UNIT 3: Antennas and Wave Propagation – Comprehensive Short Notes

I. Fundamental Radiation Theory

Retarded Potentials

• Concept: Solutions to Maxwell's equations for time-varying sources that account for the finite propagation speed \( c \) of EM waves. Unlike static potentials (instantaneous action), retarded potentials depend on source conditions at an earlier "retarded time" \( t_r = t - \frac{R}{c} \), where \( R \) is distance from source to observation point.

• Derivation (Sinusoidal Sources): For a current density \( \mathbf{J}(\mathbf{r}', t) = \mathbf{J}(\mathbf{r}') e^{j\omega t} \), the vector potential \( \mathbf{A} \) is:

$$ \mathbf{A}(\mathbf{r}, t) = \frac{\mu}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}') e^{j\omega (t - R/c)}}{R} dV' $$

The \( e^{-j\omega R/c} \) term is the **retardation factor**.

• Significance: Fundamental to solving radiation problems. It allows calculation of fields at a distance by integrating contributions from all source points, each delayed by the travel time. It is the starting point for deriving fields of all antennas.

[!TIP] Exam Focus: Be prepared to state the retarded potential equation and explain why the retardation factor \( e^{-j\omega R/c} \) is physically essential for radiation.

Elementary Radiators: Hertzian Dipole

• Definition: An infinitesimal dipole (length \( \Delta l \ll \lambda \)) carrying a uniform, sinusoidal current \( I = I_0 e^{j\omega t} \). Also called an alternating current element.

• Radiated Fields (in spherical coordinates, \( \theta \) from z-axis):

$$ E_\theta = j\frac{I_0 \Delta l \eta \beta^2}{4\pi r} \sin\theta \left(1 + \frac{1}{j\beta r}\right) e^{-j\beta r} $$

$$ H_\phi = j\frac{I_0 \Delta l \beta^2}{4\pi r} \sin\theta \left(1 + \frac{1}{j\beta r}\right) e^{-j\beta r} $$

$$ E_r = \frac{I_0 \Delta l \eta \beta^2}{2\pi r^2} \cos\theta \left(1 + \frac{1}{j\beta r}\right) e^{-j\beta r} $$

where \( \beta = 2\pi/\lambda \), \( \eta = \sqrt{\mu/\varepsilon} \).

• Field Regions:

  • Near-field (Reactive): \( r \ll \lambda \). \( E_\theta \propto 1/r^2 \), \( E_r \propto 1/r^3 \). Fields are predominantly reactive (stored energy), 90° out of phase.

  • Far-field (Radiation): \( r \gg \lambda \). \( 1/r \) terms dominate. \( E_\theta \) and \( H_\phi \) are in phase, transverse (TEM), and \( |E|/|H| = \eta \). This is the radiation field.

• Radiated Power & Radiation Resistance:

$$ P_{rad} = \frac{I_0^2 (\Delta l)^2 \eta \beta^4}{12\pi} = \frac{I_0^2 (\Delta l)^2 \eta}{3\lambda^2} $$

Radiation Resistance: \( R_r = \frac{2P_{rad}}{I_0^2} = \frac{80\pi^2 (\Delta l / \lambda)^2}{\text{}} \approx 197 \left(\frac{\Delta l}{\lambda}\right)^2 \, \Omega \).

[!TIP] Common Pitfall: Forgetting that the Hertzian dipole's radiation resistance is extremely small (e.g., ~0.08 Ω for \( \Delta l = \lambda/100 \)), requiring impedance matching for efficient transmission.

Field Regions & Fraunhofer Distance

• Near-field (Fresnel) Region: \( 0 < r < \frac{2D^2}{\lambda} \). Field structure is complex; reactive fields dominate. Angular field distribution is dependent on distance \( r \).

• Far-field (Fraunhofer) Region: \( r > \frac{2D^2}{\lambda} \). \( D \) is largest antenna dimension.

  • Fields are transverse (\( E_\theta, H_\phi \)).

  • Wavefronts are spherical but locally planar.

  • Angular field distribution is independent of \( r \).

  • \( |E| \propto 1/r \), \( |H| \propto 1/r \).

  • Importance: This is the region where antenna patterns are measured and used. The far-field approximation simplifies analysis by neglecting \( 1/r^2 \) and \( 1/r^3 \) terms.

Reciprocity Theorem

• Statement: If a current \( I_1 \) at terminal 1 of Antenna A produces a voltage \( V_2 \) at terminal 2 of Antenna B, then the same current \( I_1 \) at terminal 2 of Antenna B will produce the same voltage \( V_2 \) at terminal 1 of Antenna A.

$$ \frac{V_2}{I_1} \bigg|_{\text{A→B}} = \frac{V_1}{I_2} \bigg|_{\text{B→A}} $$

• Proof Basis: Based on Lorentz reciprocity theorem for EM fields, assuming linear, isotropic media and no time-varying materials.

• Significance for Antennas:

  1. Transmission and Reception Patterns are Identical: The radiation pattern of an antenna when transmitting is the same as its receiving pattern.

  2. Mutual Impedance Symmetry: \( Z_{12} = Z_{21} \) for two-port antenna networks.

  3. Justifies using the same antenna for transmitting and receiving measurements.

II. Antenna Parameters and Characteristics

Radiation Patterns

• 3D Pattern: Complete spatial distribution of radiated power.

• Principal Plane Patterns: Cuts through the 3D pattern in the E-plane (contains electric field vector & max radiation) and H-plane (contains magnetic field vector & max radiation, orthogonal to E-plane).

• Polar Plot: Common 2D representation (r = function of \( \theta \) or \( \phi \)).

• Beamwidth:

  • Half-Power Beamwidth (HPBW): Angle between half-power (-3 dB) points on main lobe.

  • First-Null Beamwidth (FNBW): Angle between first nulls on either side of main lobe.

  • Relation to Directivity: For a uniformly illuminated rectangular aperture, \( D \approx \frac{41253}{\text{HPBW}_\text{E} \times \text{HPBW}_\text{H}} \) (in deg²). Higher directivity → narrower beamwidth.

$$ P_r = P_t G_t(\theta_t,\phi_t) G_r(\theta_r,\phi_r) \left( \frac{\lambda}{4\pi R} \right)^2 $$

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Efficiency & Effective Length

• Radiation Efficiency (\( \eta_r \)): \( \eta_r = \frac{P_{rad}}{P_{in}} = \frac{R_r}{R_r + R_{loss}} \). \( R_{loss} \) includes ohmic, dielectric, mismatch losses.

• Mismatch Efficiency (\( \eta_m \)): \( \eta_m = 1 - |\Gamma|^2 \), where \( \Gamma = \frac{Z_{in} - Z_0^*}{Z_{in} + Z_0} \). Overall \( \eta = \eta_r \eta_m \).

• Effective Length (\( h_e \)): For a receiving antenna, relates open-circuit voltage \( V_{oc} \) to incident field \( E_{inc} \): \( V_{oc} = E_{inc} h_e \). For a Hertzian dipole, \( h_e = \frac{\Delta l}{2} \). For a half-wave dipole, \( h_e \approx \frac{\lambda}{\pi} \).

Polarization

• Linear: \( E \)-vector oscillates along a line.

• Circular: \( E \)-vector rotates with constant magnitude. Right-Hand (RHC) or Left-Hand (LHC).

• Elliptical: General case. Axial Ratio (AR) = Major axis / Minor axis. AR = 1 for circular, ∞ for linear.

• Polarization Mismatch Loss: \( \eta_{pol} = |\hat{\rho}_t \cdot \hat{\rho}_r^*|^2 \), where \( \hat{\rho} \) is polarization unit vector.

Effect of Ground on Antenna Patterns

• Image Theory: Perfectly conducting ground (\( \sigma \to \infty \)) replaced by image of antenna with opposite current phase (for vertical dipole) or same phase (for horizontal dipole).

• Impact on Vertical Patterns:

  • Vertical Dipole over Perfect Ground: Pattern is doughnut-shaped in upper hemisphere only (image cancels below ground). Directivity doubles (from 1.5 to 3 dBi).

  • Horizontal Dipole over Perfect Ground: Main lobe tilts upward; pattern becomes asymmetric.

• Real Ground (Finite \( \sigma, \varepsilon \)):

  • Reflection coefficient \( \Gamma \) at air-ground interface is complex and frequency-dependent.

  • Conductivity (\( \sigma \)): Higher \( \sigma \) → better conductor → closer to perfect ground behavior.

  • Permittivity (\( \varepsilon_r \)): Affects reflection phase.

  • Roughness: Scatters energy, reduces coherence, degrades pattern.

  • Result: Pattern tilts, gain decreases, and side lobes may appear due to imperfect cancellation.

III. Fundamental Antenna Types

Dipole Antennas

• Half-Wave Dipole (\( l = \lambda/2 \)):

  • Current Distribution: \( I(z) = I_0 \cos(\beta z) \), \( |z| \leq l/2 \).

  • Radiation Pattern: Similar to Hertzian dipole but more directional. \( E_\theta \propto \frac{\cos(\frac{\pi}{2} \cos\theta)}{\sin\theta} \). First nulls at \( \theta = \arccos(2/\pi) \approx 50.3^\circ \). HPBW ≈ 78°.

  • Input Impedance (at center): \( Z_{in} \approx 73 + j42.5 \, \Omega \). Radiation resistance \( R_r \approx 73 \, \Omega \).

• Quarter-Wave Monopole (\( l = \lambda/4 \)) over perfect ground plane:

  • Operation: Image theory makes it equivalent to half-wave dipole in free space. Current distribution same as half-wave dipole's upper half.

  • Pattern: Hemispherical (above ground only). Directivity = 2 × D_{half-dipole} ≈ 5.15 dBi (vs. isotropic).

  • Input Impedance: \( Z_{in} \approx \frac{1}{2} Z_{in,\lambda/2} \approx 36.5 + j21.25 \, \Omega \).

  • Applications: Mobile antennas, vehicle-mounted, mast radiators.

• Short Dipole (\( l \ll \lambda \)): Behaves like Hertzian dipole. Very low radiation resistance (\( \propto (l/\lambda)^2 \)), high capacitive reactance. Used in RFID, receiving loops.

Folded Dipole

• Construction: Two parallel dipoles (often \( \lambda/2 \)) connected at ends, fed at the center of one.

• Current Distribution: Equal magnitude currents in both arms, same direction.

• Impedance Transformation: For symmetric folded dipole (two identical wires), \( Z_{in} \approx 4 \times Z_{in,\text{single}} \). For half-wave single dipole \( Z_{in} \approx 73 \, \Omega \), folded dipole \( Z_{in} \approx 292 \, \Omega \). Useful for matching to 300 Ω twin-lead.

• Bandwidth: Slightly wider than simple dipole.

Turnstile Antenna

• Geometry: Two half-wave dipoles mounted perpendicularly, fed with 90° phase difference (one 0°, other 90°).

• Operation: Generates circular polarization (RHC or LHC depending on phase sequence) in directions normal to the plane of the dipoles.

• Pattern: Omnidirectional in the plane perpendicular to the dipole axes (like a single dipole), but circularly polarized.

• Applications: Satellite communications (e.g., GPS, satellite TV), where signal polarization may rotate.

IV. Antenna Arrays

Pattern Multiplication Principle

• Statement: The total radiation pattern \( F(\theta,\phi) \) of an array of identical elements is the product of the element pattern \( f_e(\theta,\phi) \) and the array factor \( AF(\theta,\phi) \):

$$ F(\theta,\phi) = f_e(\theta,\phi) \cdot AF(\theta,\phi) $$

• Application: Simplifies analysis. Element pattern accounts for single-element directivity; array factor accounts for geometry and excitation.

Linear Arrays

• Array Factor (AF) for \( N \) isotropic elements along z-axis, spacing \( d \), progressive phase shift \( \alpha \):

$$ AF(\theta) = \sum_{n=0}^{N-1} I_n e^{j(n\psi + \alpha)} \quad \text{where} \quad \psi = \beta d \cos\theta $$

For **uniform amplitude** (\( I_n = I_0 \)) and **phase**:

$$ AF(\theta) = I_0 \frac{\sin\left(\frac{N\psi}{2}\right)}{\sin\left(\frac{\psi}{2}\right)} e^{j\left(\frac{(N-1)\psi}{2} + \alpha\right)} $$

• Schelkunoff Unit Circle Method: Graphical method to find array factor zeros by plotting \( z = e^{j\psi} \) on unit circle. The polynomial \( \sum I_n z^n = 0 \) gives \( \psi \) values for nulls.

• Beam Steering (Scanning): To scan main lobe to angle \( \theta_0 \), set progressive phase shift:

$$ \alpha = -\beta d \cos\theta_0 $$

This compensates the natural phase progression.

Array Synthesis

• Binomial Array: Amplitudes follow binomial coefficients (\( I_n \propto \binom{N-1}{n} \)). No side lobes (theoretical), but wide main lobe and high taper efficiency loss.

• Dolph-Chebyshev Array: Uses Chebyshev polynomials to achieve equi-ripple side lobes at a specified level. Minimizes main lobe width for a given side lobe level. Optimal for low side lobes.

• Taylor Distribution: For continuous apertures, provides a compromise between Chebyshev (low side lobes) and uniform (narrow main lobe). Uses a modified sinc function with adjustable parameter \( \bar{n} \) controlling side lobe decay.

Parasitic Arrays: Yagi-Uda Antenna

• Construction:

  • Driven Element: Typically a half-wave dipole.

  • Reflector: One element, slightly longer (\( \approx 5\% \)) than driven, placed behind it (\( \approx 0.15\lambda - 0.25\lambda \)).

  • Directors: One or more elements, slightly shorter (\( \approx 5\% \)) than driven, placed in front (\( \approx 0.1\lambda - 0.2\lambda \) spacing).

• Operation Principle:

  • Reflector: Inductive reactance, reflects energy forward.

  • Directors: Capacitive reactance, phases the wavefront from driven element to reinforce radiation in forward direction (like a lens).

• Result: High forward gain (typically 7-15 dBi), narrow beamwidth, high front-to-back ratio.

• Applications: TV reception, point-to-point links, amateur radio.

Planar Arrays

• Configuration: Elements arranged on a 2D grid (rectangular, circular). Separately controlled amplitude/phase in x and y directions.

• Pattern: Product of two linear array factors: \( AF_{total} = AF_x(\theta,\phi) \cdot AF_y(\theta,\phi) \).

• Horizontal Patterns in Broadcast: For AM/FM broadcast towers, planar arrays are used to shape the horizontal (azimuth) pattern.

  • Purpose: To concentrate signal in the desired service area (city) and minimize interference in other directions (as per FCC/ITU regulations).

  • Control: Achieved by varying amplitudes and phases of tower elements in the circular array.

V. Broadband and Traveling Wave Antennas

Log-Periodic Antenna (LPA)

• Structure: Tooth-comb of dipoles of varying lengths, connected alternately to two parallel boom lines. Shortest dipole at the apex (feed point), longest at the end.

• Frequency-Independent Principle: The antenna's geometry is self-similar. The electrical length of each element is a function of the log-periodic ratio \( \tau = \frac{l_{n+1}}{l_n} = \frac{d_{n+1}}{d_n} \). The input impedance and pattern repeat every frequency decade. Active region (where element is resonant) moves along the boom as frequency changes.

• Radiation Mechanism: Only ~2-3 adjacent elements are active at any frequency. The active region acts like a phased array.

• Advantages: Very wide bandwidth (10:1 to 20:1), constant gain and SWR over band.

• Applications: TV reception (wideband), EMC testing, wideband communications.

Long Wire Antenna (Traveling Wave)

• Construction: A single wire, length \( l \gg \lambda \), terminated in a resistive load \( R_T \approx Z_0 \) (characteristic impedance of wire, ~400-600 Ω) to minimize reflections.

• Current Distribution: Traveling wave (non-standing wave): \( I(z) = I_0 e^{-j\beta z} \), \( 0 \leq z \leq l \). Amplitude decays slowly if well-terminated.

• Radiation Pattern:

  • Broadside (perpendicular to wire): Maximum.

  • End-fire (along wire): Minimum.

  • Pattern resembles a single lobe in the forward direction. Beamwidth narrows as \( l/\lambda \) increases.

  • Gain: Increases with length, up to ~10-12 dBi for \( l \sim 10\lambda \).

• Design Considerations: Termination critical to avoid standing waves (which create bidirectional pattern). Height above ground affects pattern.

Rhombic Antenna

• Construction: Four equal-length wires arranged in a rhombus (diamond) shape, supported by tall masts. One pair of opposite corners are feed point and termination (with \( R_T \approx Z_0 \)).

• Operation: A traveling wave antenna. Wave travels along one side, reflects at termination, travels back along opposite side. With proper termination, net pattern is a single, strong forward lobe.

• Advantages: High gain (10-15 dBi), wide bandwidth, unidirectional, robust.

• Disadvantages: Large land area, requires large masts, fixed direction.

• Applications: HF (3-30 MHz) point-to-point communications, shortwave broadcasting.

V-Antenna

• Construction: Two long wires (each \( \approx \lambda/2 \) to \( 2\lambda \)) joined at an apex with a feed line, forming a V-shape with angle \( 2\alpha \).

• Directional Properties: Main lobe is along the bisector of the V. The V-angle controls beamwidth: smaller angle → narrower beam, higher gain.

• Applications: Directional receiving (e.g., for radio direction finding), portable HF antennas.

VI. Aperture and Reflector Antennas

Horn Antennas

• Types:

  • Pyramidal: Rectangular cross-section, flare in E- and H-planes.

  • Conical: Circular cross-section, axisymmetric.

  • Corrugated: Conical horn with circumferential grooves. Provides nearly uniform phase, low cross-polarization, wide bandwidth.

• Gain & Aperture Efficiency:

$$ G = \frac{4\pi A_e}{\lambda^2} = \eta_a \frac{4\pi A_p}{\lambda^2} $$

where \( A_p \) is **physical aperture area**, \( \eta_a \) is **aperture efficiency** (typically 0.5-0.7 for simple horns, >0.9 for corrugated).

• Gain Measurement (Gain Comparison Method): Connect two identical horns (transmit/receive) at known distance \( R \). Measure received power \( P_r \). Gain of AUT:

$$ G_{AUT} = G_{ref} + 10\log\left(\frac{P_r}{P_{r,ref}}\right) + 20\log\left(\frac{4\pi R}{\lambda}\right) $$

Parabolic Reflector Antennas

• Geometry & Focusing:

  • Paraboloid of Revolution: \( z = \frac{r^2}{4f} \), where \( f \) is focal length.

  • Prime Focus: Feed at focus \( F \). Simple but feed blockage.

  • Cassegrain: Subreflector (hyperbolic) at focus reflects to feed at vertex. No feed blockage, shorter focal length, higher gain.

• Aperture Blockage: The feed, support struts, and subreflector obscure part of the main aperture.

  • Effects: Reduces effective area, increases side lobes, distorts phase front.

  • Mitigation: Use offset-fed reflectors (no blockage), minimize strut thickness.

• Feed Illumination & Taper: Feed pattern should uniformly illuminate the aperture. Taper (decrease toward edges) reduces side lobes but lowers aperture efficiency. Trade-off: -10 dB edge taper is common.

Slot Antennas & Babinet's Principle

• Babinet's Principle: The fields from a slot in an infinite, perfectly conducting plane are related to the fields from its complementary strip (thin conductor) in an infinite plane, with the same excitation.

  • \( E_{slot} \leftrightarrow H_{strip} \)

  • \( H_{slot} \leftrightarrow -E_{strip} \)

  • \( Z_{slot} = \frac{\eta^2}{4 Z_{strip}} \)

• Rectangular Slot in Infinite Ground Plane:

  • Dimensions: Length \( L \approx \lambda/2 \), width \( W \ll \lambda \).

  • Excitation: Often fed by a coaxial probe or microstrip line underneath.

  • Radiation Pattern: Similar to a dipole but with E- and H-planes swapped. For a \( \lambda/2 \) slot, the E-plane (plane of slot) has a broadside pattern \( \propto \cos(\frac{\pi}{2}\cos\phi)/\sin\phi \), H-plane is omnidirectional.

  • Polarization: Linear, with \( E \)-vector parallel to the slot's long dimension.

• Applications: Array elements (slotted waveguide arrays), aircraft antennas (flush-mounted), microwave circuits.

Lens Antennas

• Types:

  • Dielectric Lens: Made of low-loss, high-\( \varepsilon_r \) material (e.g., polystyrene). Slows waves passing through lens, causing phase delay → focuses like optical lens.

  • Metasurface (Artificial) Lens: Thin, uses sub-wavelength structures to impart phase shift. Lightweight, conformal.

• Focusing Action: Corrects spherical wavefront from feed to plane wave at aperture. Can also produce beam scanning by moving feed or using electronic lens.

Flat Sheet & Corner Reflectors

• Flat Sheet Reflector: A flat, conducting plate placed behind a dipole (at \( \lambda/4 \) distance).

  • Pattern: Bidirectional (front and back), but front gain increased by ~3 dB (like monopole over ground).

  • Use: Simple gain enhancement, corner reflectors are more common.

• Corner Reflector:

  • Construction: Two flat plates meeting at an angle \( \alpha \) (typically 90°), with dipole at apex.

  • Operation: Multiple images create a highly directional pattern in the bisecting plane.

  • Gain: \( G \approx 8\pi \frac{A_p}{\lambda^2} \sin\alpha \) (for \( \alpha = 90^\circ \), \( G \approx 10-12 \, \text{dBi} \) for reasonable size).

  • Applications: TV antennas, radar reflectors, low-gain satellite antennas.

VII. Microstrip and Printed Antennas

Construction & Operation

• Basic Structure: Patch (conducting foil, usually copper) on a dielectric substrate (\( \varepsilon_r, h \)) backed by a ground plane.

• Resonant Mechanism: The patch acts as a resonant cavity with magnetic walls on sides. Dominant mode is TM₀₀ (or \( \text{TM}_{1,0} \) for rectangular patch). Fields vary sinusoidally in \( x \) (length), constant in \( y \) (width).

• Fringing Fields: Due to thin substrate (\( h \ll \lambda \)), fields fringe at edges, making the patch electrically longer than physical length \( L \).

Design Equations (Rectangular Patch, \( \text{TM}_{10} \) Mode)

• Width \( W \) (for \( \varepsilon_r \) effective):

$$ W = \frac{c}{2f_r\sqrt{\frac{\varepsilon_r+1}{2}}} \quad (\text{for } W \text{ not too narrow}) $$

• Effective Dielectric Constant:

$$ \varepsilon_{re} = \frac{\varepsilon_r+1}{2} + \frac{\varepsilon_r-1}{2}\left(1 + 12\frac{h}{W}\right)^{-1/2} $$

• Length \( L \) (including fringing extension \( \Delta L \)):

$$ \Delta L = 0.412h \frac{(\varepsilon_{re}+0.3)(W/h+0.264)}{(\varepsilon_{re}-0.258)(W/h+0.8)} $$

$$ L = \frac{c}{2f_r\sqrt{\varepsilon_{re}}} - 2\Delta L $$

• Example (RT/Duroid 6010.2): \( \varepsilon_r = 10.2 \), \( h = 0.127 \, \text{cm} \), \( f_r = 2 \, \text{GHz} \).

  • \( W \approx \frac{3\times10^{10}}{2\times2\times10^9}\sqrt{\frac{10.2+1}{2}} \approx 1.19 \, \text{cm} \)

  • \( \varepsilon_{re} \approx 6.8 \), \( \Delta L \approx 0.15 \, \text{cm} \)

  • \( L \approx \frac{3\times10^{10}}{2\times2\times10^9\sqrt{6.8}} - 2\times0.15 \approx 0.91 \, \text{cm} \)

Feeding Techniques

• Microstrip Line: Simple, but introduces spurious radiation and impedance mismatch if not matched.

• Coaxial Probe: Easy, good for thick substrates. Probe inductance limits bandwidth.

• Aperture Coupling: Feed line on opposite side of ground plane, coupled through a slot. Isolates feed and patch, reduces spurious radiation.

• Proximity Coupling: Feed line placed under patch, separated by substrate. Low loss, wide bandwidth.

• Other: Stripline feed, etc.

Advantages & Limitations

VIII. Helical Antennas

Normal Mode

• Operation: Circumference \( C \approx \lambda/10 \), spacing \( S \approx 0.1\lambda \). Current is mostly axial (along helix axis), with small helical component.

• Radiation Pattern: Omnidirectional in plane perpendicular to axis (like a dipole). Linear polarization (E-vector parallel to axis).

• Input Impedance: Purely resistive, ~140 Ω.

• Applications: HF communications, where broadband, simple, vertically polarized antenna needed.

Axial Mode

• Operation: \( C \approx \lambda \), \( S \approx 0.25\lambda \) (pitch angle \( \alpha \approx 12^\circ-14^\circ \)). Current has large helical component.

• Radiation Mechanism: Helix acts as a slow-wave structure. Radiation is maximum in axial direction (end-fire).

• Polarization: Circular (RHC or LHC depending on helix handedness and current direction).

• Gain: \( G \approx 15 \left( \frac{C}{\lambda} \right)^2 n_s \) (for \( n_s \) turns). Can be very high.

• Design Parameters:

  • Circumference: \( C \approx \lambda \) (for best axial mode).

  • Spacing: \( S \approx \lambda/4 \).

  • Pitch angle: \( \alpha = \tan^{-1}(S/(\pi C)) \approx 13^\circ \).

• Applications: Satellite communications (e.g., GPS, satellite TV), where circular polarization is essential to mitigate Faraday rotation.

IX. Radio Wave Propagation

Ground Wave Propagation

• Components:

  • Surface Wave: Guided by Earth's surface, follows curvature. Dominant at LF/MF.

  • Space Wave: Direct + ground-reflected waves. Dominant at VHF/UHF.

• Transition: At higher frequencies, surface wave attenuates rapidly; propagation becomes line-of-sight (LOS) via space wave.

• Effect of Terrain:

  • Conductivity (\( \sigma \)): Higher \( \sigma \) (sea water) → lower attenuation.

  • Permittivity (\( \varepsilon_r \)): Affects reflection coefficient.

  • Roughness: Causes scattering, increases attenuation.

  • Result: Sea water best for ground wave; desert worst; hills/forests cause diffraction loss.

Sky Wave Propagation (Ionospheric)

• Ionospheric Layers (daytime): D (60-90 km, absorbs LF), E (90-140 km), F (150-400 km, splits into F1/F2). At night, D/E vanish, F merges into F-layer.

• Electron Density Profile: \( N_e(h) \) increases with height up to a peak (F2 max), then decreases. Determines critical frequency \( f_c \) for each layer: \( f_c = 9\sqrt{N_{e,\max}} \) (Hz, \( N_e \) in m⁻³).

• Refractive Index \( n \): \( n = \sqrt{1 - \frac{81 N_e}{f^2}} \). Decreases with height (gradient \( dn/dh < 0 \)), causing bending of rays.

• Virtual Height \( h' \): Apparent reflection height if ray path were straight. \( h' > h \) (true height).

• Critical Frequency \( f_c \): Highest frequency that will be reflected back vertically (\( \theta_i = 0^\circ \)).

• Maximum Usable Frequency (MUF): Highest frequency that can be used for a given path (via ionosphere) for a specific angle of incidence \( \theta_i \).

$$ \text{MUF} = \frac{f_c}{\cos\theta_i} = f_c \sec\theta_i $$

*   **Relation to Skip Distance** \( d \):

$$ d = 2 R_E \left( \sqrt{\left(\frac{\text{MUF}}{f_c}\right)^2 - 1} \right) \quad (\text{for flat Earth approx}) $$

    where \( R_E \) is Earth radius. **Higher MUF → longer skip distance** for same \( f_c \).

• Lowest Usable Frequency (LUF): Lowest frequency that provides acceptable signal strength. Limited by ionospheric absorption (D-layer) and noise.

• Skip Zone: Distance between end of ground wave and beginning of first sky wave return. No signal.

• Total Internal Reflection: When ray from denser (lower) to rarer (higher) ionosphere hits at angle > critical angle, it reflects back.

• Mode Conversion: Ordinary (O) and extraordinary (X) rays due to Earth's magnetic field.

Space Wave Propagation (Tropospheric)

• Components:

  • Direct Wave: LOS.

  • Ground-Reflected Wave: From Earth's surface.

  • Result: Two-ray model with possible constructive/destructive interference.

• Super Refraction:

  • Cause: Negative refractive index gradient (\( dn/dh < 0 \)) steeper than standard (\( -4 \times 10^{-5} \, \text{m}^{-1} \)). Due to temperature inversion or dry air over moist.

  • Effect: Rays bend more than Earth's curvature → radio horizon extended. Signals can be received beyond LOS. Can cause ducting if gradient very steep.

• Tropospheric Scattering:

  • Mechanism: Turbulence in troposphere causes small-scale irregularities in \( N \) (refractivity). Scatters energy forward and sideways.

  • Result: Beyond-horizon propagation (up to ~500 km). Wideband, but high path loss (~100 dB). Used for microwave links where LOS not possible.

• Tropospheric Ducting:

  • Formation: Very strong super refraction (\( dn/dh \ll 0 \)) creates a duct (waveguide) where rays are trapped and guided.

  • Conditions: Common over oceans, in temperature inversions.

  • Effect: VHF/UHF/microwave signals can propagate thousands of km with low loss. Causes unexpected long-distance TV/FM reception.

Environmental Effects

X. Advanced Topics and Design Tools

Tapered Apertures

• Concept: Instead of uniform amplitude across aperture (\( A(\rho) = \text{const} \)), apply a taper (e.g., Taylor, cosine) that reduces amplitude toward edges.

• Advantages:

  • Reduces side lobe level significantly.

  • Improves beam shape (more Gaussian).

• Disadvantages:

  • Increases main lobe beamwidth (wider beam).

  • Reduces aperture efficiency (less total power).

• Application: Used in reflector antennas (illumination taper from feed), aperture antennas, and array synthesis (Taylor distribution).

Numerical Methods for Antenna Analysis

• Method of Moments (MoM): Solves integral equations (e.g., electric field integral equation - EFIE). Best for wire antennas, surfaces. Basis functions on conductor. Computationally efficient for moderate size.

• Finite Element Method (FEM): Solves differential equations (Helmholtz) by meshing volume. Handles complex materials (dielectrics, inhomogeneous). Good for microstrip, dielectric antennas. Requires truncation boundary (ABC/PML).

• Finite Difference Time Domain (FDTD): Time-domain, grids entire volume. Simple, explicit. Good for wideband, transient problems, complex geometries. Requires fine mesh (\( \lambda/10 \)).

• Role: Design optimization, prototyping, analyzing non-standard geometries, predicting performance where analytical solutions fail. Software: HFSS (FEM), CST (FDTD), FEKO (MoM/MLFMM).

Antenna Feeding Structures & Impedance Matching

• Feeding Types:

  • Balanced: Two terminals with equal/opposite currents (e.g., dipole). Requires balun (balanced-to-unbalanced transformer) when using coaxial cable.

  • Unbalanced: One terminal connected to ground (e.g., monopole, patch with ground plane). Compatible with coaxial.

• Common Feeds: Coaxial probe, microstrip line, waveguide probe, two-wire transmission line, gamma match.

• Matching Techniques:

  • Baluns: For dipole/arrays (e.g., choke balun, transmission-line balun).

  • Stubs: Series/shunt quarter-wave transformer, single-stub matching.

  • Lumped Elements: L-network, π-network.

  • Tapered Lines: Exponential, Klopfenstein taper for broadband matching.

Special Concepts

• Virtual Height \( h' \): In ionospheric propagation, the apparent height from which a wave appears to be reflected if path were straight. \( h' = h + \Delta h \), where \( \Delta h \) accounts for bending. Determined from ionograms (height vs. frequency).

• Radius of Curvature of Ray Path \( r_c \): In a medium with refractive index \( n(h) \) varying with height, the ray bends with radius:

$$ r_c = \frac{n}{\left| \frac{dn}{dh} \right|} \quad (\text{for small angles}) $$

For standard atmosphere (\( dn/dh \approx -4 \times 10^{-5} \, \text{m}^{-1} \)), \( r_c \approx 4R_E/3 \).

• Oscillating Electric Dipole: The fundamental source of radiation. Hertzian dipole is its idealized form. Its fields (derived from retarded potentials) form the building block for all antenna analysis via superposition.