Antennas and wave Propagation (EC-602) - Unit 1 Short Notes

I. FUNDAMENTAL ANTENNA THEORY

Retarded Potentials

• Concept: Generalization of static potentials to time-varying sources, accounting for finite propagation speed \(c\).

• Derivation for sinusoidal sources: For current density \(\mathbf{J}(\mathbf{r}', t) = \mathbf{J}(\mathbf{r}') e^{j\omega t}\), the vector potential is

$$\mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}') e^{j\omega (t - |\mathbf{r}-\mathbf{r}'|/c)}}{|\mathbf{r}-\mathbf{r}'|} dV'$$

Similarly for scalar potential \(\phi\) with charge density.

• Significance: Enables calculation of radiated fields from arbitrary current distributions; essential for antenna analysis.

• Comparison with static potential: Static potentials assume instantaneous action at a distance (\(e^{j\omega t}\) factor only), while retarded potentials include delay \(|\mathbf{r}-\mathbf{r}'|/c\).

[!TIP] Retarded potentials are the foundation for deriving fields from any antenna; always use \(t_r = t - R/c\) with \(R = |\mathbf{r}-\mathbf{r}'|\).

Radiation from Elementary Sources

• Oscillating electric dipole (Hertzian dipole): Infinitesimal antenna of length \(dl \ll \lambda\) carrying current \(I_0\).

• Field components (far-field):

$$E_\theta = j \frac{\eta I_0 dl k}{4\pi r} \sin\theta \, e^{-jkr}$$

$$H_\phi = j \frac{I_0 dl k}{4\pi r} \sin\theta \, e^{-jkr}$$

where \(\eta = \sqrt{\mu_0/\varepsilon_0}\), \(k = 2\pi/\lambda\), \(r\) is distance.

• Power radiated:

$$P_{rad} = \frac{\pi \eta I_0^2 (dl)^2}{3 \lambda^2} = \frac{I_0^2 (dl)^2}{6\pi \eta} \left(\frac{2\pi}{\lambda}\right)^2$$

• Dependence: \(P_{rad} \propto (dl/\lambda)^2\); for fixed current, higher frequency (smaller \(\lambda\)) increases radiation.

[!TIP] Hertzian dipole fields are transverse in far-field; \(E_\theta\) and \(H_\phi\) are in phase and related by \(\eta\).

Antenna Parameters

• Radiation pattern: 3D plot of field/power vs direction. Cuts: E-plane (contains \(\mathbf{E}\) and \(\mathbf{r}\)), H-plane (contains \(\mathbf{H}\) and \(\mathbf{r}\)).

• Beamwidth: Half-power beamwidth (HPBW) or first-null beamwidth (FNBW).

• Directivity:

$$D = \frac{4\pi}{\Omega_A}$$

where \(\Omega_A\) = beam solid angle. For narrow beams, \(D \approx \frac{4\pi}{\theta_{BP} \phi_{BP}}\) (radians).

• Gain: \(G = \epsilon D\), where \(\epsilon\) = efficiency. In dB: \(G_{dBi} = 10 \log_{10} G\).

• Efficiency: \(\epsilon = \frac{R_{rad}}{R_{rad} + R_{loss}}\).

• Effective area: \(A_e = \frac{G \lambda^2}{4\pi}\).

• Effective length: \(h_e = \frac{V_{oc}}{E_{inc}}\) for open-circuit voltage.

• Relation: \(G = \frac{4\pi A_e}{\lambda^2}\).

[!TIP] Directivity is theoretical max; gain accounts for losses. Effective area relates receiving to transmitting gain.

Field Regions

• Fraunhofer distance: \(\boxed{R = \frac{2l^2}{\lambda}}\) where \(l\) = largest antenna dimension.

[!TIP] For pattern measurements, ensure \(r > R\); near-field scanners require transformation to far-field.

Far-Field Approximation

• Conditions: \(r \gg \lambda\) and \(r \gg l^2/\lambda\); also angular extent small.

• Assumptions:

  1. \(1/r\) dependence,

  2. Transverse fields (\(E_r, H_r \approx 0\)),

  3. Wave impedance \(\eta\).

• Importance: Simplifies field expressions; pattern depends only on angle, not \(r\). Basis for array theory and gain measurements.

[!TIP] Far-field fields are spherical waves locally plane; used in Friis transmission equation.

Ground Effects on Antennas

• Impact of ground: Conductivity \(\sigma\) and permittivity \(\varepsilon_r\) determine reflection coefficient.

• Reflection coefficients (for plane wave at angle \(\theta\) from normal):

  • Vertical polarization (E in plane of incidence):

$$\Gamma_v = \frac{\varepsilon_r \sin\theta - \sqrt{\varepsilon_r - \cos^2\theta}}{\varepsilon_r \sin\theta + \sqrt{\varepsilon_r - \cos^2\theta}}$$

• Horizontal polarization:

$$\Gamma_h = \frac{\sin\theta - \sqrt{\varepsilon_r - \cos^2\theta}}{\sin\theta + \sqrt{\varepsilon_r - \cos^2\theta}}$$

• Image theory: For perfect electric conductor (PEC), ground replaced by image antenna.

  • Vertical antenna: image in-phase (current same direction).

  • Horizontal antenna: image out-of-phase (current opposite).

• Effect on vertical patterns: For vertical antennas, ground reflection can enhance low-angle radiation (constructive interference) or cause nulls (destructive), depending on height and ground properties.

[!TIP] For AM broadcast (vertical monopoles), ground conductivity critical for low-angle radiation; poor ground increases losses.

II. WIRE ANTENNAS

Half-Wave Dipole

• Current distribution: \(I(z) = I_0 \cos(\beta z)\) for \(|z| \leq \lambda/4\), where \(\beta = 2\pi/\lambda\).

• Radiation pattern:

$$E_\theta \propto \frac{\cos\left(\frac{\pi}{2} \cos\theta\right)}{\sin\theta}$$

• E-plane: figure-8 (doughnut), H-plane: circle.

• Input impedance: \(Z_{in} \approx 73 + j42.5\ \Omega\) at resonance.

• Radiation resistance: \(R_{rad} \approx 73\ \Omega\).

• Influence of length:

  • Slightly longer: higher resistance, capacitive reactance.

  • Slightly shorter: lower resistance, inductive reactance.

  • Directional properties: as length increases, pattern splits into multiple lobes (e.g., 1.5λ dipole has 4 lobes).

[!TIP] Half-wave dipole is reference for gain (2.15 dBi); length tolerance ±5% acceptable.

Quarter-Wave Monopole

• Operation over ground plane: Uses image theory; equivalent to half-wave dipole in upper hemisphere.

• Radiation pattern: Same as half-wave dipole but only above ground; power radiated half of dipole.

• Fields: For perfect ground, \(E_\theta\) same as dipole for \(\theta \leq 90^\circ\), zero below.

• Input impedance: \(Z_{in} \approx 36.5 + j21.25\ \Omega\) (half of dipole).

• Applications: Mobile phones, FM broadcast (with ground plane).

[!TIP] Monopole gain is 2.15 dBi relative to isotropic; relative to dipole, gain is 0 dBD.

Long Wire Antennas (Traveling Wave)

• Construction: Wire length \(l \gg \lambda\), terminated with resistance \(R_T \approx 300-800\ \Omega\).

• Current distribution: Traveling wave, \(I(z) = I_0 e^{-j\beta z}\) (decaying if matched).

• Radiation pattern: End-fire along wire direction; maximum at angle \(\theta \approx \arccos(\lambda/l)\) for \(l \gg \lambda\).

  • Beverage antenna: long wire above ground, unidirectional, low angle radiation.

• Influence of length: Longer wire → narrower beam, higher gain (up to ~10 dBi for \(l \approx 10\lambda\)).

[!TIP] Long wire antennas are unidirectional due to termination; avoid reflections for pure traveling wave.

Folded Dipole

• Structure: Two parallel dipoles connected at ends; fed at center of one.

• Feed impedance transformation: Input impedance ≈ \(4 \times\) single dipole impedance (for equal diameters). So ~300 Ω for half-wave.

• Current distribution: Equal currents in both arms; voltage across feed point doubled.

• Applications: Broadband feed for Yagi-Uda, FM reception (300 Ω balanced line).

[!TIP] Folded dipole bandwidth ~15% wider than simple dipole due to increased impedance.

Other Wire Antennas

• Turnstile antenna: Two crossed dipoles, 90° phase shift → circular polarization.

• Rhombic antenna: Four-wire diamond, traveling wave with termination; broadside pattern, gain ~15 dBi.

• V-antenna: V-shaped wire, angle ~90°; broadside pattern, moderate gain.

III. ANTENNA ARRAYS

Array Fundamentals

• Array factor (AF): For \(N\) elements with currents \(I_n\), positions \(\mathbf{d}_n\), phases \(\beta_n\):

$$AF(\theta,\phi) = \sum_{n=1}^N I_n e^{j(k \mathbf{\hat{r}} \cdot \mathbf{d}_n + \beta_n)}$$

• Pattern multiplication theorem: Total pattern = element pattern × array factor (if elements identical and similarly oriented).

• Element patterns vs. array factor: Element pattern accounts for single antenna radiation; array factor accounts for spatial arrangement.

Linear Arrays

• Broadside: Maximum radiation perpendicular to array axis; phase progression \(\alpha = 0\).

• End-fire: Maximum radiation along array axis; phase progression \(\alpha = \mp k d\) (for forward/backward).

• Phased arrays: Steer beam by progressive phase \(\alpha = -k d \cos\theta_0\) (for array along \(z\), broadside at \(\theta_0=90^\circ\)).

• Element spacing \(d\):

  • \(d \leq \lambda\) avoids grating lobes (multiple maxima).

  • Grating lobes appear if \(d > \lambda\) and \(|\alpha| < k d\).

[!TIP] For scanning, maximum scan angle \(\theta_{max}\) without grating lobes: \(d \leq \frac{\lambda}{1 + |\sin\theta_0|}\).

Array Synthesis Methods

• Binomial array: Amplitude coefficients from binomial expansion \((1+x)^{N-1}\); no side lobes but wide main beam.

• Taylor distribution: Continuous approximation to reduce side lobes; parameter \(\bar{n}\) controls lobe level.

• Schelkunoff unit circle method: Place zeros of AF on unit circle in complex plane; nulls at \(\Phi = kd\cos\theta + \alpha\). Example: zeros at \(\Phi = 90^\circ,180^\circ,270^\circ\) → array coefficients from polynomial.

Yagi-Uda Antenna

• Construction: Driven element (half-wave dipole), reflector (longer, ~5% longer), directors (shorter, ~5% shorter).

• Pattern formation: Reflector induces reverse current, reinforcing forward wave; directors phase-advance waves, enhancing forward gain.

• Design parameters:

  • Reflector length: \(0.5\lambda\) to \(0.52\lambda\)

  • Director lengths: decreasing from driven element.

  • Spacing: reflector ~0.15–0.2λ, directors ~0.3–0.4λ.

• Gain: 7–15 dBi depending on number of directors.

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

[!TIP] Yagi gain increases with directors but with diminishing returns; typically 10–12 directors for ~15 dBi.

Log-Periodic Antenna

• Structure: Toothed elements of varying lengths, fed transverely along boom. Scale factor \(\tau = \frac{L_{n+1}}{L_n} < 1\), angle \(\alpha\).

• Frequency-independent principle: Geometry self-similar; input impedance and pattern repeat every decade.

• Wideband characteristics: Operates over 2:1 or more bandwidth; gain ~7–10 dBi.

• Applications: Wideband TV reception, EMC measurements.

[!TIP] LPDA feed at apex; active region where element length ≈ λ/2 shifts with frequency.

IV. APERTURE ANTENNAS

Horn Antennas

• Types: Pyramidal (rectangular), conical, sectoral (E-plane or H-plane).

• Gain calculation:

$$G = \frac{4\pi A_e}{\lambda^2} = \eta \left(\frac{\pi D}{\lambda}\right)^2$$

for circular aperture; for rectangular, \(A_e = a b\). Efficiency \(\eta \approx 0.5–0.6\).

• Feeding: Waveguide, coax-to-waveguide transition.

[!TIP] Horn gain measured by comparison method; directivity from aperture field distribution.

Slot Antennas

• Babinet’s principle: Complementary antennas (slot in infinite ground vs. dipole) have identical \(E\) and \(H\) fields interchanged, with

$$\mathbf{E}_{slot} = \frac{\lambda^2}{4\pi} \mathbf{H}_{dipole}, \quad \mathbf{H}_{slot} = -\frac{\lambda^2}{4\pi} \mathbf{E}_{dipole}$$

• Rectangular slot in infinite ground: Slot dimensions \(a \times b\); fed by probe or waveguide.

• Radiation pattern: Similar to dipole but with H- and E-planes swapped.

$$E_\phi \propto \frac{\sin\left(\frac{\pi a}{\lambda} \sin\theta \cos\phi\right)}{\frac{\pi a}{\lambda} \sin\theta \cos\phi} \cdot \frac{\sin\left(\frac{\pi b}{\lambda} \sin\theta \sin\phi\right)}{\frac{\pi b}{\lambda} \sin\theta \sin\phi}$$

[!TIP] Slot antennas are used in arrays and aircraft (flush-mounted).

Parabolic Reflector Antennas

• Geometry: Paraboloid of revolution; focal point \(f = D^2/(16C)\) where \(C\) = depth.

• Focusing principle: Parallel rays reflect to feed (prime focus) or to secondary (Cassegrain).

• Feed mechanisms:

  • Prime focus: feed at focal point.

  • Cassegrain: subreflector (hyperbolic) reflects to feed behind vertex.

• Aperture efficiency: \(\eta = \eta_{spillover} \times \eta_{taper} \times \eta_{blockage}\).

• Tapered apertures: Aperture illumination tapered (e.g., cosine) reduces side lobes at cost of gain.

• Aperture blockage: Feed and subreflector block part of aperture, reducing gain and increasing side lobes.

[!TIP] Typical efficiency 50–70%; Cassegrain reduces spillover but adds blockage.

Lens Antennas

• Types: Dielectric lens (Luneburg, hemispherical), zoned lens (reduces thickness).

• Operation: Refracts plane wave to feed or vice versa; focal length \(f = \frac{R}{2(\sqrt{\varepsilon_r}-1)}\) for hemispherical.

• Applications: Microwave systems, radar, where reflectors impractical.

Flat Sheet and Corner Reflectors

• Flat sheet: Flat conducting plate; acts as aperture when fed by dipole; pattern bidirectional.

• Corner reflector: Two flat plates at 90°; unidirectional pattern, gain ~10–12 dBi.

• Directivity: Increases with size; edge effects cause diffraction.

V. PRINTED AND PLANAR ANTENNAS

Microstrip (Patch) Antennas

• Rectangular patch design:

  • Width \(W\):

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

• Effective dielectric constant:

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

• Length \(L\) (for resonance):

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

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

• Substrate parameters: \(\varepsilon_r\) and height \(h\) affect bandwidth (\(BW \propto h/\varepsilon_r\)) and efficiency.

• Feeding techniques:

  • Microstrip line (impedance matching via inset).

  • Coaxial probe (vertical feed).

  • Aperture coupling (slot between ground and patch).

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

• Limitations: Narrowband (typically 1–5%), low efficiency (due to surface waves, dielectric loss), spurious feed radiation.

[!TIP] Patch length ≈ \(\lambda_0/(2\sqrt{\varepsilon_{eff}})\); increase \(h\) for wider bandwidth but higher surface waves.

Planar Arrays

• Configuration: 2D grid of elements (rectangular, circular).

• Pattern synthesis: Separable into two linear arrays; control both azimuth and elevation patterns.

• Applications: Phased array radars, satellite communications, beam steering.

VI. WAVE PROPAGATION MECHANISMS

Ground Wave Propagation

• Components:

  • Surface wave: Bound to ground, follows Earth’s curvature; dominant at low frequencies (MF, HF).

  • Space wave: Direct + ground-reflected; dominant at VHF/UHF.

• Transition: At higher frequencies, space wave dominates; surface wave attenuates rapidly.

• Influence of terrain:

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

  • Curvature: limits range; horizon distance \(d \approx \sqrt{2h_t} + \sqrt{2h_r}\) (km) with heights in meters.

• Attenuation:

$$A \approx 8.7 \alpha \ d$$

(dB) where \(\alpha\) = attenuation constant (dB/km), depends on frequency and ground.

[!TIP] Ground wave used for AM broadcast; over seawater, range > 1000 km at 1 MHz.

Sky Wave Propagation (Ionospheric)

• Ionospheric layers: D (60–90 km), E (90–150 km), F (150–500 km, splits into F1/F2 daytime).

• Critical frequency:

$$f_c = 9\sqrt{N_{max}}\ \text{MHz}$$

where \(N_{max}\) = maximum electron density (electrons/m³).

• Maximum Usable Frequency (MUF):

$$\text{MUF} = \frac{f_c}{\cos\theta}$$

where \(\theta\) = angle of incidence at ionosphere.

• Skip distance: Minimum ground distance with no signal;

$$d_{skip} = 2(R + h) \sin\theta \approx 2\sqrt{2Rh}$$

for small \(\theta\), \(h\) = virtual height.

• Lowest Usable Frequency (LUF): Determined by D-layer absorption; increases with solar activity.

• Virtual height: Apparent reflection height from ionogram: \(h' = \frac{c \Delta t}{2}\).

• Derivation of MUF-skip relation:

  From geometry: \(\sin\theta = \frac{d}{2(R+h)}\), and \(\cos\theta = \sqrt{1 - \sin^2\theta}\). MUF = \(f_c / \cos\theta\).

[!TIP] Optimum working frequency (OWF) ≈ 0.85 MUF to avoid fading.

Tropospheric Propagation

• Refraction: Standard atmosphere: refractive index \(n = 1 + 40.3 \times 10^{-6} N / f^2\)? Actually, for radio, \(n \approx 1 + \frac{79.5 P}{T} \times 10^{-6}\) (P in hPa, T in K). Gradient \(dn/dh \approx -4 \times 10^{-7}\)/m → Earth curvature equivalent radius \(R_e/(1+R_e dn/dh) \approx 4/3 R_e\).

• Super refraction: \(dn/dh\) more negative → ray curvature > Earth curvature; extends radio horizon.

• Subrefraction: \(dn/dh\) less negative or positive → ray curvature < Earth curvature; reduces horizon.

• Tropospheric scattering: Turbulence causes scattering; enables beyond-horizon reception (VHF/UHF).

• Duct propagation:

  • Surface-based duct: \(dn/dh < -157 \times 10^{-6}\)/m (evaporation duct).

  • Elevated duct: temperature inversion.

  • Traps waves, low loss over hundreds of km (microwave).

[!TIP] Ducting common over oceans; causes anomalous propagation for radar.

Environmental Effects

• Atmospheric conditions: Humidity, precipitation cause absorption (especially >10 GHz).

• Terrain and obstacles: Diffraction over hills, reflection from buildings → multipath fading.

• Fading:

  • Fast fading (multipath, wavelength scale).

  • Slow fading (shadowing, terrain scale).

• Impact: Signal strength fluctuations, delay spread, polarization changes.

VII. SPECIAL TOPICS & DESIGN TOOLS

Helical Antenna

• Normal mode: Circumference \(C \approx \lambda\), pitch \(S \ll \lambda\); broadside radiation, circular polarization.

• Axial mode: \(C \approx \lambda\), \(S \approx \lambda/4\); end-fire, high gain (10–15 dBi), circular polarization.

  • Axial ratio: \(AR \approx \sqrt{1 + \left(\frac{2S}{\lambda}\right)^2}\) for \(C=\lambda\).

• Applications: Satellite communications (axial mode), VHF/UHF (normal mode).

Numerical Tools for Antenna Analysis

• HFSS: Finite element method (FEM); high-frequency structures.

• CST: Time domain solver; transient analysis.

• NEC: Method of Moments (MoM); wire antennas, ground planes.

• Applications: Design, optimization, near-to-far field transformation, impedance matching.

Reciprocity Theorem

• Statement: For two antennas A and B,

$$Z_{AB} = Z_{BA}$$

where \(Z_{AB}\) = open-circuit voltage at A due to current in B.

• Proof: From Lorentz reciprocity: \(\int_V (\mathbf{J}_1 \cdot \mathbf{E}_2 - \mathbf{J}_2 \cdot \mathbf{E}_1) dV = 0\).

• Implication: Transmitting pattern of antenna A equals receiving pattern of antenna A.

Antenna Gain and Effective Area

• Detailed derivation:

  From Friis: \(P_r = P_t G_t G_r \left(\frac{\lambda}{4\pi R}\right)^2\).

  For receiving, \(P_r = \frac{|V_{oc}|^2}{2R_{rad}}\) and \(V_{oc} = E_{inc} h_e\).

  Also, \(G = \frac{4\pi A_e}{\lambda^2}\) from power conservation.

• Measurement techniques:

  • Gain: comparison with standard horn, or using Friis with known distance.

  • Effective area: measure gain, then compute.

VIII. FREQUENTLY ASKED DERIVATIONS & SHORT NOTES

(Integrated in relevant sections above)

• Derivation of radiated fields from Hertzian dipole: In Radiation from Elementary Sources.

• Power radiated by quarter-wave monopole: In Quarter-Wave Monopole (half of half-wave dipole).

• Radiation pattern of rectangular slot: In Slot Antennas (Babinet’s principle).

• Relation between MUF and skip distance: In Sky Wave Propagation (geometry derivation).

• Pattern multiplication and earth’s effect on vertical patterns: In Array Fundamentals and Ground Effects.

• Taylor synthesis: In Array Synthesis Methods.

• Schelkunoff unit circle: In Array Synthesis Methods (null placement).

• Microstrip patch dimensions: In Microstrip Antennas (formulas).

• Horn antenna gain calculation: In Horn Antennas (aperture efficiency).

• Critical frequency, virtual height, maximum coverage: In Sky Wave Propagation (\(f_c = 9\sqrt{N_{max}}\), \(h' = c\Delta t/2\), \(d_{max} \approx 2\sqrt{2Rh}\)).

[!EXAM TIP] Past papers emphasize derivations: retarded potential, Hertzian dipole fields, monopole power, MUF-skip, slot pattern, array synthesis. Practice these step-by-step with clear assumptions (far-field, sinusoidal sources). For numerical problems, always state formulas first (e.g., \(G = 4\pi A_e/\lambda^2\), MUF = \(f_c/\cos\theta\)).