Accelerator Magnets-1
Wednesday 2 August, 2006 in YangZhou , 14:00– 17:15
Ching-Shiang Hwang (黃清鄉)
(cshwang@nsrrc.org.tw)
National Synchrotron Radiation Research Center (NSRRC), Hsinchu
Outlines
•
Introduction to lattice magnets
•
The magnet features of various
accelerator magnets
•
Magnet code for field calculation
•
Magnet design and construction
•
Magnet parameters
•
Example of accelerator magnet
Introduction
1.
This talk will introduce the basic accelerator magnet type of
electromagnets in the synchrotron radiation source and discuss their
properties and roles.
2.
The main accelerator magnets are, dipole magnet for electron bending,
quadrupole magnet for electron focusing, and sextupole magnet for
controlling electron beam’s chromaticity.
3.
Magnet design should consider both in terms of physics of the
components and the engineering constraints in practical circumstances.
4.
The use of code for predicting flux density distributions and the iterative
techniques used for pole face and coil design.
5.
What parameters for accelerator magnet are required to design the
magnet?
6.
An example of SRRC magnet system will be described and discussed.
The magnet features of various accelerator magnets
‹ Lorentz fource
Although, the electric field appears in the Lorentz forces, but a
magnetic field of one Tesla gives the same bending force as an
electric field of 300 million Volts per meter for relativistic particles
with velocity v≈c.
∗
Magnets in storage ring
Dipole
-
bending and radiation
Quadrupole
-
focusing or defocusing
Sextupole
-
chromaticity correction
Corrector
-
small angle correction
The magnet features of various accelerator magnet
‹ Magnetic field features of lattice magnets
∞ ( B ( z ) + i A ( z ))( x + iy )m + 1
m
m
Φ ( x , y ,z ) = ∑
m=0
(m + 1)
Two sets of orthogonal multipoles with amplitude constants Bm and Am
which represent the Normal and Skew multipoles components. The fields
strength can be derived as
r
r
B ( x , y , z ) = −∇Φ ( x , y , z )
Therefore, the general magnetic field equation including only the most
commonly used upright multipole components is given by
A3
⋅ ( 3 x 2 y − y 3 ) + ....
6
B2
B
⋅ ( x 2 − y 2 ) + 3 ⋅ ( x 3 − 3 x y 2 )+ ....
B y = B o + B1 ⋅ x +
2
6
B x = A 1 ⋅ y + A 2 ⋅ xy +
where B0,B1(A1), B2(A2),and B3(A3) denote the harmonic field strength
components and call it the normal (skew) dipole,quadrupole,sextupole and
so on.
Magnet for accelerator lattice system
‹ Magnetic spherical harmonics derived from Maxwell’s equations.
v v
v v v
Maxwell’s equations for magneto-statics: ∇
,
⋅B = 0
∇×H = j
No current excitation: j = 0
v
v
Then we can use : B = − ∇ Φ M
Therefore the Laplace’s equation can be obtained as: ∇2Φ = 0
where ΦM is the magnetic scalar potential.
Thinking the two dimensional case (constant in the z direction) and solving for
cylindrical coordinates (r,θ):
∞
(
Φ M = (E + F θ )(G + H ln r )∑ a n r n cos n θ + b n r n sin n θ + a n r − n cos n θ + b n r − n sin n θ
n =1
In practical magnetic applications, scalar potential becomes:
(
Φ M = ∑ a n r n cos nθ + b n r n sin nθ
)
n
with n integral and an, bn a function of geometry.
This gives components of magnetic flux density:
(
B r = ∑ na n r n −1 cos nθ + nb n r n −1 sin nθ
n
)
(
B θ = ∑ − na n r n −1 sin nθ + nb n r n −1 cos nθ
n
)
)
Dipole Magnet-1
‹ Dipole field on dipole magnet expressed by n=1 case.
Cylindrical
B r = a 1 cos θ + b1 sin θ
Cartesian
B x = b1 y
Bx = a1
B θ = −a 1 sin θ + b1 cos θ
B y = b1
Φ M = a 1 r cos θ + b1 r sin θ
Φ M = a 1 x + b1 y
B y = a1x
Φ M = (a 1 + b1 )xy
So, a1 = 0, b1 ≠ 0 gives vertical dipole field:
y
y
.
ΦM= constant
x
x
B
Pure dipole magnet
Combine function
magnet
Dipole Magnet-2
b1 = 0, a1 ≠ 0 gives horizontal dipole field (which is about rotated
By
By
x
Separate function dipole
x
Combine function dipole
π
)
2 ×1
Quadrupole Magnet-1
‹ Quadrupole field given by n = 2 case.
Cylindrical
Cartesian
B r = 2a 2 r cos 2θ + 2b 2 r sin 2θ
B x = 2(a 2 x + b 2 y )
B θ = −2a 2 r sin 2θ + 2b 2 r cos 2θ
B Y = 2(− a 2 y + b 2 x )
Φ M = a 2 r 2 cos 2θ + b 2 r 2 sin 2θ
Φ M = a 2 x 2 − y 2 + 2b 2 xy
(
)
These are quadrupole distributions, with a2 = 0 giving ‘normal’ quadrupole
field.
y
y
ΦM = - C
ΦM = - C
ΦM = + C
ΦM = + C
x
ΦM = + C
x
ΦM = - C
ΦM = + C
ΦM = - C
normal
skew
Quadrupole Magnet-2
Then b2 = 0 gives ‘skew’ quadrupole fields (which is the above rotated by
Bx
By
y
x
Normal term
Skew term
π
)
2× 2
Sextupole Magnet-1
‹ Sextupole field given by n = 3 case.
Cylindrical
Cartesian
(
)
B r = 3a 3 r 2 cos 3θ + 3b 3 r 2 sin 3θ
B x = 3a 3 x 2 − y 2 + 6b 3 xy
B θ = −3a 3 r 2 sin 3θ + 3b 3 r 2 cos 3θ
B y = −6a 3 xy + 3b 3 x 2 − y 2
Φ M = 3a 3 r 3 cos 3θ + 3b 3 r 3 sin 3θ
(
)
(
(
)
Φ M = a 3 x 3 − 3y 2 x + b 3 3yx 2 − y 3
y
y
ΦM = - C
ΦM = - C
ΦM = + C
ΦM = + C
ΦM = + C
x
B
ΦM = + C
x
ΦM = - C
ΦM = - C
ΦM = - C
ΦM = + C
ΦM = + C
Normal
Skew
)
Sextupole Magnet-2
(
For a3 = 0, b3 ≠ 0, By ∝ x2, B y = 3b 3 x 2 − y 2
) give normal sextupole
For a3 ≠ 0, b3 = 0, give skew sextupole (which is about rotated
By
By
x
Normal
& skew
y
π
)
2×3
Error of quadrupole magnet
Practical quadrupole. The shape of the pole
surfaces does not exactly correspond to a true
hyperbola and the poles have been truncated
laterally to provide space for coils.
n= 2(2m+1) m=0,1,2,3 -----, n is the allow term
for the quadrupole magnet
(1) Sextupole (n=3)
: a=b=c≠d, mechanical asymmetric
-----Forbidden term
(2) Octupole (n=4)
: A=B, a=d, b=c, but a≠b
-----Forbidden term
(3) Decapole (n=5)
: Tilt one pole piece
-----Forbidden term
(4) Dodecapole (n=6)
: a. lateral truncation
b. a=b=c=d, A=B≠2R
-----Allow term
(5) 20-pole (n=10)
: a. lateral truncation
b. a=b=c=d, A=B≠2R
-----Allow term
Error of sextupole magnet
n= 3(2m+1) m=0,1,2,3 -----, n is the allow term for the sextupole magnet
If dipole field (n=1) changed, the field for the allow term of dipole (n=2m+1) would also be
changed.
The dipole perturbation in a
sextupole field generated by
geometrical asymmetry (a=b<c).
The dipole and skew octupole perturbation
in a sextupole field generated by the bad
coil (1 to 4 is bad).
(1) Octupole (n=4)
: a=b≠c, mechanical asymmetric
-----Forbidden term
(2) Decapole (n=5)
: a=b=c≠2R
-----Forbidden term
(3) 18-pole (n=9)
: a. lateral truncation
b. a=b=c≠2R
-----Allow term
(4) 30-pole (n=15)
: a. lateral truncation
b. a=b=c=d, A=B≠2R
-----Allow term
Ideal pole shapes are equal magnetic line
v v
At the steel boundary, with no currents in the steel:
∇×H = 0
Apply Stoke’s theorem to a closed loop enclosing the boundary:
v
v
v
v
v
∫∫ (∇ × H )⋅ dS = ∫ H ⋅ d l
B
dl
Hence around
v the
v loop:
∫ H ⋅ dl = 0
dl
Steel, µ = ∞
Air
But for infinite permeability in the steel: H = 0;
Therefore outside the steel H = 0 parallel to the boundary.
Therefore B in the air adjacent to the steel is normal to the steel
surface at all points on the surface should be equal. Therefore
v
v
B
=
−
∇
Φ M , the steel surface is an iso-scalar-potential line.
from
Contour equations of ideal pole shape
For normal (ie not skew) fields pole profile:
Dipole:
y = ± g/2;
Quadrupole:
xy = ± R2/2;
(g is interpole gap).
(R is inscribed radius).
This is the equation of hyperbola which is a natural shape for the
pole in a quadrupole.
Sextupole:
3x2y – y3 = ± R3 (R is inscribed radius).
Symmetry constraints in normal dipole, quadurpole
and sextupole geometries
Magnet
Dipole
Quadrupole
Sextupole
Symmetry
Constraint
φ(θ) = -φ(2π-θ)
All an = 0;
φ(θ) = φ(π-θ)
bn non-zero only for:
n=(2j-1)=1,3,5,etc; (2n)
φ(θ) = -φ(π-θ)
bn = 0 for all odd n;
φ(θ) = -φ(2π-θ)
All an = 0;
φ(θ) = φ(π/2-θ)
bn non-zero only for:
n=2(2j-1)=2,6,10,etc; (2n)
φ(θ) = -φ(2π/3-θ)
bn = 0 for all n not multiples of 3;
φ(θ) = -φ(4π/3-θ)
φ(θ) = -φ(2π-θ)
All an = 0;
φ(θ) = φ(π/3-θ)
bn non-zero only for:
n=3(2j-1)=3,9,15,etc. (2n)
Magneto-motive force in a magnetic circuit
Stoke’s theorem for vector A:
v
v v
∫ A ⋅ dl = ∫∫ (∇ × A )⋅ dS
Apply this to:
v v v
∇×H = j
Then for any magnetic circuit:
v v
∫ H ⋅ d l = NI
NI is total Amp-turns through loop dS.
A
dS
dl
Ampere-turns in a dipole
NI/2
B is approx constant round loop l & g,
and
Hiron = Hair / µ;
Bair = µ0 NI / (g + l/µ);
l
g
NI/2
µ >> 1
g, and l/µ are the reluctance of the gap
and the iron.
Ignoring iron reluctance:
NI = B g /µ0
Ampere-turns in quadrupole magnet
Quadruple has pole equation:
xy = R 2 / 2
On x axes
By = gx,
g is gradient (T/m).
At large x (to give vertical field B):
(
x
)
NI = B ⋅ l = (gx ) R 2 / 2 x / µ 0
ie
NI = gR 2 / 2µ 0
(per pole)
B
Ampere-turns in sextupole magnet
Sextupole has pole equation:
3x 2 y − y 3 = ± R 3
On x axes
By = gsx2, gs is sextupole strength (T/m2).
At large x (to give vertical field ):
⎛ R3
NI = B ⋅ l = g s x ⋅ ⎜⎜ 2
⎝ 3x µ 0
2
⎞ gsR 3
⎟=
⎟ 3µ
0
⎠
Magnet geometries and pole shim
‘C’ Type:
Advantages:
Easy access for installation;
Classic design for field measurement;
Disadvantaged:
Pole shims needed;
Asymmetric (small);
Less rigid;
‘H’ Type:
Advantages:
Symmetric of field features;
More rigid;
Disadvantages:
Also needs shims;
Access problems for installation.
Not easy for field measurement.
Shim Detail
Magnet geometries and pole shim
Quadrupole and Sextupole Type:
Shim
Effect of shim and pole width on distribution
Magnet chamfer for avoiding field saturation at
magnet edge
‹
Chamfering at both sides of dipole magnet to
compensate for the sextupole components and the
allow harmonic terms.
‹
Chamfering at both sides of the magnet edge to
compensate for the 12-pole (18-pole) on quadrupole
(sextupole) magnets and their allow harmonic terms.
‹
Chamfering means to cut the end pole along 45° on
the longitudinal axis (z-axis) to avoid the field
saturation at magnet edge.
Square ends
* display non linear effects (saturation);
* give no control of radial distribution in the fringe
region.
y
Saturation
z
Chamfered ends
* define magnetic length more precisely;
* prevent saturation;
* control transverse distribution;
* prevent flux entering iron normal to lamination (vital
for ac magnets).
y
z
Magnet end chamber or shim on lattice magnet
21
Without Shim at 1.3 GeV
With 2 Shims at 1.3 GeV
Sextupole strength ( T/m )
16
11
6
1
-4
-9
-14
-1.00 -0.75 -0.50 -0.25
0.25
0.50
0.75
1.00
s-axis (m)
Dipole end shim
Dipole end shim
Without
Shim
With 5-5 Shims
at 1.5
GeVat 1.5 GeV
no chamfer
Dodecapole strength (T/m**5)
21
16
Sextupole strength ( T/m )
0.00
11
6
1
-4
-9
chamfer
150000
50000
-50000
-150000
-250000
-350000
-14
-1.00
-0.75
-0.50
-0.25
0.00
0.25
s-axis (m)
Dipole end shim
0.50
0.75
1.00
-0.3
-0.2
-0.1
0.0
0.1
z-axis(m)
Quadrupole end chamber
0.2
0.3
Magnet end chamber & shim at the pole end
Quadrupole integral field strength
distribution w & w/o end chamber
Dipole integral field strength
distribution w & w/o end shim
The trade-off of current density in coils
j = NI/S
where:
j is the current density,
S the cross section area of copper in the coil;
NI is the required Amp-turns.
EC = G (NI)2/S
therefore:
EC = (G NI) j
Here:
EC is energy loss in coil conductor,
G is a geometrical constant factor and as function of µ.
Then, for constant NI and energy will depend on current j.
Magnet capital costs (coil & yoke materials and construction, assembly,
power supply testing and transport) vary as the size of the magnet ie
as 1/j.
Variation of magnet parameters with N and j (fixed NI)
I ∝ 1/N,
Rmagnet ∝ N2 j,
Vmagnet ∝ N j, Power ∝ j
R (Ω)=ρCu(L(m)/A(m2)=1.7x10-8 (L/A) where L and A are the
total length and cross section l area of the coil, respectively.
Ampere-turn determination
Large N (low current)
Small N (high current)
y Small, neat terminals.
y Large, bulky terminals
y Thin interconnections-low
cost & flexible.
y Thick, expensive interconnection, high cost.
y More insulation layers in coil,
hence larger coil, increased
assembly costs.
y High percentage of copper in
coil. More efficient use of
volume.
y High voltage power supplysafety problems.
y High current power supplygreater losses.
y High inductance & resistance
y Low inductance & resistance
Flux at magnet gap
b
g
ΦM ≈ Bgap (b + 2g) L
L is magnet length
Residual field in gapped magnets
In a continuous ferro-magnetic core, residual field is determined by the
remanence Br. In a magnet with a gap having reluctance much greater
than that of the core, the residual field is determined by the coercivity
HC.
With no current in the external coil, the total integral of field H around
core and gap is zero.
Thus, if Hg is the gap, l and g are the path lengths in core and gap
respectively:
B
v
v
v
v
v
v
H
⋅
d
l
=
H
⋅
d
l
+
H
⋅
d
g
∫ C
∫ C
∫ g =0
l
so,
g
Br
B residual = −µ 0 H C l / g
H
-HC
Judgment method of field quality
For central pole region:
Dipole: calculate and plot
B y ( x ) − B y (0 )
B y (x )
in pure dipole or
B y ( x ) − (B y (0 ) + g x )
By (x )
in
combine function dipole magnet, g is gradient field in dipole magnet
g (x ) − g(0 )
B y ( x ) − [bo + b1 x]
dB y (x )
= g(x ) and
Quadrupole: calculate and plot
and
By (x )
g (0 )
dx
Sextupole: calculate and plot
d 2 B y (x )
dx
2
= S(x ) and
For integral good field region:
B y ( x ) − [bo + b1 x + b 2 x 2 ]
By (x )
and S(x ) − S(0 )
∫ B (x )ds − ∫ B (0)ds or ∫ B (x )ds − [∫ B (0)ds + ∫ gxds]
∫ B (0)ds
∫ B (0)ds
∫ g (x )ds − ∫ g (0)ds
(
)
B
x
ds
−
[
(
b
+
b
x
)
ds
]
∫
∫
Quadurpole:
and
∫ g (0)ds
(
)
B
x
ds
∫
Sextuple: ∫ B(x )ds − [ ∫ (b + b x + b x )ds]
and
∫ S(x )ds − ∫ S(0)ds
∫ B(x )ds
∫ S(0)ds
Dipole:
y
y
y
y
y
y
o
o
1
1
2
2
S(0 )
Comparison of four commonly used magnet
computation codes
Advantages
Disadvantages
MAGNET:
* Quick to learn, simple to use;
* Only 2D predictions;
* Small(ish) cpu use;
* Batch processing only-slows down problem turn-round
time;
* Fast execution time;
* Inflexible rectangular lattice;
* Inflexible data input;
* Geometry errors possible from interaction of input data
with lattice;
* No pre or post processing;
* Poor performance in high saturation;
POISSON:
* Similar computation as MAGNET;
* Harder to learn;
* Interactive input/output possible;
* Only 2D predictions.
* More input options with greater flexibility;
* Flexible lattice eliminates geometery errors;
* Better handing of local saturation;
* Some post processing available.
Comparison of four commonly used magnet
computation codes
Advantages
Disadvantages
TOSCA:
* Full three dimensional package;
* Training course needed for familiarization;
* Accurate prediction of distribution and
strength in 3D;
* Expensive to purchase;
* Extensive pre/post-processing;
* Large computer needed.
* Multipole function calcula-tion
* Large use of memory.
* For static & DC & AC field calculation
* Run in PC or workstation
* Cpu time is hours for non-linear 3D problem.
RADIA:
* Full three dimensional package
* Larger computer needed;
* Accurate prediction of distribution and
strength in 3D
* Large use of memory;
* Careful to make the segmentation;
* With quick-time to view and rotat 3D structure
* DC field calculation;
* Easy to build model with mathematic
* Fast calculation
* Run in PC or MAC
* Can down load from ESRF website.
Magnets for injection and extraction system
Eddy losses in Conductors
Rectangular conductor (no cooling hole):
B
resistivity ρ;
width a;
cross section A;
in a.c. field: B sin ωt;
A
Power loss / m is:
P = ω2 B2 A a2 / (12ρ)
a
Circular conductor, diameter d:
P = ω2 B2 A d2 / (16ρ)
5 mm square copper in a 1T peak, 10 Hz a.c. field,
P = 8.5 kW / m.
Cooling formula: P = (φ ∆T)/14.33
Pw: dissipated power(kW), φ : water flow(l/min), ∆T : temperature increase(centigrade)
φ (m3/s) = NA (m2)xV(m/s) where N is the number of pancake coil for water channel
∆P(bar)=5 x 10-5x( L(m)/N) x V1.75(m/s) x(1/d1.25 (m))
∆P: pressure drop, L : coil length, V : water flow velocity, d: hole diameter of coil
Transposition of conductors
a b
c d
b
a
d
c
d c
b a
c d
a b
Standard transposition to avoid excessive eddy loss;
The conductors a, b, c, and d making up one turn are
transposed into different positions on subsequent turns in a
coil to equalize flux linkage.
Indirect cooling of a transposed, stranded conductor
in a cable used at 50 Hz
Cooling
tube.
Transposed,
stranded conductor.
Comparison between eddy and hysteresis
losses in steel
Variation with:
Eddy loss
Hysteresis loss
a.c. frequency:
Square law;
Linear;
a.c. amplitude:
Square law;
Non-linear; depends on level;
d.c. bias:
No effect;
Some increase;
Total volume of steel:
Linear;
Linear;
Lamination thickness:
Square law;
No effect;
Comparison between a low-silicon non-oriented steel, a highsilicon non-oriented steel, and a grain-oriented steel
Non-oriented lowsilicon
Non-oriented highsilicon
Grain-oriented
(along grain)
Silicon content
Low (~1%)
High (~3%)
High (~3%)
Lam. thickness
0.65 mm
0.35 mm
0.27 mm
a.c. loss (50 Hz): at
1.5 T peak
6.9 W / kg
2.25 W / kg
0.79 W / kg
µ (B = 0.05 T)
995
4,420
not quoted
µ (B = 1.5 T)
1680
990
> 10,000
µ (B = 1.8 T)
184
122
3,100
Coercivity HC
100 A / m
35 A / m
not quoted
Inductance of a dipole magnet
Inductance of a coil defined as:
Current I with N turns
Φ
L = NΦ / I
For an iron cored dipole:
Φ = BA = µ 0 nIA / (g + l / µ )
g is the length of gap path;
l is length of steel path;
A is total area of flux;
µ is permeability of steel;
Therefore, the inductance of the magnet:
Where
L M = µ 0 N 2 A / (g + l / µ )
A ≈ L(b + 2g )
L is magnetic length on longitudinal direction;
b is pole width;
g is pole gap.
Mutual inductances in series and parallel
L
L
For two coils, inductance L, on the same core, with coupling coefficients
of 1:
Inductance of coils in series = 4 L
Inductance of coils in parallel = L
Usual magnetic field biased sine wave in a fast
cycling synchrotron
( BDC + BAC )
B
BDC
t
B(t ) = B DC + B AC sin ωt
The ‘White Circuit’
LM
Magnet coil
Secondary coil
Primary coil
C LCh
transformer
LM
the standard circuit configuration
used to power fast cycling
synchrotrons. The magnets LM
are in series with secondary coil
are resonated by series capacitors
C. The auxiliary inductor LCH
provides a path for the d.c. bias.
The diagram shows a four-cell
network
C LCh
dc
ac
‘Kicker Magnet’ design
‹
In general, the injection and extraction magnet systems of
accelerators need two types of magnets – kicker and septum
magnets.
‹
The kicker magnets are used for the fast displacement of the
closed orbit temporarily.
‹
The septum magnets provide the necessary defection to the
incoming or outcoming beam to match the angle of the beam
transfer line to the accelerator orbit.
Ferrite Core
beam
Conductors
Ceramic chamber – non-magnetic
material
Simplified diagram of delay line kicker magnet and
power supply with matched surge impedance Z0
Power
,C
LSupply
1
1
Magnet
T
dc
Surge impedance:
Pulse transit time:
L2 , C2
Termin
-ation
Z0
Z0 = √(L/C);
τ = √(LC);
Advantages and disadvantages of using matched ‘delay-line’
type kicker magnet and power supply circuit
Advantages:
Disadvantages:
Very rapid rise times (<0.1 µs);
Volts on power supply are twice the pulse
voltage;
Good quality flat top on square current
pulse;
Pulse voltage (10 s of kV) present on
magnet throughout pulse;
Matched connection allows supply to be
remote from magnet;
Complex and expensive magnet;
Reverse volts on valve are not large;
Difficulty with terminating resistor
manufacture;
Alternative kicker magnet circuits using ‘lumped
inductance’ type magnets
R
L
dc
Z0
R = Z0
dc
L
Alternative septum magnet designs
Eddy current design
Eddy current
screen
Beam
Storage beam
Kicker beam
Single or multi turn coil
Criteria relating to eddy current septum magnets
Skin depth in material:
resistivity ρ,
permeability µ,
frequency ω, is given by:
δ = √ (2 ρ / ωµµ0 )
Example:
Screen thickness (at beam)
1 mm;
Screen thickness (elsewhere)
10 mm;
Excitation pulse
25 µs,
half sinewave;
Skin depth (copper, 20 kHz)
0.45 mm,
Injected beam pulse length
< 1 µs.
Features of alternative septum magnet designs
Conventional
Eddy current
Excitation
d.c. or low frequency pulse;
pulse:
waveform frequency > 10 kHz;
low trpetition rate;
Coil
single turn including front
septum;
single or multi-turn on backleg
allows large cross section;
Septum cooling
complex water cooling in
thermal contact with septum;
heat generated in dhield
conducted to base plate;
Yoke material
conventional steel;
high-frequency material (ferrite
or radio metal).
Criteria relating to choice of material in a high
frequency magnet
Transformer:
Inductance = µµ0 n2 A/ l;
l is length of magnetic circuit,
A is cross section area of flux;
Magnet:
Inductance = µ0 n2 A/ (g+ l / µ); g is gap height.
ie L (magnet) << L (transformer).
Losses appear as resistance in parallel with inductance; they are
therefore much less significant in a magnet.
Skin-depth effects modified by gap:
The gap also decreases the magnetic effect of eddy currents in the
laminations, which control the penetration of flux into the steel:
δ ≈ √ { 2 ρ g / ωµ0 ( l + g ) }
Use of magnetic material at higher than recommended
frequencies
Application:
Type of septum:
Eddy current;
Excitation:
25 µs half sine-wave;
Effective frequency:
20 kHz
Material used:
Nickel Iron Alloy
Lamination thickness:
0.1 mm
Skin depth (nickel-iron µ ~ 5,000)
~ 0.01 mm
Lamination manufacturer’s data:
Max frequency quoted
in µ and loss curves: 400 Hz
Max recommended
operational frequency: 1 kHz
Different kind of magnet at SRRC
Reference
[1]Neil, Marks, CERN Accelerator School, fifth general accelerator physics
course, Editor by S. Turner, CERN 94-01, 26 January,1994, Vol. II.
[2]Handbook of accelerator physics and engineering, World Scientific
publishing Co. Pte. Ltd., edited by Alexander Wu Chao and Maury Tigner
(1998).
[3]Jack Tanabe, Lectures of conventional magnet design, USPAS, Tucson,
Arizona, January, 2000.
[4]Synchrotron Radiation Sources- A Primer, World Scientific publishing
Co. Pte.
[5]C. H. Chang, H. H. Chen, C. S. Hwang, G. J. Hwang, and P. K. Tseng,
1994, July, "Design and Performance of the SRRC Quadrupole Magnets",
IEEE Trans. on magn., 30, No. 4, (1994) 2241-2244.
[6]C. H. Chang, C. S. Hwang, C. P. Hwang, G. J. Hwang, and P. K. Tseng,
1994, July, "Design, Construction and Measurement of the SRRC
Sextupole Magnets", IEEE Trans. on magn., 30, No. 4, (1994) 2245-2248.
(SCI)
Accelerator Magnets-2
Wednesday 2 August, 2006 in YangZhou , 14:00– 17:15
Ching-Shiang Hwang (黃清鄉)
(cshwang@nsrrc.org.tw)
National Synchrotron Radiation Research Center (NSRRC), Hsinchu
Outlines
•
Introduction to magnet measurement
•
Classification of field measurement
method and system
•
Procedure of field quality control and
analysis
•
Example of accelerator magnet
measurement
Classification of field measurement and analysis methods
1. The fluxmeter method
This method, which is based on the induction law, is the oldest of the currently rsed methods for
magnetic measurement, but it can be very precise.
a. Search coil
It suit for point field measurement
b. Rotating harmonic coil
It suit for analysis the harmonic field measurement
c. Moving or a fixed stretched wire
To measure the static or varying magnet flux field
d. Helmholtz coil
To measure the homogeneous field or the three-dimension magnetic moment of permanent
magnet.
2. The Hall effect method
Advantage: High resolution of point field and inhomogeneous field measurement
Disadvantage: not high accuracy and easy be influenced by environment.
3. Nuclear Magnetic Resonance (NMR) method
Advantage: not only for calibration purposes, but also for high precision field mapping
Disadvantage: can not measure inhomogeneous field.
4. Fluxgate magnetometer
Advantage: offering a linear measurement and well suited for static operation.
Disadvantage: restricted to low field.
Measurement methods: accuracies and ranges
Data analysis on the Hall probe system
The mapping method measure either the vertical field By(x,y) or the horizontal field
By(x,y). To do the harmonic field analysis, we record the position (x,y,s) and the
field By(x,y), and then integrated the field along the s orbits. For a normal magnet,
the magnet features depend on the magnet pole face and its magnetic field
equation. This can be solved by the Laplace’s equation in the two dimensional (x,y)
plane. The magnetic field equation can be expressed as
n
m
B y (x , y ) = ∑ ∑ N n , m H n , N x
n =1 m = 0
N n ,m
S n ,m
n −2m
y
2m
n
m
+ ∑ ∑ S n ,m H n ,S x n − 2 m −1 y 2 m −1
n =1 m = 0
m
(
− 1) n!
=
(n − 2m )!(2m )!
m
(
− 1) n!
=
(n − 2m − 1)!(2m + 1)!
(1)
(2)
(3)
Data analysis on the Hall probe system
Where Nn,m and Sn,m are the weighting factors and n-2m≥0, n-2m-1≥0. The field strength
Hn,N is the normal multipole strength and Hn,S is the skew multipole strength and n
represent the harmonic numbers. Each point of the harmonic field on the integrated
harmonic field. Also equation (1) can be transferred to be a cylindrical coordinate
equation as follows: The vertical field By(x,y) can be represented as
B y (r, θ ) = ∑ H nm r n [Sin (nθ − α nm )]
m ,n
(
H nm = H 2n , N + H 2n ,S
)
(4)
1/ 2
α nm = Sin −1 (H n , N / H nm )
(5)
(6)
Where Hnm is the combined harmonic field strength, αnm is the orientation angle and θ
is the phase angle. From the magnetic field equation (1), the skew dipole field is a
unknown term, unless the horizontal field Bx(x,y) is measured. We take the mapping data
of the magnetic field strength Bx(x,y,s) or By(x,y,s) and substitute this in equation (1). The
harmonic field strength Hn,N and Hn,S can be obtained by doing a non-linear least square
fit. We can substitute the results in equation (5) and (6) to obtain the combined harmonic
field strength Hnm with an orientation angle αnm.
The rotating coil method
The measurement philosophy
⎛ r
Bθ (r ,θ ) = ∑ Bref ⎜
⎜r
m =1
⎝ ref
∞
⎞
⎟
⎟
⎠
m −1
[Sin(mθ − α m )]
Single loop asymmetric coil in a
multipole field.
The tangent field of Bθ was put
into FFT analysis to obtain the
multipole components.
(7)
The procedure of field quality control for the mass production
The sequential main task for series field measurements
1) precise alignment (need some monument)
2) The excitation current cycling to overcome the hysteresis effects
3) Measure
(a) the centerfield By(x,y)
(b) integrated field ∫ By(x,y)ds
(c) field magnetic length BL/B0
(d) the tilt ∆θ=∫Bx(x,y)ds/∫By(x,y)ds
(e) the higher central and integrated multipoles of normal term and skew
term.
4) to find the field quality
(a) the distribution of ∆(∫By(x,y)ds)/∫By(x,y)ds
(b) the good field region
(c) the comparison results between the tolerance and measurement
(d) the statistic distribution for mass production magnet
5) to make sure
(a) the influence of the vicinity material
(b) the effect of µr and HC
(c) the effect of eddy current
(d) the effect of an initial magnetization
The esults of field measurement NSRRC dipole magnet
Dipole magnet shown in Fig1- 8
Fig1 The field deviation as a function of x of
the two dimensional theoretical calculation, and
the measurement results of the combined
function bending magnet. Where the field
deviation definition is
∆B / B = [B( x ) − (a1 + a2 x )] / B( x )
Fig2 NSRRC magnet measure dipole data
for magnet field effective length BL/B0
The results of field measurement and quality of SRRC magnet
Fig3 The gradient field deviation of the 2-D
calculation and measurement results at the
magnetic center.
Fig4
After and before 2-2 shims
measurement results of the integrated
field deviation
The results of field measurement SRRC dipole magnet
Fig5 The dipole field distribution of the radial
and lamination mapping along the beam
direction
Fig6 The gradient field distribution of radial
and lamination mapping along the beam
direction
The results of field measurement SRRC dipole magnet
Fig7 The sextupole distribution of the radial
and lamination mapping along the beam
direction
Fig8 The sextupole field distribution along the
beam direction of the after and before 2-2
shims measurement results
Dipole end shim
Sextupole strength ( T/m )
16
Without Shim at 1.3 GeV
With 2 Shims at 1.3 GeV
16
11
6
1
-4
-9
-14
-1.00 -0.75 -0.50 -0.25
0.00
0.25
s-axis (m)
0.50
Without
Shim
at 1.5 GeV
With 5-5 Shims
at 1.5
GeV
21
Sextupole strength ( T/m )
21
0.75
1.00
11
6
1
-4
-9
-14
-1.00
-0.75
-0.50
-0.25
0.00
0.25
s-axis (m)
0.50
0.75
1.00
The results of field measurement SRRC quadrupole magnet
Quadrupole magnet shown in Fig9-10
Fig9 The field deviation as a function of
transverse direction of the 2-D and
measurement results on the midplane
Fig10 The deviation of integrated field
with different thickness chamfering
sextupole strength (T/m**2)
dipole strength (T)
Multipole components of quadrupole magnet
0.0005
-0.0005
-0.0015
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
z-axis (m)
0.45
0.35
0.25
0.15
0.05
-0.05
-0.15
-0.3
-0.2
-0.1
0.0
0.1
0.03
Dodecapole strength (T/m**5)
skew qua. strength (T/m)
no chamfer
0.02
0.01
0.00
-0.01
-0.02
-0.3
-0.2
-0.1
0.0
0.3
Forbidden term
Forbidden term
-0.03
0.2
z-axis (m)
0.1
z-axis (m)
Forbidden term
0.2
0.3
chamfer
150000
50000
-50000
-150000
-250000
-350000
-0.3
-0.2
-0.1
0.0
0.1
z-axis(m)
First allow term
0.2
0.3
The results of field measurement and quality of sextupole magnet
Sextupole magnet shown in Fig11 to Fig14
Fig11 The field deviation of the calculated
and the measured results.
Fig12 The deviation of the integrated
field strength distribution.
Forbidden terms of sextupole magnet
Quadrupole distribution
Dipole distribution
0.0016
0.00
0.0014
-0.02
0.0010
Field strength (T/m)
Field strength (T)
0.0012
0.0008
0.0006
0.0004
0.0002
-0.04
-0.06
0.0000
-0.08
-0.0002
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
-0.25
zaxis (m)
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.10
0.15
0.20
0.25
zaxis (m)
Sextupole distribution
Octupole distribution
20
0
10
Field strength (T/m^3)
Field strength (T/m^2)
-50
-100
-150
-200
0
-10
-20
-250
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
zaxis (m)
0.10
0.15
0.20
0.25
-30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
zaxis (m)
Allow terms of sextupole magnet
18-pole distribution
30-pole distribution
5.00E+009
4.00E+009
1.50E+014
2.00E+009
Field strength (T/m^14)
Field strength (T/m^8)
3.00E+009
1.00E+009
0.00E+000
-1.00E+009
-2.00E+009
1.45E+014
1.40E+014
1.35E+014
1.30E+014
-3.00E+009
-4.00E+009
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
zaxis (m)
0.10
0.15
0.20
0.25
1.25E+014
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
zaxis (m)
0.10
0.15
0.20
0.25
The results of field measurement and quality of sextupole magnet
Fig13
The relationship between the
integrated sextupole field strength and the
excitation current.
Fig14 The calculated and measured field
distribution of the horizontal and vertical
steering coils in the sextupole magnet.
Quadrupole magnet measured by the rotating coil and
Hall probe system
Dipole magnet measured by the Hall probe mapping system