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Dr. Hugh Blanton ENTC 3331

ENTC 3331 RF Fundamentals. Dr. Hugh Blanton ENTC 3331. Fields and Waves. VECTORS and VECTOR CALCULUS. VECTORS. Today’s Class will focus on:. vectors - description in 3 coordinate systems. vector operations - DOT & CROSS PRODUCT. vector calculus - AREA and VOLUME INTEGRALS.

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Dr. Hugh Blanton ENTC 3331

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  1. ENTC 3331 RF Fundamentals Dr. Hugh Blanton ENTC 3331

  2. Fields and Waves VECTORS and VECTOR CALCULUS Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 2

  3. VECTORS Today’s Class will focus on: • vectors - description in 3 coordinate systems • vector operations - DOT & CROSS PRODUCT • vector calculus - AREA and VOLUME INTEGRALS Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 3

  4. Choice is based on symmetry of problem VECTOR REPRESENTATION 3 PRIMARY COORDINATE SYSTEMS: • RECTANGULAR • CYLINDRICAL • SPHERICAL Examples: Sheets - RECTANGULAR Wires/Cables - CYLINDRICAL Spheres - SPHERICAL Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 4

  5. Orthogonal Coordinate Systems: (coordinates mutually perpendicular) z P(x,y,z) Cartesian Coordinates y P (x,y,z) x Rectangular Coordinates z z P(r, θ, z) Cylindrical Coordinates P (r, Θ, z) y r x θ z P(r, θ, Φ) Spherical Coordinates θ r P (r, Θ, Φ) y x Φ Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 5 Page 108

  6. Parabolic Cylindrical Coordinates (u,v,z) • Paraboloidal Coordinates (u, v, Φ) • Elliptic Cylindrical Coordinates (u, v, z) • Prolate Spheroidal Coordinates (ξ, η, φ) • Oblate Spheroidal Coordinates (ξ, η, φ) • Bipolar Coordinates (u,v,z) • Toroidal Coordinates (u, v, Φ) • Conical Coordinates (λ, μ, ν) • Confocal Ellipsoidal Coordinate (λ, μ, ν) • Confocal Paraboloidal Coordinate (λ, μ, ν) Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 6

  7. Parabolic Cylindrical Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 7

  8. Paraboloidal Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 8

  9. Elliptic Cylindrical Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 9

  10. Prolate Spheroidal Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 10

  11. Oblate Spheroidal Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 11

  12. Bipolar Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 12

  13. Toroidal Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 13

  14. Conical Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 14

  15. Confocal Ellipsoidal Coordinate Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 15

  16. Confocal Paraboloidal Coordinate Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 16

  17. z z Cartesian Coordinates P(x,y,z) P(x,y,z) P(r, θ, Φ) θ r y x y x Φ Cylindrical Coordinates P(r, θ, z) Spherical Coordinates P(r, θ, Φ) z z P(r, θ, z) y r x θ Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 17

  18. z y x VECTOR NOTATION VECTOR NOTATION: Rectangular or Cartesian Coordinate System Dot Product (SCALAR) Cross Product (VECTOR) Magnitude of vector Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 18

  19. Cartesian Coordinates ( x, y, z) Vector representation z z1 Z plane Magnitude of A x plane y plane Az y1 y Ay Position vector A Ax x1 x Base vector properties Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 19 Page 109

  20. Cartesian Coordinates z Dot product: Az y Cross product: Ay Ax x Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 20 Back Page 108

  21. z r P z x f y VECTOR REPRESENTATION: CYLINDRICAL COORDINATES UNIT VECTORS: Cylindrical representation uses: r ,f , z Dot Product (SCALAR) Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 21

  22. z P q r x f y VECTOR REPRESENTATION: SPHERICAL COORDINATES UNIT VECTORS: Spherical representation uses: r ,q , f Dot Product (SCALAR) Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 22

  23. z y x VECTOR REPRESENTATION: UNIT VECTORS Rectangular Coordinate System Unit Vector Representation for Rectangular Coordinate System The Unit Vectors imply : Points in the direction of increasing x Points in the direction of increasing y Points in the direction of increasing z Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 23

  24. z r P z f x y VECTOR REPRESENTATION: UNIT VECTORS Cylindrical Coordinate System The Unit Vectors imply : Points in the direction of increasing r Points in the direction of increasing j Points in the direction of increasing z Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 24

  25. z P q r f y x VECTOR REPRESENTATION: UNIT VECTORS Spherical Coordinate System The Unit Vectors imply : Points in the direction of increasing r Points in the direction of increasing q Points in the direction of increasing j Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 25

  26. VECTOR REPRESENTATION: UNIT VECTORS Summary RECTANGULAR Coordinate Systems CYLINDRICAL Coordinate Systems SPHERICAL Coordinate Systems NOTE THE ORDER! r,f, z r,q ,f Note: We do not emphasize transformations between coordinate systems Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 26

  27. Unit is in “meters” METRIC COEFFICIENTS 1. Rectangular Coordinates: When you move a small amount in x-direction, the distance is dx In a similar fashion, you generate dy and dz Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 27

  28. Differential quantities: Length: Area: Volume: Cartesian Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 28 Page 109

  29. y df r x METRIC COEFFICIENTS 2. Cylindrical Coordinates: Differential Distances: Distance = r df ( dr, rdf, dz ) Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 29

  30. z P q r r sinq f y x y df x METRIC COEFFICIENTS 3. Spherical Coordinates: Differential Distances: Distance = r sinq df ( dr, rdq, r sinq df ) Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 30

  31. METRIC COEFFICIENTS Representation of differential length dl in coordinate systems: rectangular cylindrical spherical Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 31

  32. dy dx y 6 2 3 7 x AREA INTEGRALS • integration over 2 “delta” distances Example: AREA = = 16 Note that: z = constant In this course, area & surface integrals will be on similar types of surfaces e.g. r =constant or f = constant or q = constant et c…. Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 32

  33. SURFACE NORMAL Representation of differential surface element: Vector is NORMAL to surface Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 33

  34. DIFFERENTIALS FOR INTEGRALS Example of Line differentials or or Example of Surface differentials or Example of Volume differentials Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 34

  35. r radial distance in x-y plane Φ azimuth angle measured from the positive x-axis Z Cylindrical Coordinates ( r, θ, z) A1 Vector representation Base Vectors Magnitude of A Base vector properties Position vector A Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 35 Back Pages 109-112

  36. Cylindrical Coordinates Dot product: A B Cross product: Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 36 Back Pages 109-111

  37. Cylindrical Coordinates Differential quantities: Length: Area: Volume: Pages 109-112 Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 37

  38. Spherical Coordinates (R, θ, Φ) Vector representation Magnitude of A Position vector A Base vector properties Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 38 Back Pages 113-115

  39. Spherical Coordinates Dot product: A B Cross product: Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 39 Back Pages 113-114

  40. Spherical Coordinates Differential quantities: Length: Area: Volume: Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 40 Back Pages 113-115

  41. Cartesian to Cylindrical Transformation Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 41 Back Page 115

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