Antennas and wave Propagation (EC-602) - Unit 2 Short Notes

I. FUNDAMENTAL RADIATION THEORY & ANTENNA PARAMETERS

Retarded Potentials & Radiation from Elementary Sources

• Concept: The retarded potential accounts for the finite propagation speed ($c$) of EM waves. The potential at a point $\mathbf{r}$ and time $t$ depends on the source distribution at an earlier retarded time $$\displaystyle t_r = t - \frac{|\mathbf{r} - \mathbf{r}'|}{c} $$.

  [!TIP] Core idea: Effects are not instantaneous. This is the foundation for all radiation calculations.

• Liénard-Wiechert Potentials: General solution for potentials from a moving point charge.

• Oscillating Electric Dipole (Hertzian Dipole): An infinitesimal current element $I\,dl$ with sinusoidal current $$\displaystyle I = I_0 e^{j\omega t} $$.

  • Far-field ($r \gg \lambda$) Radiated Fields:

$$ \mathbf{E} \approx \frac{j\eta I_0 dl e^{-jkr}}{4\pi r} \sin\theta \, \hat{\theta} $$

$$ \mathbf{H} \approx \frac{j I_0 dl e^{-jkr}}{4\pi r} \sin\theta \, \hat{\phi} $$

    where $$\displaystyle \eta = \sqrt{\mu_0/\varepsilon_0} $$, $$\displaystyle k = 2\pi/\lambda $$.

*   **Time-Averaged Radiated Power**:

$$ P_{\text{rad}} = \frac{\eta I_0^2 (dl)^2}{12\pi^2} \left( \frac{2\pi}{\lambda} \right)^2 = \frac{\pi}{3} \eta I_0^2 \left( \frac{dl}{\lambda} \right)^2 \boxed{P_{\text{rad}} \propto (I_0 dl)^2 \cdot f^2} $$

    Power $\propto$ square of frequency and square of dipole moment.

Antenna Characterization & Parameters

Near-field and Far-field Regions

• Reactive Near-field (Fresnel): $$\displaystyle r < 0.62\sqrt{D^3/\lambda} $$ (D = largest antenna dim.). Field energy stored, not radiating. $\mathbf{E}$ & $\mathbf{H}$ out of phase, complex impedance.

• Radiating Far-field (Fraunhofer): $$\displaystyle r > \frac{2D^2}{\lambda} $$. Fields are:

  1. Transverse ($$\displaystyle E_\theta, H_\phi $$ only).

  2. $$\displaystyle |\mathbf{E}|/|\mathbf{H}| = \eta $$ (intrinsic impedance of medium).

  3. Angularly independent of $r$.

  4. $\mathbf{E}$ & $\mathbf{H}$ in phase.

  [!TIP] Far-field approximation assumes $r \gg \lambda$ and $r \gg D$, allowing $|\mathbf{r}-\mathbf{r}'| \approx r - \hat{r}\cdot\mathbf{r}'$ in phase term and $1/|\mathbf{r}-\mathbf{r}'| \approx 1/r$ in amplitude.

Antenna Analysis Principles

• Pattern Multiplication Theorem: Radiation pattern of an array = Array Factor (AF) × Element Pattern.

  • AF depends on geometry, amplitude, phase of array.

  • Element pattern is pattern of a single antenna.

• Babinet's Principle: For complementary apertures (e.g., slot & dipole) in infinite conducting plane:

$$ \mathbf{E}_{\text{slot}} = \eta_0 \hat{n} \times \mathbf{H}_{\text{dipole}}, \quad \mathbf{H}_{\text{slot}} = -\frac{\hat{n} \times \mathbf{E}_{\text{dipole}}}{\eta_0} $$

*   **Impedance relation**: $$\displaystyle Z_{\text{slot}} Z_{\text{dipole}} = \eta_0^2/4 $$.

• Equivalence Principle: Replaces actual sources with equivalent electric/magnetic surface currents on a closed surface to compute fields outside.

II. WIRE ANTENNAS & BASIC ELEMENTS

Short Dipole & Monopole

• Short Dipole ($L \ll \lambda$): Current ~ constant ($$\displaystyle I_0 $$). Radiation pattern: Figure-8 in E-plane ($\sin\theta$), omnidirectional in H-plane.

• Quarter-wave Monopole over perfect ground plane:

  • Image Theory: Replace ground with image of monopole (inverted). Creates half-wave dipole in free space.

  • Radiation Pattern: Same as half-wave dipole in upper hemisphere only. Gain = $2 \times$ gain of half-wave dipole (since power radiated only into $2\pi$ steradians) → $G \approx 5.15$ dBi (or 3.15 dBd).

  • Input Impedance: $$\displaystyle Z_{\text{in}} \approx 36.5 + j21.25\ \Omega $$ (resistance ≈ half of half-wave dipole's 73Ω).

  • Applications: Mobile/vehicle antennas, mast radiators.

Half-wave Dipole

• Current distribution: $$\displaystyle I(z) = I_0 \cos(kz) $$.

• Input Impedance: $$\displaystyle Z_{\text{in}} \approx 73 + j42.5\ \Omega $$. Standard reference antenna.

• Radiation Resistance: $$\displaystyle R_r \approx 73\ \Omega $$.

Long Wire & Travelling Wave Antennas

• Long Wire Antenna ($$\displaystyle L > \lambda $$):

  • Current: Travelling wave (not standing wave). $$\displaystyle I(z) = I_0 e^{-j\beta z} $$.

  • Radiation: Main lobe at angle $$\displaystyle \theta_m $$ where $$\displaystyle \sin\theta_m = \frac{\lambda}{L} $$ (for $L \gg \lambda$). End-fire direction.

  • Pattern: Multiple lobes (grating lobes). Number of lobes ≈ $L/\lambda$.

  • Length Influence: Longer wire → narrower main lobe, more lobes.

• V Antenna: Two long wires at angle. Main lobe along bisector.

• Rhombic Antenna: Four-wire diamond shape. Travelling wave terminated with resistor. Wideband, high gain (10-15 dBi), unidirectional. Used for HF point-to-point.

Folded Dipole & Variants

• Folded Dipole: Two parallel $\lambda/2$ dipoles connected at ends. Fed at center of one.

  • Input Impedance: $\approx 4 \times$ impedance of single dipole → $\approx 300\ \Omega$. Wider bandwidth than simple dipole.

  • Current: Equal magnitude, opposite phase in two arms.

• Turnstile Antenna (Crossed Dipoles):

  • Two $\lambda/2$ dipoles crossed at center, fed with $$\displaystyle 90^\circ $$ phase shift.

  • Axial Mode: Circularly polarized radiation along axis (broadside). Applications: Satellite comms, FM broadcasting (for CP reception).

• Log-Periodic Antenna:

  • Structure: Series of driven elements with lengths & spacing increasing by factor $\tau$ (scale factor). Feed via transmission line crossing all elements.

  • Frequency-Independent Principle: Input impedance & radiation pattern repeat periodically with $\log(f)$. Wideband (octave+).

  • Active Region: Only 1-3 elements radiate effectively at a given frequency.

III. ANTENNA ARRAYS

Array Fundamentals & Analysis

• Array Factor (AF) for N isotropic sources with spacing $d$, progressive phase $\beta$:

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

• Broadside Array: $$\displaystyle \beta = 0 $$, main lobe at $$\displaystyle \theta = 90^\circ $$.

• End-fire Array: $$\displaystyle \beta = \pm kd $$, main lobe at $$\displaystyle \theta = 0^\circ $$ or $$\displaystyle 180^\circ $$. Hansen-Woodyard condition for optimum: $$\displaystyle \beta = \pm (kd + \delta) $$ with $\delta \approx 0.226(kd)$.

• Scanning: To steer beam to $$\displaystyle \theta_0 $$, set $$\displaystyle \beta = -kd \cos\theta_0 $$.

Array Synthesis Methods

• Schelkunoff Unit Circle Method: Graphical synthesis. Plot array coefficients on unit circle. Zeros of AF correspond to roots on circle. Enables design for null placement.

Practical Wire Arrays

• Yagi-Uda Antenna:

  • Elements: 1 driven (dipole), 1 reflector (longer, behind), multiple directors (shorter, in front).

  • Mechanism: Reflector induces current with phase lag → forward reinforcement. Directors induce current with phase lead → forward reinforcement.

  • Pattern: Unidirectional, high gain (7-15 dBi), narrow bandwidth (≈5%).

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

  [!TIP] Key: Reflector length $\approx 0.5\lambda$, spacing $\approx 0.15-0.2\lambda$. Director lengths $\approx 0.45-0.48\lambda$, spacing decreasing.

• Log-Periodic: Already covered.

Planar & Aperture Distributions

• Uniform Aperture: $$\displaystyle J(\mathbf{r}') = \text{constant} $$. Narrowest beamwidth but high first sidelobe ($-13.2$ dB).

• Tapered Aperture (e.g., Taylor, Chebyshev): Amplitude tapers to edges.

  • Advantage: Reduced sidelobes.

  • Disadvantage: Wider main lobe and reduced aperture efficiency.

• Horizontal Patterns in Broadcast Arrays:

  • AM broadcast towers use phased arrays to shape horizontal pattern.

  • Goal: Concentrate radiation in service area, minimize radiation in undesired directions (e.g., to avoid interference).

  • Control: By adjusting amplitudes & phases of multiple towers.

IV. APERTURE & REFLECTOR ANTENNAS

Horn Antennas

• Types: Pyramidal (rectangular), Conical, Sectoral (E-plane/H-plane), Corrugated (low cross-pol, wideband).

• Radiation Mechanism: Aperture field is TE$$\displaystyle _{10} $$ mode from waveguide. Phase error across aperture limits gain.

• Aperture Efficiency ($$\displaystyle \eta_a $$): $$\displaystyle \eta_a = \eta_r \eta_p \eta_s $$ (spillover, phase, blockage efficiencies). Typical $$\displaystyle \eta_a \approx 0.5-0.6 $$.

• Gain: $$\displaystyle G = \frac{4\pi A_e}{\lambda^2} = \eta_a \frac{4\pi A_{\text{phys}}}{\lambda^2} $$.

• Gain Measurement (Gain Comparison): Compare received power with standard gain horn at same distance.

Parabolic Reflector Antennas

• Principle: Parallel rays from feed reflect off parabola → converge at focus (or vice versa). Phase error zero on-axis.

• Feed Mechanisms:

  • Axial (Prime Focus): Feed at focus. Simple, large aperture blockage by feed & supports.

  • Cassegrain: Subreflector (hyperbolic) reflects to feed at vertex. Lower blockage, higher gain.

  • Gregorian: Ellipsoidal subreflector. Larger focal length, less sensitive to feed illumination.

• Aperture Blockage: Caused by feed, subreflector, supports.

  • Effects: Reduces gain, increases sidelobes, distorts pattern.

  • Mitigation: Use transparent subreflectors, offset feeds (Offset Parabola).

• Surface Accuracy: RMS surface error $\sigma$ causes gain reduction: $$\displaystyle \eta_s \approx e^{-(4\pi\sigma/\lambda)^2} $$ (Ruze's formula).

Slot & Complementary Antennas

• Slot Antenna: Slot in conducting plane. Radiates when fed (e.g., waveguide).

• Babinet's Principle Applied:

  • Slot in infinite ground plane $$\displaystyle \leftrightarrow $$ Dipole of complementary shape in free space.

  • Radiation Pattern: Slot pattern is identical to dipole pattern but with $\mathbf{E}$ & $\mathbf{H}$ interchanged, and $\theta$ & $\phi$ swapped.

  • Rectangular Slot ($a \times b$):

    • $$\displaystyle E_\phi $$ (co-pol) $$\displaystyle \propto \frac{\sin(kb\sin\theta\cos\phi/2)}{(kb\sin\theta\cos\phi/2)} \cdot \frac{\sin(ka\sin\theta\sin\phi/2)}{(ka\sin\theta\sin\phi/2)} $$

    • Pattern similar to thin dipole ($b \ll \lambda$, $a \approx \lambda/2$).

• Microstrip (Patch) Antenna:

  • Construction: Metallic patch on dielectric substrate over ground plane.

  • Radiation Mechanism: Fringing fields at edges. Cavity model: Patch acts as resonant cavity.

  • Feeding: Microstrip line, probe, aperture coupling.

  • Advantages: Low profile, conformal, lightweight, inexpensive, easy to integrate.

  • Limitations: Narrow bandwidth (1-5%), low efficiency (due to substrate losses), low power handling.

  [!TIP] RGPV focus: Design equations (Transmission Line Model):

  • Width: $$\displaystyle W = \frac{c}{2f_r\sqrt{\frac{\varepsilon_r+1}{2}}} $$

  • Effective dielectric: $$\displaystyle \varepsilon_{\text{eff}} = \frac{\varepsilon_r+1}{2} + \frac{\varepsilon_r-1}{2}(1+12h/W)^{-1/2} $$

  • Length: $$\displaystyle L = \frac{c}{2f_r\sqrt{\varepsilon_{\text{eff}}}} - 2\Delta L $$, $$\displaystyle \Delta L \approx 0.412h\frac{(\varepsilon_{\text{eff}}+0.3)(W/h+0.264)}{(\varepsilon_{\text{eff}}-0.258)(W/h+0.8)} $$

V. SPECIALIZED & PRINTED ANTENNAS

Microstrip Antennas (Patch) - Details

• Substrate Parameters: Higher $$\displaystyle \varepsilon_r $$ → smaller size, narrower bandwidth, more surface waves. Lower $h$ → narrower bandwidth, higher field concentration.

• Polarization: Linear (along width). Circular via sequential rotation or perturbed square patch.

• Arrays: Series/parallel feeds, corporate feeds for high gain.

Helical Antenna

• Normal Mode ($L \approx N\lambda/10$, $D \approx \lambda/10$):

  • Current ~ uniform, radiation similar to short dipole.

  • Broadside pattern (perpendicular to helix axis).

  • Applications: Low-gain, compact antennas for handheld devices.

• Axial Mode ($L \approx N\lambda$, $D \approx 0.1-0.3\lambda$, $C \approx \lambda$):

  • Current Travelling wave along helix.

  • Radiation: Maximum along helix axis (end-fire).

  • Circular Polarization (sense determined by winding direction).

  • Gain: $$\displaystyle G \approx 15 \left( \frac{C}{\lambda} \right)^{2.5} \left( \frac{N\lambda}{L} \right) $$ (Balmain's formula).

  • Bandwidth: Wide (octave+). Applications: Satellite comms, deep-space.

Lens Antennas

• Types:

  • Dielectric Lens: Slow wave medium, focuses like optical lens.

  • Zoned Lens: Reduces thickness & weight by removing annular sections.

  • Artificial Lens (Metamaterial): Uses subwavelength structures.

• Principle: Corrects phase of spherical wave from feed to planar wavefront.

• Advantages: High gain, low spillover loss.

• Disadvantages: Heavy (dielectric), expensive, narrowband (dielectric constant variation).

VI. GROUND EFFECTS & ENVIRONMENTAL IMPACT

Effect of Ground on Antenna Patterns

• Image Theory:

  • Perfect Electric Conductor (PEC): Image current same magnitude, opposite direction for vertical dipole; same direction for horizontal dipole.

  • Imperfect Ground: Complex reflection coefficient $$\displaystyle \Gamma = |\Gamma|e^{j\psi} $$ depends on angle of incidence, conductivity $\sigma$, permittivity $$\displaystyle \varepsilon_r $$.

• Impact on Vertical Patterns:

  • Vertical Dipole (over ground): Pattern is combination of direct + ground-reflected waves.

  • Low-angle radiation (critical for ground wave, sky wave) is strongly affected by ground conductivity.

  • Good ground ($\sigma$ high, e.g., sea water): High reflection, constructive interference at low angles → elevated low-angle lobes.

  • Poor ground ($\sigma$ low, e.g., dry sand): Low reflection, destructive interference → depressed low-angle lobes.

  [!TIP] For vertical polarization, ground reflection coefficient $$\displaystyle \Gamma_\parallel $$ is negative for grazing incidence → phase reversal → pattern null at horizon for perfect ground.

• Horizontal Dipole: Image same phase → pattern reinforced at low angles.

General Environmental Effects

• Atmosphere: Refractive index decreases with height → bending (refraction) of rays downward (standard refraction).

• Terrain: Hills/buildings cause diffraction, scattering, shadowing.

• Obstacles: Attenuate signals, cause multipath.

VII. RADIO WAVE PROPAGATION MECHANISMS

Ground Wave Propagation

• Components:

  • Direct Wave: Line-of-sight.

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

  • Surface Wave: Bound to earth's surface, follows curvature. Dominant for LF/MF.

• Transition Surface ↔ Space Wave: At some distance, surface wave becomes negligible compared to space wave (direct + reflected). This transition distance depends on frequency, antenna height, ground properties.

• Effect of Terrain: Smooth, conductive ground (sea) → longer transition, better propagation. Rough, poor ground → shorter transition, higher attenuation.

Sky Wave (Ionospheric) Propagation

• Ionosphere Layers: D (60-90 km, daytime only, absorbs LF), E (90-120 km), F (150-400 km, splits into F1/F2 day).

• Critical Frequency ($$\displaystyle f_c $$): Max frequency that will be reflected back vertically.

$$ f_c = 9\sqrt{N_{\text{max}}} \ \text{Hz}, \quad N_{\text{max}} = \text{peak electron density (m}^{-3}\text{)} $$

• Virtual Height ($h'$): Apparent height from which wave appears reflected (due to bending).

• MUF (Maximum Usable Frequency): Max frequency that can be used for a given path via a specific layer at angle $$\displaystyle \theta_i $$.

$$ \boxed{\text{MUF} = f_c \sec\theta_i} $$

where $$\displaystyle \theta_i $$ is angle of incidence at ionosphere.

• Skip Distance ($$\displaystyle d_s $$): Minimum distance from transmitter where sky wave returns to earth.

$$ d_s = 2h' \tan\theta_i $$

• Relation between MUF and Skip Distance:

$$ \boxed{d_s = 2h' \sqrt{\left(\frac{\text{MUF}}{f_c}\right)^2 - 1}} $$

> [!TIP] Derivation: From MUF = $$\displaystyle f_c \sec\theta_i $$, so $$\displaystyle \tan\theta_i = \sqrt{\sec^2\theta_i - 1} = \sqrt{(\text{MUF}/f_c)^2 - 1} $$. Then $$\displaystyle d_s = 2h' \tan\theta_i $$.

• LUF (Lowest Usable Frequency): Min frequency that can be received with acceptable signal-to-noise ratio. Limited by atmospheric noise at lower frequencies.

• Day/Night: D-layer disappears at night → lower absorption → better HF propagation. F-layer higher at night → lower MUF.

Space Wave & Tropospheric Propagation

• Direct + Ground-Reflected: Resultant pattern depends on antenna heights, phase difference.

• Super Refraction: Refractive index decreases rapidly with height ($$\displaystyle dN/dh < -157 $$ N-units/km). Rays bend more than normal → extends radio horizon.

• Sub Refraction: $$\displaystyle dN/dh > -157 $$ N-units/km. Rays bend less → reduces radio horizon.

• Tropospheric Scattering: Turbulence in troposphere causes forward scattering of UHF/SHF signals beyond horizon. Scatter volume at common volume of TX/RX beams. Applications: Beyond-line-of-sight comms (100-500 km).

• Tropospheric Ducts:

  • Surface Duct: Strong inversion near ground (e.g., over cool water). Traps waves, propagates with low loss over hundreds of km.

  • Elevated Duct: Inversion at 100s of meters. Common over oceans.

  • Evaporation Duct: Over water, humidity gradient forms duct ~10-40 m high.

• Radius of Curvature of Ray Path:

  In a stratified atmosphere with refractive index $n(h)$ decreasing linearly: $$\displaystyle n(h) = n_0 - \alpha h $$.

$$ \boxed{R = \frac{1}{\alpha} = \frac{n_0}{\left| \frac{dn}{dh} \right|}} $$

For standard atmosphere, $$\displaystyle R \approx 4/3 \times R_{\text{earth}} $$ (effective Earth radius factor $$\displaystyle k = 4/3 $$).

VIII. NUMERICAL TOOLS & ADVANCED TOPICS

Numerical Methods for Antenna Analysis

Other Specialized Topics (Brief)

• Flat Sheet & Corner Reflectors: Flat sheet (planar reflector) increases gain of dipole by ~2-3 dB. Corner reflector (two flat sheets at $$\displaystyle 90^\circ $$) gives higher gain (~8-10 dB) and unidirectional pattern.

• Feeding Structures: Probe (conducting pin), aperture (slot), microstrip line, etc. Choice affects bandwidth, cross-pol.

• V Antenna: Already covered in long wire.

• Parabolic Reflector Design: Key considerations: f/D ratio (focal length/diameter) affects spillover, blockage, phase error. Typical $f/D \approx 0.3-0.5$. Surface tolerance $$\displaystyle \sigma < \lambda/50 $$ for high efficiency.