Download 3 - CERN Indico

Unit 3
Field harmonics, Taylor expansions,
and pure multipoles
Ezio Todesco
European Organization for Nuclear Research (CERN)
E. Todesco, Milano Bicocca January-February 2016
QUESTIONS
How to express the magnetic field shape and uniformity
The magnetic field is a continuous, vectorial quantity
Can we express it, and its shape, using a finite number of coefficients ?
Yes: field harmonics
But not everywhere …
We are talking about electromagnets:
What is the field we can get from a current line ?41° 49’ 55” N – 88 ° 15’ 07” W
40° 53’ 02” N – 72 ° 52’ 32” W
The Biot-Savart law !
What are the components of the magnetic field shape ?
What are the beam dynamics requirements ?
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 2
CONTENTS
1. Definition of field harmonics
2. Field harmonics of a current line
3. Validity limits of field harmonics
4. Beam dynamics requirements on field harmonics
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 3
1. FIELD HARMONICS: MAXWELL EQUATIONS
Maxwell equations for magnetic field
B x B y B z
B 


0
x
y
z
  B   0 J   0 0
E
t
In absence of charge and magnetized material
 B y B z B z B x B x B y 
0
  B  

,

,

y x
z y
x 
 z
If
James Clerk Maxwell,
Scottish
(13 June 1831 – 5 November 1879)
B z
 0 (constant longitudinal field), then
z
Bx By

0
x
y
E. Todesco, Milano Bicocca January-February 2016
Bx By

0
y
x
Unit 3 - 4
1. FIELD HARMONICS: ANALYTIC FUNCTIONS
A complex function of complex variables is analytic if it
coincides with its power series

f ( z )   Cn z
n 1
n 1

f x ( x, y )  if y ( x, y )   Cn ( x  iy ) n 1
on a domain D !
( x, y )  D
n 1
Note: domains are usually a painful part, we talk about it later
A necessary and sufficient condition to be analytic is that
 f x f y
 x  y  0

f
 f x  y  0
 y x
called the Cauchy-Riemann conditions
E. Todesco, Milano Bicocca January-February 2016
Augustin Louis Cauchy
French
(August 21, 1789 – May 23, 1857)
Unit 3 - 5
1. DEFINITION OF FIELD HARMONICS
If
B z
0
z
 f x f y
 x  y  0

f
 f x  y  0
 y x
By
Bx

0
x
y
Maxwell gives
By
y

Bx
0
x
and therefore the function By+iBx is analytic

n1
By ( x, y )  iBx ( x, y )   Cn x  iy 
n1
( x, y )  D
Georg Friedrich Bernhard Riemann,
German
(November 17, 1826 - July 20, 1866)
where Cn are complex coefficients
Advantage: we reduce the description of a function from R2
to R2 to a (simple) series of complex coefficients
Attention !! We lose something (the function outside D)
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 6
1. DEFINITION OF FIELD HARMONICS

n 1
By ( x, y )  iBx ( x, y )   Cn x  iy 
n 1
 C0  C1 ( x  iy )  ...
( x, y )  D
Each coefficient corresponds to a “pure” multipolar field
A dipole
A quadrupole
[from P. Schmuser et al, pg. 50]
A sextupole
Magnets usually aim at generating a single multipole
Dipole, quadrupole, sextupole, octupole, decapole, dodecapole …
Combined magnets: provide more components at the same time (for
instance dipole and quadrupole) – more common in low energy
rings, resistive magnets – one superconducting example: JPARC
(Japan)
Unit 3 - 7
E. Todesco, Milano Bicocca January-February 2016
1. DEFINITION OF FIELD HARMONICS
n1

By ( x, y )  iBx ( x, y )   Cn x  iy 
n1
n1

  Bn  iAn x  iy 
n1
The field harmonics are rewritten as
 x  iy 

B y  iB x  10 B1  (bn  ia n )
 R 
n 1
 ref 
4

n 1
We factorize the main component (B1 for dipoles, B2 for quadrupoles)
We introduce a reference radius Rref to have dimensionless
coefficients
We factorize 10-4 since the deviations from ideal field are 0.01%
The coefficients bn, an are called normalized multipoles
bn are the normal, an are the skew (adimensional)
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 8
1. DEFINITION OF FIELD HARMONICS
 x  iy 

B y  iB x  10 B1  (bn  ia n )
 R 
n 1
 ref 
4

n 1
Reference radius is usually chosen as 2/3 of
the aperture radius
This is done to have numbers for the multipoles
that are not too far from 1
y
40
-
Some wrong ideas about reference radius
Wrong statement 1: “the expansion is valid up
to the reference radius”
+
-
R ref
+
0
The reference radius has no physical meaning, it
is as choosing meters of mm
Wrong statement 2: “the expansion is done
around the reference radius”
A power series is around a point, not around a
circle. Usually the expansion is around the origin
E. Todesco, Milano Bicocca January-February 2016
-4 0
0
r
+
-
+
x
-4 0
Unit 3 - 9
40
1. FIELD HARMONICS: LINEARITY
40
40
40
C' n
C'' n
c' n
c ' n +c'' n
=C' n +C'' n
-
c'' n
-
+
-
+
-
+
+
0
0
0
-4 0
0
40
-4 0
0
40
+
+
-
-
-4 0
0
40
+
+
-
-
-4 0
-4 0
-4 0
Linearity of coefficients (very important)
Non-normalized coefficients are additive
Normalized coefficients are not additive
Cn Cn'  Cn'' Cn' Cn''
'
''
cn 
 '



c

c
n
n
B1 B1  B1'' B1' B1''
Normalization gives handy (and physical) quantities, but some
drawbacks – pay attention !!
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 10
CONTENTS
1. Definition of field harmonics
2. Field harmonics of a current line
3. Validity limits of field harmonics
4. Beam dynamics requirements on field harmonics
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 11
2. FIELD OF A CURRENT LINE
Field given by a current line (Biot-Savart law)
Differential form (international system)
r
Infinite current line
A factor two is given by the atan integration
Félix Savart,
French
(June 30, 1791-March 16, 1841)
r
Field in a centre of a circular loop, radius r
r
E. Todesco, Milano Bicocca January-February 2016
Jean-Baptiste Biot,
French
(April 21, 1774 – February 3, 1862)
Unit 3 - 12
2. FIELD OF A CURRENT LINE:
COMPLEX NOTATION
Field given by a current line (Biot-Savart law)
Infinite current line
y
Complex notation
40
z 0 =x 0 +iy 0
B=B y +iB x
0
-4 0
0
x
40
Félix Savart,
French
(June 30, 1791-March 16, 1841)
z=x+iy
Using the relation
-4 0
We obtain the compact very useful notation
E. Todesco, Milano Bicocca January-February 2016
Jean-Baptiste Biot,
French
(April 21, 1774 – February 3, 1862)
Unit 3 - 13
2. FIELD HARMONICS OF A CURRENT LINE
Field given by a current line (Biot-Savart law)
y
B( z )  By ( z )  iBx ( z )
I 0
I
1
B( z ) 
 0
2 ( z  z0 )
2z0 1  z
z0
using
40
z 0 =x 0 +iy 0
B=B y +iB x
0
-4 0
x
0
40
z=x+iy
-4 0

1
2
3
 1  t  t  t  ...   t n1
1 t
n 1
t  1 !!!
Félix Savart,
French
(June 30, 1791-March 16, 1841)
we get
I 0
B( z )  
2z0
 z
 

n 1  z 0 

E. Todesco, Milano Bicocca January-February 2016
n 1
I 0

2z0
 Rref


n 1  z 0




n 1
 x  iy 


 R 
 ref 
n 1
Jean-Baptiste Biot,
French
(April 21, 1774 – February 3, 1862)
Unit 3 - 14
2. FIELD HARMONICS OF A CURRENT LINE
Now we can compute the multipoles of a current line at z0
I 0
B( z )  
2z0
 z
 

n 1  z 0 

n 1
I 0

2z0
 Rref


n 1  z 0

 x  iy 

B y  iB x  10 B1  (bn  ia n )
 R 
n 1
 ref 
4

 x  iy 


 R 
 ref 
I 010  Rref

bn  ia n  
2z0 B1  z0
4



n 1
x  iy  z0
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
E. Todesco, Milano Bicocca January-February 2016
n 1
n 1
¦bn+ian¦
I0  1 
B1  
Re 
2
 z0 



n 1
5
10
Multipole order n
15
Unit 3 - 15
2. FIELD HARMONICS OF A CURRENT LINE
Multipoles given by a current line decay with the order
10



n 1
 I  010 
 Rref


ln  bn  ia n   ln
 n ln 
 2R B 
ref 1 
 z0

4
1
¦bn+ian¦
I 010  Rref

bn  ia n  
2z0 B1  z0
4

  p  nq


0.1
0.01
0.001
0
5
10
Multipole order n
15
The slope of the decay is the logarithm of (Rref/|z0|)
At each order, the multipole decreases by a factor Rref /|z0|
The decay of the multipoles tells you the ratio Rref /|z0|, i.e. where
is the coil w.r.t. the reference radius –
like a radar … one can detect assembly errors in the magnet
through magnetic field shape
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 16
2. FIELD HARMONICS OF A CURRENT LINE
Multipoles given by a current line decay with the order
10



n 1
 I  010 
 Rref


ln  bn  ia n   ln
 n ln 
 2R B 
ref 1 
 z0

4
1
¦bn+ian¦
I 010  Rref

bn  ia n  
2z0 B1  z0
4

  p  nq


0.1
0.01
0.001
0
5
10
Multipole order n
15
The semilog scale is the natural way to plot multipoles
This is the point of view of Biot-Savart
But usually specifications are on a linear scale
In general, multipoles must stay below one or a fraction of units – see later
This explains why only low order multipoles, in general, are relevant
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 17
CONTENTS
1. Definition of field harmonics
Multipoles
2. Field harmonics of a current line
The Biot-Savart law
3. Validity limits of field harmonics
4. Beam dynamics requirements on field harmonics
Random and skew components
Specifications (targets) at injection and high field
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 18
3. VALIDITY LIMITS OF FIELD HARMONICS
When we expand a function in a power series we lose
something

1
 1  t  t 2  t 3  ...   t n1
1 t
n 1
y
t 1
Example 1. In t=1/2, the function is
1
1

2
1 1/ 2 1/ 2
40
0
-4 0
0
x
40
and using the series one has
-4 0
2
3
1 1 1
1         ...  1  0.5  0.25  0.125  1.875  ...
2 2 2
not bad … with 4 terms we compute the function within 7%
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 19
3. VALIDITY LIMITS OF FIELD HARMONICS
When we expand a function in a power series we lose
something

1
2
3
 1  t  t  t  ...   t n1
1 t
n 1
y
40
t 1
Ex. 2. In t=1, the function is infinite
1
1
 
11 0
and using the series one has
1  1  1  1  ...  1  1  1  1  ...
2
0
-4 0
0
x
40
-4 0
3
which diverges … this makes sense
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 20
3. VALIDITY LIMITS OF FIELD HARMONICS
When we expand a function in a power series we lose
something

1
2
3
 1  t  t  t  ...   t n1
1 t
n 1
y
40
t 1
Ex. 3. In t=-1, the function is well defined
1
1

11 2
0
-4 0
0
x
40
BUT using the series one has
-4 0
1  (1)   1   1  ...  1  1  1  1  ...
2
3
even if the function is well defined, the series does not work: we are
outside the convergence radius
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 21
3. VALIDITY LIMITS OF FIELD HARMONICS
When we expand a function in a power series we lose
something

1
2
3
 1  t  t  t  ...   t n1
1 t
n 1
y
40
t 1
Ex. 3. In t=-1, the function is well defined
1
1

11 2
0
-4 0
0
x
40
BUT using the series one has
-4 0
1  (1)   1   1  ...  1  1  1  1  ...
2
3
even if the function is well defined, the series does not work: we are
outside the convergence radius
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 22
3. VALIDITY LIMITS OF FIELD HARMONICS
If we have a circular aperture, the field harmonics
expansion relative to the center is valid within the aperture
40
60
-
-
+
+
+
+
0
0
-4 0
0
40
+
+
-4 0
0
+
+
-
-
40
-
-
-4 0
-6 0
For other shapes, the expansion is valid over a circle that
touches the closest current line
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 23
3. VALIDITY LIMITS OF FIELD HARMONICS
Field harmonics in the heads
Harmonic measurements are done with rotating coils of a given
length (see unit 21) – they give integral values over that length
If the rotating coil extremes are in a region where the field does not
vary with z, one can use the 2d harmonic expansion for the integral
2d field harmonics
B (T)
B (T)
2d field harmonics
-1
-1
0
length (m)
15
0
length (m)
15
B (T)
If the rotating coil extremes are in a region where the field vary with z,
one cannot use the 2d harmonic expansion for the integral
no 2d field harmonics
One has to use a more
complicated expansion
-1
0
E. Todesco, Milano Bicocca January-February 2016
length (m)
15
Unit 3 - 24
CONTENTS
1. Definition of field harmonics
2. Field harmonics of a current line
3. Validity limits of field harmonics
4. Beam dynamics requirements on field harmonics
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 25
4. BEAM DYNAMICS REQUIREMENTS
Main component of the dipoles (field)
This is ensuring the orbit along the ring
Typically a absolute knowledge of the magnetic field within 0.1%
A spread between magnets of the order of 0.1%
A reproducibility of the order of 0.01%
We will see that the tolerances needed for building superconducting
magnets naturally guarantee these levels
Cables are positioned within 0.05 mm, apertures of the order of 25 mm,
this gives 0.2% error in field, and similar values for spread
For reproducibility it is critical to cycle the magnets
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 26
4. BEAM DYNAMICS REQUIREMENTS
Main component of the quadrupoles (gradient)
The tune has to be controlled within 0.001
Tune is proportional to quadrupole main component
If the total tune is large (for instance 60 in the LHC) even 0.01%
variation of quadrupole force is visible
Corrector elements and feedback solve the problem – magnets alone
cannot reach this level
In general a absolute knowledge of quadrupole main component
within 0.2% is achievable, a spread of 0.1%, and a reproducibility of
0.01%
Note that accelerators with larger number of cells (larger tune) become
more difficult
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 27
4. BEAM DYNAMICS REQUIREMENTS
Sextupolar components
Sextupole gives chromaticity – to be controlled on a edge of a cliff
(positive but smaller than 10, negative values make the beam
unstable)
Dipoles have sextupolar components that needs to be controlled
within 0.1 units (normalized multipoles)
Example: in the LHC 1 units of b3 in the dipoles give 40 chromaticity
units – so one need to know and control b3 within 0.05 units,
This is within reach of measurement systems, and with proper
precycling reproducibility can be guaranteed
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 28
4. BEAM DYNAMICS REQUIREMENTS
High order mutipoles
Rule of thumb (just to give a zero order idea): field harmonics have
to be of the order of 0.1 to 1 unit
¦bn+ian¦
Higher order are ignored in beam dynamics codes (in LHC up to order
11 only)
Note that spec is rather flat, but multipoles are decaying !! – Therefore in
principle higher orders cannot be a problem
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
E. Todesco, Milano Bicocca January-February 2016
5
10
Multipole order n
15
Unit 3 - 29
SUMMARY
We outlined the Maxwell equations for the magnetic field
We showed how to express the magnetic field in terms of
field harmonics
Compact way of representing the field
Biot-Savart: multipoles decay with multipole order as a power law
Attention !! Validity limits and convergence domains
We outlined some issues about the beam dynamics
specifications on harmonics
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 30
COMING SOON
Coming soon …
It is useful to have magnets that provide pure field harmonics
How to build a pure field harmonic (dipole, quadrupole …) [pure
enough for the beam …] with a superconducting cable ? Which
field/gradient can be obtained ?
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 31
REFERENCES
On field harmonics
A. Jain, “Basic theory of magnets”, CERN 98-05 (1998) 1-26
Classes given by A. Jain at USPAS
P. Schmuser, Ch. 4
On convergence domains of analytic functions
Hardy, “Divergent series”, first chapter (don’t go further)
On the field model
A.A.V.V. “LHC Design Report”, CERN 2004-003 (2004) pp. 164-168.
N. Sammut, et al. “Mathematical formulation to predict the harmonics
of the superconducting Large Hadron Collider magnets”
Phys. Rev. ST Accel. Beams 9 (2006) 012402.
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 32
ACKOWLEDGEMENTS
A. Jain for discussions about reference radius, multipoles in heads,
vector potential
S. Russenschuck for discussions about vector potential
G. Turchetti for teaching me analytic functions and divergent series, and
other complicated subjects in a simple way
www.wikipedia.org for most of the pictures
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 33
1. DEFINITION OF FIELD HARMONICS
Vector potential
Since   B  0 one can always define a vector potential A such that
 A B
The vector potential is not unique (gauge invariance): if we add the
gradient of any scalar function, A'  A  f it still satisfies
  A'    A    f    A  B
Scalar potential
In the regions free of charge and magnetic material   B  0
Therefore in this case one can also define a scalar potential (such as
for gravity)  g  B
One can prove that A  ig is an analytic function in a
region free of charge and magnetic material
E. Todesco, Milano Bicocca January-February 2016
Unit 3 - 34
2. FIELD HARMONICS OF A CURRENT LINE
Field given by a current line (Biot-Savart law) – vector
potential formalism …
y
B  
Az
r
1 Az
Br 
r 
40
z 0 =x 0 +iy 0
B=B y +iB x
0
-4 0
2
2
0 I  R 
 0 I   0    2  0  cos(   0 ) 
Az (  ,  )  
ln    
ln

2   0 
2 
0


0
x
40
z=x+iy
-4 0
y
z 0 =  0 exp(i  0 )
40
… and polar coordinates formalism
n 1
 I
Br   0
20
 


n 1   0



 I
B   0
20
 


n 1   0
n 1





sin n   0 
B
0
-4 0
0
x
40
z =  exp(i  )
cosn   0 
E. Todesco, Milano Bicocca January-February 2016
-4 0
Unit 3 - 35