Conformal Transformations

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Conformal Transformations

• Neil Marks,
• DLS/CCLRC,
• Daresbury Laboratory,
• Warrington WA4 4AD,
• U.K.
• Tel (44) (0)1925 603191
• Fax (44) (0)1925 603192

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Use of transformations

• This is a mathematical technique developed to
  give analytical expressions for potential
  and flux density distributions for (simple)
  geometries.
• This was the standard technique for magnet design
  before finite element analysis (f.e.a.) codes
  were developed.
• The technique is based on transforming
  geometries using functions of the complex
  variable.
• We start with a model with unknown field and
  potential distributions and transform that to a
  geometry where distributions can be analytically
  defined. The known distribution is then
  transformed back to the initial model using the
  inverse of the first transform, giving the
  required result.
• Each transformation often involves an
  intermediate step.

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The Transformations.

• We start by defining two complex planes, Z and W

All conformal transformations preserve the
angle of intersection between two curves
except at the origin and at poles of the
transforming function. As the most suitable
ideal pole to use for the known distribution
will be a pair of parallel poles extending
to plus and minus infinity, such a
function will be necessary.
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The 'Schwarz - Christoffel' Transformation

• This is one type of a conformal transform it
  transforms polygons (complete or open) in the
  Z plane to a straight line on the real
  axis in the W plane
• The transformation is given by
• dZ/dW M ( W - a1)((a1/p ) -1) (W - a2)((a2/p )
  - 1) ........
• where M is an arbitrary constant to be determined.

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Second transformation.

• A second transformation is then used to
  translate from a T plane, where the
  geometry is an infinite dipole, back into
  the W plane
• The two transformations are then combined
  to predict the distributions in the real
  magnet.

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Example a dipole end.

Transformation Z to W
dZ/dW M (W 1)1/2 ( W ) -1 Z M 2(W
1)1/2 ln (W 1)1/2 -1 - ln (W 1)1/2 1
N where N is another arbitrary constant N
0 just sets the origin in Z plane for W
-1, the above expression gives Z i M p i
g/2 so M g/2p for W gt 0, Z is real.
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Transformation T to W

• The T plane, an infinite dipole
• dT/dW M (W a )(a/p 1)
• for the above a 0 a 0
• so dT/dW M W -1
• T M ln(W) N
• again N 0 sets the origin in the Z plane
• for W -1, above expression gives Z Mip
  ig/2
• so M g/2p, giving
• T (g/2p) ln W
• W exp(2pT/g)

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Resulting equation

• We now substitute for W to give Z in terms of T
• Z (g/2p) 2 exp (2pT/g) 11/2
• ln exp(2pT/g 1 -1
• - ln exp(2pT/g 1 1
• Equipotential lines are Im (T) const
• Flux lines are Re (T) const.
• Expanding with Re and Im of T and then separating
  the real and imaginary components of Z is long
  and detailed see next slide.

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Solution

For g 1
where
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Graphical results - lines of scalar potential

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Graphical results - lines of flux

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Flux density B

• B -? f
• put T y if
• then Bx - ?f / ? x
• By - ? f / ? y
• Bx - j By - ? f/? x i ? f/? y
• from Cauchy-Riemann equations
• ? y / ? x ? f / ? y
• ? y / ? y - ? f/? x
• so Bx - iBy
• - ? f / ? x i? y / ? x
• i ? T / ? x
• hence
• B d T/ dZ
• ( dT/dW )( dW/dZ )

Flux density on x axis
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Problems

• integration is only analytical for angles of 0
  or multiples of p/2
• and a limited number of right angles
• other more complex geometries require numerical
  integration
• predicts distributions only for µ ? in the
  steel
• the technique takes no account of coils ie all
  currents are at infinity.

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The 'Rogowski' roll-off

• The classical end solution, developed originally
  in electrostatics during the study of the end
  effect for two parallel capacitor plates. The
  analysis also uses the conformal transformation
  method

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Analysis

• dZ/dW M (W 1)/W
• Z (d/?)( 1 W ln W )
• T (1/?) ln W
• so Z (d/?)( 1 exp ?T ?T )
• if T y jf
• then in the Z plane
• x (d/?)(1 (exp py)(cos ?f) ?y)
• y (d/?)( (exp ?y)(sin ?f) ?f)
• Potential is lines of const f stream lines are
  const y .

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Graphical results

• Potential lines

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Blown-up version

The central heavy line is for f 0.5. Rogowski
showed that this was the fastest changing line
along which the field intensity was monotonically
decreasing.
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Application to magnet ends

• Conclusion Recall that a high µ steel surface is
  a line of constant scalar potential. Hence, a
  magnet pole end using the f 0.5 potential line
  will see a monotonically decreasing flux density
  normal to the steel at some point where B is
  much lower, this can break to a vertical end
  line.

Magnet half gap height g/2 Centre line of
gap is y 0 Equation is y g/2
(g/?) exp ((?x/g)-1)