EE3321 ELECTROMAGENTIC FIELD THEORY

1. EE3321 ELECTROMAGENTIC FIELD THEORY Week 1 Wave Concepts Coordinate Systems and Vector Products

2. International System of Units (SI) • Length meter m • Mass kilogram kg • Time second s • Current Ampere A • Temperature Kelvin K • Newton = kg m/s2 • Coulomb = A s • Volt = (Newton /Coulomb) m Dr. Benjamin C. Flores

3. Standard prefixes (SI) Dr. Benjamin C. Flores

4. Exercise • The speed of light in free space is c = 2.998 x 105 km/s. Calculate the distance traveled by a photon in 1 ns. Dr. Benjamin C. Flores

5. Propagating EM wave • Characteristics • Amplitude • Phase • Angular frequency • Propagation constant • Direction of propagation • Polarization • Example • E(t,z) = Eocos (ωt – βz) ax Dr. Benjamin C. Flores

6. Forward and backward waves • Sign Convention - βz propagation in +z direction + βz propagation in –z direction Which is it? a) forward traveling b) backward traveling Dr. Benjamin C. Flores

7. Partial reflection • This happens when there is a change in medium Dr. Benjamin C. Flores

8. Standing EM wave • Characteristics • Amplitude • Angular frequency • Phase • Polarization • No net propagation • Example • E(t,z) = A cos (ωt ) cos( βz) ax Dr. Benjamin C. Flores

9. Complex notation • Recall Euler’s formula exp(jφ) = cos (φ) + j sin (φ) Dr. Benjamin C. Flores

10. Exercise • Calculate the magnitude of exp(jφ) = cos (φ)+ j sin (φ) • Determine the complex conjugate of exp(j φ) Dr. Benjamin C. Flores

11. Traveling wave complex notation • Let φ = ωt – βz • Complex field Ec(t, z) = A exp [j(ωt – βz)] ax = A cos(ωt – βz) ax + j A sin(ωt – βz) ax E(z,t) = Real { Ec(t, z) } Dr. Benjamin C. Flores

12. Standing wave complex notation • E = A exp[ j(ωt – βz) + A exp[ j(ωt + βz) = A exp(jωt) [exp(–jβz) + exp(+jβz)] = 2A exp(jωt) cos(βz) • E = 2A[cos(ωt) + j sin (ωt) ] cos(βz) • Re { E } = 2A cos(ωt) cos(βz) • Im { E } = 2A sin(ωt) cos(βz) Dr. Benjamin C. Flores

13. Exercise Show that E(t) = A exp(jωt) sin(βz) can be written as the sum of two complex traveling waves. Hint: Recall that j2 sin(φ) = exp (j φ) – exp(– j φ) Dr. Benjamin C. Flores

14. Transmission line/coaxial cable •  Voltage wave • V = Vocos (ωt – βz) • Current wave • I = Iocos (ωt – βz) • Characteristic Impedance • ZC = Vo / Io • Typical values: 50, 75 ohms Dr. Benjamin C. Flores

15. RADAR • Radio detection and ranging Dr. Benjamin C. Flores

16. Time delay • Let r be the range to a target in meters • φ = ωt – βr = ω[ t – (β/ω)r ] • Define the phase velocity as v = β/ω • Let τ = r/v be the time delay • Then φ = ω (t – τ) • And the field at the target is Ec(t, τ) = A exp [jω( t – τ )] ax Dr. Benjamin C. Flores

17. Definition of coordinate system • A coordinate system is a system for assigning real numbers (scalars) to each point in a 3-dimensional Euclidean space. • Systems commonly used in this course include: • Cartesian coordinate system with coordinates x (length), y (width), and z (height) • Cylindrical coordinate system with coordinates ρ (radius on x-y plane), φ (azimuth angle), and z (height) • Spherical coordinate system with coordinates r (radius or range), Ф (azimuth angle), and θ (zenith or elevation angle) Dr. Benjamin C. Flores

18. Definition of vector • A vector (sometimes called a geometric or spatial vector) is a geometric object that has a magnitude, direction and sense. Dr. Benjamin C. Flores

19. Direction of a vector • A vector in or out of a plane (like the white board) are represented graphically as follows: • Vectors are described as a sum of scaled basis vectors (components): Dr. Benjamin C. Flores

20. Cartesian coordinates Dr. Benjamin C. Flores

21. Principal planes Dr. Benjamin C. Flores

22. Unit vectors • ax = x = i • ay = y = j • az = z = k • u = A/|A| Dr. Benjamin C. Flores

23. Handedness of coordinate system Left handed Right handed Dr. Benjamin C. Flores

24. Are you smarter than a 5th grader? • Euclidean geometry studies the relationships among distances and angles in flat planes and flat space. • true • false • Analytic geometry uses the principles of algebra. • true • false Dr. Benjamin C. Flores

25. Cylindrical coordinate system Φ = tan-1 y/x ρ2= x2 + y2 Dr. Benjamin C. Flores

26. Vectors in cylindrical coordinates • Any vector in Cartesian can be written in terms of the unit vectors in cylindrical coordinates: • The cylindrical unit vectors are related to the Cartesian unit vectors by: Dr. Benjamin C. Flores

27. Spherical coordinate system Φ = tan-1 y/x θ = tan-1 z/[x2 + y2]1/2 r2 = x2 + y2 + z2 Dr. Benjamin C. Flores

28. Vectors in spherical coordinates • Any vector field in Cartesian coordinates can be written in terms of the unit vectors in spherical coordinates: • The spherical unit vectors are related to the Cartesian unit vectors by: Dr. Benjamin C. Flores

29. Dot product • The dot product (or scalar product) of vectors a and b is defined as • a · b = |a| |b| cosθ where • |a| and |b| denote the length of a and b • θ is the angle between them. Dr. Benjamin C. Flores

30. Exercise • Let a = 2x + 5y + z and b = 3x – 4y + 2z. • Find the dot product of these two vectors. • Determine the angle between the two vectors. Dr. Benjamin C. Flores

31. Cross product • The cross product (or vector product) of vectors a and b is defined as a x b = |a| |b| sin θn where • θ is the measure of the smaller angle between a and b (0° ≤ θ ≤ 180°), • a and b are the magnitudes of vectors a and b, • and n is a unit vector perpendicular to the plane containing a and b. Dr. Benjamin C. Flores

32. Cross product Dr. Benjamin C. Flores

33. Exercise • Consider the two vectors a= 3x + 5y + 7z and b = 2x – 2y – 2z • Determine the cross product c = a x b • Find the unit vector n of c Dr. Benjamin C. Flores

34. Homework • Read all of Chapter 1, sections 1-1, 1-2, 1-3, 1-4, 1-5, 1-6 • Read Chapter 3, sections 3-1, 3-2, 3-3 • Solve end-of-chapter problems 3.1, 3.3, 3.5 , 3.7, 3.19, 3.21, 3.25, 3.29 Dr. Benjamin C. Flores