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AEiE0602 Wave propagation and antenna

Displacement current and Maxwell's equations

  • Displacement current density in a dielectric is:
\[ J_d=\frac{\partial D}{\partial t}. \]
  • It completes Ampere's law for time-varying fields and allows electromagnetic wave propagation in vacuum.

Maxwell equations in point form:

\[ \nabla\cdot D=\rho_f \]
\[ \nabla\cdot B=0 \]
\[ \nabla\times E=-\frac{\partial B}{\partial t} \]
\[ \nabla\times H=J+\frac{\partial D}{\partial t} \]

Maxwell equations in integral form are:

\[ \oint_S D\cdot dS=Q_{\text{enclosed}} \]
\[ \oint_S B\cdot dS=0 \]
\[ \oint_C E\cdot dl=-\frac{d}{dt}\int_S B\cdot dS \]
\[ \oint_C H\cdot dl=\int_S J\cdot dS+ \frac{d}{dt}\int_S D\cdot dS. \]

The closed contour \(C\) bounds the oriented surface \(S\). The first two equations are closed-surface flux laws; the last two are circulation laws. Ampere-Maxwell includes both conduction current and displacement current.

Here \(\rho_f\) is free volume-charge density and \(J\) is free conduction-current density.

Plane-wave propagation

  • A uniform plane wave has field variation mainly along one direction and fields transverse to propagation.
  • In free space, \(E\), \(H\), and direction of propagation are mutually perpendicular.
  • Wave speed in free space:
\[ c=\frac{1}{\sqrt{\mu_0\epsilon_0}}\approx3\times10^8\ \text{m/s}. \]
  • Free-space intrinsic impedance:
\[ \eta_0=\sqrt{\frac{\mu_0}{\epsilon_0}}\approx377\ \Omega. \]

Known ambiguity control:

  • In free space, \(E/B=c\) because \(B=\mu_0H\).
  • In free space, \(E/H=\eta_0\approx377\ \Omega\).
  • Never interchange these two ratios.

Lossless dielectric, lossy dielectric, and good conductor

Medium Key trait Propagation cue
Lossless dielectric Conductivity \(\sigma=0\) or negligible No attenuation ideally
Lossy dielectric Finite \(\sigma\) Both attenuation and phase change
Good conductor \(\sigma\gg\omega\epsilon\) Strong attenuation, shallow skin depth

Attenuation and phase constants govern amplitude decay and phase advance.

For a good conductor,

\[ \delta=\sqrt{\frac{2}{\omega\mu\sigma}} =\frac{1}{\sqrt{\pi f\mu\sigma}}. \]

Increasing \(f\), \(\mu\), or \(\sigma\) decreases skin depth. Temperature can affect \(\delta\) indirectly by changing \(\sigma\), and sometimes \(\mu\); with material properties held constant, frequency is the direct variable in the formula.

Reflection at boundaries

  • Reflection occurs when wave impedance changes at a boundary.
  • At normal incidence, reflection coefficient for electric field is:
\[ \Gamma=\frac{\eta_2-\eta_1}{\eta_2+\eta_1}. \]
  • Transmission coefficient depends on both media impedances.
  • Oblique incidence introduces polarization distinctions such as TE and TM relative to the plane of incidence.

Perfect-conductor cue:

  • Tangential electric field at a perfect conductor surface is zero.
  • Strong reflection occurs from a perfect conductor.

Rectangular waveguide and modes

  • A waveguide confines electromagnetic waves.
  • Rectangular waveguide supports TE and TM modes, but not TEM mode in a hollow single-conductor guide.
  • TE means transverse electric: \(E_z=0\) along the guide axis.
  • TM means transverse magnetic: \(H_z=0\) along the guide axis.
  • Dominant mode in rectangular waveguide is TE\(_{10}\).

Antenna radiation and parameters

  • An antenna converts guided electromagnetic energy to radiated energy and vice versa.
  • Radiation pattern shows directional distribution of radiated power.
  • Gain combines directivity with efficiency.
  • Directivity measures concentration of radiation relative to isotropic radiation.
  • Bandwidth is the useful frequency range meeting performance limits.
  • Polarization is the orientation of the electric field of the radiated wave.

Antenna classes in the syllabus

Type Recognition cue
Isotropic antenna Ideal point source radiating equally in all directions
Omni-directional antenna Uniform or near-uniform in one plane, not all 3D directions
Dipole Basic resonant wire antenna, often half-wave dipole
Directional antenna Concentrates radiation in preferred directions
Travelling-wave antenna Current wave travels along structure with reduced standing-wave behavior

Trap:

  • Isotropic antenna is ideal and theoretical.
  • Omni-directional does not mean equal radiation in every 3D direction.

Source uncertainty: the syllabus phrase “Antenna Radiation, Theorem and Parameters” does not name the theorem. If it intends reciprocity, the relevant rule is that an antenna in a linear reciprocal medium has corresponding transmit and receive characteristics; do not treat that identification as confirmed until the official detailed source is found.

Wave-and-antenna examples

  1. If a stem asks the dominant mode of rectangular waveguide, choose TE\(_{10}\).
  2. If the ratio is \(E/H\) in free space, use about \(377\ \Omega\).
  3. If the ratio is \(E/B\), use \(c\).

AEiE0602 revision box

  • Maxwell completion: displacement current matters for time-varying fields.
  • \(c\approx3\times10^8\) m/s, \(\eta_0\approx377\ \Omega\).
  • Free-space ratios: \(E/B=c\), \(E/H=\eta_0\).
  • Rectangular waveguide dominant mode: TE\(_{10}\).
  • Isotropic is ideal; omni-directional is practical and plane-uniform.