1、外文翻译 2006 IEEE COMPEL Workshop, Rensselaer Polytechnic Institute, Troy, NY, USA, July 16-19, 2006 A Preliminary Investigation of Computer-Aided Schwarz-Christoffel Transformation for Electric Machine Design and Analysis Timothy C. OConnell and Philip T. Klein Grainger Center for Electric Machinery a
2、nd Electro mechanics Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign 1406 W. Green St. Urbana, IL 61801-2918 USA Abstract-Analternative method to finiteelement analysis (FEA)for electricmachine design and analysis is presented that applies Schwarz-Christo
3、ffel(SC)conformal mapping theoryusing the SC Toolbox for MATLAB that has appeared in the previous literature. In this method, a two-dimensional (2D) developed machine cross-section domain is mappedvia SCtransformation to a concentric cylinder domain where solutions for the electromagnetic (EM) field
4、s are known. These solutions are mapped back tothe original domain, thussolving theoriginal problem. All mapping is done via the SC Toolbox. Examples are given in which the procedure is used to calculate the EM field in the air gap of and the force on the rotor ofvarious 2D developed machine cross-s
5、ections. The numerical accuracy of the results is verified by comparing the solutions as the air gap gets small with magnetic equivalent circuit (MEC)-derived co-energy solutions. I. INTRODUCTION The most general electric machine design problem can be described as followsgiven a set of desired machi
6、ne output characteristics,find the optimum machine geometry, materials, and input source characteristics that will achieve these goals. This is a formidable problem in its most general form, especially -considering the recent increase in theavailability of inverters, exotic permanent magnet (PM) mat
7、erials, and low-cost, precision manufacturing. Usually several - assumptions and basic a priori design decisions must be made to render the problem tractable. A standard technique is to use basic machine theory to generate a rough design which might include the type of machine (synchronous, inductio
8、n, PM, etc.), the number of poles, and the materials. The base design is then analyzed and refined in an iterative process using FEA software until an acceptable match to the desired output is found. While FEA is a powerful analysis tool that is fairly easy to use and widely available in a number of
9、 commercial software packages, its utility in design is less obvious. Using FEA, it is oftendifficult to see the relationships between various input and outputparameters without extensive and time-consuming iterations.Frequently, the necessary accuracy needed for agiven problem cannot be achieved wi
10、thout unreasonable computer run times. Thus, an alternative to FEA, more suitedto design, can be auseful addition to the machine designersrepertoire.This paper investigates the utility of the MATLAB SC Toolbox, a free add-on toolbox developed by T. Driscoll 1-3 that automates the process of calculat
11、ing SC maps. The SC method is a 2D complex analysis tool that allows one to circumventmany of the difficulties encountered when solving a boundary value problem on a domain defined by a complicated geometry. Using a complex conformal mapping from the problem domain to a simpler domain, one can more
12、easily solve the problem, and then map the solution back to the original geometry. The key to successfully applying the SC method is to find the correct mapping between domains. The SC Toolbox makes this step much easier than it has been previously and thus allows further exploration into the merits
13、 of SC mapping as a viable machine design tool. The torque (force) on the moveable member of an electric machine is usually found by applying either the Coulomb virtual work (CVW) method 4, 5 or the Maxwell stress tensor (MST) method 6 to the EM fields 7-10. In either method, the force is found as t
14、he product of field terms； thus, any errors in the calculated fields are compounded when force is computed. In addition, the useful forces in an electric machine are typically concentrated at sharp corners (i.e. at pole teeth corners) where FEA solutions are least accurate. When using an FEA field s
15、olution, both CVW and MST are sensitive to the mesh choice because the fields are interpolated between a finite numbers of solution points at the mesh nodes. In contrast, SC conformal mapping theory can calculate an accurate solution for the fields at every point. No interpolation is necessary. Inhe
16、rently, the accuracy of the SC mapping does not suffer at sharp- corners. The SC Toolbox makes it much easier than previously possible to find the EM fields accurately using conformal mapping. With an accurate field solution force calculation also becomes more accurate and easier to implement. The u
17、tility of SC mapping is explored here for the following reasons: (i) its implementation is much easier than previously owing to the introduction of the SC Toolbox； (ii) it can produce an accurate field solution at every point that does not suffer near sharp corners； (iii) the solutions allow an II.
18、BACKGROUND Early machine designers quickly saw the difficulties of solving the EM - field equations in electric machines. In Teslas seminal induction machine paper 11 there arealmost no equations. Much of his design was based on a fundamental understanding of the field interactions necessary to crea
19、te motion Behr end 12 developed several graphical arose when designers attempted to analyze pole-pieces, slots, and teeth, all of which significantly changed the boundary conditions and field distributions in machines. Conformal mapping, of which the SC method is a subset, was used by Carter in an a
20、ttempt to better understand these problems 13, 14. However, due to the mathematical -complexities he encountered, Carter recommended against using SC mapping for all but the simplest cases.Signers subsequently used measurements and observations to develop empirical design equations； equivalent circu
21、it models, magnetic equivalent circuits (MEC) and graphical methods were all commonplace by 1916 15. The success of these methods is wellestablished. In 1929, Hague 16 noted there was insufficient literature describing fundamental machine interactions from Maxwells equations. He presented a theory f
22、or determining the EM fields of current-carrying conductors in the air gap of various iron geometries, the solution of which describes the operation of electrical machines. Hague found it surprising that only one person before him (Searle, in 1898 17) had considered this problem. While Hagues work was important in bridging the gap between fundamental field-based machine analysis and circuit-based models, it did not present a useful alternative to the
