Unit 1: Introduction to Antennas + Electromagnetics Review

Transcription

1 Unit 1: Introduction to Antennas + Electromagnetics Review Antennas ENGI 7811 Khalid El-Darymli, Ph.D., EIT Dept. of Electrical and Computer Engineering Faculty of Engineering and Applied Science Memorial University of Newfoundland St. John's, Newfoundland, Canada Winter 2017 K. El-Darymli Winter /71

2 Outline 1 Introduction 2 Maxwell's Eqns & Related Formulae Maxwell's Equations Constitutive Relationships Miscellaneous Relations Some Propagation Parameters Time-Harmonic Fields Boundary Conditions 3 Scalar & Vector Potentials Review New Material on Potentials 4 Radiation Mechanism Antenna, a Circuit Point of View Two-Wire Antennas Helpful Animation and Applets 5 Ackgt K. El-Darymli Winter /71

3 Antennas act as transducers associated with the region of transition between guided wave structures and free space, or vice versa. K. El-Darymli Winter /71

4 Antennas act as transducers associated with the region of transition between guided wave structures and free space, or vice versa. The guiding structure could be, for example, a two-wire transmission line or a waveguide (hollow pipe) leading from a transmitter or receiver to the antenna itself. K. El-Darymli Winter /71

5 Antennas act as transducers associated with the region of transition between guided wave structures and free space, or vice versa. The guiding structure could be, for example, a two-wire transmission line or a waveguide (hollow pipe) leading from a transmitter or receiver to the antenna itself. Generally, the antennas are made of good-conducting material and are designed to have dimensions and shape conducive to radiating or receiving electromagnetic (e-m) energy in an ecient manner. K. El-Darymli Winter /71

6 Antennas act as transducers associated with the region of transition between guided wave structures and free space, or vice versa. The guiding structure could be, for example, a two-wire transmission line or a waveguide (hollow pipe) leading from a transmitter or receiver to the antenna itself. Generally, the antennas are made of good-conducting material and are designed to have dimensions and shape conducive to radiating or receiving electromagnetic (e-m) energy in an ecient manner. As we shall see, the antenna structure may take many dierent forms: e.g., wires, horns, slots, microstrips, reectors, and combinations of these. K. El-Darymli Winter /71

7 Antennas act as transducers associated with the region of transition between guided wave structures and free space, or vice versa. The guiding structure could be, for example, a two-wire transmission line or a waveguide (hollow pipe) leading from a transmitter or receiver to the antenna itself. Generally, the antennas are made of good-conducting material and are designed to have dimensions and shape conducive to radiating or receiving electromagnetic (e-m) energy in an ecient manner. As we shall see, the antenna structure may take many dierent forms: e.g., wires, horns, slots, microstrips, reectors, and combinations of these. While we shall be able to examine many important basic characteristics using mathematics appearing earlier in the programme, in most practical situations antenna design must be carried out using sophisticated software i.e., ecient numerical techniques and packages must be employed. K. El-Darymli Winter /71

8 Antennas act as transducers associated with the region of transition between guided wave structures and free space, or vice versa. The guiding structure could be, for example, a two-wire transmission line or a waveguide (hollow pipe) leading from a transmitter or receiver to the antenna itself. Generally, the antennas are made of good-conducting material and are designed to have dimensions and shape conducive to radiating or receiving electromagnetic (e-m) energy in an ecient manner. As we shall see, the antenna structure may take many dierent forms: e.g., wires, horns, slots, microstrips, reectors, and combinations of these. While we shall be able to examine many important basic characteristics using mathematics appearing earlier in the programme, in most practical situations antenna design must be carried out using sophisticated software i.e., ecient numerical techniques and packages must be employed. The main purpose of this course is to introduce the basics so that future exposure to the engineering software will be meaningful. K. El-Darymli Winter /71

9 Antennas act as transducers associated with the region of transition between guided wave structures and free space, or vice versa. The guiding structure could be, for example, a two-wire transmission line or a waveguide (hollow pipe) leading from a transmitter or receiver to the antenna itself. Generally, the antennas are made of good-conducting material and are designed to have dimensions and shape conducive to radiating or receiving electromagnetic (e-m) energy in an ecient manner. As we shall see, the antenna structure may take many dierent forms: e.g., wires, horns, slots, microstrips, reectors, and combinations of these. While we shall be able to examine many important basic characteristics using mathematics appearing earlier in the programme, in most practical situations antenna design must be carried out using sophisticated software i.e., ecient numerical techniques and packages must be employed. The main purpose of this course is to introduce the basics so that future exposure to the engineering software will be meaningful. For Antennas simulations in this course, you're encouraged to use the Student Edition of FEKO, K. El-Darymli Winter /71

10 Antennas act as transducers associated with the region of transition between guided wave structures and free space, or vice versa. The guiding structure could be, for example, a two-wire transmission line or a waveguide (hollow pipe) leading from a transmitter or receiver to the antenna itself. Generally, the antennas are made of good-conducting material and are designed to have dimensions and shape conducive to radiating or receiving electromagnetic (e-m) energy in an ecient manner. As we shall see, the antenna structure may take many dierent forms: e.g., wires, horns, slots, microstrips, reectors, and combinations of these. While we shall be able to examine many important basic characteristics using mathematics appearing earlier in the programme, in most practical situations antenna design must be carried out using sophisticated software i.e., ecient numerical techniques and packages must be employed. The main purpose of this course is to introduce the basics so that future exposure to the engineering software will be meaningful. For Antennas simulations in this course, you're encouraged to use the Student Edition of FEKO, K. El-Darymli Winter /71

11 Introduction Maxwell's Eqns & Related Formulae Scalar & Vector Potentials Radiation Mechanism Ackgt Timeline Ref.: W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 3rd edition, 2013, John Wiley & Sons, Inc. Pre-modern civilization K. El-Darymli (up to 2 million years ago) Winter /71

12 Introduction Maxwell's Eqns & Related Formulae Scalar & Vector Potentials Radiation Mechanism Ackgt Timeline Ref.: W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 3rd edition, 2013, John Wiley & Sons, Inc. Pre-modern civilization (up to 2 million years ago) Acoustical communications: Drums K. El-Darymli Winter /71

13 Introduction Maxwell's Eqns & Related Formulae Scalar & Vector Potentials Radiation Mechanism Ackgt Timeline Ref.: W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 3rd edition, 2013, John Wiley & Sons, Inc. Pre-modern civilization (up to 2 million years ago) Acoustical communications: Drums Optical communications: Smoke signals, ags K. El-Darymli Winter /71

14 Introduction Maxwell's Eqns & Related Formulae Scalar & Vector Potentials Radiation Mechanism Ackgt Timeline Ref.: W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 3rd edition, 2013, John Wiley & Sons, Inc. Pre-modern civilization (up to 2 million years ago) Acoustical communications: Drums Optical communications: Smoke signals, ags This 14th century BC image of Akhenaton is the rst known image that depicts that light travels in a straight line. [Ref.: K. El-Darymli Winter /71

15 Timeline 1844 TelegraphThe beginning of electronic communication K. El-Darymli Winter /71

16 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse K. El-Darymli Winter /71

17 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse 1864 Maxwell's equationsprinciples of radio waves and the electromagnetic spectrum K. El-Darymli Winter /71

18 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse 1864 Maxwell's equationsprinciples of radio waves and the electromagnetic spectrum James Clerk Maxwell K. El-Darymli Winter /71

19 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse 1864 Maxwell's equationsprinciples of radio waves and the electromagnetic spectrum James Clerk Maxwell 1866 First lasting transatlantic telegraph cable K. El-Darymli Winter /71

20 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse 1864 Maxwell's equationsprinciples of radio waves and the electromagnetic spectrum James Clerk Maxwell 1866 First lasting transatlantic telegraph cable 1876 TelephoneWireline analog communication over long distance K. El-Darymli Winter /71

21 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse 1864 Maxwell's equationsprinciples of radio waves and the electromagnetic spectrum James Clerk Maxwell 1866 First lasting transatlantic telegraph cable 1876 TelephoneWireline analog communication over long distance Alexander Bell K. El-Darymli Winter /71

22 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse 1864 Maxwell's equationsprinciples of radio waves and the electromagnetic spectrum James Clerk Maxwell 1866 First lasting transatlantic telegraph cable 1876 TelephoneWireline analog communication over long distance Alexander Bell 1887 First Antenna K. El-Darymli Winter /71

23 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse 1864 Maxwell's equationsprinciples of radio waves and the electromagnetic spectrum James Clerk Maxwell 1866 First lasting transatlantic telegraph cable 1876 TelephoneWireline analog communication over long distance Alexander Bell 1887 First Antenna Heinrich Hertz K. El-Darymli Winter /71

24 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse 1864 Maxwell's equationsprinciples of radio waves and the electromagnetic spectrum James Clerk Maxwell 1866 First lasting transatlantic telegraph cable 1876 TelephoneWireline analog communication over long distance Alexander Bell 1887 First Antenna Heinrich Hertz 1897 First practical wireless (radio) systems K. El-Darymli Winter /71

25 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse 1864 Maxwell's equationsprinciples of radio waves and the electromagnetic spectrum James Clerk Maxwell 1866 First lasting transatlantic telegraph cable 1876 TelephoneWireline analog communication over long distance Alexander Bell 1887 First Antenna Heinrich Hertz 1897 First practical wireless (radio) systems Guglielmo Marconi K. El-Darymli Winter /71

26 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse 1864 Maxwell's equationsprinciples of radio waves and the electromagnetic spectrum James Clerk Maxwell 1866 First lasting transatlantic telegraph cable 1876 TelephoneWireline analog communication over long distance Alexander Bell 1887 First Antenna Heinrich Hertz 1897 First practical wireless (radio) systems Guglielmo Marconi 1901 First transatlantic radio K. El-Darymli Winter /71

27 Timeline 1844 TelegraphThe beginning of electronic communication Samuel Morse 1864 Maxwell's equationsprinciples of radio waves and the electromagnetic spectrum James Clerk Maxwell 1866 First lasting transatlantic telegraph cable 1876 TelephoneWireline analog communication over long distance Alexander Bell 1887 First Antenna Heinrich Hertz 1897 First practical wireless (radio) systems Guglielmo Marconi 1901 First transatlantic radio Guglielmo Marconi K. El-Darymli Winter /71

28 Timeline 1920 First broadcast radio station K. El-Darymli Winter /71

29 Timeline 1920 First broadcast radio station World War II Development of radar; horn, reector, and array antennas K. El-Darymli Winter /71

30 Timeline 1920 First broadcast radio station World War II Development of radar; horn, reector, and array antennas 1950s Broadcast television in wide use K. El-Darymli Winter /71

31 Timeline 1920 First broadcast radio station World War II Development of radar; horn, reector, and array antennas 1950s Broadcast television in wide use 1960s Satellite communications and ber optics K. El-Darymli Winter /71

32 Timeline 1920 First broadcast radio station World War II Development of radar; horn, reector, and array antennas 1950s Broadcast television in wide use 1960s Satellite communications and ber optics 1980s Wireless reinvented with widespread use of cellular telephones K. El-Darymli Winter /71

33 Timeline 1920 First broadcast radio station World War II Development of radar; horn, reector, and array antennas 1950s Broadcast television in wide use 1960s Satellite communications and ber optics 1980s Wireless reinvented with widespread use of cellular telephones K. El-Darymli Winter /71

34 Single-Element Antennas Wire antennas (examples) (a) Monopole antenna. (b) Dipole antenna. (c) Loop antenna. (d) Helical antenna. K. El-Darymli Winter /71

35 Single-Element Antennas Examples of monopole type antennas used in cellular and cordless telephones, walkie-talkies, and CB radios (taken from Balanis). K. El-Darymli Winter /71

36 Single-Element Antennas Examples of monopole type antennas used in cellular and cordless telephones, walkie-talkies, and CB radios (taken from Balanis). The monopoles used in these units are either stationary or reactable/telescopic. K. El-Darymli Winter /71

37 Single-Element Antennas Aperture antennas (examples) K. El-Darymli Winter /71

38 Single-Element Antennas Aperture antennas (examples) Corner reector (example from Balanis) K. El-Darymli Winter /71

39 Single-Element Antennas This 500 meter Aperture Spherical Telescope (FAST) in china is the world's largest single-aperture telescope (courtesy of: Xinhua). K. El-Darymli Winter /71

40 Single-Element Antennas Lens antennas with index n > 1 (examples from Balanis) K. El-Darymli Winter /71

41 Single-Element Antennas Lens antennas with index n > 1 (examples from Balanis) Lens antennas with index n < 1 (examples from Balanis) K. El-Darymli Winter /71

42 Single-Element Antennas An example for a microstrip (or printed) monopole antenna (courtesy of: Amitec) K. El-Darymli Winter /71

43 Antenna Arrays Log-periodic array (courtesy of: BAZ Spezialantennen, Germany) K. El-Darymli Winter /71

44 Antenna Arrays Patch antenna array (courtesy of: Amitec) K. El-Darymli Winter /71

45 Antenna Arrays Triangular array of dipoles used as a sectoral base-station antenna for mobile communication (taken from Balanis). K. El-Darymli Winter /71

46 Antenna Arrays Very Large Array, National Radio Astronomy Observatory, Socorro County, USA (courtesy of: K. El-Darymli Winter /71

47 The Electromagnetic Spectrum Taken from: W. L. Stutzman et al., Antenna Theory and Design, 3rd edition, 2013, John Wiley & Sons, Inc. K. El-Darymli Winter /71

48 The Electromagnetic Spectrum Microwave bands: K. El-Darymli Winter /71

49 Maxwell's Equations Outline 1 Introduction 2 Maxwell's Eqns & Related Formulae Maxwell's Equations Constitutive Relationships Miscellaneous Relations Some Propagation Parameters Time-Harmonic Fields Boundary Conditions 3 Scalar & Vector Potentials Review New Material on Potentials 4 Radiation Mechanism Antenna, a Circuit Point of View Two-Wire Antennas Helpful Animation and Applets 5 Ackgt K. El-Darymli Winter /71

50 Maxwell's Equations We have previously seen that to properly describe any time-varying electromagnetic phenomenon, the following may be invoked: In point form Ẽ = B t (1.1) = H + J D (1.2) t = ρ (1.3) D v = 0 (1.4) B where the subscript has been used to represent any time variation. Equations (1.1) - (1.4) are referred to as Maxwell's equations. K. El-Darymli Winter /71

51 Maxwell's Equations We have previously seen that to properly describe any time-varying electromagnetic phenomenon, the following may be invoked: In point form Ẽ = B t (1.1) = H + J D (1.2) t = ρ (1.3) D v = 0 (1.4) B where the subscript has been used to represent any time variation. Equations (1.1) - (1.4) are referred to as Maxwell's equations. Of course, Ẽ E ( r,t) Electric eld intensity in V/m. H ( r,t) Magnetic eld intensity in A/m. H D ( r,t) Electric ux density in C/m D 2. B ( r,t) Magnetic ux density in Wb/m B 2 or T (tesla). r position vector as measured from some origin to a eld or observation point. t time. J ( r,t) current density in A/m J 2. ρ v charge density in C/m3. K. El-Darymli Winter /71

52 Maxwell's Equations We have previously seen that to properly describe any time-varying electromagnetic phenomenon, the following may be invoked: In point form Ẽ = B t (1.1) = H + J D (1.2) t = ρ (1.3) D v = 0 (1.4) B where the subscript has been used to represent any time variation. Equations (1.1) - (1.4) are referred to as Maxwell's equations. Of course, Ẽ E ( r,t) Electric eld intensity in V/m. H ( r,t) Magnetic eld intensity in A/m. H D ( r,t) Electric ux density in C/m D 2. B ( r,t) Magnetic ux density in Wb/m B 2 or T (tesla). r position vector as measured from some origin to a eld or observation point. t time. J ( r,t) current density in A/m J 2. ρ v charge density in C/m3. The last two quantities are source terms or supports for the eld quantities Ẽ,, H, and D B. K. El-Darymli Winter /71

53 Maxwell's Equations Equation (1.1) is Faraday's law; K. El-Darymli Winter /71

54 Maxwell's Equations Equation (1.1) is Faraday's law; Equation (1.2) is a modication of Ampères law with t D being the so-called displacement current density (in A/m 2 of course); K. El-Darymli Winter /71

55 Maxwell's Equations Equation (1.1) is Faraday's law; Equation (1.2) is a modication of Ampères law with t D being the so-called displacement current density (in A/m 2 of course); Equation (1.3) is Gauss' law (electric); and K. El-Darymli Winter /71

56 Maxwell's Equations Equation (1.1) is Faraday's law; Equation (1.2) is a modication of Ampères law with t D being the so-called displacement current density (in A/m 2 of course); Equation (1.3) is Gauss' law (electric); and Equation (1.4) is Gauss' law (magnetic) and it precludes the possibility of magnetic monopoles i.e., the B eld lines do not terminate on a magnetic charge. K. El-Darymli Winter /71

57 Maxwell's Equations Develop Maxwell's Equations in Integral Form Recall (for a vector eld A): K. El-Darymli Winter /71

58 Maxwell's Equations Develop Maxwell's Equations in Integral Form Recall (for a vector eld A): K. El-Darymli Winter /71

59 Maxwell's Equations Develop Maxwell's Equations in Integral Form Recall (for a vector eld A): K. El-Darymli Winter /71

60 Maxwell's Equations Develop Maxwell's Equations in Integral Form Recall (for a vector eld A): (a) Geometry in (I) (b) Geometry in (II) K. El-Darymli Winter /71

61 Maxwell's Equations Develop Maxwell's Equations in Integral Form Recall (for a vector eld A): (a) Geometry in (I) (b) Geometry in (II) Using (I) and (II), equations (1.1) to (1.4) may be written in integral form. K. El-Darymli Winter /71

62 Maxwell's Equations Equation (1.1): K. El-Darymli Winter /71

63 Maxwell's Equations Equation (1.2): K. El-Darymli Winter /71

64 Maxwell's Equations Equation (1.3): Equation (1.4): K. El-Darymli Winter /71

65 Constitutive Relationships Outline 1 Introduction 2 Maxwell's Eqns & Related Formulae Maxwell's Equations Constitutive Relationships Miscellaneous Relations Some Propagation Parameters Time-Harmonic Fields Boundary Conditions 3 Scalar & Vector Potentials Review New Material on Potentials 4 Radiation Mechanism Antenna, a Circuit Point of View Two-Wire Antennas Helpful Animation and Applets 5 Ackgt K. El-Darymli Winter /71

66 Constitutive Relationships In addition to Maxwell's equations, we have the following constitutive relationships: D = ε Ẽ (1.5) B = µ H (1.6) K. El-Darymli Winter /71

67 Constitutive Relationships In addition to Maxwell's equations, we have the following constitutive relationships: D = ε Ẽ (1.5) B = µ H (1.6) where ε, measured in F/m, is referred to as the permittivity of the medium in which the eld exists, and µ, in H/m, is the permeability of the medium. K. El-Darymli Winter /71

68 Constitutive Relationships In addition to Maxwell's equations, we have the following constitutive relationships: D = ε Ẽ (1.5) B = µ H (1.6) where ε, measured in F/m, is referred to as the permittivity of the medium in which the eld exists, and µ, in H/m, is the permeability of the medium. For homogeneous, isotropic media, ε and µ are simply scalars. In our problems, this will always be true (at least, it will be considered to be true). K. El-Darymli Winter /71

69 Constitutive Relationships In addition to Maxwell's equations, we have the following constitutive relationships: D = ε Ẽ (1.5) B = µ H (1.6) where ε, measured in F/m, is referred to as the permittivity of the medium in which the eld exists, and µ, in H/m, is the permeability of the medium. For homogeneous, isotropic media, ε and µ are simply scalars. In our problems, this will always be true (at least, it will be considered to be true). Whenever free space is being considered, ε and µ take on the special notation and values given as K. El-Darymli Winter /71

70 Constitutive Relationships In addition to Maxwell's equations, we have the following constitutive relationships: D = ε Ẽ (1.5) B = µ H (1.6) where ε, measured in F/m, is referred to as the permittivity of the medium in which the eld exists, and µ, in H/m, is the permeability of the medium. For homogeneous, isotropic media, ε and µ are simply scalars. In our problems, this will always be true (at least, it will be considered to be true). Whenever free space is being considered, ε and µ take on the special notation and values given as ε 0 = π F/m, and µ 0 = 4π 10 7 H/m. K. El-Darymli Winter /71

71 Constitutive Relationships In addition to Maxwell's equations, we have the following constitutive relationships: D = ε Ẽ (1.5) B = µ H (1.6) where ε, measured in F/m, is referred to as the permittivity of the medium in which the eld exists, and µ, in H/m, is the permeability of the medium. For homogeneous, isotropic media, ε and µ are simply scalars. In our problems, this will always be true (at least, it will be considered to be true). Whenever free space is being considered, ε and µ take on the special notation and values given as ε 0 = π F/m, and µ 0 = 4π 10 7 H/m. In general, ε=ε 0 (1+χ e ) and µ=µ 0 (1+χ m ) where χ e and χ m are the electric and magnetic susceptibility, respectively. The former was encountered in Term 5 and the latter in Term 6. K. El-Darymli Winter /71

72 Constitutive Relationships Using the constitutive relationships and the forms of ε and µ as given, equations (1.5) and (1.6) become K. El-Darymli Winter /71

73 Constitutive Relationships Using the constitutive relationships and the forms of ε and µ as given, equations (1.5) and (1.6) become D = ε 0 Ẽ + ε0 χ e Ẽ = ε0 Ẽ + P K. El-Darymli Winter /71

74 Constitutive Relationships Using the constitutive relationships and the forms of ε and µ as given, equations (1.5) and (1.6) become and D = ε 0 Ẽ + ε0 χ e Ẽ = ε0 Ẽ + P K. El-Darymli Winter /71

75 Constitutive Relationships Using the constitutive relationships and the forms of ε and µ as given, equations (1.5) and (1.6) become = ε 0 Ẽ + ε0 χ e Ẽ = ε0 Ẽ + D P and ( B =µ 0 H +µ 0 χ m H =µ 0 + H M ) where P is called the polarization (due to bound e-m charges) and M is called the magnetization (due to bound currents). K. El-Darymli Winter /71

76 Constitutive Relationships Using the constitutive relationships and the forms of ε and µ as given, equations (1.5) and (1.6) become = ε 0 Ẽ + ε0 χ e Ẽ = ε0 Ẽ + D P and ( B =µ 0 H +µ 0 χ m H =µ 0 + H M ) where P is called the polarization (due to bound e-m charges) and M is called the magnetization (due to bound currents). Recall also, the notion of: (1) relative permittivity: ε R = ε/ε 0 and (2) relative permeability: µ R = µ/µ 0. K. El-Darymli Winter /71

77 Miscellaneous Relations Outline 1 Introduction 2 Maxwell's Eqns & Related Formulae Maxwell's Equations Constitutive Relationships Miscellaneous Relations Some Propagation Parameters Time-Harmonic Fields Boundary Conditions 3 Scalar & Vector Potentials Review New Material on Potentials 4 Radiation Mechanism Antenna, a Circuit Point of View Two-Wire Antennas Helpful Animation and Applets 5 Ackgt K. El-Darymli Winter /71

78 Miscellaneous Relations Besides equations (1.1) to (1.6), we have the following useful results: Continuity of Current (or Conservation of Charge) J = ρ v t (1.7) K. El-Darymli Winter /71

79 Miscellaneous Relations Besides equations (1.1) to (1.6), we have the following useful results: Continuity of Current (or Conservation of Charge) J = ρ v t (1.7) Convection Current Density where v is the charge velocity. J cnv = ρ v v (1.8) K. El-Darymli Winter /71

80 Miscellaneous Relations Ohm's Law J = σ Ẽ (1.9) where σ is conductivity in mhos per metre ( /m) or siemens per metre (S/m). K. El-Darymli Winter /71

81 Miscellaneous Relations Ohm's Law J = σ Ẽ (1.9) where σ is conductivity in mhos per metre ( /m) or siemens per metre (S/m). Lorentz Force Equation F = Q ( E + v B ) (1.10) where Q is charge in coulombs and F is force in newtons. K. El-Darymli Winter /71

82 Some Propagation Parameters Outline 1 Introduction 2 Maxwell's Eqns & Related Formulae Maxwell's Equations Constitutive Relationships Miscellaneous Relations Some Propagation Parameters Time-Harmonic Fields Boundary Conditions 3 Scalar & Vector Potentials Review New Material on Potentials 4 Radiation Mechanism Antenna, a Circuit Point of View Two-Wire Antennas Helpful Animation and Applets 5 Ackgt K. El-Darymli Winter /71

83 Some Propagation Parameters Phase Velocity, in general, v p = 1 µε (1.11) K. El-Darymli Winter /71

84 Some Propagation Parameters Phase Velocity, in general, v p = 1 µε (1.11) or for free space, c = 1 µ0 ε 0 (1.12) where c= m/s. K. El-Darymli Winter /71

85 Some Propagation Parameters Phase Velocity, in general, v p = 1 µε (1.11) or for free space, c = 1 µ0 ε 0 (1.12) where c= m/s. Also, v p = f λ (1.13) where f is the frequency in hertz (Hz) and λ is wavelength in metres. K. El-Darymli Winter /71

86 Some Propagation Parameters Wave Number and Wave Vector We dene a wave number k for lossless media as k = 2π λ = ω µε (1.14) where k is in radians per metre and the radian frequency ω = 2πf. In the case of the lossless medium, k is equivalent to the β of the Term 6 course. K. El-Darymli Winter /71

87 Some Propagation Parameters Wave Number and Wave Vector We dene a wave number k for lossless media as k = 2π λ = ω µε (1.14) where k is in radians per metre and the radian frequency ω = 2πf. In the case of the lossless medium, k is equivalent to the β of the Term 6 course. If the medium is lossy, k may be complex and we dene jk = α + jβ (1.15) In this case, k is referred to as the complex propagation constant and jk is the same as γ of the Term 6 course. The quantity α is the attenuation coecient in nepers per metre (Np/m). In the lossy case, β = 2π λ. K. El-Darymli Winter /71

88 Some Propagation Parameters Wave Number and Wave Vector We dene a wave number k for lossless media as k = 2π λ = ω µε (1.14) where k is in radians per metre and the radian frequency ω = 2πf. In the case of the lossless medium, k is equivalent to the β of the Term 6 course. If the medium is lossy, k may be complex and we dene jk = α + jβ (1.15) In this case, k is referred to as the complex propagation constant and jk is the same as γ of the Term 6 course. The quantity α is the attenuation coecient in nepers per metre (Np/m). In the lossy case, β = 2π λ. For plane wave propagation in lossless isotropic media, we may dene a wave vector, k, such that k = k = 2π λ, and the direction of k is the direction of wave energy ow. K. El-Darymli Winter /71

89 Some Propagation Parameters In Term 6 we also arrived at the Helmholtz equation (point form, time-harmonic elds with no source), K. El-Darymli Winter /71

90 Some Propagation Parameters Intrinsic Impedance Finally, we dene the intrinsic impedance, η, (in ohms) of a medium in which a waveeld exists as η = µ ε (1.16) K. El-Darymli Winter /71

91 Some Propagation Parameters Intrinsic Impedance Finally, we dene the intrinsic impedance, η, (in ohms) of a medium in which a waveeld exists as η = µ ε (1.16) which for free space becomes It may be noted that η 0 120π Ω. µ0 η 0 = (1.17) ε 0 K. El-Darymli Winter /71

92 Time-Harmonic Fields Outline 1 Introduction 2 Maxwell's Eqns & Related Formulae Maxwell's Equations Constitutive Relationships Miscellaneous Relations Some Propagation Parameters Time-Harmonic Fields Boundary Conditions 3 Scalar & Vector Potentials Review New Material on Potentials 4 Radiation Mechanism Antenna, a Circuit Point of View Two-Wire Antennas Helpful Animation and Applets 5 Ackgt K. El-Darymli Winter /71

93 Time-Harmonic Fields We discovered previously that one solution to Maxwell's equations was indicative of plane waves travelling through a medium whose properties are stipulated by the values of ε, µ, σ, and so on. In arriving at this conclusion, we started with time-harmonic (sinusoidal) elds. K. El-Darymli Winter /71

94 Time-Harmonic Fields We discovered previously that one solution to Maxwell's equations was indicative of plane waves travelling through a medium whose properties are stipulated by the values of ε, µ, σ, and so on. In arriving at this conclusion, we started with time-harmonic (sinusoidal) elds. Recall, for a time harmonic eld, Ã, of the form à = A 0 cos(ωt + φ), that K. El-Darymli Winter /71

95 Time-Harmonic Fields We discovered previously that one solution to Maxwell's equations was indicative of plane waves travelling through a medium whose properties are stipulated by the values of ε, µ, σ, and so on. In arriving at this conclusion, we started with time-harmonic (sinusoidal) elds. Recall, for a time harmonic eld, Ã, of the form à = A 0 cos(ωt + φ), that Ã(r,t) à = Re { As e jωt} (1.18) where A s is the phasor form of the eld. K. El-Darymli Winter /71

96 Time-Harmonic Fields We discovered previously that one solution to Maxwell's equations was indicative of plane waves travelling through a medium whose properties are stipulated by the values of ε, µ, σ, and so on. In arriving at this conclusion, we started with time-harmonic (sinusoidal) elds. Recall, for a time harmonic eld, Ã, of the form à = A 0 cos(ωt + φ), that Ã(r,t) à = Re { As e jωt} (1.18) where A s is the phasor form of the eld. Also, recall that the time derivative / t transforms to jω in the phasor domain. K. El-Darymli Winter /71

97 Time-Harmonic Fields We discovered previously that one solution to Maxwell's equations was indicative of plane waves travelling through a medium whose properties are stipulated by the values of ε, µ, σ, and so on. In arriving at this conclusion, we started with time-harmonic (sinusoidal) elds. Recall, for a time harmonic eld, Ã, of the form à = A 0 cos(ωt + φ), that Ã(r,t) à = Re { As e jωt} (1.18) where A s is the phasor form of the eld. Also, recall that the time derivative / t transforms to jω in the phasor domain. From now on, since it is generally to be understood almost everywhere IN THIS COURSE that the elds are time-harmonic, we shall drop the s subscript on the phasor and use simply the form A. K. El-Darymli Winter /71

98 Time-Harmonic Fields Equations (1.1)(1.4) in phasor form become E = jω B (1.19) H = J + jω D (1.20) D = ρv (1.21) B = 0 (1.22) K. El-Darymli Winter /71

99 Time-Harmonic Fields Equations (1.1)(1.4) in phasor form become E = jω B (1.19) H = J + jω D (1.20) D = ρv (1.21) B = 0 (1.22) In these equations, for free space, J = 0 and ρ v = 0. K. El-Darymli Winter /71

100 Time-Harmonic Fields Equations (1.1)(1.4) in phasor form become E = jω B (1.19) H = J + jω D (1.20) D = ρv (1.21) B = 0 (1.22) In these equations, for free space, J = 0 and ρ v = 0. ASIDE: Many texts use Ẽ, etc. to denote phasors. K. El-Darymli Winter /71

101 Boundary Conditions Outline 1 Introduction 2 Maxwell's Eqns & Related Formulae Maxwell's Equations Constitutive Relationships Miscellaneous Relations Some Propagation Parameters Time-Harmonic Fields Boundary Conditions 3 Scalar & Vector Potentials Review New Material on Potentials 4 Radiation Mechanism Antenna, a Circuit Point of View Two-Wire Antennas Helpful Animation and Applets 5 Ackgt K. El-Darymli Winter /71

102 Boundary Conditions There are many instances in which e-m energy impinges a boundary between two electromagnetically distinct media. For example, K. El-Darymli Winter /71

103 Boundary Conditions There are many instances in which e-m energy impinges a boundary between two electromagnetically distinct media. For example, In general, for two media as shown, where ˆn is the unit normal to the boundary or interface, the following important relationships hold: K. El-Darymli Winter /71

104 Boundary Conditions There are many instances in which e-m energy impinges a boundary between two electromagnetically distinct media. For example, In general, for two media as shown, where ˆn is the unit normal to the boundary or interface, the following important relationships hold: (1) The tangential E-eld ( E T ) is continuous across the boundary; i.e., ˆn E 1 = ˆn E 2 E T1 = E T2. K. El-Darymli Winter /71

105 Boundary Conditions There are many instances in which e-m energy impinges a boundary between two electromagnetically distinct media. For example, In general, for two media as shown, where ˆn is the unit normal to the boundary or interface, the following important relationships hold: (1) The tangential E-eld ( E T ) is continuous across the boundary; i.e., ˆn E 1 = ˆn E 2 E T1 = E T2. What does this imply if medium 2 is a perfect conductor? K. El-Darymli Winter /71

106 Boundary Conditions There are many instances in which e-m energy impinges a boundary between two electromagnetically distinct media. For example, In general, for two media as shown, where ˆn is the unit normal to the boundary or interface, the following important relationships hold: (1) The tangential E-eld ( E T ) is continuous across the boundary; i.e., ˆn E 1 = ˆn E 2 E T1 = E T2. What does this imply if medium 2 is a perfect conductor? (2) If no surface current density ( K) exists on the boundary, then the tangential H-eld ( HT ) is continuous across the boundary i.e., ˆn H 1 = ˆn H 2 H T1 = H T2 ; else H T is discontinuous by an amount equal to K; i.e., [ ˆn H1 ] H 2 = K. K. El-Darymli Winter /71

107 Boundary Conditions (3) The normal component of the D-eld is discontinuous by an amount equal to the surface charge density, ρ s, on the boundary; i.e., [ ˆn D1 ] D 2 = ρ s. K. El-Darymli Winter /71

108 Boundary Conditions (3) The normal component of the D-eld is discontinuous by an amount equal to the surface charge density, ρ s, on the boundary; i.e., [ ˆn D1 ] D 2 = ρ s. (What's the implication if medium 2 is a perfect conductor? The answer can also be arrived at by applying Gauss' law.) K. El-Darymli Winter /71

109 Boundary Conditions (3) The normal component of the D-eld is discontinuous by an amount equal to the surface charge density, ρ s, on the boundary; i.e., [ ˆn D1 ] D 2 = ρ s. (What's the implication if medium 2 is a perfect conductor? The answer can also be arrived at by applying Gauss' law.) K. El-Darymli Winter /71

110 Boundary Conditions (3) The normal component of the D-eld is discontinuous by an amount equal to the surface charge density, ρ s, on the boundary; i.e., [ ˆn D1 ] D 2 = ρ s. (What's the implication if medium 2 is a perfect conductor? The answer can also be arrived at by applying Gauss' law.) K. El-Darymli Winter /71

111 Boundary Conditions (3) The normal component of the D-eld is discontinuous by an amount equal to the surface charge density, ρ s, on the boundary; i.e., [ ˆn D1 ] D 2 = ρ s. (What's the implication if medium 2 is a perfect conductor? The answer can also be arrived at by applying Gauss' law.) (4) The normal component of the B-eld is continuous across the boundary; i.e., ˆn B 1 = ˆn B 2. K. El-Darymli Winter /71

112 Review Outline 1 Introduction 2 Maxwell's Eqns & Related Formulae Maxwell's Equations Constitutive Relationships Miscellaneous Relations Some Propagation Parameters Time-Harmonic Fields Boundary Conditions 3 Scalar & Vector Potentials Review New Material on Potentials 4 Radiation Mechanism Antenna, a Circuit Point of View Two-Wire Antennas Helpful Animation and Applets 5 Ackgt K. El-Darymli Winter /71

113 Review In earlier electromagnetics courses, it was observed that implementing constructs referred to as potentials facilitated the calculation of the E- and B-eld quantities. K. El-Darymli Winter /71

114 Review In earlier electromagnetics courses, it was observed that implementing constructs referred to as potentials facilitated the calculation of the E- and B-eld quantities. In general, it was found that the potential integrals were easier to calculate than the eld expressions appearing, for example, in Coulomb's law (static electric eld) or in the Biot-Savart law (steady magnetic eld). K. El-Darymli Winter /71

115 Review In earlier electromagnetics courses, it was observed that implementing constructs referred to as potentials facilitated the calculation of the E- and B-eld quantities. In general, it was found that the potential integrals were easier to calculate than the eld expressions appearing, for example, in Coulomb's law (static electric eld) or in the Biot-Savart law (steady magnetic eld). Furthermore, on determining the potentials, the E and B elds were readily found using derivatives rather than integrals. K. El-Darymli Winter /71

116 Review In earlier electromagnetics courses, it was observed that implementing constructs referred to as potentials facilitated the calculation of the E- and B-eld quantities. In general, it was found that the potential integrals were easier to calculate than the eld expressions appearing, for example, in Coulomb's law (static electric eld) or in the Biot-Savart law (steady magnetic eld). Furthermore, on determining the potentials, the E and B elds were readily found using derivatives rather than integrals. Recall the following geometry for the specication of a eld point P(x,y,z): K. El-Darymli Winter /71

117 Review In earlier electromagnetics courses, it was observed that implementing constructs referred to as potentials facilitated the calculation of the E- and B-eld quantities. In general, it was found that the potential integrals were easier to calculate than the eld expressions appearing, for example, in Coulomb's law (static electric eld) or in the Biot-Savart law (steady magnetic eld). Furthermore, on determining the potentials, the E and B elds were readily found using derivatives rather than integrals. Recall the following geometry for the specication of a eld point P(x,y,z): r position vector for observation or eld point, P. K. El-Darymli Winter /71

118 Review In earlier electromagnetics courses, it was observed that implementing constructs referred to as potentials facilitated the calculation of the E- and B-eld quantities. In general, it was found that the potential integrals were easier to calculate than the eld expressions appearing, for example, in Coulomb's law (static electric eld) or in the Biot-Savart law (steady magnetic eld). Furthermore, on determining the potentials, the E and B elds were readily found using derivatives rather than integrals. Recall the following geometry for the specication of a eld point P(x,y,z): r position vector for observation or eld point, P. r position vector for points in the source region. K. El-Darymli Winter /71

119 Review In earlier electromagnetics courses, it was observed that implementing constructs referred to as potentials facilitated the calculation of the E- and B-eld quantities. In general, it was found that the potential integrals were easier to calculate than the eld expressions appearing, for example, in Coulomb's law (static electric eld) or in the Biot-Savart law (steady magnetic eld). Furthermore, on determining the potentials, the E and B elds were readily found using derivatives rather than integrals. Recall the following geometry for the specication of a eld point P(x,y,z): r position vector for observation or eld point, P. r position vector for points in the source region. r r is as shown. K. El-Darymli Winter /71

120 Review Electrostatic Field The source consists of electric charges. It was discovered that if a scalar potential, say Φ, existed at P, then the electric eld was simply given by E ( r) = Φ( r) (1.23) K. El-Darymli Winter /71

121 Review Electrostatic Field The source consists of electric charges. It was discovered that if a scalar potential, say Φ, existed at P, then the electric eld was simply given by E ( r) = Φ( r) (1.23) Note that Φ is the V of Term 6 and ρ v ( r Φ( r) = V ( r) = v )dv 4πε r r (1.24) where (in Cartesian coordinates),...dv...dx dy dz and ρ v is the v z y x charge density. Equation (1.24) is the solution to Poisson's equation 2 Φ = ρ v ε (1.25) K. El-Darymli Winter /71

122 Review Electrostatic Field The source consists of electric charges. It was discovered that if a scalar potential, say Φ, existed at P, then the electric eld was simply given by E ( r) = Φ( r) (1.23) Note that Φ is the V of Term 6 and ρ v ( r Φ( r) = V ( r) = v )dv 4πε r r (1.24) where (in Cartesian coordinates),...dv...dx dy dz and ρ v is the v z y x charge density. Equation (1.24) is the solution to Poisson's equation 2 Φ = ρ v ε (1.25) Note that equation (1.24) could be also written explicitly as a 3-dimensional spatial convolution: x ρ v (x,y,z )dx dy dz Φ(x,y,z) = z y 4πε (x x ) 2 + (y y ) 2 + (z z ) 2 = ρ v (x,y,z) 1 3d ε 4π r (1.26) K. El-Darymli Winter /71

123 Review Electrostatic Field The source consists of electric charges. It was discovered that if a scalar potential, say Φ, existed at P, then the electric eld was simply given by E ( r) = Φ( r) (1.23) Note that Φ is the V of Term 6 and ρ v ( r Φ( r) = V ( r) = v )dv 4πε r r (1.24) where (in Cartesian coordinates),...dv...dx dy dz and ρ v is the v z y x charge density. Equation (1.24) is the solution to Poisson's equation 2 Φ = ρ v ε (1.25) Note that equation (1.24) could be also written explicitly as a 3-dimensional spatial convolution: x ρ v (x,y,z )dx dy dz Φ(x,y,z) = z y 4πε (x x ) 2 + (y y ) 2 + (z z ) 2 = ρ v (x,y,z) 1 3d ε 4π r (1.26) with r = x 2 + y 2 + z 2 while 3d represents a 3-dimensional convolution. K. El-Darymli Winter /71

124 Review Electrostatic Field The source consists of electric charges. It was discovered that if a scalar potential, say Φ, existed at P, then the electric eld was simply given by E ( r) = Φ( r) (1.23) Note that Φ is the V of Term 6 and ρ v ( r Φ( r) = V ( r) = v )dv 4πε r r (1.24) where (in Cartesian coordinates),...dv...dx dy dz and ρ v is the v z y x charge density. Equation (1.24) is the solution to Poisson's equation 2 Φ = ρ v ε (1.25) Note that equation (1.24) could be also written explicitly as a 3-dimensional spatial convolution: x ρ v (x,y,z )dx dy dz Φ(x,y,z) = z y 4πε (x x ) 2 + (y y ) 2 + (z z ) 2 = ρ v (x,y,z) 1 3d ε 4π r (1.26) with r = x 2 + y 2 + z 2 while 3d represents a 3-dimensional convolution. Recall: K. El-Darymli Winter /71

125 Review Magnetostatic Field The source consists of a steady current. We have seen that steady currents (i.e., dc) produce steady magnetic elds. In our analysis we introduced a vector potential, A, dened as B = A (1.27) K. El-Darymli Winter /71

126 Review Magnetostatic Field The source consists of a steady current. We have seen that steady currents (i.e., dc) produce steady magnetic elds. In our analysis we introduced a vector potential, A, dened as B = A (1.27) Making the substitution into Maxwell's equations and invoking the Coulomb gauge ( A=0) which we proved had to be true for steady elds as a result of there being no ( / t) terms it was shown that 2 A = µ0 J (1.28) K. El-Darymli Winter /71

127 Review Magnetostatic Field The source consists of a steady current. We have seen that steady currents (i.e., dc) produce steady magnetic elds. In our analysis we introduced a vector potential, A, dened as B = A (1.27) Making the substitution into Maxwell's equations and invoking the Coulomb gauge ( A=0) which we proved had to be true for steady elds as a result of there being no ( / t) terms it was shown that 2 A = µ0 J (1.28) On comparing (1.28) with (1.25), while keeping an eye on (1.24) and (1.26), we may immediately write that A( r) = µ 0 J( r )dv v 4π r r = µ 0 J(x,y,z) 3d 1 4π r (1.29) K. El-Darymli Winter /71

128 Review Magnetostatic Field The source consists of a steady current. We have seen that steady currents (i.e., dc) produce steady magnetic elds. In our analysis we introduced a vector potential, A, dened as B = A (1.27) Making the substitution into Maxwell's equations and invoking the Coulomb gauge ( A=0) which we proved had to be true for steady elds as a result of there being no ( / t) terms it was shown that 2 A = µ0 J (1.28) On comparing (1.28) with (1.25), while keeping an eye on (1.24) and (1.26), we may immediately write that A( r) = µ 0 J( r )dv v 4π r r = µ 0 J(x,y,z) 3d 1 4π r (1.29) Generally, equation (1.27) and (1.29) together give a simpler way of calculating the magnetic ux density, B, (or, equivalently, the magnetic eld intensity, H) than is available via the Biot-Savart law which contains a cross product. K. El-Darymli Winter /71

129 New Material on Potentials Outline 1 Introduction 2 Maxwell's Eqns & Related Formulae Maxwell's Equations Constitutive Relationships Miscellaneous Relations Some Propagation Parameters Time-Harmonic Fields Boundary Conditions 3 Scalar & Vector Potentials Review New Material on Potentials 4 Radiation Mechanism Antenna, a Circuit Point of View Two-Wire Antennas Helpful Animation and Applets 5 Ackgt K. El-Darymli Winter /71

130 New Material on Potentials So far, we have considered only the non-time-varying eld. However, in antenna theory, the elds are time varying. K. El-Darymli Winter /71

131 New Material on Potentials So far, we have considered only the non-time-varying eld. However, in antenna theory, the elds are time varying. In fact, as we shall elaborate, the production of radiated energy requires charges to be accelerating. K. El-Darymli Winter /71

132 New Material on Potentials So far, we have considered only the non-time-varying eld. However, in antenna theory, the elds are time varying. In fact, as we shall elaborate, the production of radiated energy requires charges to be accelerating. Therefore, the steady eld forms for the vector and scalar potentials must be revisited for this new case. K. El-Darymli Winter /71

133 New Material on Potentials So far, we have considered only the non-time-varying eld. However, in antenna theory, the elds are time varying. In fact, as we shall elaborate, the production of radiated energy requires charges to be accelerating. Therefore, the steady eld forms for the vector and scalar potentials must be revisited for this new case. We shall assume time-harmonic sources and elds. K. El-Darymli Winter /71

134 New Material on Potentials So far, we have considered only the non-time-varying eld. However, in antenna theory, the elds are time varying. In fact, as we shall elaborate, the production of radiated energy requires charges to be accelerating. Therefore, the steady eld forms for the vector and scalar potentials must be revisited for this new case. We shall assume time-harmonic sources and elds. From equation (1.19), K. El-Darymli Winter /71

135 New Material on Potentials So far, we have considered only the non-time-varying eld. However, in antenna theory, the elds are time varying. In fact, as we shall elaborate, the production of radiated energy requires charges to be accelerating. Therefore, the steady eld forms for the vector and scalar potentials must be revisited for this new case. We shall assume time-harmonic sources and elds. From equation (1.19), Suppose K. El-Darymli Winter /71

136 New Material on Potentials So far, we have considered only the non-time-varying eld. However, in antenna theory, the elds are time varying. In fact, as we shall elaborate, the production of radiated energy requires charges to be accelerating. Therefore, the steady eld forms for the vector and scalar potentials must be revisited for this new case. We shall assume time-harmonic sources and elds. From equation (1.19), Suppose Then, K. El-Darymli Winter /71

137 New Material on Potentials So far, we have considered only the non-time-varying eld. However, in antenna theory, the elds are time varying. In fact, as we shall elaborate, the production of radiated energy requires charges to be accelerating. Therefore, the steady eld forms for the vector and scalar potentials must be revisited for this new case. We shall assume time-harmonic sources and elds. From equation (1.19), Suppose Then, (Since the curl of the gradient is always zero). K. El-Darymli Winter /71

138 New Material on Potentials So far, we have considered only the non-time-varying eld. However, in antenna theory, the elds are time varying. In fact, as we shall elaborate, the production of radiated energy requires charges to be accelerating. Therefore, the steady eld forms for the vector and scalar potentials must be revisited for this new case. We shall assume time-harmonic sources and elds. From equation (1.19), Suppose Then, (Since the curl of the gradient is always zero). However, if E = 0, there cannot be a time-varying B-eld (see equation (1.1) and remember that the curl is a spatial operator). K. El-Darymli Winter /71

139 New Material on Potentials So far, we have considered only the non-time-varying eld. However, in antenna theory, the elds are time varying. In fact, as we shall elaborate, the production of radiated energy requires charges to be accelerating. Therefore, the steady eld forms for the vector and scalar potentials must be revisited for this new case. We shall assume time-harmonic sources and elds. From equation (1.19), Suppose Then, (Since the curl of the gradient is always zero). However, if E = 0, there cannot be a time-varying B-eld (see equation (1.1) and remember that the curl is a spatial operator). Since the discussion has now turned to time-varying elds it is clear that our old (i.e., steady eld) scalar potential cannot be used. K. El-Darymli Winter /71

140 New Material on Potentials The details are a little more complicated this time, but the starting point is our observation that a time-varying B-eld is still solenoidal or divergenceless, i.e., K. El-Darymli Winter /71

141 New Material on Potentials The details are a little more complicated this time, but the starting point is our observation that a time-varying B-eld is still solenoidal or divergenceless, i.e., Therefore, a vector potential, A, is still generally dened by (1.27) B = A. K. El-Darymli Winter /71

142 New Material on Potentials The details are a little more complicated this time, but the starting point is our observation that a time-varying B-eld is still solenoidal or divergenceless, i.e., Therefore, a vector potential, A, is still generally dened by (1.27) B = A. However, we cannot invoke the Coulomb gauge ( A = 0) and expect A to be useful in determining time-varying elds. K. El-Darymli Winter /71

143 New Material on Potentials The details are a little more complicated this time, but the starting point is our observation that a time-varying B-eld is still solenoidal or divergenceless, i.e., Therefore, a vector potential, A, is still generally dened by (1.27) B = A. However, we cannot invoke the Coulomb gauge ( A = 0) and expect A to be useful in determining time-varying elds. What to do?!! Stay tuned (if you are still tuned)! K. El-Darymli Winter /71

144 New Material on Potentials The details are a little more complicated this time, but the starting point is our observation that a time-varying B-eld is still solenoidal or divergenceless, i.e., Therefore, a vector potential, A, is still generally dened by (1.27) B = A. However, we cannot invoke the Coulomb gauge ( A = 0) and expect A to be useful in determining time-varying elds. What to do?!! Stay tuned (if you are still tuned)! Using equation (1.27) in E = jω B K. El-Darymli Winter /71

145 New Material on Potentials The details are a little more complicated this time, but the starting point is our observation that a time-varying B-eld is still solenoidal or divergenceless, i.e., Therefore, a vector potential, A, is still generally dened by (1.27) B = A. However, we cannot invoke the Coulomb gauge ( A = 0) and expect A to be useful in determining time-varying elds. What to do?!! Stay tuned (if you are still tuned)! Using equation (1.27) in E = jω B we get E = K. El-Darymli Winter /71

146 New Material on Potentials The details are a little more complicated this time, but the starting point is our observation that a time-varying B-eld is still solenoidal or divergenceless, i.e., Therefore, a vector potential, A, is still generally dened by (1.27) B = A. However, we cannot invoke the Coulomb gauge ( A = 0) and expect A to be useful in determining time-varying elds. What to do?!! Stay tuned (if you are still tuned)! Using equation (1.27) in E = jω B we get which implies E = (1.30) K. El-Darymli Winter /71

147 New Material on Potentials Since the curl of a gradient is zero, equation (1.30) implies that (1.31) where Φ is a scalar potential. (Note that the allows us to write E = Φ when A is not time-varying: recall jω t and à dependence.) t =0 when A has no time K. El-Darymli Winter /71

148 New Material on Potentials Since the curl of a gradient is zero, equation (1.30) implies that (1.31) where Φ is a scalar potential. (Note that the allows us to write E = Φ when A is not time-varying: recall jω t and à dependence.) For time-harmonic elds, equation (1.31) gives t =0 when A has no time E = jω A Φ (1.32) and it is seen that the E eld depends on both the scalar and vector potential. K. El-Darymli Winter /71

149 New Material on Potentials Since the curl of a gradient is zero, equation (1.30) implies that (1.31) where Φ is a scalar potential. (Note that the allows us to write E = Φ when A is not time-varying: recall jω t and à dependence.) For time-harmonic elds, equation (1.31) gives t =0 when A has no time E = jω A Φ (1.32) and it is seen that the E eld depends on both the scalar and vector potential. Now, (1.27) and (1.32) satisfy E = jω B, and. B = 0 automatically, since that's how we started. K. El-Darymli Winter /71

150 New Material on Potentials Since the curl of a gradient is zero, equation (1.30) implies that (1.31) where Φ is a scalar potential. (Note that the allows us to write E = Φ when A is not time-varying: recall jω t and à dependence.) For time-harmonic elds, equation (1.31) gives t =0 when A has no time E = jω A Φ (1.32) and it is seen that the E eld depends on both the scalar and vector potential. Now, (1.27) and (1.32) satisfy E = jω B, and. B = 0 automatically, since that's how we started. What about the remaining Maxwell equations? K. El-Darymli Winter /71

151 New Material on Potentials Since the curl of a gradient is zero, equation (1.30) implies that (1.31) where Φ is a scalar potential. (Note that the allows us to write E = Φ when A is not time-varying: recall jω t and à dependence.) For time-harmonic elds, equation (1.31) gives t =0 when A has no time E = jω A Φ (1.32) and it is seen that the E eld depends on both the scalar and vector potential. Now, (1.27) and (1.32) satisfy E = jω B, and. B = 0 automatically, since that's how we started. What about the remaining Maxwell equations? (1.33) K. El-Darymli Winter /71

152 New Material on Potentials Since the curl of a gradient is zero, equation (1.30) implies that (1.31) where Φ is a scalar potential. (Note that the allows us to write E = Φ when A is not time-varying: recall jω t and à dependence.) For time-harmonic elds, equation (1.31) gives t =0 when A has no time E = jω A Φ (1.32) and it is seen that the E eld depends on both the scalar and vector potential. Now, (1.27) and (1.32) satisfy E = jω B, and. B = 0 automatically, since that's how we started. What about the remaining Maxwell equations? (1.33) (1.34) K. El-Darymli Winter /71

153 New Material on Potentials Since the curl of a gradient is zero, equation (1.30) implies that (1.31) where Φ is a scalar potential. (Note that the allows us to write E = Φ when A is not time-varying: recall jω t and à dependence.) For time-harmonic elds, equation (1.31) gives t =0 when A has no time E = jω A Φ (1.32) and it is seen that the E eld depends on both the scalar and vector potential. Now, (1.27) and (1.32) satisfy E = jω B, and. B = 0 automatically, since that's how we started. What about the remaining Maxwell equations? (1.33) (1.34) Since (1.33) and (1.34) must also be satised by (1.27) and (1.32), it looks like picking the proper forms of A and Φ could be quite messy! K. El-Darymli Winter /71

154 New Material on Potentials Substituting from equations (1.27) and (1.32) into (1.33) gives. (1.35) K. El-Darymli Winter /71

155 New Material on Potentials Substituting from equations (1.27) and (1.32) into (1.33) gives. (1.35) Invoking the vector identity A = ( A) 2 A, equation (1.35) becomes 2 A+ω 2 µε A= (( ) ). A + jωµεφ -µ J. K. El-Darymli Winter /71

156 New Material on Potentials Substituting from equations (1.27) and (1.32) into (1.33) gives. (1.35) Invoking the vector identity A = ( A) 2 A, equation (1.35) becomes 2 A+ω 2 µε A= (( ) ). A + jωµεφ -µ J. Recalling k = ω µε, (( ) ) 2 A + k 2 A =. A + jωµεφ µ J. (1.36) K. El-Darymli Winter /71

157 New Material on Potentials Substituting from equations (1.27) and (1.32) into (1.33) gives. (1.35) Invoking the vector identity A = ( A) 2 A, equation (1.35) becomes 2 A+ω 2 µε A= (( ) ). A + jωµεφ -µ J. Recalling k = ω µε, (( ) ) 2 A + k 2 A =. A + jωµεφ µ J. (1.36) We have argued in Term 6 that to completely specify a vector, both the curl and divergence are required. K. El-Darymli Winter /71

158 New Material on Potentials Substituting from equations (1.27) and (1.32) into (1.33) gives. (1.35) Invoking the vector identity A = ( A) 2 A, equation (1.35) becomes 2 A+ω 2 µε A= (( ) ). A + jωµεφ -µ J. Recalling k = ω µε, (( ) ) 2 A + k 2 A =. A + jωµεφ µ J. (1.36) We have argued in Term 6 that to completely specify a vector, both the curl and divergence are required. (It may be seen that the curl alone is not enough to uniquely dene A since ( A + λ) = A for any scalar function λ). K. El-Darymli Winter /71

159 New Material on Potentials Substituting from equations (1.27) and (1.32) into (1.33) gives. (1.35) Invoking the vector identity A = ( A) 2 A, equation (1.35) becomes 2 A+ω 2 µε A= (( ) ). A + jωµεφ -µ J. Recalling k = ω µε, (( ) ) 2 A + k 2 A =. A + jωµεφ µ J. (1.36) We have argued in Term 6 that to completely specify a vector, both the curl and divergence are required. (It may be seen that the curl alone is not enough to uniquely dene A since ( A + λ) = A for any scalar function λ). Of course, the curl is specied here by A = B... but what are we to do about the divergence ( A) in (1.36)? K. El-Darymli Winter /71

160 New Material on Potentials Substituting from equations (1.27) and (1.32) into (1.33) gives. (1.35) Invoking the vector identity A = ( A) 2 A, equation (1.35) becomes 2 A+ω 2 µε A= (( ) ). A + jωµεφ -µ J. Recalling k = ω µε, (( ) ) 2 A + k 2 A =. A + jωµεφ µ J. (1.36) We have argued in Term 6 that to completely specify a vector, both the curl and divergence are required. (It may be seen that the curl alone is not enough to uniquely dene A since ( A + λ) = A for any scalar function λ). Of course, the curl is specied here by A = B... but what are we to do about the divergence ( A) in (1.36)? In electro/magnetostatics we invoked the Coulomb gauge, which we have said is not a good plan for time-varying elds. K. El-Darymli Winter /71

161 New Material on Potentials Substituting from equations (1.27) and (1.32) into (1.33) gives. (1.35) Invoking the vector identity A = ( A) 2 A, equation (1.35) becomes 2 A+ω 2 µε A= (( ) ). A + jωµεφ -µ J. Recalling k = ω µε, (( ) ) 2 A + k 2 A =. A + jωµεφ µ J. (1.36) We have argued in Term 6 that to completely specify a vector, both the curl and divergence are required. (It may be seen that the curl alone is not enough to uniquely dene A since ( A + λ) = A for any scalar function λ). Of course, the curl is specied here by A = B... but what are we to do about the divergence ( A) in (1.36)? In electro/magnetostatics we invoked the Coulomb gauge, which we have said is not a good plan for time-varying elds. With a view to eliminating the gradient on the R.H.S. of (1.36), we DEFINE AND USE the Lorenz gauge:. A = jωµεφ (1.37) K. El-Darymli Winter /71

162 New Material on Potentials Substituting from equations (1.27) and (1.32) into (1.33) gives. (1.35) Invoking the vector identity A = ( A) 2 A, equation (1.35) becomes 2 A+ω 2 µε A= (( ) ). A + jωµεφ -µ J. Recalling k = ω µε, (( ) ) 2 A + k 2 A =. A + jωµεφ µ J. (1.36) We have argued in Term 6 that to completely specify a vector, both the curl and divergence are required. (It may be seen that the curl alone is not enough to uniquely dene A since ( A + λ) = A for any scalar function λ). Of course, the curl is specied here by A = B... but what are we to do about the divergence ( A) in (1.36)? In electro/magnetostatics we invoked the Coulomb gauge, which we have said is not a good plan for time-varying elds. With a view to eliminating the gradient on the R.H.S. of (1.36), we DEFINE AND USE the Lorenz gauge:. A = jωµεφ (1.37) This is a good choice as, in retrospect, it is seen to lead to all of the proper results for the elds. K. El-Darymli Winter /71

163 New Material on Potentials On using the Lorenz gauge, equation (1.36) becomes and from (1.32), (1.34) and (1.37) 2 A + k 2 A = µ J (1.38) 2 Φ + k 2 Φ = ρ v ε (1.39) K. El-Darymli Winter /71

164 New Material on Potentials On using the Lorenz gauge, equation (1.36) becomes and from (1.32), (1.34) and (1.37) 2 A + k 2 A = µ J (1.38) 2 Φ + k 2 Φ = ρ v ε (1.39) Equations (1.38) and (1.39) are the inhomogeneous Helmholtz equations for vector and scalar potentials. K. El-Darymli Winter /71

165 New Material on Potentials On using the Lorenz gauge, equation (1.36) becomes and from (1.32), (1.34) and (1.37) 2 A + k 2 A = µ J (1.38) 2 Φ + k 2 Φ = ρ v ε (1.39) Equations (1.38) and (1.39) are the inhomogeneous Helmholtz equations for vector and scalar potentials. We note that for the static case, in which k = 0 since ω = 0, (1.38) and (1.39) reduce to the proper forms given by (1.25) and (1.28). K. El-Darymli Winter /71

166 New Material on Potentials On using the Lorenz gauge, equation (1.36) becomes and from (1.32), (1.34) and (1.37) 2 A + k 2 A = µ J (1.38) 2 Φ + k 2 Φ = ρ v ε (1.39) Equations (1.38) and (1.39) are the inhomogeneous Helmholtz equations for vector and scalar potentials. We note that for the static case, in which k = 0 since ω = 0, (1.38) and (1.39) reduce to the proper forms given by (1.25) and (1.28). It is possible to develop solutions to (1.38) and (1.39) by analogy to the static case (of course, they may be solved rigorously also). K. El-Darymli Winter /71

167 New Material on Potentials On using the Lorenz gauge, equation (1.36) becomes and from (1.32), (1.34) and (1.37) 2 A + k 2 A = µ J (1.38) 2 Φ + k 2 Φ = ρ v ε (1.39) Equations (1.38) and (1.39) are the inhomogeneous Helmholtz equations for vector and scalar potentials. We note that for the static case, in which k = 0 since ω = 0, (1.38) and (1.39) reduce to the proper forms given by (1.25) and (1.28). It is possible to develop solutions to (1.38) and (1.39) by analogy to the static case (of course, they may be solved rigorously also). We will not give a rigorous solution in this course. Rather, let's write down the answers and observe some of the properties which seem to make intuitive sense. K. El-Darymli Winter /71

168 New Material on Potentials The answers are very important because they eventually lead to the time-varying elds produced by time-harmonic sources. We get and A( r) = µ 0 J( r )e jk r r dv v 4π r r ρ v ( r Φ( r) = v )e jk r r dv 4πε r r (1.40) (1.41) K. El-Darymli Winter /71

169 New Material on Potentials The answers are very important because they eventually lead to the time-varying elds produced by time-harmonic sources. We get and Note that A( r) = µ 0 J( r )e jk r r dv v 4π r r ρ v ( r Φ( r) = v )e jk r r dv 4πε r r (1.40) (1.41) K. El-Darymli Winter /71

170 New Material on Potentials The answers are very important because they eventually lead to the time-varying elds produced by time-harmonic sources. We get and Note that A( r) = µ 0 J( r )e jk r r dv v 4π r r ρ v ( r Φ( r) = v )e jk r r dv 4πε r r (1.40) (1.41) Source to the Observation (eld) point. K. El-Darymli Winter /71

171 New Material on Potentials The answers are very important because they eventually lead to the time-varying elds produced by time-harmonic sources. We get and Note that A( r) = µ 0 J( r )e jk r r dv v 4π r r ρ v ( r Φ( r) = v )e jk r r dv 4πε r r (1.40) (1.41) Source to the Observation (eld) point. Since E and B can be determined from A via equations (1.27), (1.33), and (1.38), let's consider (1.40) in detail: K. El-Darymli Winter /71

172 New Material on Potentials Observations of Similarities Between (1.29) and (1.40) 1. Mathematically, at points removed from the source (i.e., r r ), the static and time-varying forms dier by the presence of a phase term, e jk r r in the latter. K. El-Darymli Winter /71

173 New Material on Potentials Observations of Similarities Between (1.29) and (1.40) 1. Mathematically, at points removed from the source (i.e., r r ), the static and time-varying forms dier by the presence of a phase term, e jk r r in the latter. 1 (Note that this is similarly the case for the Φ's of equations (1.26) and (1.41)). K. El-Darymli Winter /71

174 New Material on Potentials Observations of Similarities Between (1.29) and (1.40) 1. Mathematically, at points removed from the source (i.e., r r ), the static and time-varying forms dier by the presence of a phase term, e jk r r in the latter. 1 (Note that this is similarly the case for the Φ's of equations (1.26) and (1.41)). 2 That is, physically, as compared to the static case, the A viewed at the eld position r is phase-delayed by an amount determined by the distance R = r r as shown: K. El-Darymli Winter /71

175 New Material on Potentials Observations of Similarities Between (1.29) and (1.40) 1. Mathematically, at points removed from the source (i.e., r r ), the static and time-varying forms dier by the presence of a phase term, e jk r r in the latter. 1 (Note that this is similarly the case for the Φ's of equations (1.26) and (1.41)). 2 That is, physically, as compared to the static case, the A viewed at the eld position r is phase-delayed by an amount determined by the distance R = r r as shown: 3 That is, the phase delay is determined by the distance from the source to the observer. K. El-Darymli Winter /71

176 New Material on Potentials 2. If we take (1.40) to the time domain, we would see that à at r is determined by the state of the source at time (R/c) seconds earlier. K. El-Darymli Winter /71

177 New Material on Potentials 2. If we take (1.40) to the time domain, we would see that à at r is determined by the state of the source at time (R/c) seconds earlier. Of course, (R/c) is simply the time necessary for a phenomenon travelling at the speed of light to cover the distance R. For this reason the potentials in (1.40) and (1.41) are referred to as retarded potentials. K. El-Darymli Winter /71

178 New Material on Potentials 2. If we take (1.40) to the time domain, we would see that à at r is determined by the state of the source at time (R/c) seconds earlier. Of course, (R/c) is simply the time necessary for a phenomenon travelling at the speed of light to cover the distance R. For this reason the potentials in (1.40) and (1.41) are referred to as retarded potentials. 3. The amplitude of A decreases as 1/R. K. El-Darymli Winter /71

179 New Material on Potentials 2. If we take (1.40) to the time domain, we would see that à at r is determined by the state of the source at time (R/c) seconds earlier. Of course, (R/c) is simply the time necessary for a phenomenon travelling at the speed of light to cover the distance R. For this reason the potentials in (1.40) and (1.41) are referred to as retarded potentials. 3. The amplitude of A decreases as 1/R. We have, then, the following procedure for nding the E and B elds due to a source: K. El-Darymli Winter /71

180 New Material on Potentials It may be noted that Jdv could be replaced by KdS or Idl ˆl for surface and line currents, respectively, and the triple integral in A would accordingly reduce to a double or single integral. K. El-Darymli Winter /71

181 New Material on Potentials It may be noted that Jdv could be replaced by KdS or Idl ˆl for surface and line currents, respectively, and the triple integral in A would accordingly reduce to a double or single integral. The important point is that A is derived from the current source, no matter what that source might be. K. El-Darymli Winter /71

182 New Material on Potentials It may be noted that Jdv could be replaced by KdS or Idl ˆl for surface and line currents, respectively, and the triple integral in A would accordingly reduce to a double or single integral. The important point is that A is derived from the current source, no matter what that source might be. We are nally in a position to begin our discussion of radiated energy due to time-varying sources. K. El-Darymli Winter /71

183 New Material on Potentials It may be noted that Jdv could be replaced by KdS or Idl ˆl for surface and line currents, respectively, and the triple integral in A would accordingly reduce to a double or single integral. The important point is that A is derived from the current source, no matter what that source might be. We are nally in a position to begin our discussion of radiated energy due to time-varying sources. In what follows, the sources are antenna currents. K. El-Darymli Winter /71

184 Antenna, a Circuit Point of View Outline 1 Introduction 2 Maxwell's Eqns & Related Formulae Maxwell's Equations Constitutive Relationships Miscellaneous Relations Some Propagation Parameters Time-Harmonic Fields Boundary Conditions 3 Scalar & Vector Potentials Review New Material on Potentials 4 Radiation Mechanism Antenna, a Circuit Point of View Two-Wire Antennas Helpful Animation and Applets 5 Ackgt K. El-Darymli Winter /71

185 Antenna, a Circuit Point of View From a circuit point of view, a transmitting antenna behaves like an equivalent impedance that dissipates the power transmitted. K. El-Darymli Winter /71

186 Antenna, a Circuit Point of View From a circuit point of view, a transmitting antenna behaves like an equivalent impedance that dissipates the power transmitted. The transmitter is equivalent to a generator. K. El-Darymli Winter /71

187 Antenna, a Circuit Point of View From a circuit point of view, a transmitting antenna behaves like an equivalent impedance that dissipates the power transmitted. The transmitter is equivalent to a generator. [Taken from: Ulaby, Fawwaz T., Eric Michielssen, and Umberto Ravaioli. "Fundamentals of Applied Electromagnetics, 6th edition, 2010, Boston, Massachussetts: Prentice Hall.] K. El-Darymli Winter /71

188 Antenna, a Circuit Point of View A receiving antenna behaves like a generator with an internal impedance corresponding to the antenna equivalent impedance. K. El-Darymli Winter /71

189 Antenna, a Circuit Point of View A receiving antenna behaves like a generator with an internal impedance corresponding to the antenna equivalent impedance. The receiver represents the load impedance that dissipates the time average power generated by the receiving antenna (taken from Ulaby). K. El-Darymli Winter /71

190 Two-Wire Antennas Outline 1 Introduction 2 Maxwell's Eqns & Related Formulae Maxwell's Equations Constitutive Relationships Miscellaneous Relations Some Propagation Parameters Time-Harmonic Fields Boundary Conditions 3 Scalar & Vector Potentials Review New Material on Potentials 4 Radiation Mechanism Antenna, a Circuit Point of View Two-Wire Antennas Helpful Animation and Applets 5 Ackgt K. El-Darymli Winter /71

191 Two-Wire Antennas Antennas are in general reciprocal devices, which can be used both as transmitting and as receiving elements. K. El-Darymli Winter /71

192 Two-Wire Antennas Antennas are in general reciprocal devices, which can be used both as transmitting and as receiving elements. The basic principle of operation of an antenna is easily understood starting from a two=wire transmission line, terminated by an open circuit (taken from Ulaby). K. El-Darymli Winter /71

193 Two-Wire Antennas Imagine to bend the end of the transmission line, forming a dipole antenna. K. El-Darymli Winter /71

194 Two-Wire Antennas Imagine to bend the end of the transmission line, forming a dipole antenna. Because of the change in geometry, there is now an abrupt change in the characteristic impedance at the transition point, where the current is still continuous. K. El-Darymli Winter /71

195 Two-Wire Antennas Imagine to bend the end of the transmission line, forming a dipole antenna. Because of the change in geometry, there is now an abrupt change in the characteristic impedance at the transition point, where the current is still continuous. The dipole leaks electromagnetic energy into the surrounding space, therefore it reects less power than the original open circuit the standing wave pattern on the transmission line is modied (taken from Ulaby). K. El-Darymli Winter /71

196 Two-Wire Antennas Current distribution on linear dipoles (taken from Balanis) K. El-Darymli Winter /71

197 Two-Wire Antennas Current variation as a function of time for λ/2 dipole (taken from Balanis) K. El-Darymli Winter /71

198 Two-Wire Antennas Source, transmission lines, antenna and detachment of electric eld lines: K. El-Darymli Winter /71

199 Two-Wire Antennas Source, transmission lines, antenna and detachment of electric eld lines: Antenna and electric eld lines (taken from Balanis) K. El-Darymli Winter /71

200 Two-Wire Antennas Antenna and free space wave (taken from Balanis) K. El-Darymli Winter /71

201 Two-Wire Antennas In the space surrounding the dipole we have an electric eld. At zero frequency (dc bias), xed electrostatic eld lines connect the metal elements of the antenna with circular symmetry. K. El-Darymli Winter /71

202 Two-Wire Antennas In the space surrounding the dipole we have an electric eld. At zero frequency (dc bias), xed electrostatic eld lines connect the metal elements of the antenna with circular symmetry. Antenna and electric eld lines (taken from Ulaby). K. El-Darymli Winter /71

203 Two-Wire Antennas At higher frequency, the current oscillates in the wires and the eld emanating from the dipole changes periodically. The eld lines propagate away from the dipole and form closed loops (taken from Ulaby). K. El-Darymli Winter /71

204 Two-Wire Antennas At higher frequency, the current oscillates in the wires and the eld emanating from the dipole changes periodically. The eld lines propagate away from the dipole and form closed loops (taken from Ulaby). K. El-Darymli Winter /71