By W.J. Gibbs
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Extra resources for Conformal Transformations in Electrical Engineering
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. The material is ideal in the sense that it is infinitely expansible and infinitely compressible without its changing character and without there being constraints of any kind.
This result enables us to consider the trans- formation of configurations or shapes consisting of straight lines and angles. It follows from the previous section that any small figure, say the small triangle shown in the z plane Fig. 23 (a), remains a small triangle when transformed to the w plane Fig. 23 (b), but that it is either enlarged or diminished by the transformation and it is also turned through an angle. Fig. 23 shows the result of a typical transformation. 4) is zero or infinite. Y D 4 U (a) Fm.
5) gives ,920 ( 2- Therefore since the real and imaginary parts in this equation must separately be zero a+ay20 . 8 Therefore the real part and the imaginary part of any function of the complex variable x+jy automatically satisfies Laplace's equation. These real and imaginary functions are called conjugate functions. e. where X = O + j o and z = x+jy X=f(z) 25 CONFORMAL TRANSFORMATIONS we know that and . . . 9) 0 V20 The functions 0 and 0 are conjugate functions. 9), page 10. 9), 0 is identified with potential and called the potential function, the conjugate function b is called the stream function or the flux function.