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Conformal Transformations

Conformal Transformations. Neil Marks, DLS/CCLRC, Daresbury Laboratory, Warrington WA4 4AD, U.K. Tel: (44) (0)1925 603191 Fax: (44) (0)1925 603192. Use of transformations.

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Conformal Transformations

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

  2. 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.

  3. y W = f(Z) y Z x x 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.

  4. a1 a1 Z W a2 a3 a2 a3 a1 a2 a3 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.

  5. f W T B g/2 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.

  6. W = - ig/2 W = -1, 3p/2 W = 0, a = 0. g/2 W =  0 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 > 0, Z is real.

  7. W = -  W = -1 i g / 2 W = 0; a = 0 W =  W = 1 0 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)

  8. Resulting equation • We now substitute for W to give Z in terms of T: • Z = (g/2p) [ 2 {exp (2pT/g) + 1}1/2 • + 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.

  9. Solution: For g = 1: where

  10. Graphical results - lines of scalar potential

  11. Graphical results - lines of flux

  12. 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:

  13. 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.

  14. 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:

  15. 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 .

  16. Graphical results • Potential lines:

  17. 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.

  18. 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)

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