Motor design using the Schwarz-Christoffel transformation

1
Motor design using the Schwarz-Christoffel
transformation

• Tim C. OConnell
• Doctoral Student Advisor P.T. Krein
• University of Illinois at Urbana-Champaign
• Power Affiliates Program
• Friday, May 12th, 2006

2
Overview

• Machine design overview
• Schwarz-Christoffel (SC) mapping
• Application to motor design
• Example
• Conclusions

3
Overview

• Machine design overview
• Schwarz-Christoffel (SC) mapping
• Application to motor design
• Example
• Conclusions

4
Machine design overview

• Shape
• Used to steer the flux
• Materials
• Affect efficiency, weight, acoustic properties,
  manufacturability, cost
• Sources
• Characteristics and placements of currents
• Types and placement of permanent magnets

5
Machine design overview
Standard Methods

• Equivalent circuit models
• Magnetic circuit models
• Finite element analysis (FEA)

6
Overview

• Machine design overview
• Schwarz-Christoffel (SC) mapping
• Application to motor design
• Example
• Conclusions

7
SC mapping
What is a Schwarz-Christoffel map?
w f(z)
8
SC mapping
Can be used to map electomagnetic field lines
from one polygon to another.
9
SC mapping
Thm Fundamental Theorem of Schwarz-Christoffel
Mapping
10
SC mapping

• Numerical solution required for
• Calculating SC integral parameters
• Calculating the SC integral
• Inverting the map

11
SC mapping
Historical milestones machine design with SC
mapping

• 1820s Gauss idea of conformal mapping
• 1867-90 Schwarz and Christoffel discover SC
  formula and variants
• 1900-01 F.W. Carter uses SC mapping for field
  between poles
• I by no means recommend that one should go to
  the trouble of using these somewhat difficult
  formulae in average practical cases Carter
• 1980 Trefethen SCPACK FORTRAN program
• 1996 Driscoll SC Toolbox for Matlab
• 1998 Improvements to SC Toolbox algorithms

12
Overview

• Machine design overview
• Schwarz-Christoffel (SC) mapping
• Application to motor design
• Example
• Conclusions

13
Application to motor design
Design goal Calculate the electromagnetic fields
and corresponding rotor torques/forces for a
given geometry and set of materials and sources
14
Application to motor design
Assumptions

1. 2D developed machine cross-section
2. Air gap is a polygon (no curves) with n vertices
3. Linear magnetics
4. Periodic boundary condition (BC) at polygon edges
5. Finite, discrete currents as sources

wn
P
Periodic BC
Periodic BC
w1
15
Application to motor design
w u iv
iy
iv
w u iv
z exp(-iw)
x
w i log(z)
u
16
Application to motor design
w-plane
z f-1(w)
w f(z)

• Hcc known in infinite series form due to Hague
  circa 1930.
• Periodic BC automatically enforced

17
Application to motor design
SC Toolbox for MATLAB

• Released in 1996
• Calculates SC integral parameters
• Computes forward and inverse maps
• Computes derivative of maps (easier)
• Graphical and object-oriented user interfaces

18
Application to motor design
Force/torque calculation

• Maxwell Stress Tensor (MST) method
• Integrate the MST around a closed path
• Highly path- and element-dependent for FEA
• Ideal for SC solution
• No path dependence

19
Overview

• Machine design overview
• Schwarz-Christoffel (SC) mapping
• Application to motor design
• Example
• Conclusions

20
Examples
periodic BC
periodic BC

• 40- vertex air gap polygon
• 2 coils

21
Examples
Solve for Hcc using Hagues analytic solution.
µr 100µ0
µs 100µ0
22
Examples
23
Examples
MST Force Calculation
Integrate tangential and normal force densities
around closed path
24
Examples
25
Examples
26
Overview

• Machine design overview
• Schwarz-Christoffel (SC) mapping
• Application to motor design
• Example
• Conclusions

27
Conclusions

• Design benefits
• accurate field and force calculation
• may be possible to design in conc. cylinder
  domain
• Can be useful when fields near sharp corners of
  poles and teeth are needed to high accuracy
• Promising technique due to its accuracy