UNIT 4: ANTENNAS AND WAVE PROPAGATION - SHORT NOTES

1. FOUNDATIONAL ANTENNA THEORY & RADIATION MECHANISMS

Retarded Potential

• Concept: Accounts for finite propagation speed ($c$) of EM waves. The potential at point $P$ at time $t$ depends on source values at an earlier retarded time $$\displaystyle t_r = t - R/c $$, where $R$ is distance from source to $P$.

• Derivation for Sinusoidal Sources: For a current element $$\displaystyle I(z')e^{j\omega t} $$ along $z'$, vector potential $\mathbf{A}$ is:

$$\mathbf{A}(z,t) = \frac{\mu}{4\pi} \int \frac{I(z') e^{j\omega (t - R/c)}}{R} \,dz' \, \mathbf{\hat{z}}$$

where $R \approx r - z'\cos\theta$ for $r \gg$ element length. This leads to radiation fields.

• Significance: Fundamental to solving radiation problems; separates near-field (quasi-static) and radiation-field terms.

• Approaches to Solve Radiation Problems:

  1. Retarded Potential Method: Direct integration.

  2. Vector Potential + Wave Equation: Solve Helmholtz equation.

  3. Huygens' Principle: Aperture integration.

[!TIP] Exam Focus: Retarded potential is the bridge between static fields and radiation. Be ready to derive $\mathbf{A}$ for a short dipole and identify the $1/r$ term as the radiation term.

Oscillating Electric Dipole / Current Element

• Model: Infinitesimal dipole (Hertzian dipole): length $dl \ll \lambda$, carrying uniform current $$\displaystyle I_0 $$.

• Radiated Fields (in spherical coordinates, $\theta$-polarized):

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

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

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

• Far-field ($r \gg \lambda$): Terms in $1/(\beta r)$ vanish → $$\displaystyle E_\theta \propto 1/r $$, $$\displaystyle H_\phi \propto 1/r $$.

• Power Radiation:

$$P_{rad} = \frac{I_0^2 (\beta l)^2}{12\pi\eta} \quad \text{(for } \beta l \ll 1\text{)}$$

Power $$\displaystyle \propto (I_0 f)^2 $$ (since $$\displaystyle \beta = 2\pi f/c $$).

• Radiation Resistance:

$$R_r = \frac{2P_{rad}}{I_0^2} = \frac{80\pi^2 (dl/\lambda)^2}{}$$

\boxed{R_r \approx 80\pi^2 (dl/\lambda)^2 , \Omega}

[!TIP] Common Pitfall: The $$\displaystyle 1/r^2 $$ and $$\displaystyle 1/r^3 $$ terms are reactive (store energy), not radiative. Only the $1/r$ terms carry time-average power to infinity.

Near-field and Far-field Regions

• Definitions (for antenna of max dimension $D$):

  • Reactive Near-field ($$\displaystyle r < 0.62\sqrt{D^3/\lambda} $$): Reactive fields dominate ($$\displaystyle E \propto 1/r^3 $$, $$\displaystyle 1/r^2 $$). Field pattern not well-defined.

  • Fresnel (Radiation) Near-field ($$\displaystyle 0.62\sqrt{D^3/\lambda} < r < 2D^2/\lambda $$): Both reactive and radiative fields present. Field structure complex, varies with $r$.

  • Fraunhofer (Far-field) ($$\displaystyle r > 2D^2/\lambda $$): Pure radiative field ($E,H \propto 1/r$). Angular field distribution independent of $r$. Pattern measurement region.

• Key Difference: Near-field has significant $E$- and $H$-phase difference and $E/H$ ratio ≠ $\eta$; Far-field has $E \perp H$, $$\displaystyle |E|/|H| = \eta $$, in-phase.

[!TIP] Rule of Thumb: Far-field starts at $$\displaystyle r_{ff} = \frac{2D^2}{\lambda} $$. For $$\displaystyle D = 1\,m $$, $$\displaystyle \lambda=0.1\,m $$, $$\displaystyle r_{ff} = 20\,m $$.

Far-field Approximation

• Significance: Simplifies antenna analysis. In far-field:

  • $R \approx r$ in denominator.

  • $$\displaystyle e^{-j\beta R} \approx e^{-j\beta r} e^{j\beta \hat{\mathbf{r}}\cdot\mathbf{r}'} $$ (phase approximation).

  • $\mathbf{\hat{R}} \approx \mathbf{\hat{r}}$ (direction approximation).

• Conditions: $r \gg \lambda$ and $$\displaystyle r \gg D^2/\lambda $$. Enables pattern multiplication and array factor concepts.

• Simplifications: Radiated fields transverse ($$\displaystyle E_\theta, H_\phi $$ only for z-axis dipole). Power flow radial ($\mathbf{S} \propto \mathbf{\hat{r}}$).

Reciprocity Theorem

• Statement: If a current $$\displaystyle I_1 $$ at antenna 1 produces field $$\displaystyle E_2 $$ at antenna 2, then same current $$\displaystyle I_1 $$ at antenna 2 produces same field $$\displaystyle E_1 $$ at antenna 1. Mathematically:

$$\oint_S (\mathbf{E}_1 \times \mathbf{H}_2 - \mathbf{E}_2 \times \mathbf{H}_1) \cdot d\mathbf{S} = 0$$

• Implication for Antennas: Transmitting and receiving patterns are identical for same antenna. Also, mutual impedance $$\displaystyle Z_{12} = Z_{21} $$.

• Proof Sketch: Apply vector identity $$\displaystyle \nabla \cdot (\mathbf{E}_1 \times \mathbf{H}_2 - \mathbf{E}_2 \times \mathbf{H}_1) = \mathbf{H}_2 \cdot (\nabla \times \mathbf{E}_1) - \mathbf{E}_1 \cdot (\nabla \times \mathbf{H}_2) - (\mathbf{E}_1 \leftrightarrow \mathbf{E}_2) $$. Use Maxwell's equations and integrate over closed surface; surface integral → 0 if sources enclosed or fields vanish at infinity.

Antenna Parameters & Definitions

[!TIP] Directivity vs Gain: Directivity is theoretical max; Gain is real (includes losses). For lossless antenna, $$\displaystyle G = D $$.

2. WIRE ANTENNAS

Dipole Antennas

• Half-wave Dipole ($$\displaystyle l = \lambda/2 $$):

  • Current Distribution: $$\displaystyle I(z) = I_0 \cos(\beta z) $$ (sinusoidal, zero at ends).

  • Radiation Pattern: Figure-8 in E-plane ($$\displaystyle \phi=0 $$ cut), omnidirectional in H-plane. Max at $$\displaystyle \theta=90^\circ $$.

  • Directivity: $D \approx 1.64$ (2.15 dBi). $$\displaystyle R_r \approx 73\,\Omega $$.

  • Length Influence: As $l$ increases beyond $\lambda/2$, main beam splits, grating lobes appear. For $l \approx 1.2\lambda$, back lobe develops.

• Short Dipole ($l \ll \lambda$): Same pattern as infinitesimal dipole (sin²θ). $$\displaystyle R_r \propto (l/\lambda)^2 $$.

• Infinitesimal Dipole: Theoretical limit; $$\displaystyle R_r \approx 80\pi^2 (dl/\lambda)^2\,\Omega $$.

[!TIP] Pattern Shape: Dipole E-plane pattern $\propto \sin\theta$. Half-wave is standard reference.

Quarter-wave Monopole

• Working Principle: Half dipole over perfect ground plane. Uses image theory: ground plane replaced by image monopole (current opposite phase) → full $\lambda/2$ dipole in free space.

• Radiation Pattern: Same as half-wave dipole but only above ground plane (hemispherical). Ground plane must be large ($$\displaystyle > \lambda $$).

• Power Radiated: Half of $\lambda/2$ dipole → $$\displaystyle P_{rad} = \frac{I_0^2 \cdot 73}{2} \Omega $$ (if $$\displaystyle R_r $$ same).

• Applications: Vehicle antennas, mast radiators for AM broadcast.

[!TIP] Impedance: Monopole over perfect ground: $$\displaystyle R_r \approx 36.5\,\Omega $$ (half of dipole). Over real ground, $$\displaystyle R_r $$ decreases due to losses.

Folded Dipole

• Construction: Two parallel $\lambda/2$ dipoles connected at ends; fed at center of one.

• Current Distribution: $$\displaystyle I_1 $$ (feed arm), $$\displaystyle I_2 $$ (other arm). For symmetric case, $$\displaystyle I_1 = I_0 $$, $$\displaystyle I_2 = -I_0 $$ → balanced.

• Impedance: $$\displaystyle R_{in} \approx 4 \times R_{r,dipole} \approx 292\,\Omega $$ (for lossless, equal arms). High impedance useful for matching to twin-lead (300 $\Omega$).

• Bandwidth: Slightly wider than simple dipole due to increased thickness effect.

Long Wire Antenna / Traveling Wave Antennas

• Radiation from Wire of Length $l$: Current is traveling wave (not standing wave). $$\displaystyle I(z) = I_0 e^{-j\beta z} $$ (for $$\displaystyle z>0 $$).

• Pattern Characteristics:

  • Pattern Function: $$\displaystyle F(\theta) \propto \frac{\cos(\frac{\beta l}{2} \cos\theta) - \cos(\frac{\beta l}{2})}{\sin\theta} $$

  • For $l \gg \lambda$, main lobe at $$\displaystyle \theta \approx 90^\circ $$ (broadside) with cardioid shape if terminated in matched load.

  • End-fire if open-circuited (standing wave component).

• Length Influence:

  • $$\displaystyle l = n\lambda/2 $$: Resonant, high $$\displaystyle R_r $$, but pattern multi-lobed.

  • $l \approx \lambda$ to $2\lambda$: Good end-fire gain (5–8 dBi).

  • $l \gg \lambda$: Narrow beam, high directivity, but inefficient feed.

• V-Antenna: Two long wires at angle $\alpha$; pattern tilted by $\alpha/2$.

[!TIP] Long Wire vs Dipole: Long wire has traveling wave → unidirectional pattern; dipole has standing wave → bidirectional.

Rhombic Antenna (Short Notes)

• Structure: Four long wires in diamond shape, terminated with resistors.

• Operation: Traveling wave along each leg; main lobe along rhombus axis.

• Advantages: High gain (15–20 dBi), wide bandwidth, unidirectional.

• Disadvantages: Low efficiency (termination loss), large size.

Turnstile Antenna

• Construction: Two orthogonal half-wave dipoles fed $$\displaystyle 90^\circ $$ out-of-phase.

• Radiation: Circularly polarized (CP) when axes crossed at $$\displaystyle 90^\circ $$ and fed in quadrature. Omnidirectional in horizontal plane for CP.

• Applications: FM broadcasting, satellite communication (CP reception).

3. APERTURE & REFLECTOR ANTENAS

Horn Antennas

• Types:

  • E-plane horn: Flared in E-plane.

  • H-plane horn: Flared in H-plane.

  • Pyramidal horn: Flared in both planes (most common).

• Gain Calculation:

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

where $$\displaystyle A_{phys} = a \times b $$ (aperture dimensions), $$\displaystyle \eta_a $$ = efficiency (0.5–0.7).

For pyramidal horn, $$\displaystyle \eta_a \approx 0.51 $$ (approx).

• Aperture Efficiency: $$\displaystyle \eta_a = \eta_{illum} \cdot \eta_{spill} \cdot \eta_{block} $$ (illumination, spillover, blockage).

Parabolic Reflector Antenna

• Principle: Parallel rays from feed reflect off paraboloid → converge at focus (reverse for transmitting). Phase equality along aperture.

• Feed Mechanisms:

  • Axial feed: Feed at focus; simple but blocks aperture.

  • Offset feed: Feed off-axis; no blockage, lower sidelobes.

  • Cassegrain: Subreflector (hyperbolic) reflects to feed behind dish.

• Design Considerations:

  • Illumination Taper: Edge illumination ~ -10 dB to -15 dB for good efficiency/sidelobe trade-off.

  • Spillover: Feed radiation missing reflector → increases noise, reduces efficiency.

  • Aperture Blockage: Feed/support structures shadow aperture → reduces gain, increases sidelobes.

• Gain: $$\displaystyle G \approx \eta_a \left(\frac{\pi D}{\lambda}\right)^2 $$ (for circular dish, $D$ diameter).

[!TIP] Blockage Effect: Central blockage reduces gain by factor $$\displaystyle \left(1 - (d/D)^2\right)^2 $$ (d = blockage diameter) and raises sidelobes.

Slot Antennas

• Concept: Aperture in conducting plane. Radiates from slot edges.

• Babinet's Principle: Fields of complementary structures (slot vs dipole) related:

$$\mathbf{E}_{slot} \propto \mathbf{H}_{dipole}, \quad \mathbf{H}_{slot} \propto -\mathbf{E}_{dipole}$$

Impedances: $$\displaystyle Z_{slot} Z_{dipole} = \eta^2/4 $$.

• Rectangular Slot in Infinite Ground Plane:

  • Radiation Pattern: Same as dipole orthogonal to slot. For $a \times b$ slot ($a$ along $x$, $b$ along $y$), E-plane cut ($$\displaystyle \phi=0 $$) pattern:

$$E_\theta \propto \frac{\sin\left(\frac{\beta a}{2} \sin\theta\right)}{\frac{\beta a}{2} \sin\theta} \cdot \cos\left(\frac{\beta b}{2} \cos\theta\right)$$

• Directivity: $$\displaystyle D \approx \frac{4ab}{\lambda^2} $$ (for $a,b \gg \lambda$).

• Applications: Slotted waveguide arrays (radar), aircraft wing slots.

Lens Antennas (Definitions)

• Principle: Dielectric lens (or metal-plate lens) focuses radiation from feed like optical lens.

• Types: Dielectric lens (Luneburg, hemispherical), metal-plate lens (E-plane, H-plane).

• Advantages: Low loss, good scan capability.

• Disadvantages: Heavy, bulky, frequency-sensitive (dielectric constant dispersion).

Tapered Apertures vs Uniform Apertures

• Uniform Aperture: Constant amplitude across aperture → high gain but high sidelobes (~ -13.2 dB for rectangular).

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

  • Advantages: Reduces sidelobes significantly (Taylor: -20 to -30 dB).

  • Disadvantages: Slightly lower gain, wider main beam.

• Tapering Functions:

  • Binomial: $A(x) \propto \binom{N}{k}$; sidelobes eliminated, but wide beam.

  • Taylor: Optimized for specified sidelobe level; near-optimum gain for given sidelobe.

[!TIP] Trade-off: Sidelobe reduction always costs gain and beamwidth. Taylor distribution is popular for radar (low clutter).

4. ARRAY ANTENNAS & PATTERN SYNTHESIS

Antenna Arrays Fundamentals

• Pattern Multiplication Theorem: Total pattern = element pattern × array factor.

$$F_{total}(\theta,\phi) = f_{element}(\theta,\phi) \cdot F_{array}(\theta,\phi)$$

• Array Factor for $N$ identical elements:

$$F_{array} = \sum_{n=1}^N I_n e^{j \psi_n}$$

where $$\displaystyle \psi_n = \beta \mathbf{\hat{r}} \cdot \mathbf{d}_n + \alpha_n $$ (phase due to position $$\displaystyle d_n $$ and excitation phase $$\displaystyle \alpha_n $$).

Linear Array Design

• Broadside Array: Maxima at $$\displaystyle \theta = 90^\circ $$ (perpendicular to array axis).

  • Condition: All elements in phase ($$\displaystyle \alpha_n = 0 $$) and uniform spacing $d$.

  • Array Factor: $$\displaystyle F \propto \frac{\sin(N \psi/2)}{\sin(\psi/2)} $$, $$\displaystyle \psi = \beta d \cos\theta $$.

• End-fire Array: Maxima at $$\displaystyle \theta = 0^\circ $$ or $$\displaystyle 180^\circ $$ (along array axis).

  • Condition: Progressive phase $$\displaystyle \alpha = -\beta d $$ (or $+\beta d$) to steer beam along axis.

  • Array Factor: $$\displaystyle F \propto \frac{\sin(N (\psi+\beta d)/2)}{\sin((\psi+\beta d)/2)} $$.

• Beam Scanning (off broadside by $$\displaystyle \theta_0 $$):

  • Required phase progression: $$\displaystyle \alpha = -\beta d \cos\theta_0 $$.

  • Example: Scan to $$\displaystyle 30^\circ $$ off broadside, $$\displaystyle d = 3\,cm $$, $$\displaystyle f=64\,kHz $$ → $$\displaystyle \lambda = c/f \approx 4687\,m $$, $\beta d \approx 0.00004$ rad → $\alpha \approx 0$ (nearly broadside). Note: For HF, $d \ll \lambda$ so scanning limited.

• Directivity vs Array Length: $$\displaystyle D \approx \frac{2L}{\lambda} $$ for broadside (end-fire: $$\displaystyle D \approx \frac{4L}{\lambda} $$), where $$\displaystyle L = Nd $$.

[!TIP] Grating Lobes: Occur when $$\displaystyle d > \lambda $$ and scanning. Condition to avoid: $$\displaystyle d \leq \frac{\lambda}{1+|\sin\theta_0|} $$.

Schelkunoff Unit Circle Method

• Polynomial Representation: Array factor $$\displaystyle F(z) = \sum_{n=0}^{N-1} I_n z^n $$, where $$\displaystyle z = e^{j\psi} $$.

• Root Placement: Zeros of $F(z)$ on unit circle correspond to null directions.

• Design Example: For $$\displaystyle N=4 $$, zeros at $$\displaystyle \psi = 90^\circ, 180^\circ, 270^\circ $$ → $$\displaystyle z = j, -1, -j $$. Solve polynomial → excitation coefficients.

• Advantage: Graphical synthesis; ensures nulls at specified angles.

Binomial Array

• Excitation Coefficients: Binomial coefficients (Pascal's triangle): $$\displaystyle I_n = \binom{N-1}{n} $$.

• Pattern: No sidelobes (except main lobe), but wide main beam.

• Application: When sidelobe suppression is critical (e.g., low interference).

Taylor Synthesis

• For Sum Patterns: Approximate ideal pattern with specified sidelobe level.

• Method: Start with continuous distribution $$\displaystyle I(x) = I_0 \frac{\sin(\pi x)}{\pi x} \prod_{n=1}^{\bar{n}-1} \frac{1 - (x/n)^2}{1 - (x/\bar{n})^2} $$, where $\bar{n}$ controls sidelobe level.

• Discrete Approximation: Sample continuous distribution for array elements.

• Result: Near-optimum gain for given sidelobe level; first $\bar{n}-1$ sidelobes equal, rest decay.

Planar Arrays

• Configuration: Elements on 2D grid (rectangular, circular).

• Pattern: $$\displaystyle F(\theta,\phi) = f_{element}(\theta,\phi) \cdot F_x(\theta,\phi) \cdot F_y(\theta,\phi) $$ (separable).

• Requirements: Control both azimuth and elevation patterns; spacing $$\displaystyle d_x, d_y \leq \lambda/2 $$ to avoid grating lobes in both planes.

• Applications: Phased array radars, base station antennas.

5. WIDEBAND & SPECIAL ANTENNAS

Log-Periodic Antenna

• Structure: Series of driven and parasitic elements (toothed) along boom; lengths and spacings follow $$\displaystyle \tau = \frac{L_{n+1}}{L_n} < 1 $$, $$\displaystyle \sigma = \frac{s_n}{2L_n} \approx 0.05–0.2 $$.

• Frequency-Independent Principle: Input impedance and radiation pattern repeat every $\log f$ cycle. Self-similar structure.

• Operation: At any frequency, elements near $\lambda/2$ are active; others are too long/short → broadband.

• Bandwidth: Limited by smallest/largest elements; can exceed 10:1.

• Applications: TV reception, EMC testing, wideband communication.

[!TIP] Key Parameter: $\tau$ (scale factor) determines bandwidth: $$\displaystyle \text{BW} = \frac{f_{max}}{f_{min}} = \tau^{-(N-1)} $$, $N$ = number of elements.

Helical Antenna

• Normal Mode ($D \ll \lambda$, $C \approx \lambda$):

  • Radiation Pattern: Figure-8 along helix axis (like dipole).

  • Applications: HF communications (broadside radiation).

• Axial Mode ($D \approx \lambda/3$, $C \approx \lambda$, $L \approx n\lambda$):

  • Radiation: Circularly polarized wave along helix axis (right-hand or left-hand CP).

  • Gain: $$\displaystyle G \approx 15 \left(\frac{C}{\lambda}\right)^2 n $$ (approx). For $$\displaystyle C=\lambda $$, $$\displaystyle n=10 $$, $G \approx 15$ (11.8 dBi).

  • Half-power beamwidth: $$\displaystyle \text{HPBW} \approx \frac{52^\circ \lambda^{3/2}}{C \sqrt{n}} $$ (narrower with more turns).

  • Dimensions: $$\displaystyle C = \pi D \approx \lambda $$, $L \approx n\lambda$, pitch angle $$\displaystyle \alpha \approx 12^\circ–14^\circ $$.

Microstrip Antennas

• Construction: Patch (conducting), substrate (dielectric), ground plane.

• Radiation Mechanism: Fringing fields at patch edges; patch acts as $\lambda/2$ resonator.

• Feeding Techniques: Microstrip line, coaxial probe, aperture coupling, proximity coupling.

• Rectangular Patch Design Equations (for $$\displaystyle W > h $$):

  • Width $W$:

$$W = \frac{c}{2f_r} \sqrt{\frac{2}{\varepsilon_r + 1}}$$

\boxed{W = \frac{c}{2f_r} \sqrt{\frac{2}{\varepsilon_r + 1}}}

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

• Length $L$:

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

where $$\displaystyle \Delta L \approx 0.412h \frac{(\varepsilon_{eff}+0.3)(W/h+0.264)}{(\varepsilon_{eff}-0.258)(W/h+0.8)} $$

• Advantages: Low profile, conformal, lightweight, low cost, easy integration with circuits.

• Limitations: Narrow bandwidth (1–5%), low efficiency (due to dielectric/surface wave losses), low gain (6–8 dBi), spurious feed radiation.

[!TIP] Bandwidth Limitation: Primarily due to high $Q$ of cavity. Thicker substrate ($h \uparrow$) and lower $$\displaystyle \varepsilon_r $$ increase bandwidth.

Yagi-Uda Antenna

• Construction:

  • Driven element: $\lambda/2$ dipole (or folded dipole).

  • Reflector: $\lambda/2$ dipole, ~5% longer than driven, placed $\lambda/4$ behind, passive (short-circuited).

  • Directors: $\lambda/2$ dipoles, ~5% shorter than driven, spaced $\lambda/3$–$\lambda/4$ apart, passive.

  • Boom: Non-conductive support.

• Role of Reflector: Reflects forward radiation, increases forward gain, reduces backward radiation.

• Role of Directors: Phase acceleration; each director re-radiates with phase leading driven element → constructive interference in forward direction.

• Design Example (6 elements, 12 dB gain at 200 MHz):

  • $$\displaystyle f=200\,\text{MHz} \Rightarrow \lambda = 1.5\,m $$.

  • Driven: $$\displaystyle 0.5\lambda = 0.75\,m $$.

  • Reflector: $$\displaystyle 0.51\lambda = 0.765\,m $$, spacing $$\displaystyle 0.15\lambda = 0.225\,m $$.

  • Directors (4): lengths $0.48\lambda, 0.46\lambda, 0.44\lambda, 0.42\lambda$; spacings $0.15\lambda, 0.25\lambda, 0.35\lambda, 0.4\lambda$ (typical).

  • Gain $\approx 10–12$ dBi.

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

[!TIP] Design Trend: More directors → higher gain but diminishing returns; spacing ~0.3λ optimal.

6. PROPAGATION MECHANISMS (WAVE PROPAGATION)

Ground Wave Propagation

• Transition: Surface wave (bound to earth) → space wave (refracted upward) as height increases.

• Effect of Ground:

  • Conductivity $\sigma$: Higher $\sigma$ (sea water) → lower attenuation, longer range.

  • Permittivity $$\displaystyle \varepsilon_r $$: Affects wave tilt and attenuation.

  • Terrain: Smooth (water) → low loss; rough (hilly) → scattering, diffraction loss.

• Vertical Pattern Influence: Ground reflection creates wave tilt; for vertical polarization, ground wave stronger at low angles.

Sky Wave Propagation (Ionospheric)

• Ionosphere as Variable Refractive Index Medium: Plasma frequency $$\displaystyle f_p = 9\sqrt{N_e} $$ ($$\displaystyle N_e $$ = electron density). Refractive index $$\displaystyle n = \sqrt{1 - (f_p/f)^2} $$. Decreases with height → bends rays downward.

• Critical Frequency $$\displaystyle f_c $$: Max frequency reflected vertically ($$\displaystyle \theta_i=0 $$). $$\displaystyle f_c = 9\sqrt{N_{e,max}} $$ (for layer).

• Virtual Height $h'$: Apparent reflection height assuming straight-line propagation. $$\displaystyle h' > h_{true} $$ due to refraction.

• Maximum Usable Frequency (MUF):

  • Definition: Highest frequency that can be reflected for given path and angle.

  • Derivation: From Snell's law for ionosphere: $$\displaystyle n \sin\theta_i = \text{constant} $$. At reflection, $$\displaystyle n=0 $$ → $$\displaystyle \sin\theta_i = n_{0} = f_c/f $$.

$$\boxed{f_{MUF} = \frac{f_c}{\cos\theta_i}}$$

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

• Relation to Skip Distance $d$:

$$\boxed{d = 2 h' \tan\theta_i = 2 h' \sqrt{\left(\frac{f_{MUF}}{f_c}\right)^2 - 1}}$$

• Lowest Usable Frequency (LUF): Lowest frequency that provides acceptable signal-to-noise ratio; limited by atmospheric noise and absorption.

• Multi-hop Propagation: Signal reflects multiple times (earth-ionosphere) → longer distances.

[!TIP] MUF Calculation: Given $$\displaystyle f_c $$ and $$\displaystyle \theta_i $$, $$\displaystyle f_{MUF} = f_c / \cos\theta_i $$. For $$\displaystyle \theta_i=60^\circ $$, $$\displaystyle f_{MUF}=2f_c $$. Skip distance increases with MUF.

Space Wave Propagation

• Line-of-sight (LOS): Direct path; limited by Earth's curvature.

  • Radio Horizon: $$\displaystyle d \approx 4.12(\sqrt{h_t} + \sqrt{h_r}) $$ km ($h$ in meters).

• Beyond Horizon:

  • Super Refraction (Ducting): Refractive index decreases rapidly with height → rays bend more than Earth's curvature. Can trap waves in ducts (evaporation ducts over sea, surface ducts). Enables propagation beyond LOS (up to 500–1000 km).

  • Tropospheric Scattering: Turbulence in troposphere scatters energy; allows communication beyond horizon (up to 500 km). Scatter volume concept; requires high power, large antennas.

• Effects: Ducting → strong signals, fading; scattering → weak, fluctuating signals.

Tropospheric Propagation

• Troposphere Structure: Up to ~10 km; refractive index decreases with height (standard atmosphere: $n-1 \propto 1/h$).

• Ducts:

  • Surface-based duct: Strong inversion near surface.

  • Elevated duct: Inversion layer aloft.

• Use in Microwave: Ducting can cause anomalous propagation (beyond LOS) for 1–10 GHz; also causes interference.

7. ENVIRONMENTAL & PRACTICAL EFFECTS ON PROPAGATION

Effects of Atmosphere, Terrain, and Obstacles

• Atmosphere:

  • Refraction: Bends rays downward (effective Earth radius $k \approx 4/3$).

  • Absorption: By oxygen (60 GHz), water vapor (22 GHz, 183 GHz).

  • Scattering: By rain, fog (above 10 GHz).

• Terrain:

  • Roughness: Causes diffuse scattering; reduces specular reflection.

  • Slopes: Changes effective antenna height.

• Obstacles (buildings, vegetation):

  • Diffraction: Around edges; knife-edge model.

  • Penetration loss: Through walls, trees (frequency-dependent).

Earth’s Effect on Antenna Patterns

• Vertical Pattern Distortion: Ground reflection combines with direct wave → interference pattern.

• Image Theory (perfect ground):

  • Replace ground by image antenna (current opposite phase for vertical dipole, same phase for horizontal dipole).

  • Total field = direct + reflected (with phase change: $$\displaystyle 180^\circ $$ for vertical pol, $$\displaystyle 0^\circ $$ for horizontal pol over perfect conductor).

• Real Ground: Finite conductivity → reflection coefficient $$\displaystyle \Gamma < 1 $$, phase not exactly $$\displaystyle 0^\circ $$ or $$\displaystyle 180^\circ $$. Reduces gain, tilts pattern.

• Monopole Example: Over perfect ground, pattern $$\displaystyle \propto \cos(\frac{\beta h}{2} \cos\theta) $$; over real ground, lower gain, higher elevation angle for max.

Broadcast Antenna Arrays

• Horizontal Patterns: Controlled by array of towers (AM: vertical polarization; FM: horizontal).

• AM Broadcast Arrays:

  • Directional Arrays: Use towers with different excitations to shape horizontal pattern → protect other stations at night (skywave interference).

  • Pattern: Cardioid, multi-lobed.

• FM Broadcast Arrays:

  • Often circularly polarized; arrays for specific coverage.

• Coverage Area Optimization: Maximize signal in target area, minimize in others. Use superposition of array factors.

8. ADVANCED TOPICS & DESIGN TOOLS

Numerical Tools for Antenna Analysis

• Method of Moments (MoM): Integral equation solution; good for wires, surfaces. Basis functions, impedance matrix.

• Finite Element Method (FEM): Differential equation; good for complex dielectrics, cavities. Mesh generation.

• Finite Difference Time Domain (FDTD): Time-domain; broadband, transient analysis. Yee lattice.

• Software:

  • HFSS (High-Frequency Structure Simulator): FEM/MoM, 3D.

  • CST Microwave Studio: FDTD, integral equation.

  • NEC (Numerical Electromagnetics Code): MoM for wires.

• Application: Predict patterns, impedance, SAR, etc.

Antenna Gain Measurement

• Standard Gain Horn Method: Compare unknown antenna to calibrated gain horn.

$$G_{unknown} = G_{horn} + 10\log_{10}\left(\frac{P_{un}}{P_{horn}}\right) - 10\log_{10}\left(\frac{A_{un}}{A_{horn}}\right) - \text{losses}$$

or using Friis: $$\displaystyle G_{un} = G_{horn} + 10\log_{10}(P_{un}/P_{horn}) - 10\log_{10}((4\pi R/\lambda)^2) - L $$.

• Example: Given oscillator power, attenuator reading, distance → compute gain.

Virtual Height

• Definition: Apparent reflection height assuming straight-line propagation. $$\displaystyle h' > h_{true} $$ due to refraction.

• Measurement: Using ionosonde; send pulse, measure round-trip time $t$ → $$\displaystyle h' = c t / 2 $$.

• Use: In MUF and skip distance calculations.

V-Antenna

• Definition: Two long wire antennas joined at an angle $\alpha$ (typically $$\displaystyle 90^\circ $$–$$\displaystyle 120^\circ $$).

• Pattern: Main lobe tilted from broadside by $\alpha/2$.

• Application: HF communication, where broadside pattern not desired.

Flat Sheet and Corner Reflectors

• Flat Sheet Reflector: Flat conducting plate placed $\lambda/4$ behind dipole → increases gain by ~6 dB, narrows beam.

• Corner Reflector: Two flat sheets at $$\displaystyle 90^\circ $$ (or $$\displaystyle 60^\circ $$); dipole at apex. Gain $\approx 8–10$ dBi, unidirectional, wide bandwidth.

• Principle: Image in reflector(s) creates array effect; spacing $\lambda/4$ gives in-phase addition.

END OF UNIT 4 NOTES
Focus on High-Frequency topics: Retarded Potential, Oscillating Dipole, Dipoles, Long Wire, Yagi, Microstrip, Parabolic Reflector, Sky Wave (MUF/Skip), Ground Wave, Space Wave.