ElECTROMAGNETIC FIELDS AND WAVES COORDINATE SYSTEMS AND TRANSFORMATION (MODULE 2) 1 Module Learning Outcomes • Define Electromagnetics (EM) • Solve problems involving spherical coordinates. • Solve problems involving cylindrical coordinates. • Transform from one coordinate system to another. 2 INTRODUCTION • In general, the physical quantities we shall be dealing with in EM are functions of space and time. • In order to describe the spatial variations of the quantities, we must be able to define all points uniquely in space in a suitable manner. • A point or vector can be represented in any curvilinear coordinate system, which may be orthogonal or non-orthogonal. 3 INTRODUCTION CONTINUED • An orthogonal system is one in which the coordinate surfaces are mutually perpendicular. • Nonorthogonal systems are hard to work with, and they are of little or no practical use. • Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the circular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal. 4 Cartesian Coordinates (x, y, z) A point P can be represented as (x, y, z) as illustrated in figure 1.1. The ranges of the coordinate variables x, y, z are: −∞ < π₯ < ∞ −∞ < π¦ < ∞ −∞ < π§ < ∞ A vector A in Cartesian (otherwise known as rectangular) coordinates can be written as: (π΄π₯, π΄π¦ , π΄π§, ) or π΄π₯ ππ + π΄π¦ ππ + π΄π§ ππ 5 Circular Cylindrical Coordinates (π, π, π) A vector A in cylindrical coordinates can be written as: π΄π , π΄π , π΄π§ ππ π΄π ππ + π΄π ππ + π΄π§ ππ The variables ranges from: Radial distance: 0 ≤ π < ∞ Azimuthal angle: 0 ≤ π < 2π Z-axis : −∞ < π§ < ∞ The magnitude of A is: π΄ = π¨π 2 + π¨π 2 + π¨π 1 2 2 6 Circular Cylindrical Coordinates (π, π, π) ππ . ππ = ππ . ππ = ππ . ππ = π ππ . ππ = ππ . ππ = ππ . ππ = π ππ × ππ = ππ ππ × ππ = ππ ππ × ππ = ππ 7 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE CYLINDRICAL COORDINATE SYSTEM Cartesian to Cylindrical π₯2 + π¦2 π¦ −1 ∅ = π‘ππ π₯ π§=π§ π= • This is for point-to-point transformation 8 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE CYLINDRICAL COORDINATE SYSTEM Cylindrical to Cartesian π₯ = ππππ ∅ π¦ = ππ ππ∅ π§=π§ • This is for point-to-point transformation 9 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE CYLINDRICAL COORDINATE SYSTEM The unit vector relationship between the Cartesian coordinate system and the Cylindrical coordinate system: ππ₯ = πΆππ ∅ππ − πππ∅π∅ ππ¦ = πππ∅ππ + πΆππ ∅π∅ π π§ = ππ§ 10 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE CYLINDRICAL COORDINATE SYSTEM The unit vector relationship between the cylindrical coordinate system and the Cartesian coordinate system: ππ = πΆππ ∅ππ₯ + πππ∅ππ¦ π∅ = −πππ∅ππ₯ + πΆππ ∅ππ¦ π π§ = ππ§ 11 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE CYLINDRICAL COORDINATE SYSTEM In matrix form, we write the transformation of vector A from π΄π₯ , π΄π¦ , π΄π§ to π΄π , π΄∅ , π΄π§ as: π΄π πΆππ ∅ πππ∅ 0 π΄∅ = −πππ∅ πΆππ ∅ 0 0 0 1 π΄π§ π΄π₯ π΄π¦ π΄π§ This equation is used for vector-to-vector transformation. 12 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE CYLINDRICAL COORDINATE SYSTEM In matrix form, we write the transformation of vector A from π΄π , π΄∅ , π΄π§ to π΄π₯ , π΄π¦ , π΄π§ as: π΄π₯ πΆππ ∅ −πππ∅ 0 π΄π¦ = πππ∅ πΆππ ∅ 0 π΄π§ 0 0 1 π΄π π΄∅ π΄π§ This equation is used for vector-to-vector transformation. 13 SPHERICAL COORDINATES (r, π½, ∅) A vector A in spherical coordinates may be written as: π΄π , π΄π , π΄π ππ π΄π ππ + π΄π ππ½ + π΄π ππ The variables ranges from: Radius of the Sphere: 0 ≤ π < ∞ Colatitude: 0≤π≤π Azimuthal angle: 0 ≤ π < 2π The magnitude of A is: π΄ = π¨π 2 + π¨π½ 2 + π¨∅ 1 2 2 14 SPHERICAL COORDINATES (r, π½, ∅) ππ . ππ = ππ½ . ππ½ = ππ . ππ = π ππ . ππ½ = ππ½ . ππ = ππ . ππ = π ππ × ππ½ = ππ ππ½ × ππ = ππ ππ × ππ = ππ½ 15 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE SPHERICAL COORDINATE SYSTEM Cartesian to Spherical π= π₯2 + π¦2 + π§2 π = π‘ππ−1 ∅= π₯2 + π¦2 π§ π‘ππ−1 π¦ π₯ These equations are used for point-to-point transformation 16 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE SPHERICAL COORDINATE SYSTEM Spherical to Cartesian π₯ = ππ ππππππ ∅ π¦ = ππ ππππ ππ∅ π§ = ππππ π These equations are used for point-to-point transformation 17 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE SPHERICAL COORDINATE SYSTEM The unit vector relationship between the cartesian coordinate system and the spherical coordinate system: ππ₯ = πππππΆππ ∅ππ + πΆππ ππΆππ ∅ππ − πππ∅π∅ ππ¦ = πππππππ∅ππ + πΆππ ππππ∅ππ + πΆππ ∅π∅ ππ§ = πΆππ ππ − ππππππ 18 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE SPHERICAL COORDINATE SYSTEM The unit vector relationship between the spherical coordinate system and the Cartesian coordinate system: ππ = πππππΆππ ∅ππ₯ + πππππππ∅ππ¦ + πΆππ πππ§ ππ = πΆππ ππΆππ ∅ππ₯ + πΆππ ππππ∅ππ¦ − ππππππ§ π∅ = −πππ∅ππ₯ + πΆππ ∅ππ¦ 19 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE SPHERICAL COORDINATE SYSTEM In matrix form, we write the transformation of vector A from π΄π₯ , π΄π¦ , π΄π§ to π΄π , π΄π , π΄∅ as: π΄π πππππΆππ ∅ πππππππ∅ πΆππ π π΄π = πΆππ ππΆππ ∅ πΆππ ππππ∅ −ππππ π΄∅ −πππ∅ πΆππ ∅ 0 π΄π₯ π΄π¦ π΄π§ This equation is used for vector-to-vector transformation. 20 RELATIONSHIP BETWEEN THE CARTESIAN COORDINATE SYSTEM AND THE SPHERICAL COORDINATE SYSTEM In matrix form, we write the transformation of vector A from π΄π , π΄π , π΄∅ to π΄π₯ , π΄π¦ , π΄π§ as: π΄π₯ πππππΆππ ∅ πΆππ ππΆππ ∅ −πππ∅ π΄π¦ = πππππππ∅ πΆππ ππππ∅ πΆππ ∅ π΄π§ πΆππ π −ππππ 0 π΄π π΄π π΄∅ This equation is used for vector-to-vector transformation. 21 THE DISTANCE BETWEEN TWO POINTS IN THE THREE COORDINATE SYSTEMS • Cartesian π 2 = π₯2 − π₯1 2 − π¦2 − π¦1 2 + π§2 − π§1 2 • Cylindrical π 2 = π22 + π12 − 2π1 π2 πΆππ ∅2 − ∅1 + π§2 − π§1 • Spherical 2 π 2 = π22 + π12 − 2π1 π2 πΆππ π2 πΆππ π1 − 2π1 π2 ππππ2 ππππ1 πΆππ ∅2 − ∅1 22 ASSIGNMENT Obtain the point and vector transformation relationship between cylindrical and spherical coordinates (PROBLEM 2.16) on Page 56 of Elements of Electromagnetics by N.O. Sadiku (2018). 23
