1. Electromagnetic Waves Chapter 1

2. WHY STUDY ?? In ancient time –Why do paper clip get attracted to rod rubbed with silk –What causes lightening –Why do colors get separated when light beam passes through a prism Modern days –How we receive TV/radio signals –Why is radio signal good in some corner of the room but not other –Why do cell phones have signal fluctuation 2

3. 3 Chapter Outlines Chapter 1 Electromagnetic Waves  Maxwell’s Equations  Ampere-Maxwell Correction  EM Wave Fundamental and Equations  EM Wave Propagation in Different Media  EM Wave Reflection and Transmission at Normal or Oblique Incidence  Waveguide Field Equations  Rectangular Waveguides

4. 4 Introduction In your previous experience in studying electromagnetic, you have learned about and experimented with electrostatics and magnetostatics … concentrating on static, or time invariant electromagnetic fields (EM Fields). Henceforth, we shall examine situations where electric and magnetic fields are dynamic or time varying !!

5. 5 Where : In static EM Fields, electric and magnetic fields are independent each other, but in dynamic field both are interdependent. Time varying EM Fields, represented by E(x,y,z,t) and H(x,y,z,t) are of more practical value than static EM Fields. In time varying fields, it usually due to accelerated charges or time varying currents. Introduction (Cont’d..)

6. 6 In summary:  Stationary charges  electrostatic fields  Steady currents  magnetostatic fields  Time varying currents  electromagnetic fields or waves Introduction (Cont’d..)

7. Maxwell Equations Below is the generalized forms of Maxwell Equations: Maxwell EquationsPoint or Differential Form Integral Form Gauss’s Law Gauss’s Law for Magnetic Field Faraday’s Law Ampere’s Circuital Law

8. Maxwell Equations (Cont’d..) It is worthwhile to mention other equations that go hand in hand with Maxwell’s equations. Lorent’z Force Equation Constitutive Relations Current Continuity Equation

9. Circuit - Field relations: Field Relation Circuit Relation Maxwell Equations (Cont’d..)

10. Displacement Current We recall from Ampere’s Circuital Law for static field, ‘c’ subscript is used to identify it as a conduction current density, which related to electric field Ohm’s Law by: But divergence of curl of a vector is identically zero,

11. Displacement Current (Cont’d..) The current continuity equation, We see that the static form of Ampere’s Law is clearly invalid for time varying fields since it violates the law of current continuity. It was resolved by Maxwell introduction which what we called displacement current density, Where J d is the rate of change of the electric flux density,

12. Displacement Current (Cont’d..) The insertion of J d was one of the major contribution of Maxwell. Without J d term, electromagnetic wave propagation (e.g. radio or TV waves) would be impossible. At low frequencies, J d is usually neglected compared with J c. But at radio frequencies, the two terms are comparable. Therefore,

13. Displacement Current (Cont’d..) By applying the divergence of curl, rearrange, integrate and apply Stoke’s Theorem, we can get the integral form of Ampere’s circuital Law: Do you really understand this displacement current?? Only formula and formula…????

14. Displacement Current (Cont’d..) To have clear understanding of displacement current, consider the simple capacitor circuit of figure below. A sinusoidal voltage source is applied to the capacitor, and from circuit theory we know the voltage is related to the current by the capacitance. i(t) here is the conduction current.

15. Displacement Current (Cont’d..) Consider the loop surrounding the plane surface S 1. By static form of Ampere’s Law, the circulation of H must be equal to the current that cuts through the surface. But, the same current must pass through S 2 that passes between the plates of capacitor.

16. Displacement Current (Cont’d..) But, there is no conduction current passes through an ideal capacitor, (where J=0, due to σ=0 for an ideal dielectric ) flows through S 2. This is contradictory in view of the fact that the same closed path as S 1 is used. But to resolve this conflict, the current passing through S 2 must be entirely a displacement current, where it needs to be included in Ampere’s Circuital Law. So we obtain the same current for either surface though it is conduction current in S 1 and displacement current in S 2.

17. Displacement Current (Cont’d..) The ratio of conduction current magnitude to the displacement current magnitude is called loss tangent, where it is used to measure the quality of the dielectric  good dielectric will have very low loss tangent. Other example for physical meaning: J i = current source J c = conducted current through resistor J d =displacement current through dielectric material m i = magnetic current source m d =displacement magnetic current

18. Lossless TEM Waves Let’s use Maxwell’s equations to study the relationship between the electric and magnetic field components of an electromagnetic wave. Consider an x-polarized wave propagating in the +z direction in some ideal medium characterized by µ and ε, with σ = 0. Electromagnetic wave as x-polarized means that the E field vector is always pointing in the x or –x direction. Choose σ = 0 to make medium lossless for simplicity, as given by:

19. Lossless TEM Waves (Cont’d..) A plot of the equation E(z,0) = E 0 cos(  z)a x at 10 MHz in free space with E 0 = 1 V/m.

20. Lossless TEM Waves (Cont’d..) Upon application of Maxwell equations, we would also find the magnetic field propagates in the +z direction. But the field is always normal or perpendicular to the electric field vector  the wave is said to propagate in a transverse electromagnetic wave mode, or TEM. TEM Waves has no E field or H field components along the direction of propagation. We can apply Faraday’s Law, to the propagating electric field equation previously. Solve the right hand side and the left hand side to get H !!

21. Lossless TEM Waves (Cont’d..) Where,

22. We could see that the time varying E is the only source of H, if no conduction current given that can also generate H  Thus C must be zero. Plot of the equation H(z,0) = (  E 0 /  ) cos(–  z)a y at 10 MHz in free space with E 0 = 1 V/m along with the lighter plot of E(z,0). The amplitudes of E and H are related by Maxwell equations. Lossless TEM Waves (Cont’d..)

23. We can apply Ampere’s Circuital Law, to the propagating magnetic field equation previously. By considering lossless characteristic, taking the curl of H, equating the equation and then integrate it, we could get : And propagation velocity relation: to get

24. 24 Example 1 Suppose in free space that: E(z,t) = 5.0 e - 2zt a x V/m.  Is the wave lossless?  Find H(z,t).

25. Solution to Example 1 Since the wave has an attenuation term (e- 2zt ) it is clearly not lossless. To find H, Therefore,

26. Solution to Example 1 (Cont’d..) This integral is solved by parts where we let We arrive at:

27. EM Wave Fundamental & equations In free space, the constitutive parameters are σ = 0, µ r = 1, ε r = 1, so the Ampere’s Law and Faraday’s Law equations become : If there is some point in space a source of time varying E field, a H field is induced in the surrounding region. As this H field also changing with time, it in turn induces an E field. Energy is pass back and forth between E and H fields as they radiate away from the source at the speed of light.

28. EM Wave Fundamental & equations (Cont’d..) The EM waves radiates spherically, but at a remote distance away from the source they resemble uniform plane wave. In a uniform plane wave, the E and H fields are orthogonal, or transverse to the direction of propagation (to propagate in TEM mode ).

29. We will briefly review some of the fundamental features of waves before employing them in the study of electromagnetic. Consider time harmonic waves, represented by sine waves, rather than transient waves (pulses or step functions), generally as: The E field is a function of position (z) and time (t). It is always pointing in +x or –x direction  x-polarized wave. EM Wave Fundamental & equations (Cont’d..)

30. Where, Initial amplitude at z = 0 exponential terms  attenuation Angular frequency Phase constant Phase shift And important relations:

31. EM Wave Fundamental & equations (Cont’d..) E(0,t) = E x a x = E 0 cos(  t)a x.E(0,t) =E x a x = E 0 cos(  t +  )a x. E(z, 0) = E 0 cos(–  z)a x.E(z, 0) = E 0 e –  z cos (–  z)a x.

32. EM Wave Fundamental & equations (Cont’d..) Use Maxwell’s equations to derive formulas governing EM wave propagation. Consider that the medium is free of any charge, where: And linear, isotropic, homogeneous, and time invariant (simple media), whereby the Maxwell’s equation can be rewritten as :

33. EM Wave Fundamental & equations (Cont’d..) Take curl of both sides of Faraday’s Law, Consider position derivative acting on a time derivative in a homogeneous material, Exchange the Faraday’s Law for the curl of H Invoking a vector identity,

34. EM Wave Fundamental & equations (Cont’d..) So now we have : But, medium is charge free, so divergence of E is zero, Helmholtz Wave Equation for E. This can be broken up into three vector equations ( E x, E y and E z )

35. 35 Task 1 By starting with Maxwell’s equations for simple and charge free media, derive the Helmholtz Wave Equation for magnetic field, H

36. EM Wave Fundamental & equations (Cont’d..) But, with interest on Helmholtz equations for time harmonic fields, the previous Helmholtz wave equation becomes : because Generally written in the form: Where the propagation constant, γ, with real part is attenuation and an imaginary part is phase constant,

37. EM Wave Fundamental & equations (Cont’d..) The value of α and β in terms of the material’s constitutive parameters:

38. EM Wave Fundamental & equations (Cont’d..) The Helmholtz wave equation for time harmonic magnetic fields, H : With loss of generality, we assume that x polarized traveling in the +a z direction, where E s has only an x component with function of z, since for plane wave that the fields do not vary in transverse direction, where in this case is xy plane. Then :

39. By substituting this equation into the Helmholtz wave equation for E field, it becomes: EM Wave Fundamental & equations (Cont’d..) Hence, With first and second term is zero :

40. EM Wave Fundamental & equations (Cont’d..) This is a scalar wave equation, a linear homogeneous differential equation. A possible solution for this equation is : If we let, then, So, this equation, becomes: Which has two solutions,

41. EM Wave Fundamental & equations (Cont’d..) The general solution is linear superposition of these two: For the represents the E field amplitude of +z traveling wave at z=0, by reinserting the vector, multiply by time factor apply Euler’s identity and use the propagation constant to get :

42. EM Wave Fundamental & equations (Cont’d..) For the represents the E field amplitude of negative z traveling wave at z=0, and similarly by previous approach to get : The general instantaneous solution is superposition of these two solutions above. or generally as :

43. EM Wave Fundamental & equations (Cont’d..) Evaluating the curl of E to find: We can solve for H, If we assume that the wave propagates along +a z and H s has only an y component, as what we had assume for E s The magnetic field can be found by applying Faraday’s Law :

44. EM Wave Fundamental & equations (Cont’d..) We would have been led to the expression, By making comparison, we can find a relationship between and, or and intrinsic impedance (in ohms)

45. EM Wave Fundamental & equations (Cont’d..) By plugging in the equation of intrinsic impedance into equation of magnetic field, we could get: Where E and H are out of phase by θ n at any instant of time due to the complex intrinsic impedance of the medium. Thus at any time, E leads H or H lags E by θ n. The ratio of magnitude of conduction current density to displacement current density in a lossy medium is: loss tangent

46. EM Wave Fundamental and equations (Cont’d..) Tan δ is used to determine how lossy a medium is, i.e. Good (lossless or perfect) dielectric if Good conductor if

47. EM Wave Fundamental and equations (Cont’d..) Basically, we can use Fleming’s Left Hand Rule to determine the E, H and propagation direction ; Propagation direction (first finger) H direction (second finger) E direction (thumb)

48. EM Wave Fundamental and equations (Cont’d..) By knowing the EM wave’s direction of propagation, given as unit vector a p, is the same as the cross product of E s with unit vector, a E and H s with unit vector a H : And also, a pair of simple formulas can be derived: In addition to properties of cross product,

49. EM Wave Fundamental and equations (Cont’d..) Representation of waves; in (a), the wave travels in the a p = +a z direction and has E s = E 0 + e –  z a x and H s = (E 0 + /  )e –  z a y. In (b), the wave travels in the a p = –a z direction and has E s = E 0 – e  z a x along with H s = –(E 0 – /  )e  z a y.

50. 50 Example 2 Suppose in free space : H(x,t) = 100 cos(2π x 10 7 t – βx + π/4) a z mA/m. Find E(x,t).

51. Solution to Example 2 We could find: So then, Since free space is stated, and then

52. 52 Example 3 Suppose: E (x,y,t) = 5 cos(π x 10 6 t – 3.0x + 2.0y) a z V/m. Find :  H (x,y,t).  The direction of propagation, a p

53. Solution to Example 3 We could find: Assume nonmagnetic material and therefore have: So that,

54. Solution to Example 3 (Cont’d..) The direction of propagation : Where, And with the exponential terms canceling in the top and bottom of the equation for a p, we have:

55. EM Wave Propagation In Different Media We will consider time harmonic field propagating in different types of media :-  Lossless, Charge-Free  Dielectrics  Conductors Lossless, Charge - Free Charge free, ρ v =0, medium has zero conductivity, σ=0. This is the case where waves traveling in vacuum or free space (free of any charges). Perfect dielectric is also considered as lossless media.

56. EM Wave Propagation In Different Media (Cont’d..) Evaluate the propagation constant, So, Where, Since, the signal does not attenuate as it travels  lossless medium. The propagation velocity,

57. EM Wave Propagation In Different Media (Cont’d..) Evaluate the intrinsic impedance, For lossless materials, E and H are always in phase. Again, intrinsic impedance of free space

58. 58 In a lossless, nonmagnetic material with : ε r = 16, and H = 100 cos(ωt – 10y) a z mA/m. Determine :  The propagation velocity  The angular frequency  The instantaneous expression for the electric field intensity. Example 4

59. Solution to Example 4 The propagation velocity: The angular frequency: From given H field :

60. So, the time harmonic H field is: Solution to Example 4 (Cont’d..) Where, Finally, the instantaneous expression for E field is:

61. Dielectric Treating a dielectric as lossless is often a good approximation, but all dielectrics are to some degree lossy  finite conductivity, polarization loss etc. With finite conductivity, the E field gives rise to conduction current density results in power dissipation. Thus, it will give a complex permittivity, complex propagation constant with attenuation constant greater than zero. The intrinsic impedance is also complex, resulting a phase difference between E and H fields. EM Wave Propagation In Different Media (Cont’d..)

62. Conductor In any decent conductor, the loss tangent, σ/ωε>>1 or σ>>ωε so that σ ≈ ∞, so that: Therefore, where interior brackets becomes:

63. EM Wave Propagation In Different Media (Cont’d..) The intrinsic impedance approximated by: By considering leading to: and also.. E leads H by 45 0

64. EM Wave Propagation In Different Media (Cont’d..) A wave from air (free space) penetrates rapidly in a good conductor, with wavelength clearly much shorter. In a good conductor, the large attenuation means the penetration depth can be quite small, confining the fields near the surface or skin, of the conductor  skin depth. A large attenuation means the fields cannot penetrate far into the conductor.

65. 65 Example 5 In a nonmagnetic material, E(z,t) = 10 e -200z cos(2π x 10 9 t - 200z) a x mV/m. Find H(z,t)

66. Solution to Example 5 Since α = β, the media is a good metal, with µ r = 1 we have: We could also find the intrinsic impedance,

67. Solution to Example 5 (Cont’d) So, to calculate H, The instantaneous expression for the magnetic field intensity.

68. EM Wave Reflection & Transmission at Normal Incidence What happens when a EM wave is incident on a different medium? E.g. Light wave incident with mirror, most of it gets reflected but a portion gets transmitted (rapidly attenuating in the silver backing of the mirror. Consider a plane wave that are normally incident which means the planar boundary separating the two media is perpendicular to the wave’s propagation direction. Generally, consider a time harmonic x-polarized electric field incident from medium 1 (µ r1, ε r1, σ r1 ) to medium 2 (µ r2, ε r2, σ r2 )

69. EM Wave Reflection & Transmission at Normal Incidence (Cont’d..) With incident field:

70. EM Wave Reflection & Transmission at Normal Incidence (Cont’d..) We have the following sets of equations: Incident Fields Reflected Fields Transmitted Fields The E field intensities at z=0

71. The boundary conditions: Applying these boundary conditions at z=0 to get: Reflection Coefficient Transmission Coefficient EM Wave Reflection & Transmission at Normal Incidence (Cont’d..) Try this!!

72. EM Wave Reflection & Transmission at Normal Incidence (Cont’d..) By comparison, Consider a special case when medium 1 is a perfect dielectric (lossless,σ 1 =0) and medium 2 is a perfect conductor (σ 2 = ∞). For this case, showing that the wave is totally reflected  fields in perfect conductor must vanish, so there can be no transmitted wave, E 2 = 0. The totally reflected wave combines with the incident wave to form a standing wave  it stands and does not travel, it consists of two traveling waves E i and E r of equal amplitudes but in opposite directions.

73. EM Wave Reflection & Transmission at Normal Incidence (Cont’d..) Standing wave pattern for an incident wave in a lossless medium reflecting off a second medium at z=0 where  = 0.5.

74. 74 Example 6 A uniform planar waves is normally incident from media 1 (z 0, σ = 0, µ r = 8.0, ε r = 2.0). Calculate the reflection and transmission coefficients seen by this wave.

75. Solution to Example 6 The reflection coefficient ; This leads to: and the transmission coefficient,

76. 76 Example 7 Suppose media 1 (z 0) has ε r = 16. The transmitted magnetic field intensity is known to be: H t = 12 cos (ωt - β 2 z) a y mA/m. Determine the instantaneous value of the incident electric field.

77. Solution to Example 7 We know that, From transmitted H field, we could find the transmitted E field,

78. Since we know the relation between transmitted E field and incident E field, Thus, Solution to Example 7 (Cont’d..)

79. EM Wave Reflection & Transmission at Oblique Incidence A uniform plane waves traveling in the a i direction is obliquely incident from medium 1 onto medium 2. Plane of incidence  plane containing both a normal to the boundary and the incident’s wave propagation. In figure, the propagation direction is a i and the normal is a z, so the plane incidence is the x z plane. The angle of incidence, reflection and transmission is the angle that makes the field a normal to the boundary.

80. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..) When EM Wave in plane wave form obliquely incident on the boundary, it can be decomposed into:  Perpendicular Polarization, or transverse electric (TE) polarization  The E Field is perpendicular or transverse to the plane of incidence.  Parallel Polarization, or transverse magnetic (TM) polarization  The E Field is parallel to the plane of incidence, but the H Field is transverse. We need to decompose into its TE and TM components separately, and once the reflected an the transmitted fields for each polarization determined, it can be recombined for final answer.

81. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..) TE Polarization

82. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..) For TE polarization, the fields are summarized as follows: Incident Fields Reflected Fields Transmitted Fields

83. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..) By applying Snell’s Law, We could get: Try to solve this!!!

84. TM Polarization EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)

85. By similar geometric arguments, TM fields are: Incident Fields Reflected Fields Transmitted Fields

86. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..) By applying Snell’s Law and employing the boundary conditions, we could get the following expressions relating the field amplitudes:

87. EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..) For TM polarizations, there exists an incidence angle at which all of the wave is transmitted into the second medium  Brewster Angle, θ i = θ BA, where: When a randomly polarized wave such as light is incident on a material at the Brewster angle, the TM polarized portion is totally transmitted but a TE component is partially reflected.

88. 88 Example 8 A 100 MHz TE polarized wave with amplitude 1.0 V/m is obliquely incident from air (z 0). The angle of incidence is 40 . Calculate: (a) the angle of transmission, (b) the reflection and transmission coefficients, (c) the incident, reflected and transmitted for E fields.

89. Solution to Example 8 (a) (b)

90. Solution to Example 8 (Cont’d) (c) For incident field: Thus, For reflected field:

91. Leading to: Thus, Solution to Example 8 (Cont’d) Finally for transmitted field:

92. To get: Therefore, Solution to Example 8 (Cont’d)

93. Electromagnetic Waves End