus

excitation current we can consider

Br = Bref

I

I 0 (9)

where Bref is the field corresponding to the reference current

I 0 It will be computed starting from the matrix in Table I

with n = 11 rows (corresponding to n measuring positions)

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 46

TABLE IITHE REFERENCE FIELD COEFFICIENT AND ITS STANDARD DEVIATION FOR SEVERAL POSITION AND CURRENTS

Bref (i j)[T]x[m] I II III IV V VI VII Bref std -005 01 215 0118 4 0 1174 011 71 01166 01 148 0110 5 0 1166 28 882

-004 00 779 0075 0 0 0739 007 35 00730 00 717 0069 0 0 0734 23 549

-003 00 441 0041 5 0 0403 003 99 00395 00 387 0037 1 0 0402 19 108

-002 00 205 0018 0 0 0167 001 63 00160 00 155 0014 7 0 0168 16 246-001 00 066 0004 2 0 0030 000 26 00023 00 020 0001 7 0 0032 14 414

000 00000 00000 00000 00000 00000 00000 00000 00000 00000

001 00085 00060 00047 00043 00041 00037 00035 00050 14566

002 00240 00215 00203 00199 00196 00191 00183 00204 15962

003 00496 00469 00457 00452 00450 00441 00425 00456 19052

004 00849 00819 00809 00805 00802 00789 00761 00805 23282

005 01305 01273 01264 01261 01257 01239 01195 01256 28915

Fig 8 Polynomial approximation versus measured values

and m = 7 columns (corresponding to m excitation currents)

which contain the n times m measured data with

Bref ij = Brij

I 0I j

(10)

The average value of these reference coefficients for each

measuring position is presented in the Table II (where is

included also its standard deviation)

If the current is considered smaller or equal to I 0 the

results are presented the Table III For currents lower than

the reference current the error of linear model is acceptablebeing under 11 So we can conclude that the reference

current do not saturates the magnetic core

TABLE IIITHE REFERENCE FIELD COEFFICIENT AND ITS STANDARD DEVIATION FOR

CURRENTS LOWER THAN I 0

Bref (i j)[T]

I II III IV Bref std

-005 01215 01184 01174 01171 01186 10153

-004 00779 00750 00739 00735 00750 10141

-003 00441 00415 00403 00399 00415 09693

-002 00205 00180 00167 00163 00179 09579

-001 00066 00042 00030 00026 00041 09357

000 00000 00000 00000 00000 00000 000000

001 00085 00060 00047 00043 00059 09583

002 00240 00215 00203 00199 00214 09532

003 00496 00469 00457 00452 00468 09978

004 00849 00819 00809 00805 00821 10287005 01305 01273 01264 01261 01276 10487

The errors under 3 in the Table II validate the linear model

(10) also for higher currents Therefore the radial flux density

of the characterized electromagnet along horizontal radius

within its aperture has following variation

Br(r I ) = f 1048616 r

r01048617 middot I

I 0 =983131

c21048616 r

r010486172

+ c41048616 r

r010486174

+

+c6

1048616 r

r0

10486176

+ c8

1048616 r

r0

10486178

+ c10

1048616 r

r0

104861710983133

I

I 0

(11)

TABLE IVFITTING WITH EVEN POLYNOMIALS

Degree II IV VI VIII X

c2 01208 01158 01093 00976 006939

c4 mdash 0005962 002744 01062 04759

c6 mdash mdash -001515 -01611 -1586

c8 mdash mdash mdash 00789 213

c10 mdash mdash mdash mdash -09678

suma 01208 01218 01216 01216 01215

sse 5048e minus 6 2037e minus 6 1141e minus 6 5333e minus 6 1642e minus 6

RMSE 0001005 00007136 00006167 00005164 4052 middot

10minus9

Fig 9 Even polynomials

D Least-squares method

The experimental data sets preprocessed by averaging val-

ues measured at symmetric positions were fitted with even

polynomials by using the least squares method The most

reliable model is that having 6th degree At higher grades

the coefficients do not decrease monotonically with the order

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 56

probably due to measurement error and numerical instability

results can be seen in the Table IV and in Fig 9 Similar

results can be seen in the Table V and in Fig 10 where the

results are obtained by using measurements in all 11 positive

and negative positions without previous averaging

Fig 10 Fitting of the 11 experimental values

TABLE VFITTING THE 11 EXPERIMENTAL VALUES WITH EVEN POLYNOMIALS

Degree II IV VI VIII X

c2 01208 01158 01093 00976 006939

c4 mdash 0005962 002744 01062 04759

c6 mdash mdash -001515 -01611 -1586

c8 mdash mdash mdash 00789 213

c10 mdash mdash mdash mdash -09678

suma 01208 01218 01216 01216 01215sse 9712e minus 8 911e minus 5 8931e minus 5 8809e minus 5 8703e minus 5

RMSE 0003116 0003182 0003341 0003547 0003808

IV THE FIELD DISTRIBUTION IN THE APERTURE BASED

ON PERIPHERAL MEASUREMENTS

Fig 11 The 30 degree slice extracted from circular aperture

A slice of 30 degree is extracted from circular aperture

disk (Fig 11) The boundaries of this sub-domain D are

bull C 1 on symmetry axe of the pole

bull C 2 on the polar inter-axes

bull C 3 on the aperture border with radius r0 = 50mm

Magnetic field satisfies the following boundary conditions

bull (c1) Bn = 0 on C 1 because the magnetic field lines are

tangential along this radial direction

bull (c2) H t = 0 on C 2 because the magnetic field lines are

perpendicular to this radial direction

bull (c3) Bn = Br known - in this procedure the radial flux

density is measured on C 3 - the aperture periphery

If we will consider that the angle ϕ originates from C 1 thenfrom (5) and (c1) follows that all coefficients Bn are zero

From (c2) and (5) we have

An cos983080nπ

6

983081 = 0 rArr

rArr (2k + 1)π

2 =

nπ

6 rArr n = 3(2k + 1)

(12)

For any value of the index n which is not a multiple of

three times odd numbers An = 0 Therefore the magnetic

scalar potential in aperture has only cos harmonics with orders

3 9 15 and the radial component has polynomial variation

vs radius of orders 2 8 14

Br(xprime) =sum852008

Anxprimenminus1 cos(nϕ)852009

n = 3 9 15 (13)

For ϕ = 0 it results following variation on C 1

Br(xprime) =sum

(Anxprimenminus1) =

= A3xprime2 + A9xprime8 + middot middot middot = c2xprime2 + c8xprime8 + middot middot middot(14)

The presence of non-zero terms c4xprime4 c6xprime6 and c10xprime10 in (7)

has three possible explanations

bull structural asymmetries

bull measurement errors

bull computational errors

V THE FIELD DISTRIBUTION IN THE APERTURE BASED ON

RADIAL MEASUREMENTS

If these errors would be zero then the coefficients c2 c8 c14extracted from the measurement determining the Fourier co-

efficients of harmonic solution A3 = c2 A9 = c8 A15 = c14

With them using (5) can be determining the variation of the

flux density vector field throughout the aperture Magnetic

field distribution in the electromagnetrsquos aperture can be deter-

mined with (5) where coefficients An Bn can be computed

by using the model (11) extracted from experimental data

It should be determined the coefficients c2 c8 and c14 from

the experimental data looking for the the best approximationof a polynomial with these coefficients (ascending order

2 8 14)

Unfortunately the conditioning number of high polynomial

interpolation is extremely high According Gautschrsquos theorem

in our case this number has the approximate value κ = (1 +radic 2)n+1 = 16238 for n = 10 [9]

This means that a disturbance of 1 in experimental data

is amplified ten thousand times in the computed polynomial

approximation This behavior makes totally irrelevant the

numerical result

In this approach it was actually solved the Laplace equation

with Dirichlet conditions on C 1 and C 2 Moreover there are

not boundary conditions on C 3 but instead an additional Neu-

mann condition was imposed on C 1 Although this problem

has an unique solution it is not well formulated problem

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 66

because it is numerically unstable [10] Moreover this problem

is very similar with Hadamardrsquos famous counterexample [11]

Unlike this problem if Dirichlet or Neumann the boundary

conditions are imposed in each point of the domainrsquos border

D including on C 3 as in section V then the problem is well-

formulated in Hadamard sense Consequently its conditioning

number is not greater than 1 [10]

V I CONCLUSIONS

The field problem based on peripheral measurements is well

conditioned because small changes in boundary data are not

amplified and the corresponding variations of the field in any

point within the aperture D has a relative error not greater

than the relative measurement error

It should be noted that from practical point of view the

effort to make measurements is the same With the same

number of measurements the error in determining the fieldinside aperture is much higher when measurement are used

along radius than in the case of using circular measurements

According to (14) the best polynomial approximation with

three terms is

Br(xprime) = c2xprime2 + c8xprime8 + c14xprime14 (15)

Their coefficients are represented in Table VI and its graphic

is depicted in Figure 12 However as it was proven this is not

a reliable result because its conditioning number is 551614

TABLE VIFITTING WITH THE POLYNOMIAL Br(xprime) = c2x

prime2 + c8xprime8 + c14x

prime14

c2 01161

c8 002028

c14 -001479

sum 012159

sse 1910 middot 10minus6

RMSE 00007978

Fig 12 Fitting of experimental values with the polynomial Br(xprime) =c2x

prime2 + c8xprime8 + c14x

prime14

REFERENCES

[1] S Russenschuck Field Computation for Accelerator Magnets Analyti-cal and Numerical Methods for Electromagnetic Design and Optimiza-tion John Wiley and Sons May 11 2010

[2] L Walckiers ldquoMagnetic measurement with coils and wiresrdquo CERN-2010-004 pp 357ndash385 March 2011 httparxivorgabs11043784v1

[3] A Temnykh Y Levashovb and Z Wolf ldquoA study of undulator magnetscharacterization using the vibrating wire techniquerdquo Stanfort - LCLS-TN-10-2

[4] A K Jain Measurements of Field Quality Using HarmonicCoils httpwwwbnlgovmagnetsstaffguptascmag-courseuspas01 AJ01HarmonicCoil_Slidespdf

[5] D Dan and D Ioan ldquoEroarea de metoda icircn determinarea experimen-tal analitica a cacircmpului unui electromanget pentru acceleratoarele departiculerdquo SNET12 Bucharest 2012 httpsnetelthpubrosnet2012

[6] ICPE-CA ldquoElectromagneti si surse de alimentare pentru proiectul fair- poveste de succes pentru icpe - cardquo 2012

[7] I G Shabac Matematici speciale 2 Bucuresti Romacircnia EdituraDidactica si Pedagogica 1965

[8] ICPE-CA Institutul National de Cercetare Proiectare pentru Ingineriaelectric˘ a httpwwwicpe-carorohttpwwwicpe-caroro

[9] K A Gallivan ldquoPolynomial interpolationrdquo 2013[10] S Zaglmayr High Order Finite Element Methods for Electromagnetic

Field Computation Thesis - Linz Univ 2006[11] R Corless Elementary Partial Differential Equations - Hadamardrsquos

example httpwwwapmathsuwoca 1998

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 36

TABLE IEXPERIMENTAL DATA OF Br[T] [ 8]

x[m] I 1 = 50A I 2 = 100A I 3 = 200A I 4 = 300A I 5 = 400A I 6 = 500A I 7 = 580A-005 0020242 0039452 0078291 0117118 0155458 0191329 0213700

-004 0012976 0024993 0049257 0073475 0097344 0119578 0133432

-003 0007351 0013833 0026891 0039899 0052654 0064424 0071698

-002 0003409 0006005 0011156 0016300 0021297 0025759 0028406

-001 0001107 0001415 0002014 0002574 0003061 0003299 0003318000 0000457 0000109 0000648 0001424 0002211 0003188 0003892

001 0001410 0002012 0003157 0004308 0005432 0006240 0006703

002 0004000 0007174 0013530 0019865 0026131 0031800 0035338

003 0008261 0015618 0030443 0045250 0059941 0073532 0082093

004 0014156 0027311 0053920 0080519 0106886 0131478 0147069

005 0021755 0042434 0084248 0126067 0167577 0206448 0230987

Fig 5 Radial magnetic flux density vs current

Fig 6 Radial magnetic flux density vs radius

the first value is perturbed by an obvious error it was replaced

with Br = 0 the correct value due to the symmetry

B Polynomial approximation

The spatial variation of magnetic field Brminusexp for the

reference current (6) was fitted with a 10th degree even

Fig 7 Radial magnetic flux density vs radius and current

polynomial function

Brminusmod(xprime) = c2xprime2 + c4xprime4 + c6xprime6 + c8xprime8 + c10xprime10 (7)

Using CFTOOL module of MATLAB the coefficients pre-

sented in Table IV were obtained for several polynomial

degrees

A graphical approximation of 10th degree polynomial ap-

proximation compared with experimental data is given in Fig

8 Relative standard deviation estimated with

ε =maxk=111(Brminusexpk minus Brminusmodk)

maxk=111

(Brminusexpk) (8)

is 221

C Linear model

Figure 5 suggests that the magnetic core is not strongly

saturated If the magnetic field has a linear dependence versus

excitation current we can consider

Br = Bref

I

I 0 (9)

where Bref is the field corresponding to the reference current

I 0 It will be computed starting from the matrix in Table I

with n = 11 rows (corresponding to n measuring positions)

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 46

TABLE IITHE REFERENCE FIELD COEFFICIENT AND ITS STANDARD DEVIATION FOR SEVERAL POSITION AND CURRENTS

Bref (i j)[T]x[m] I II III IV V VI VII Bref std -005 01 215 0118 4 0 1174 011 71 01166 01 148 0110 5 0 1166 28 882

-004 00 779 0075 0 0 0739 007 35 00730 00 717 0069 0 0 0734 23 549

-003 00 441 0041 5 0 0403 003 99 00395 00 387 0037 1 0 0402 19 108

-002 00 205 0018 0 0 0167 001 63 00160 00 155 0014 7 0 0168 16 246-001 00 066 0004 2 0 0030 000 26 00023 00 020 0001 7 0 0032 14 414

000 00000 00000 00000 00000 00000 00000 00000 00000 00000

001 00085 00060 00047 00043 00041 00037 00035 00050 14566

002 00240 00215 00203 00199 00196 00191 00183 00204 15962

003 00496 00469 00457 00452 00450 00441 00425 00456 19052

004 00849 00819 00809 00805 00802 00789 00761 00805 23282

005 01305 01273 01264 01261 01257 01239 01195 01256 28915

Fig 8 Polynomial approximation versus measured values

and m = 7 columns (corresponding to m excitation currents)

which contain the n times m measured data with

Bref ij = Brij

I 0I j

(10)

The average value of these reference coefficients for each

measuring position is presented in the Table II (where is

included also its standard deviation)

If the current is considered smaller or equal to I 0 the

results are presented the Table III For currents lower than

the reference current the error of linear model is acceptablebeing under 11 So we can conclude that the reference

current do not saturates the magnetic core

TABLE IIITHE REFERENCE FIELD COEFFICIENT AND ITS STANDARD DEVIATION FOR

CURRENTS LOWER THAN I 0

Bref (i j)[T]

I II III IV Bref std

-005 01215 01184 01174 01171 01186 10153

-004 00779 00750 00739 00735 00750 10141

-003 00441 00415 00403 00399 00415 09693

-002 00205 00180 00167 00163 00179 09579

-001 00066 00042 00030 00026 00041 09357

000 00000 00000 00000 00000 00000 000000

001 00085 00060 00047 00043 00059 09583

002 00240 00215 00203 00199 00214 09532

003 00496 00469 00457 00452 00468 09978

004 00849 00819 00809 00805 00821 10287005 01305 01273 01264 01261 01276 10487

The errors under 3 in the Table II validate the linear model

(10) also for higher currents Therefore the radial flux density

of the characterized electromagnet along horizontal radius

within its aperture has following variation

Br(r I ) = f 1048616 r

r01048617 middot I

I 0 =983131

c21048616 r

r010486172

+ c41048616 r

r010486174

+

+c6

1048616 r

r0

10486176

+ c8

1048616 r

r0

10486178

+ c10

1048616 r

r0

104861710983133

I

I 0

(11)

TABLE IVFITTING WITH EVEN POLYNOMIALS

Degree II IV VI VIII X

c2 01208 01158 01093 00976 006939

c4 mdash 0005962 002744 01062 04759

c6 mdash mdash -001515 -01611 -1586

c8 mdash mdash mdash 00789 213

c10 mdash mdash mdash mdash -09678

suma 01208 01218 01216 01216 01215

sse 5048e minus 6 2037e minus 6 1141e minus 6 5333e minus 6 1642e minus 6

RMSE 0001005 00007136 00006167 00005164 4052 middot

10minus9

Fig 9 Even polynomials

D Least-squares method

The experimental data sets preprocessed by averaging val-

ues measured at symmetric positions were fitted with even

polynomials by using the least squares method The most

reliable model is that having 6th degree At higher grades

the coefficients do not decrease monotonically with the order

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 56

probably due to measurement error and numerical instability

results can be seen in the Table IV and in Fig 9 Similar

results can be seen in the Table V and in Fig 10 where the

results are obtained by using measurements in all 11 positive

and negative positions without previous averaging

Fig 10 Fitting of the 11 experimental values

TABLE VFITTING THE 11 EXPERIMENTAL VALUES WITH EVEN POLYNOMIALS

Degree II IV VI VIII X

c2 01208 01158 01093 00976 006939

c4 mdash 0005962 002744 01062 04759

c6 mdash mdash -001515 -01611 -1586

c8 mdash mdash mdash 00789 213

c10 mdash mdash mdash mdash -09678

suma 01208 01218 01216 01216 01215sse 9712e minus 8 911e minus 5 8931e minus 5 8809e minus 5 8703e minus 5

RMSE 0003116 0003182 0003341 0003547 0003808

IV THE FIELD DISTRIBUTION IN THE APERTURE BASED

ON PERIPHERAL MEASUREMENTS

Fig 11 The 30 degree slice extracted from circular aperture

A slice of 30 degree is extracted from circular aperture

disk (Fig 11) The boundaries of this sub-domain D are

bull C 1 on symmetry axe of the pole

bull C 2 on the polar inter-axes

bull C 3 on the aperture border with radius r0 = 50mm

Magnetic field satisfies the following boundary conditions

bull (c1) Bn = 0 on C 1 because the magnetic field lines are

tangential along this radial direction

bull (c2) H t = 0 on C 2 because the magnetic field lines are

perpendicular to this radial direction

bull (c3) Bn = Br known - in this procedure the radial flux

density is measured on C 3 - the aperture periphery

If we will consider that the angle ϕ originates from C 1 thenfrom (5) and (c1) follows that all coefficients Bn are zero

From (c2) and (5) we have

An cos983080nπ

6

983081 = 0 rArr

rArr (2k + 1)π

2 =

nπ

6 rArr n = 3(2k + 1)

(12)

For any value of the index n which is not a multiple of

three times odd numbers An = 0 Therefore the magnetic

scalar potential in aperture has only cos harmonics with orders

3 9 15 and the radial component has polynomial variation

vs radius of orders 2 8 14

Br(xprime) =sum852008

Anxprimenminus1 cos(nϕ)852009

n = 3 9 15 (13)

For ϕ = 0 it results following variation on C 1

Br(xprime) =sum

(Anxprimenminus1) =

= A3xprime2 + A9xprime8 + middot middot middot = c2xprime2 + c8xprime8 + middot middot middot(14)

The presence of non-zero terms c4xprime4 c6xprime6 and c10xprime10 in (7)

has three possible explanations

bull structural asymmetries

bull measurement errors

bull computational errors

V THE FIELD DISTRIBUTION IN THE APERTURE BASED ON

RADIAL MEASUREMENTS

If these errors would be zero then the coefficients c2 c8 c14extracted from the measurement determining the Fourier co-

efficients of harmonic solution A3 = c2 A9 = c8 A15 = c14

With them using (5) can be determining the variation of the

flux density vector field throughout the aperture Magnetic

field distribution in the electromagnetrsquos aperture can be deter-

mined with (5) where coefficients An Bn can be computed

by using the model (11) extracted from experimental data

It should be determined the coefficients c2 c8 and c14 from

the experimental data looking for the the best approximationof a polynomial with these coefficients (ascending order

2 8 14)

Unfortunately the conditioning number of high polynomial

interpolation is extremely high According Gautschrsquos theorem

in our case this number has the approximate value κ = (1 +radic 2)n+1 = 16238 for n = 10 [9]

This means that a disturbance of 1 in experimental data

is amplified ten thousand times in the computed polynomial

approximation This behavior makes totally irrelevant the

numerical result

In this approach it was actually solved the Laplace equation

with Dirichlet conditions on C 1 and C 2 Moreover there are

not boundary conditions on C 3 but instead an additional Neu-

mann condition was imposed on C 1 Although this problem

has an unique solution it is not well formulated problem

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 66

because it is numerically unstable [10] Moreover this problem

is very similar with Hadamardrsquos famous counterexample [11]

Unlike this problem if Dirichlet or Neumann the boundary

conditions are imposed in each point of the domainrsquos border

D including on C 3 as in section V then the problem is well-

formulated in Hadamard sense Consequently its conditioning

number is not greater than 1 [10]

V I CONCLUSIONS

The field problem based on peripheral measurements is well

conditioned because small changes in boundary data are not

amplified and the corresponding variations of the field in any

point within the aperture D has a relative error not greater

than the relative measurement error

It should be noted that from practical point of view the

effort to make measurements is the same With the same

number of measurements the error in determining the fieldinside aperture is much higher when measurement are used

along radius than in the case of using circular measurements

According to (14) the best polynomial approximation with

three terms is

Br(xprime) = c2xprime2 + c8xprime8 + c14xprime14 (15)

Their coefficients are represented in Table VI and its graphic

is depicted in Figure 12 However as it was proven this is not

a reliable result because its conditioning number is 551614

TABLE VIFITTING WITH THE POLYNOMIAL Br(xprime) = c2x

prime2 + c8xprime8 + c14x

prime14

c2 01161

c8 002028

c14 -001479

sum 012159

sse 1910 middot 10minus6

RMSE 00007978

Fig 12 Fitting of experimental values with the polynomial Br(xprime) =c2x

prime2 + c8xprime8 + c14x

prime14

REFERENCES

[1] S Russenschuck Field Computation for Accelerator Magnets Analyti-cal and Numerical Methods for Electromagnetic Design and Optimiza-tion John Wiley and Sons May 11 2010

[2] L Walckiers ldquoMagnetic measurement with coils and wiresrdquo CERN-2010-004 pp 357ndash385 March 2011 httparxivorgabs11043784v1

[3] A Temnykh Y Levashovb and Z Wolf ldquoA study of undulator magnetscharacterization using the vibrating wire techniquerdquo Stanfort - LCLS-TN-10-2

[4] A K Jain Measurements of Field Quality Using HarmonicCoils httpwwwbnlgovmagnetsstaffguptascmag-courseuspas01 AJ01HarmonicCoil_Slidespdf

[5] D Dan and D Ioan ldquoEroarea de metoda icircn determinarea experimen-tal analitica a cacircmpului unui electromanget pentru acceleratoarele departiculerdquo SNET12 Bucharest 2012 httpsnetelthpubrosnet2012

[6] ICPE-CA ldquoElectromagneti si surse de alimentare pentru proiectul fair- poveste de succes pentru icpe - cardquo 2012

[7] I G Shabac Matematici speciale 2 Bucuresti Romacircnia EdituraDidactica si Pedagogica 1965

[8] ICPE-CA Institutul National de Cercetare Proiectare pentru Ingineriaelectric˘ a httpwwwicpe-carorohttpwwwicpe-caroro

[9] K A Gallivan ldquoPolynomial interpolationrdquo 2013[10] S Zaglmayr High Order Finite Element Methods for Electromagnetic

Field Computation Thesis - Linz Univ 2006[11] R Corless Elementary Partial Differential Equations - Hadamardrsquos

example httpwwwapmathsuwoca 1998

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 46

TABLE IITHE REFERENCE FIELD COEFFICIENT AND ITS STANDARD DEVIATION FOR SEVERAL POSITION AND CURRENTS

Bref (i j)[T]x[m] I II III IV V VI VII Bref std -005 01 215 0118 4 0 1174 011 71 01166 01 148 0110 5 0 1166 28 882

-004 00 779 0075 0 0 0739 007 35 00730 00 717 0069 0 0 0734 23 549

-003 00 441 0041 5 0 0403 003 99 00395 00 387 0037 1 0 0402 19 108

-002 00 205 0018 0 0 0167 001 63 00160 00 155 0014 7 0 0168 16 246-001 00 066 0004 2 0 0030 000 26 00023 00 020 0001 7 0 0032 14 414

000 00000 00000 00000 00000 00000 00000 00000 00000 00000

001 00085 00060 00047 00043 00041 00037 00035 00050 14566

002 00240 00215 00203 00199 00196 00191 00183 00204 15962

003 00496 00469 00457 00452 00450 00441 00425 00456 19052

004 00849 00819 00809 00805 00802 00789 00761 00805 23282

005 01305 01273 01264 01261 01257 01239 01195 01256 28915

Fig 8 Polynomial approximation versus measured values

and m = 7 columns (corresponding to m excitation currents)

which contain the n times m measured data with

Bref ij = Brij

I 0I j

(10)

The average value of these reference coefficients for each

measuring position is presented in the Table II (where is

included also its standard deviation)

If the current is considered smaller or equal to I 0 the

results are presented the Table III For currents lower than

the reference current the error of linear model is acceptablebeing under 11 So we can conclude that the reference

current do not saturates the magnetic core

TABLE IIITHE REFERENCE FIELD COEFFICIENT AND ITS STANDARD DEVIATION FOR

CURRENTS LOWER THAN I 0

Bref (i j)[T]

I II III IV Bref std

-005 01215 01184 01174 01171 01186 10153

-004 00779 00750 00739 00735 00750 10141

-003 00441 00415 00403 00399 00415 09693

-002 00205 00180 00167 00163 00179 09579

-001 00066 00042 00030 00026 00041 09357

000 00000 00000 00000 00000 00000 000000

001 00085 00060 00047 00043 00059 09583

002 00240 00215 00203 00199 00214 09532

003 00496 00469 00457 00452 00468 09978

004 00849 00819 00809 00805 00821 10287005 01305 01273 01264 01261 01276 10487

The errors under 3 in the Table II validate the linear model

(10) also for higher currents Therefore the radial flux density

of the characterized electromagnet along horizontal radius

within its aperture has following variation

Br(r I ) = f 1048616 r

r01048617 middot I

I 0 =983131

c21048616 r

r010486172

+ c41048616 r

r010486174

+

+c6

1048616 r

r0

10486176

+ c8

1048616 r

r0

10486178

+ c10

1048616 r

r0

104861710983133

I

I 0

(11)

TABLE IVFITTING WITH EVEN POLYNOMIALS

Degree II IV VI VIII X

c2 01208 01158 01093 00976 006939

c4 mdash 0005962 002744 01062 04759

c6 mdash mdash -001515 -01611 -1586

c8 mdash mdash mdash 00789 213

c10 mdash mdash mdash mdash -09678

suma 01208 01218 01216 01216 01215

sse 5048e minus 6 2037e minus 6 1141e minus 6 5333e minus 6 1642e minus 6

RMSE 0001005 00007136 00006167 00005164 4052 middot

10minus9

Fig 9 Even polynomials

D Least-squares method

The experimental data sets preprocessed by averaging val-

ues measured at symmetric positions were fitted with even

polynomials by using the least squares method The most

reliable model is that having 6th degree At higher grades

the coefficients do not decrease monotonically with the order

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 56

probably due to measurement error and numerical instability

results can be seen in the Table IV and in Fig 9 Similar

results can be seen in the Table V and in Fig 10 where the

results are obtained by using measurements in all 11 positive

and negative positions without previous averaging

Fig 10 Fitting of the 11 experimental values

TABLE VFITTING THE 11 EXPERIMENTAL VALUES WITH EVEN POLYNOMIALS

Degree II IV VI VIII X

c2 01208 01158 01093 00976 006939

c4 mdash 0005962 002744 01062 04759

c6 mdash mdash -001515 -01611 -1586

c8 mdash mdash mdash 00789 213

c10 mdash mdash mdash mdash -09678

suma 01208 01218 01216 01216 01215sse 9712e minus 8 911e minus 5 8931e minus 5 8809e minus 5 8703e minus 5

RMSE 0003116 0003182 0003341 0003547 0003808

IV THE FIELD DISTRIBUTION IN THE APERTURE BASED

ON PERIPHERAL MEASUREMENTS

Fig 11 The 30 degree slice extracted from circular aperture

A slice of 30 degree is extracted from circular aperture

disk (Fig 11) The boundaries of this sub-domain D are

bull C 1 on symmetry axe of the pole

bull C 2 on the polar inter-axes

bull C 3 on the aperture border with radius r0 = 50mm

Magnetic field satisfies the following boundary conditions

bull (c1) Bn = 0 on C 1 because the magnetic field lines are

tangential along this radial direction

bull (c2) H t = 0 on C 2 because the magnetic field lines are

perpendicular to this radial direction

bull (c3) Bn = Br known - in this procedure the radial flux

density is measured on C 3 - the aperture periphery

If we will consider that the angle ϕ originates from C 1 thenfrom (5) and (c1) follows that all coefficients Bn are zero

From (c2) and (5) we have

An cos983080nπ

6

983081 = 0 rArr

rArr (2k + 1)π

2 =

nπ

6 rArr n = 3(2k + 1)

(12)

For any value of the index n which is not a multiple of

three times odd numbers An = 0 Therefore the magnetic

scalar potential in aperture has only cos harmonics with orders

3 9 15 and the radial component has polynomial variation

vs radius of orders 2 8 14

Br(xprime) =sum852008

Anxprimenminus1 cos(nϕ)852009

n = 3 9 15 (13)

For ϕ = 0 it results following variation on C 1

Br(xprime) =sum

(Anxprimenminus1) =

= A3xprime2 + A9xprime8 + middot middot middot = c2xprime2 + c8xprime8 + middot middot middot(14)

The presence of non-zero terms c4xprime4 c6xprime6 and c10xprime10 in (7)

has three possible explanations

bull structural asymmetries

bull measurement errors

bull computational errors

V THE FIELD DISTRIBUTION IN THE APERTURE BASED ON

RADIAL MEASUREMENTS

If these errors would be zero then the coefficients c2 c8 c14extracted from the measurement determining the Fourier co-

efficients of harmonic solution A3 = c2 A9 = c8 A15 = c14

With them using (5) can be determining the variation of the

flux density vector field throughout the aperture Magnetic

field distribution in the electromagnetrsquos aperture can be deter-

mined with (5) where coefficients An Bn can be computed

by using the model (11) extracted from experimental data

It should be determined the coefficients c2 c8 and c14 from

the experimental data looking for the the best approximationof a polynomial with these coefficients (ascending order

2 8 14)

Unfortunately the conditioning number of high polynomial

interpolation is extremely high According Gautschrsquos theorem

in our case this number has the approximate value κ = (1 +radic 2)n+1 = 16238 for n = 10 [9]

This means that a disturbance of 1 in experimental data

is amplified ten thousand times in the computed polynomial

approximation This behavior makes totally irrelevant the

numerical result

In this approach it was actually solved the Laplace equation

with Dirichlet conditions on C 1 and C 2 Moreover there are

not boundary conditions on C 3 but instead an additional Neu-

mann condition was imposed on C 1 Although this problem

has an unique solution it is not well formulated problem

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 66

because it is numerically unstable [10] Moreover this problem

is very similar with Hadamardrsquos famous counterexample [11]

Unlike this problem if Dirichlet or Neumann the boundary

conditions are imposed in each point of the domainrsquos border

D including on C 3 as in section V then the problem is well-

formulated in Hadamard sense Consequently its conditioning

number is not greater than 1 [10]

V I CONCLUSIONS

The field problem based on peripheral measurements is well

conditioned because small changes in boundary data are not

amplified and the corresponding variations of the field in any

point within the aperture D has a relative error not greater

than the relative measurement error

It should be noted that from practical point of view the

effort to make measurements is the same With the same

number of measurements the error in determining the fieldinside aperture is much higher when measurement are used

along radius than in the case of using circular measurements

According to (14) the best polynomial approximation with

three terms is

Br(xprime) = c2xprime2 + c8xprime8 + c14xprime14 (15)

Their coefficients are represented in Table VI and its graphic

is depicted in Figure 12 However as it was proven this is not

a reliable result because its conditioning number is 551614

TABLE VIFITTING WITH THE POLYNOMIAL Br(xprime) = c2x

prime2 + c8xprime8 + c14x

prime14

c2 01161

c8 002028

c14 -001479

sum 012159

sse 1910 middot 10minus6

RMSE 00007978

Fig 12 Fitting of experimental values with the polynomial Br(xprime) =c2x

prime2 + c8xprime8 + c14x

prime14

REFERENCES

[1] S Russenschuck Field Computation for Accelerator Magnets Analyti-cal and Numerical Methods for Electromagnetic Design and Optimiza-tion John Wiley and Sons May 11 2010

[2] L Walckiers ldquoMagnetic measurement with coils and wiresrdquo CERN-2010-004 pp 357ndash385 March 2011 httparxivorgabs11043784v1

[3] A Temnykh Y Levashovb and Z Wolf ldquoA study of undulator magnetscharacterization using the vibrating wire techniquerdquo Stanfort - LCLS-TN-10-2

[4] A K Jain Measurements of Field Quality Using HarmonicCoils httpwwwbnlgovmagnetsstaffguptascmag-courseuspas01 AJ01HarmonicCoil_Slidespdf

[5] D Dan and D Ioan ldquoEroarea de metoda icircn determinarea experimen-tal analitica a cacircmpului unui electromanget pentru acceleratoarele departiculerdquo SNET12 Bucharest 2012 httpsnetelthpubrosnet2012

[6] ICPE-CA ldquoElectromagneti si surse de alimentare pentru proiectul fair- poveste de succes pentru icpe - cardquo 2012

[7] I G Shabac Matematici speciale 2 Bucuresti Romacircnia EdituraDidactica si Pedagogica 1965

[8] ICPE-CA Institutul National de Cercetare Proiectare pentru Ingineriaelectric˘ a httpwwwicpe-carorohttpwwwicpe-caroro

[9] K A Gallivan ldquoPolynomial interpolationrdquo 2013[10] S Zaglmayr High Order Finite Element Methods for Electromagnetic

Field Computation Thesis - Linz Univ 2006[11] R Corless Elementary Partial Differential Equations - Hadamardrsquos

example httpwwwapmathsuwoca 1998

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 56

probably due to measurement error and numerical instability

results can be seen in the Table IV and in Fig 9 Similar

results can be seen in the Table V and in Fig 10 where the

results are obtained by using measurements in all 11 positive

and negative positions without previous averaging

Fig 10 Fitting of the 11 experimental values

TABLE VFITTING THE 11 EXPERIMENTAL VALUES WITH EVEN POLYNOMIALS

Degree II IV VI VIII X

c2 01208 01158 01093 00976 006939

c4 mdash 0005962 002744 01062 04759

c6 mdash mdash -001515 -01611 -1586

c8 mdash mdash mdash 00789 213

c10 mdash mdash mdash mdash -09678

suma 01208 01218 01216 01216 01215sse 9712e minus 8 911e minus 5 8931e minus 5 8809e minus 5 8703e minus 5

RMSE 0003116 0003182 0003341 0003547 0003808

IV THE FIELD DISTRIBUTION IN THE APERTURE BASED

ON PERIPHERAL MEASUREMENTS

Fig 11 The 30 degree slice extracted from circular aperture

A slice of 30 degree is extracted from circular aperture

disk (Fig 11) The boundaries of this sub-domain D are

bull C 1 on symmetry axe of the pole

bull C 2 on the polar inter-axes

bull C 3 on the aperture border with radius r0 = 50mm

Magnetic field satisfies the following boundary conditions

bull (c1) Bn = 0 on C 1 because the magnetic field lines are

tangential along this radial direction

bull (c2) H t = 0 on C 2 because the magnetic field lines are

perpendicular to this radial direction

bull (c3) Bn = Br known - in this procedure the radial flux

density is measured on C 3 - the aperture periphery

If we will consider that the angle ϕ originates from C 1 thenfrom (5) and (c1) follows that all coefficients Bn are zero

From (c2) and (5) we have

An cos983080nπ

6

983081 = 0 rArr

rArr (2k + 1)π

2 =

nπ

6 rArr n = 3(2k + 1)

(12)

For any value of the index n which is not a multiple of

three times odd numbers An = 0 Therefore the magnetic

scalar potential in aperture has only cos harmonics with orders

3 9 15 and the radial component has polynomial variation

vs radius of orders 2 8 14

Br(xprime) =sum852008

Anxprimenminus1 cos(nϕ)852009

n = 3 9 15 (13)

For ϕ = 0 it results following variation on C 1

Br(xprime) =sum

(Anxprimenminus1) =

= A3xprime2 + A9xprime8 + middot middot middot = c2xprime2 + c8xprime8 + middot middot middot(14)

The presence of non-zero terms c4xprime4 c6xprime6 and c10xprime10 in (7)

has three possible explanations

bull structural asymmetries

bull measurement errors

bull computational errors

V THE FIELD DISTRIBUTION IN THE APERTURE BASED ON

RADIAL MEASUREMENTS

If these errors would be zero then the coefficients c2 c8 c14extracted from the measurement determining the Fourier co-

efficients of harmonic solution A3 = c2 A9 = c8 A15 = c14

With them using (5) can be determining the variation of the

flux density vector field throughout the aperture Magnetic

field distribution in the electromagnetrsquos aperture can be deter-

mined with (5) where coefficients An Bn can be computed

by using the model (11) extracted from experimental data

It should be determined the coefficients c2 c8 and c14 from

the experimental data looking for the the best approximationof a polynomial with these coefficients (ascending order

2 8 14)

Unfortunately the conditioning number of high polynomial

interpolation is extremely high According Gautschrsquos theorem

in our case this number has the approximate value κ = (1 +radic 2)n+1 = 16238 for n = 10 [9]

This means that a disturbance of 1 in experimental data

is amplified ten thousand times in the computed polynomial

approximation This behavior makes totally irrelevant the

numerical result

In this approach it was actually solved the Laplace equation

with Dirichlet conditions on C 1 and C 2 Moreover there are

not boundary conditions on C 3 but instead an additional Neu-

mann condition was imposed on C 1 Although this problem

has an unique solution it is not well formulated problem

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 66

because it is numerically unstable [10] Moreover this problem

is very similar with Hadamardrsquos famous counterexample [11]

Unlike this problem if Dirichlet or Neumann the boundary

conditions are imposed in each point of the domainrsquos border

D including on C 3 as in section V then the problem is well-

formulated in Hadamard sense Consequently its conditioning

number is not greater than 1 [10]

V I CONCLUSIONS

The field problem based on peripheral measurements is well

conditioned because small changes in boundary data are not

amplified and the corresponding variations of the field in any

point within the aperture D has a relative error not greater

than the relative measurement error

It should be noted that from practical point of view the

effort to make measurements is the same With the same

number of measurements the error in determining the fieldinside aperture is much higher when measurement are used

along radius than in the case of using circular measurements

According to (14) the best polynomial approximation with

three terms is

Br(xprime) = c2xprime2 + c8xprime8 + c14xprime14 (15)

Their coefficients are represented in Table VI and its graphic

is depicted in Figure 12 However as it was proven this is not

a reliable result because its conditioning number is 551614

TABLE VIFITTING WITH THE POLYNOMIAL Br(xprime) = c2x

prime2 + c8xprime8 + c14x

prime14

c2 01161

c8 002028

c14 -001479

sum 012159

sse 1910 middot 10minus6

RMSE 00007978

Fig 12 Fitting of experimental values with the polynomial Br(xprime) =c2x

prime2 + c8xprime8 + c14x

prime14

REFERENCES

[1] S Russenschuck Field Computation for Accelerator Magnets Analyti-cal and Numerical Methods for Electromagnetic Design and Optimiza-tion John Wiley and Sons May 11 2010

[2] L Walckiers ldquoMagnetic measurement with coils and wiresrdquo CERN-2010-004 pp 357ndash385 March 2011 httparxivorgabs11043784v1

[3] A Temnykh Y Levashovb and Z Wolf ldquoA study of undulator magnetscharacterization using the vibrating wire techniquerdquo Stanfort - LCLS-TN-10-2

[4] A K Jain Measurements of Field Quality Using HarmonicCoils httpwwwbnlgovmagnetsstaffguptascmag-courseuspas01 AJ01HarmonicCoil_Slidespdf

[5] D Dan and D Ioan ldquoEroarea de metoda icircn determinarea experimen-tal analitica a cacircmpului unui electromanget pentru acceleratoarele departiculerdquo SNET12 Bucharest 2012 httpsnetelthpubrosnet2012

[6] ICPE-CA ldquoElectromagneti si surse de alimentare pentru proiectul fair- poveste de succes pentru icpe - cardquo 2012

[7] I G Shabac Matematici speciale 2 Bucuresti Romacircnia EdituraDidactica si Pedagogica 1965

[8] ICPE-CA Institutul National de Cercetare Proiectare pentru Ingineriaelectric˘ a httpwwwicpe-carorohttpwwwicpe-caroro

[9] K A Gallivan ldquoPolynomial interpolationrdquo 2013[10] S Zaglmayr High Order Finite Element Methods for Electromagnetic

Field Computation Thesis - Linz Univ 2006[11] R Corless Elementary Partial Differential Equations - Hadamardrsquos

example httpwwwapmathsuwoca 1998

8112019 Campo Magnetico ensayos

httpslidepdfcomreaderfullcampo-magnetico-ensayos 66

because it is numerically unstable [10] Moreover this problem

is very similar with Hadamardrsquos famous counterexample [11]

Unlike this problem if Dirichlet or Neumann the boundary

conditions are imposed in each point of the domainrsquos border

D including on C 3 as in section V then the problem is well-

formulated in Hadamard sense Consequently its conditioning

number is not greater than 1 [10]

V I CONCLUSIONS

The field problem based on peripheral measurements is well

conditioned because small changes in boundary data are not

amplified and the corresponding variations of the field in any

point within the aperture D has a relative error not greater

than the relative measurement error

It should be noted that from practical point of view the

effort to make measurements is the same With the same

number of measurements the error in determining the fieldinside aperture is much higher when measurement are used

along radius than in the case of using circular measurements

According to (14) the best polynomial approximation with

three terms is

Br(xprime) = c2xprime2 + c8xprime8 + c14xprime14 (15)

Their coefficients are represented in Table VI and its graphic

is depicted in Figure 12 However as it was proven this is not

a reliable result because its conditioning number is 551614

TABLE VIFITTING WITH THE POLYNOMIAL Br(xprime) = c2x

prime2 + c8xprime8 + c14x

prime14

c2 01161

c8 002028

c14 -001479

sum 012159

sse 1910 middot 10minus6

RMSE 00007978

Fig 12 Fitting of experimental values with the polynomial Br(xprime) =c2x

prime2 + c8xprime8 + c14x

prime14

REFERENCES

[1] S Russenschuck Field Computation for Accelerator Magnets Analyti-cal and Numerical Methods for Electromagnetic Design and Optimiza-tion John Wiley and Sons May 11 2010

[2] L Walckiers ldquoMagnetic measurement with coils and wiresrdquo CERN-2010-004 pp 357ndash385 March 2011 httparxivorgabs11043784v1

[3] A Temnykh Y Levashovb and Z Wolf ldquoA study of undulator magnetscharacterization using the vibrating wire techniquerdquo Stanfort - LCLS-TN-10-2

[4] A K Jain Measurements of Field Quality Using HarmonicCoils httpwwwbnlgovmagnetsstaffguptascmag-courseuspas01 AJ01HarmonicCoil_Slidespdf

[5] D Dan and D Ioan ldquoEroarea de metoda icircn determinarea experimen-tal analitica a cacircmpului unui electromanget pentru acceleratoarele departiculerdquo SNET12 Bucharest 2012 httpsnetelthpubrosnet2012

[6] ICPE-CA ldquoElectromagneti si surse de alimentare pentru proiectul fair- poveste de succes pentru icpe - cardquo 2012

[7] I G Shabac Matematici speciale 2 Bucuresti Romacircnia EdituraDidactica si Pedagogica 1965

[8] ICPE-CA Institutul National de Cercetare Proiectare pentru Ingineriaelectric˘ a httpwwwicpe-carorohttpwwwicpe-caroro

[9] K A Gallivan ldquoPolynomial interpolationrdquo 2013[10] S Zaglmayr High Order Finite Element Methods for Electromagnetic

Field Computation Thesis - Linz Univ 2006[11] R Corless Elementary Partial Differential Equations - Hadamardrsquos

example httpwwwapmathsuwoca 1998