Q9Antenna And Wave Propagation

(a) Critical Frequency, Critical Angle, and Relationship with Electron Density

Critical frequency (fc): the highest frequency at which a radio wave, transmitted vertically upward (at normal/zero incidence angle) toward the ionosphere, is still reflected back to earth by a given ionospheric layer, rather than penetrating through it into space — at any frequency above fc, a vertically-incident wave passes through that layer without being reflected.

Critical angle: for a wave transmitted at a frequency higher than the critical frequency, reflection can still occur, but only if the wave is incident on the ionosphere obliquely (at some angle from the vertical) rather than vertically — the critical angle is the maximum angle of incidence, measured from the vertical, at which a wave of a given frequency (specifically, at the layer's own critical frequency) is just still reflected; equivalently, for a wave of frequency exceeding fc, reflection requires the angle of incidence to exceed a corresponding minimum oblique angle.

Relationship with electron density: the critical frequency of an ionospheric layer is directly related to that layer's maximum electron density, Nmax, by the standard formula:

showing that critical frequency increases with the square root of the layer's peak electron density, since a higher electron density provides a correspondingly stronger refractive bending effect on an incident radio wave, allowing higher frequencies to still be turned back before penetrating the layer. The critical angle is likewise related to electron density (and frequency) through the secant law of oblique-incidence reflection (discussed further in the corresponding MUF answer elsewhere in this paper), since the required degree of oblique refraction needed to reflect a wave of a given frequency depends directly on how that frequency compares to the layer's critical frequency (itself set by Nmax) — a layer with higher electron density (and hence higher fc) can reflect a wider range of frequencies at a wider range of incidence angles (a larger critical angle) compared to a layer with lower electron density.

(b) Effects of Ground on Antennas; Grounded and Ungrounded Antennas

The presence of the earth's ground beneath an antenna significantly modifies its radiation pattern, input impedance, and overall performance compared to the same antenna operating in free space, primarily through the ground-reflection effect (in which a portion of the antenna's downward-directed radiation reflects off the ground surface and combines, constructively or destructively depending on height and angle, with the antenna's direct radiation, modifying the overall vertical radiation pattern) and, for a antenna deliberately connected to ground, through the antenna's interaction with the conducting (or, for real, imperfectly-conducting soil, partially-lossy) ground acting as part of its own effective radiating/current-return structure.

Grounded antennas: antennas that are deliberately and directly electrically connected to the earth/ground at their base (such as a quarter-wave monopole mounted directly on a ground plane or the actual earth surface), relying on the ground itself (in combination with the classical image theory, in which the ground acts electrically like a mirror, creating an image of the antenna's current distribution below the ground surface) to complete the effective radiating structure — a quarter-wave grounded monopole, for instance, radiates a pattern and exhibits an input impedance essentially identical to the upper half of a full half-wave dipole in free space, precisely because the ground's image effect supplies the electrically-missing lower half of the equivalent dipole structure. The quality (conductivity) of the actual ground/earth beneath a grounded antenna directly affects its performance, since real (imperfectly conducting) earth introduces ground losses that reduce radiation efficiency, motivating the common practice of installing an artificial ground system (a network of buried radial ground wires) beneath grounded monopole antennas (such as AM broadcast towers) to improve effective ground conductivity and reduce these losses.

Ungrounded antennas: antennas that form a complete, self-contained radiating structure without relying on any direct electrical connection to ground (such as a center-fed half-wave dipole mounted at some height above ground, or a horizontal Yagi-Uda antenna) — the ground beneath an ungrounded antenna still influences its performance through the ground-reflection effect on its radiation pattern (as discussed above), and through image-theory-based modification of its input impedance at low heights above ground, but the antenna does not rely on any direct galvanic ground connection to complete its own basic radiating current structure, in contrast to a grounded monopole.

Capacitance and Inductance Loading of Grounded Antennas

A physically short grounded monopole (electrically shorter than a quarter wavelength, as is common for MF/LF broadcast towers where a full quarter-wave tower would be impractically tall) presents a feed-point impedance that is predominantly capacitive reactance rather than the purely resistive impedance of a resonant quarter-wave monopole, together with a reduced radiation resistance, both of which degrade matching efficiency and reduce radiated power for a given input power unless corrected. Inductive loading compensates for this by inserting a lumped loading inductor (a loading coil) in series with the antenna, sized to cancel the antenna's own capacitive reactance and bring the total feed-point reactance to zero (resonance) at the operating frequency, restoring an easily matched, purely resistive feed impedance; the coil is commonly placed at the antenna base (base loading), partway up the tower (center loading, generally yielding a somewhat more uniform current distribution and higher effective height than base loading), or, less commonly, near the top. Capacitance (top) loading instead adds capacitive structure — a horizontal top-hat, disc, or umbrella of wires — at the top of a short vertical radiator, which increases the effective electrical length and current magnitude over the upper part of the antenna (raising its effective height and radiation resistance) without needing to increase the physical tower height, a technique long used for practical MF/LF broadcast and standard-frequency transmitting towers where the physically achievable tower height falls well short of a full quarter wavelength. Both loading techniques trade off increased structural/electrical complexity for a substantial improvement in the achievable radiation efficiency and impedance match of an antenna that is physically constrained to be electrically short.

Effect of Antenna Height Above Ground on Radiation Pattern

The height of an antenna above the ground strongly shapes its vertical-plane radiation pattern through the constructive and destructive interference between the antenna's direct radiation and its ground-reflected image, as captured by the standard height-gain (array) factor for a horizontal antenna at height h: F(theta)=2sin(betah*sin(theta)), where theta is the elevation angle measured from the horizon. At very low heights (h much less than lambda/4), this factor suppresses low-angle (near-horizon) radiation almost entirely while favoring high-angle radiation, which is generally undesirable for long-distance sky-wave or ground-wave service that depends on low-angle energy; as the height is increased toward and beyond roughly lambda/2, an increasing number of elevation lobes appear in the vertical pattern, with the lowest lobe progressively moving down toward the horizon, improving low-angle radiation useful for long-distance HF sky-wave communication (which typically requires low take-off angles for maximum single-hop range) — this is precisely why HF communication antennas intended for long-distance service are commonly mounted as high as practicable above ground (often a full wavelength or more), while antennas intended for short-range, high-angle 'NVIS' (near-vertical-incidence skywave) regional coverage are deliberately mounted low (a fraction of a wavelength), deliberately exploiting the opposite, high-angle-favoring end of the same height-dependent interference behaviour.

(c) Numerical: Range of Communication System

Given: f=60 MHz, Pt=2 kW, directive antenna field strength = 5 times that of a half-wave reference antenna, ht=60 m, hr=6 m, required field strength for satisfactory reception=100 microV/m.

For space-wave (line-of-sight, two-ray ground-reflection) propagation, the field strength at a receiver distance d, referenced to a half-wave dipole transmitting antenna, is given by the standard formula (as also derived symbolically elsewhere in this paper's companion 2019 examination paper):

Since the actual (directive) antenna's field strength is 5 times this reference value, the effective field strength formula for this problem is:

Calculating the wavelength: at f=60 MHz, lambda=c/f=(3x10^8)/(60x10^6)=5 m.

Solving for range d: setting E=100 microV/m=100x10^-6 V/m and solving for d:

Substituting ht=60, hr=6, Pt=2000 W, lambda=5, E=100x10^-6:

Result: the maximum range of the communication system is approximately 105.3 km. This result is consistent with typical VHF space-wave propagation range for the given moderate antenna heights and transmitter power, and illustrates the characteristic inverse-square (1/d^2) dependence of space-wave field strength on distance (rather than the simpler inverse-first-power 1/d dependence of a pure free-space wave), a direct consequence of the constructive/destructive ground-reflection interference mechanism underlying space-wave (line-of-sight) propagation over a reflecting earth surface.