UNIT 5: ANTENNAS AND WAVE PROPAGATION - EXAM-FOCUSED SHORT NOTES

Based on rigorous analysis of RGPV past papers (2022-2025). Tier 1 topics (appearing in 4-5 papers) are marked with [TIER 1].

I. FUNDAMENTALS OF ANTENNA RADIATION

1.1 Retarded Potential & Radiation from Current Elements [TIER 1]

• Concept: Accounts for finite propagation speed c. The potential at a point (r, θ, φ) at time t depends on the source's state at an earlier retarded time t_r = t - r/c.

• Liénard-Wiechert Potentials: For a point charge/current element.

  • Scalar potential: V = (q / (4πε₀)) * [1 / (r - (v·r̂)/c)]_ret

  • Vector potential: A = (μ₀qv / (4π)) * [1 / (r - (v·r̂)/c)]_ret

• Hertzian Dipole (Oscillating Electric Dipole): Infinitesimal dipole dl << λ, carrying current I = I₀ e^(jωt).

  • Radiated Fields (Far-field, θ-component only):

$$E_θ = \frac{jωμ₀ I_0 dl}{4πr} \sinθ \, e^{-jkr}$$

$$H_φ = \frac{j k I_0 dl}{4πr} \sinθ \, e^{-jkr} = \frac{E_θ}{η}$$

*   `k = 2π/λ`, `η = √(μ₀/ε₀) ≈ 120π Ω`.

• Power Radiated & Radiation Resistance (R_rad):

$$P_{rad} = \int_0^{2π} \int_0^π \frac{|E_θ|^2}{η} r^2 \sinθ \, dθ dφ = \frac{ω^4 μ₀ |I_0 dl|^2}{12πc} = \frac{80π^2 (I_0 dl / λ)^2}{3}$$

> **Radiation Resistance:** `R_rad = 2P_rad / |I_0|^2 = 80π² (dl/λ)²` (for center-fed dipole, `dl` is effective length).

\boxed{R_{rad} \approx 80\pi^2 \left(\frac{dl}{\lambda}\right)^2 \text{ for a short dipole}}

1.2 Antenna Field Regions & Fundamental Parameters [TIER 1]

• Field Regions:

  | Region | Distance from Antenna | Field Characteristics | | :--- | :--- | :--- | | Reactive Near-field | r < 0.62√(l³/λ) | E & H ~ 1/r³, reactive dominance, phase not constant. | | Fresnel (Near-field) | 0.62√(l³/λ) < r < 2l²/λ | E & H ~ 1/r², field structure complex, phase varies. | | Fraunhofer (Far-field) | r > 2l²/λ (and r >> λ) | Plane waves, E⊥H⊥r̂, E/H = η, pattern independent of r. |

  Far-field Condition: r >> max(λ, l²/λ). All standard pattern measurements are in far-field.

• Key Parameters:

  • Radiation Pattern: 3D/2D plot of |F(θ, φ)|² (field/ power). F(θ, φ) is pattern function (normalized).

  • Directivity (D): Ratio of max radiation intensity to average.

$$D = \frac{4π U_{max}}{P_{rad}} = \frac{4π}{\int_0^{2π}\int_0^π |F(θ, φ)|^2 \sinθ dθ dφ}$$

*   **Gain (`G`):** `G = η_e D`, where `η_e` is **antenna efficiency** (accounts for losses).

*   **Effective Aperture (`A_e`):** `A_e = (λ² G) / (4π)` for lossless. `A_e = η_a A_phys` for aperture antennas.

*   **Beamwidth:** **HPBW** (Half-Power Beamwidth), **FNBW** (First-Null Beamwidth). `D ≈ 4π / (θ_{HPBW} φ_{HPBW})` (for narrow beams).

• Friis Transmission Equation: [TIER 1]

$$P_r = P_t G_t G_r \left( \frac{\lambda}{4πr} \right)^2$$

\boxed{P_r = P_t G_t G_r \left( \frac{\lambda}{4\pi r} \right)^2}

*   `P_t`, `P_r`: Tx/Rx power.

*   `G_t`, `G_r`: Gains (including efficiency).

*   `r`: Separation, `λ`: wavelength.

• Radar Range Equation:

$$P_r = P_t G_t G_r \frac{σ λ^2}{(4π)^3 r^4}$$

`σ` = Radar Cross-Section (RCS) of target.

1.3 Reciprocity & Pattern Multiplication [TIER 1]

• Reciprocity Theorem: The transmitting pattern of an antenna is identical to its receiving pattern. S₁₂ = S₂₁ for any two antennas.

• Pattern Multiplication Principle: [TIER 1]

  • For an array of N identical elements with individual pattern F_elem(θ, φ) and Array Factor (AF) AF(θ, φ):

    Total Pattern = F_elem(θ, φ) × AF(θ, φ)

  • Effect of Earth on Vertical Patterns: Image theory applies. For a vertical antenna over perfect ground, image is in-phase (for E-plane). For horizontal antenna, image is out-of-phase. Alters vertical polarization and pattern.

II. BASIC ANTENNA TYPES & THEIR PROPERTIES

2.1 Dipole Antennas [TIER 1]

• Center-Fed Half-Wave Dipole (l = λ/2): [TIER 1]

  • Current distribution: I(z) = I₀ cos(kz), |z| ≤ l/2.

  • Radiation Pattern: Figure-8 in E-plane (θ=90° max), omnidirectional in H-plane.

  • Input Impedance: R_in ≈ 73 + j42.5 Ω (theoretical). R_rad ≈ 73 Ω.

  • Directivity: D ≈ 1.64 (2.15 dB).

• Influence of Dipole Length (l): [TIER 1]

  | Length | Radiation Pattern | Input Impedance | Directivity | | :--- | :--- | :--- | :--- | | λ/2 | Single main lobe | ~73 Ω | 1.64 (2.15 dB) | | λ | Two major lobes (θ=90°, 0°), higher gain | ~200-300 Ω (high) | ~1.8 | | 3λ/2 | Multiple lobes (grating lobes) | Complex, high | Higher but multi-lobe |

  • Feed-point position: End-fed dipole (high impedance, ~2000-4000 Ω) vs center-fed.

• Quarter-Wave Monopole: [TIER 1]

  • Working: Half dipole over perfect ground plane. Uses image theory (in-phase image).

  • Radiation Pattern: Same as λ/2 dipole but only above ground (hemispherical). D_monopole = 2 * D_dipole ≈ 3.28 (5.15 dB).

  • Input Impedance: R_in ≈ (R_dipole/2) ≈ 36.5 Ω (for λ/4).

  • Applications: Vehicle antennas, mobile comms, mast radiators.

2.2 Travelling Wave & Long Wire Antennas [TIER 1]

• Long Wire Antenna: l >> λ, terminated with resistance R_T ≈ R_rad.

  • Operation: Travelling wave (current magnitude nearly constant, phase progressive).

  • Radiation Pattern: Main lobe at angle θ_m from wire axis: cosθ_m ≈ λ/l (for l >> λ). End-fire.

  • Influence of Length: Longer wire → main lobe narrower (higher gain), closer to end-fire.

  • V-Antenna: Two long wires at angle α. Main lobe along bisector. cosθ_m ≈ (λ/l) / sin(α/2).

  • Rhombic Antenna: Four-wire diamond, terminated. Broadband, high-gain, directional.

2.3 Loop & Slot Antennas [TIER 1]

• Small Loop Antenna (C << λ): [TIER 1]

  • Magnetic Dipole Equivalent. Area A = πa² (circular).

  • Radiation Pattern: Same as λ/2 dipole but E & H fields swapped (Eφ, Hθ). Figure-8 in H-plane.

  • Radiation Resistance: R_rad = 31,200 (A/λ²)² Ω (very small, needs matching).

  • Used as: Directional finders (due to sharp nulls), RFID tags.

• Slot Antenna & Babinet's Principle: [TIER 1]

  • Babinet's Principle: Complementary antennas (aperture & obstacle) in infinite conducting screen have identical H fields when excited by same E field. E_slot = E_inc - E_dipole.

  • Rectangular Slot in Infinite Ground: [TIER 1]

    • Slot dimensions a (along x), b (along y), a > b.

    • Pattern: Same as dipole of same dimensions but E & H fields interchanged and co-polarization rotated by 90°.

    • E-field: E_θ ∝ (sin(ka sinθ cosφ) / (ka sinθ cosφ)) * (sin(kb sinθ sinφ) / (kb sinθ sinφ)) * cosθ.

  • Applications: Waveguide apertures, mobile devices (compact, conformal).

III. ANTENNA ARRAYS (Critical High-Weightage)

3.1 Array Fundamentals & Factor [TIER 1]

• Array Factor (AF): For N identical elements with amplitudes a_n, phases β_n, positions d_n.

$$AF(θ, φ) = \sum_{n=1}^N a_n e^{j(k \cdot d_n + β_n)}$$

• Uniform Linear Array (ULA): a_n = 1, d_n = n d (spacing d), progressive phase α.

$$AF(θ) = \frac{\sin\left(\frac{Nψ}{2}\right)}{\sin\left(\frac{ψ}{2}\right)} \quad \text{where} \quad ψ = kd \cosθ + α$$

*   **Broadside:** `α = 0` → max at `θ=90°`.

*   **End-fire:** `α = -kd` → max at `θ=0°` or `180°`.

• Schelkunoff Unit Circle Method: [TIER 1]

  • Represents AF as polynomial in z = e^{jψ}: AF(z) = ∑_{n=0}^{N-1} a_n z^n.

  • Zeros of AF lie on unit circle |z|=1. ψ for nulls found from roots.

  • Design: Specify null positions → find polynomial coefficients a_n → amplitude distribution.

3.2 Array Design & Synthesis [TIER 2/3]

• Binomial Array: [TIER 3]

  • Principle: Coefficients a_n = binomial coefficients C(N-1, n). No side lobes.

  • Pattern: AF(ψ) ∝ [cos(ψ/2)]^{N-1}. HPBW ≈ 2 * 0.886 * λ/(Nd).

  • Disadvantage: Large amplitude taper → low efficiency, high Q.

• Dolph-Chebyshev Array: [TIER 2]

  • Principle: Achieve specified side lobe level (SLL) with minimum array length.

  • Uses Chebyshev polynomials. SLLs are equal (for uniform spacing).

  • Design: Choose SLL → find Chebyshev polynomial order N → map to AF(ψ).

• Taylor Synthesis: [TIER 3]

  • For sum patterns. Approximates ideal pattern with n equal side lobes.

  • Uses modified sinc function. Better efficiency than Dolph-Chebyshev for large N.

3.3 Practical Array Considerations [TIER 2]

• Effect of Ground on Horizontal Patterns: [TIER 1]

  • For broadcast arrays (AM/FM), ground affects horizontal (azimuth) pattern.

  • Vertical polarization: Ground reflection causes pattern distortion (tilt, asymmetry).

  • Design: Use multiple towers with specific phasing (directional arrays) to shape ground wave coverage.

• Tapered vs Uniform Apertures: [TIER 1]

  • Uniform: Highest gain for given size, but high side lobes (~-13.2 dB).

  • Tapered (e.g., Taylor, Chebyshev): Reduces side lobes at cost of slightly lower gain and wider main beam. Trade-off: SLL vs Gain/Beamwidth.

• Aperture Blockage: [TIER 2]

  • In reflector/array systems (e.g., Cassegrain), feed/subreflector blocks part of aperture.

  • Effects: Increases side lobes, reduces gain and efficiency, distorts beam.

  • Mitigation: Use transparent subreflectors, offset feeds.

IV. SPECIFIC & BROADBAND ANTENNAS

4.1 Yagi-Uda Antenna [TIER 1]

• Construction: [TIER 1]

  1. Driven Element: λ/2 dipole (feed point).

  2. Reflector: Slightly longer (~5%), placed behind driven element (~0.15-0.2λ).

  3. Directors: Slightly shorter (~5%), placed in front (~0.1-0.2λ spacing), 1+ in number.

• Roles: [TIER 1]

  • Reflector: Inductive reactance, reflects energy forward.

  • Directors: Capacitive reactance, phased array effect, steers beam forward, increases gain.

• Design: Element lengths ≈ 0.95-0.98λ (reflector), 0.9-0.95λ (directors). Spacings 0.1λ to 0.3λ.

• Applications: TV reception (8-15 dBi gain), point-to-point links. Advantages: Simple, cheap, high gain, directional. Disadvantages: Narrowband, sensitive to length/spacing.

4.2 Log-Periodic Dipole Array (LPDA) [TIER 1]

• Working Principle & Self-Similarity: [TIER 1]

  • Structure: Dipoles of varying lengths L_n, spacings d_n, with constant ratio τ = L_{n+1}/L_n = d_{n+1}/d_n < 1.

  • Self-similar: Electrical geometry repeats scaled by 1/τ. Frequency-independent behavior over design bandwidth.

• Wideband Operation: [TIER 1]

  • At any f, only 3-5 dipoles around resonant length (L ≈ λ/2) are active (others too long/short).

  • Active region "travels" along array as f changes. Input impedance and pattern stable over f.

  • Bandwidth: B.W. ≈ (1+τ)/(1-τ). Typical τ=0.95 → ~20:1 bandwidth.

• Applications: Wideband TV reception, EMC testing, HF communications.

4.3 Helical Antenna [TIER 2]

• Normal Mode: [TIER 2]

  • C << λ, S << λ (circumference, spacing).

  • Radiation: Broadside (perpendicular to helix axis), doughnut-shaped pattern.

  • Polarization: Linear (along axis).

  • Applications: HF communications, telemetry.

• Axial Mode: [TIER 2]

  • C ≈ λ, S ≈ 0.1-0.3λ (pitch angle α ≈ 12°-14°).

  • Radiation: End-fire along helix axis.

  • Polarization: Circular (RHCP or LHCP depending on winding).

  • Gain: G ≈ 15 * (N * S/λ)² (for N turns).

  • Applications: Satellite comms, space telemetry, GPS.

4.4 Other Specific Antennas [TIER 2/3]

• Turnstile Antenna: [TIER 3]

  • Construction: Two orthogonal λ/2 dipoles fed with 90° phase difference.

  • Operation: Produces circular polarization (CP) in axial direction. Omnidirectional CP in plane perpendicular to axis.

  • Applications: Satellite ground stations (CP for satellite signals), FM broadcasting.

• Folded Dipole: [TIER 3]

  • Construction: λ/2 dipole with another parallel conductor, connected at ends (λ/4 apart).

  • Current Distribution: I₁ = I₀/2 (in dipole arms), I₂ = I₀/2 (in folded conductor). Feed point impedance: R_in ≈ 4 * R_dipole ≈ 292 Ω.

  • Advantages: Higher impedance (easy 300Ω feed), wider bandwidth (due to thicker effective conductor).

• Microstrip/Patch Antenna: [TIER 1]

  • Construction: Metallic patch (λ/2 length L, width W) on dielectric substrate (ε_r, h) over ground plane.

  • Radiation Mechanism: Fringing fields at edges. L ≈ λ/(2√ε_eff)), W ≈ λ/(2√ε_r).

  • Advantages: [TIER 1] Low profile, conformal, lightweight, cheap, easy integration with circuits.

  • Limitations: [TIER 1] Narrow bandwidth (~1-5%), low efficiency (due to dielectric/conductor losses), spurious feed radiation, sensitive to substrate.

• Horn Antenna: [TIER 2]

  • Principle: Waveguide aperture flared to reduce diffraction, match impedance.

  • Types: Pyramidal (rectangular), Conical (circular).

  • Gain: G ≈ (π d / λ)² * η_a (for d = aperture dimension, η_a ≈ 0.5-0.6).

  • Applications: Feed for reflectors, standard gain horns for measurements.

• Parabolic Reflector Antenna: [TIER 2]

  • Principle: Paraboloid reflects spherical wave from feed to plane wave (or vice versa). Aperture efficiency η_a key.

  • Feed Mechanisms:

    • Prime Focus: Feed at focal point. Simple but blocks aperture.

    • Cassegrain: Feed illuminates subreflector (hyperbolic), which reflects to main. No blockage (feed behind), higher gain.

  • Aperture Efficiency & Blockage: [TIER 1]

    • η_a = η_s * η_t * η_p * η_b (spillover, taper, phase error, blockage).

    • Blockage by feed/subreflector reduces gain, increases side lobes.

  • Applications: Satellite TV, radio astronomy, deep space comms, radar.

• Lens Antenna: [TIER 3]

  • Principle: Dielectric lens (or zoned) in front of feed to collimate waves (like optical lens).

  • Types: Dielectric lens, Metamaterial lens, Zoned lens (reduced thickness/weight).

  • Applications: Millimeter-wave systems, where reflectors are difficult.

V. WAVE PROPAGATION MECHANISMS (High-Weightage)

5.1 Ground Wave Propagation [TIER 1]

• Transition: [TIER 1]

  • Space Wave: Direct + ground-reflected waves (near transmitter, r < few λ).

  • Surface Wave: Waves hugging Earth's surface due to continuous refraction (bending) by Earth's curvature. Dominates at MF/LF.

  • Transition: At large r, surface wave becomes dominant component of ground wave.

• Effect of Terrain: [TIER 1]

  • Conductivity (σ) & Permittivity (ε): Higher σ → lower attenuation. Sea water (high σ) → longer range. Dry ground (low σ) → high attenuation.

  • Vertical Pattern: Earth acts as imperfect reflector. Image theory modifies pattern. For vertical polarization, surface wave is in-phase with direct wave (constructive). For horizontal polarization, out-of-phase (destructive) → higher attenuation. Hence, AM broadcast uses vertical polarization.

  • Frequency & Range: Attenuation ∝ √f. Lower frequencies (< 3 MHz) propagate farther (e.g., AM broadcast ~100-200 km).

5.2 Sky Wave Propagation (Ionospheric) [TIER 1]

• Ionosphere as Variable Refractive Index: [TIER 1]

  • Free electron density N → plasma frequency f_p = 9√N (Hz). N varies with altitude (D, E, F layers), time (day/night), solar activity.

  • Refractive Index: n ≈ √(1 - (f_p/f)²) < 1. Waves bend towards higher N (lower n).

• Critical Frequency (f_c): [TIER 1]

  • Max f that reflects vertically incident wave from a layer. f_c = 9√N_max (layer peak density).

• Virtual Height (h'): [TIER 3]

  • Apparent reflection height in linear propagation model. h' > h (actual layer height) due to refraction.

• MUF (Maximum Usable Frequency): [TIER 1]

  • Max f that can be reflected for a given path (angle of incidence θ). f_MUF = f_c / cosθ.

• Skip Distance (d_skip): [TIER 1]

  • Minimum distance from transmitter where sky wave returns to Earth. For single hop: d_skip ≈ 2h' tanθ.

• Derivation: MUF vs Skip Distance: [TIER 1]

  • From triangle: sinθ = d_skip / (2R_E + 2h') ≈ d_skip / (2R_E) (small θ).

  • cosθ ≈ √(1 - sin²θ) ≈ 1 - (d_skip²)/(8R_E²).

  • f_MUF = f_c / cosθ ≈ f_c (1 + d_skip²/(8R_E²)).

  \boxed{f_{MUF} \approx f_c \left(1 + \frac{d_{skip}^2}{8R_E^2}\right) \quad \text{or} \quad d_{skip} \approx \sqrt{8R_E^2 \left(\frac{f_{MUF}}{f_c} - 1\right)}}

  • R_E = Earth radius (~6370 km).

• LUF (Lowest Usable Frequency): [TIER 3] Limited by ionospheric absorption (D-layer) and noise.

• Day-Night Effects: D-layer (absorption) exists only in daytime. Night: higher f_c in F-layer, lower absorption → better long-distance HF.

5.3 Space Wave Propagation (Tropospheric) [TIER 1]

• Super-refraction (Ducting): [TIER 1]

  • Cause: Negative refractivity gradient dN/dh < -157 N-units/km (abnormal). N = (n-1)×10⁶.

  • Effect: Ray curvature > Earth's curvature. Waves trapped in duct (channel). Extends radio horizon (microwave propagation over ~500 km).

• Tropospheric Scattering: [TIER 1]

  • Mechanism: Scattering from turbulent eddies, refractive index fluctuations in lower troposphere.

  • Effect: Enables beyond-horizon propagation (~200-500 km). Wideband but high path loss (~100 dB). Used for troposcatter links.

• Effect of Atmosphere/Terrain/Obstacles: [TIER 1]

  • Atmosphere: Gaseous absorption (oxygen, water vapor peaks at 22/60 GHz). Rain/fog attenuation (microwave).

  • Terrain: Hills cause shadowing, diffraction loss. Smooth sea/desert → lower loss.

  • Obstacles (buildings, foliage): Multipath fading, shadowing, penetration loss (higher at UHF/SHF).

5.4 Special Propagation Phenomena [TIER 2]

• Tropospheric Ducting: [TIER 2]

  • Formation of evaporation duct (over sea, N gradient sharp near surface) or surface duct (inversion layer).

  • Use: Microwave propagation over sea beyond line-of-sight (e.g., ship-to-ship, coastal links). Low loss within duct.

• Scattering (General): [TIER 2]

  • Mechanism: Redirection of energy by small objects/irregularities (Rayleigh, Mie, Bragg).

  • Types: Ionospheric (forward scatter), Tropospheric, Rain scatter (microwave), Aircraft scatter.

VI. NUMERICAL & DESIGN TOOLS (Emerging Topic) [TIER 4]

• Method of Moments (MoM): Solves integral equations (e.g., electric field integral equation - EFIE). Discretizes surface currents. Accurate for conducting structures. Basis: J = ∑ I_n f_n.

• Finite Difference Time Domain (FDTD): Solves Maxwell's curl equations in time domain on a grid (Yee cell). Time-stepping. Good for broadband response, complex materials, transient analysis.

• Finite Element Method (FEM): Solves variational form of wave equation. Meshes volume with irregular elements. Excellent for dielectric structures, complex geometries.

• Role: Predict performance (pattern, impedance, SAR), optimize design, handle geometries impossible to solve analytically. Software: CST, HFSS (FEM), FEKO (MoM), XFDTD.

EXAM TIPS & COMMON PITFALLS:

1. Retarded Potential: Always emphasize delay r/c. It's the core reason for radiation from time-varying sources.

2. Far-field: Conditions r >> λ and r >> l²/λ both required. l²/λ dominates for electrically large antennas.

3. Dipole Length: λ/2 is reference. Longer dipoles develop grating lobes (multiple maxima). l = λ has max at θ=0° (end-fire) too.

4. Monopole vs Dipole: Monopole over perfect ground has half the input impedance and double the directivity of center-fed dipole. Always use image theory.

5. Pattern Multiplication: Total = Element × Array Factor. Element pattern often sinθ for dipoles (nulls on axis). Array factor determines main beam direction/null positions.

6. Yagi-Uda: Reflector longer, directors shorter than driven element. Spacing critical (~0.15-0.2λ). Gain increases with number of directors (up to point).

7. LPDA: Key is constant ratio τ. Explains wideband: active region shifts with f. τ smaller → wider bandwidth but lower gain.

8. MUF & Skip: f_MUF = f_c / cosθ. MUF increases with angle (longer path). Skip distance increases with MUF (from derived formula). Optimum frequency for a path is just below MUF.

9. Super-refraction vs Ducting: Super-refraction is bending > Earth curvature. Ducting is extreme super-refraction creating a waveguide (trapping).

10. Microstrip: Bandwidth limitation is due to high Q of resonant cavity. Thicker substrate/lower ε_r increases bandwidth but may cause spurious modes.

11. Schelkunoff Circle: Zeros of AF on unit circle. ψ for nulls from z = e^{jψ}. Amplitude distribution from polynomial coefficients.

12. Babinet's Principle: Complementary antennas have identical radiation patterns but E & H fields swapped. Slot in infinite ground has same pattern as dipole of same size.