20091102Lect28EE529

20091102Lect28EE529 - ECE 529 EMI IN MICROELECTRONICS...

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Unformatted text preview: ECE 529 EMI IN MICROELECTRONICS University at Buffalo Dr. James J. Whalen Lecture 8 2SSep96-.~---.~--- ~ -.---- - -- ~-'- .-.-- ..- - ...--~ ~ ...,-- _.-.------ .-------. ~.7 THE SPHERICAL COORDINATE SYSTEM The spherical coordinate system is also formed by three surfaces. One surface is a cone about the z axis of half angle e. The next surface is a sphere centered on the origin of radius r, and the third surface is a plane perpendicular to the xy plane and rotated about the z axis by angle in the direction from x to y as shown in Fig. 2.10. Observe that o < t:P < 21T and 0 < e < 1T. A point is defined as the intersection of these three surfaces, P(rl,e1,t:Pl)' Unit vectors an ae, and a point in the direction of increasing coordinate value. Note that all three unit vectors change direction from point to point unlike the three unit vectors of a rectangular coordinate system which are fixed in direction. The three axes are labeled r, e, and and these are ordered with the right- hand-rule such that a, X ae = a. Note that, for example, a, X a =-ae. The same symbol r is used in both the cylindrical and the spherical coordinate systems but they have different meaning. Two vectors expressed in this coordinate system at point P can be added at that point since the unit vectors of the corresponding components are parallel at that point. z 8 =8 1 cone = 1 plane r= r 1 sphere \ I", / .. \ I' / . \ I / \ 1/ (1 ~---- /...... ~. ... / ... r---------~--~y / / / / x- - - - C / J 1 Figure 2.10 The spherical coordi- nate system illustrating the unit vectors and the location of a point as the intersection of three constant-coordinate surfaces. 2.7 The- Spherical Coordinate System . 0;) Hence for A = A,, + Aeae + A.pa.p B = B,; + Beae + B.pa.p the sum is (2.21) The dot and cross products are very similar to those for the rectangular coordinate system: (2.22) (2.23) Observe that the cyclic ordering of the axis labels, r ~ e~ ~ r ~ e~ ~ ... can, like the rectangular coordinate system, be used to quickly form the cross product...
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20091102Lect28EE529 - ECE 529 EMI IN MICROELECTRONICS...

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