BNL - Superconducting Accelerator Magnets [presentation slides]

220 Pages·2003·5.19 MB·English

by  A. Jain

Harmonic Description of
2-Dimensional Fields
Animesh Jain
Brookhaven National Laboratory
Upton, New York 11973-5000, USA
US Particle Accelerator School on Superconducting Accelerator Magnets
Santa Barbara, California, June 23-27, 2003
Fields in Free Space: Scalar Potential
• ∇.B = 0 (Always true)
•
In a region free of any currents or magnetic
material, ∇ × B = 0 , and B may be written
as the gradient of a scalar potential, B = ∇Φ
m
•
The two equations above may be combined
to obtain the Laplace’s equation for the
scalar potential, Φ ,
m
2
∇ Φ = 0
m
USPAS, Santa Barbara: June 23-27, 2003 2 Animesh Jain, BNL
2-D Fields in Free Space
• B = ∇Φ and ∇2Φ = 0
m
m
•
Most accelerator magnet apertures have a
cylindrical symmetry, with a length much
larger than the aperture. In such situations,
the field away from the ends can be
considered 2-dimensional, and the general
solution can be expressed in a relatively
simple harmonic series.
USPAS, Santa Barbara: June 23-27, 2003 3 Animesh Jain, BNL
Commonly Used Coordinate System
Y-Axis (TOP)
Aperture
Users of magnetic
Currents
measurements
data may use a
B system oriented
y
B
differently, often
B
θ
B
r requiring suitable
θ
r θ transformations
B
x
of the measured
θ
harmonics.
X-Axis (RIGHT)
View from the Lead End of the Magnet
B (r,θ) = B cosθ − B sinθ B (r,θ) = B sinθ+ B cosθ
x r θ y r θ
USPAS, Santa Barbara: June 23-27, 2003 4 Animesh Jain, BNL
Solution in Cylindrical Coordinates
For no z-dependence (2-D fields),
 1  ∂  ∂Φ   1  ∂2Φ 
∇2Φ = r m +  m  = 0
     
 
m 2 2
r ∂r ∂r r ∂θ
     
 
writing Φ (r,θ) = R(r)Θ(θ), and imposing
m
the conditions
Θ(θ + 2π) = Θ(θ); R(r) = finite at r = 0
we can get the solution of the Laplace’s
equation in terms of a harmonic series.
USPAS, Santa Barbara: June 23-27, 2003 5 Animesh Jain, BNL
2-D Fields: Harmonic Series
•
Components of 2-D fields in cylindrical
coordinates:
n−1
∞  r 
∑  
B (r,θ) = C(n) sin[n(θ−α )]
r   n
R
 
n=1 ref
n−1
∞  r 
∑  
B (r,θ) = C(n) cos[n(θ−α )]
θ   n
R
 
n=1 ref
•
C(n) = Amplitude, α = phase angle of the
n
2n-pole term in the expansion.
•
R = Reference radius, arbitrary, typically
ref
chosen ~ the region of interest. C(n) scales as R n − 1
ref
USPAS, Santa Barbara: June 23-27, 2003 6 Animesh Jain, BNL
2-D Fields: Cartesian Components
•
Cartesian components of B may be written
as: n−1
∞  r 
∑  
B (r,θ) = C(n) sin[(n − 1)θ− nα ]
x   n
R
 
n=1 ref
n−1
∞  r 
∑  
B (r,θ) = C(n) cos[(n − 1)θ− nα ]
y   n
R
 
n=1 ref
•
A Complex field, B(z) = B + iB , where
y x
z = x + iy, combines the 2 equations above:
n−1
 
∞ z
[ ]
∑  
B(z) = C(n) exp(−inα )
n  
R
 
n=1 ref
USPAS, Santa Barbara: June 23-27, 2003 7 Animesh Jain, BNL
2-D Fields: Normal & Skew Terms
n−1
∞  z 
∑[ ]  
B(z) = B + iB = C(n) exp(−inα )
y x n  
R
 
n=1 ref
n−1
Simple power
may be ∞  z 
[ ]
∑  
B(z) = B + iA series, valid within
written as: n n  
R
  source-free zone.
n=1 ref
B ≡ C(n) cos(nα ) = 2n - pole NORMAL Term
n n
where:
A ≡ −C(n) sin(nα ) = 2n - pole SKEW Term
n n
In the US, the 2n-pole terms are denoted by B and A .
n–1 n-1
Sometimes, the skew terms are defined without the
negative sign, but the above form is the most common now.
USPAS, Santa Barbara: June 23-27, 2003 8 Animesh Jain, BNL
Analytic Functions of a Complex Variable
Any function of the complex variable, z, given by
F(z) = U(x,y) + i V(x,y)
is an Analytic function of z, if
 ∂U   ∂V   ∂U   ∂V 
Cauchy-Riemann
=   and   = −
   
   
Conditions.
∂x ∂y ∂y ∂x
   
   
An analytic function can be expressed as a power
series in z. This series is valid within the circle of
convergence, which extends to the nearest
singularity. Analytic function does not depend on z*.
USPAS, Santa Barbara: June 23-27, 2003 9 Animesh Jain, BNL
Analyticity of Complex Field
 ∂U   ∂V   ∂U   ∂V  Cauchy-Riemann
  =   and   = − 
   
Conditions.
∂x ∂y ∂y ∂x
       
Maxwell’s equations in source free region:
 ∂B   ∂B   ∂B   ∂B 
∇.B = 0 ⇒  y  = − x  (∇ × B) = 0 ⇒  y  =  x 
     
z
∂y ∂x ∂x ∂y
       
Maxwell’s equations = Cauchy-Riemann conditions
if we choose: U(x,y) = B (x,y) and V(x,y) = B (x,y)
y x
Thus, B(z) = B (x,y) + i B (x,y) is an analytic
y x
function of z. The analyticity is useful in dealing
with 2-D problems in magnetostatics.
USPAS, Santa Barbara: June 23-27, 2003 10 Animesh Jain, BNL