By W.J. Gibbs
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Extra resources for Conformal Transformations in Electrical Engineering
386+j0 as shown in Fig. 24 (b). e. 099 +j0 as shown in Fig. 24 (b). 386+j7r as shown in Fig. 24 (b). Thus the four boundary points in the z plane become the four boundary points shown in Fig. 24 (b). Moreover the equations show that in the X plane the points are connected by straight Therefore the shaded area in Fig. 24 (b) corresponds to the shaded annular area of Fig. 24 (a). In considering the transformations it is helpful to imagine lines. that the area enclosed by the boundaries contains an ideal elastic substance or membrane corresponding to the equipotentials and the lines of flow.
29 appears. s 4 -12 0 4 FIG. 29. w plane. 55 e 12 CHAPTER 6 The Schwarz-Christoffel Transformation IN ALL THE EXAMPLES discussed in previous chapters trans- formation equations have been set down first and the corresponding configurations in both planes have been discovered afterwards. This is hardly a satisfactory procedure for actual problems because in practice the engineer knows the two configurations, the one he is concerned with and the simpler one possessing a known solution. From these known configurations he has to determine the appropriate equation of transformation.
180 -869 Notice that at u = jrr, x = 0 giving the imaginary axis of the z plane. When plotted on the z plane the two sets of values given in Table 3 of x and y give the lower halves of two hyperbolas. The upper halves can be found by giving negative values to v and positive values less than 0-55. The two hyperbolas are shown in Fig. 19. Note that negative values of x can and 37r/2. be found by selecting values of u between 43 2Tr CONFORMAL TRANSFORMATIONS 8 4 U. 2 - 57 3 6 Y 1111 u' 6 2 u-O 4 6 8 -2 -4 6 -6 U_ Uz 2 3 -8 Confocal hyperbolas.
Conformal Transformations in Electrical Engineering by W.J. Gibbs