Q2Antenna And Wave Propagation

(a) Equivalent Circuit of Antenna

The equivalent circuit representing a transmitting antenna, as seen from its feed terminals, models the antenna as a combination of a Thevenin equivalent source (representing the transmitter/generator driving the antenna, with its own internal generator impedance Zg=Rg+jXg) connected to the antenna's own input impedance, which itself consists of three series elements: the radiation resistance Rr (a fictitious resistance representing the power that is genuinely radiated away as electromagnetic energy into space); the loss resistance RL (representing the real ohmic power dissipated as heat within the antenna's own imperfectly-conducting metal conductors and any lossy dielectric material used in its construction); and the antenna reactance Xa (representing the reactive, non-radiating stored energy in the antenna's near-field region, which must be tuned out, e.g. via a matching network, for maximum power transfer at the desired operating frequency).

For maximum power transfer (and hence maximum radiated power) from the generator to the antenna, conjugate impedance matching is required between the generator impedance Zg and the antenna's total input impedance Za=(Rr+RL)+jXa, i.e., Zg=Za* — this equivalent circuit representation allows the antenna to be treated using ordinary circuit-theory power-transfer and matching analysis, while the radiation resistance Rr specifically captures the desired radiated power, distinct from the parasitic loss represented by RL.

Polarisation

Polarisation describes the time-varying orientation and shape traced out by the tip of the antenna's radiated (or received) electric field vector, observed at a fixed point in space over one complete cycle of oscillation. Linear polarisation occurs when the electric field vector oscillates back and forth along a single fixed straight line direction (e.g., purely vertical or purely horizontal polarisation), the simplest and most common polarisation type, produced by simple linear-wire antennas such as dipoles and monopoles. Circular polarisation occurs when the electric field vector's tip traces out a perfect circle as it rotates at the wave's oscillation frequency, with the field magnitude remaining constant while its direction continuously rotates — produced by combining two orthogonal, equal-amplitude linear field components with a 90-degree phase difference between them (using, e.g., crossed dipoles or helical antennas), and classified as either right-hand or left-hand circular polarisation depending on the rotation sense. Elliptical polarisation is the general, most common real-world case, occurring when the electric field vector's tip traces out an ellipse (rather than a perfect circle or a straight line), arising when the two orthogonal field components have either unequal amplitudes, a phase difference other than exactly 90 degrees, or both — linear and circular polarisation are simply the two special limiting cases of the more general elliptical polarisation.

Front-to-Back Ratio (FBR)

The front-to-back ratio is defined as the ratio of the power radiated (or radiation intensity) in the antenna's direction of maximum radiation (the 'front' of the pattern) to the power radiated in the exactly opposite direction, 180 degrees away (the 'back' of the pattern):

FBR is usually expressed in decibels and serves as an important figure of merit for directional antennas (such as Yagi-Uda arrays, horn antennas, and reflector antennas) that are specifically intended to radiate predominantly in one direction while suppressing radiation in the opposite direction — a high FBR (typically 15-25 dB or more for well-designed directional antennas) indicates good front-direction concentration of radiated power with minimal unwanted back-lobe radiation, which is particularly important in applications like point-to-point microwave links and directional broadcast antennas where minimizing interference or reception from the rear direction is desired.

Antenna Bandwidth

Antenna bandwidth is the range of frequencies over which the antenna's performance, with respect to some specified characteristic, meets a defined standard — bandwidth may be specified with respect to input impedance/VSWR (the frequency range over which the antenna's impedance match to its feed line remains acceptable, e.g. VSWR less than 2:1), radiation pattern characteristics (the frequency range over which the pattern shape, gain, polarization, or side-lobe level remain within acceptable limits), or a combination of both — for broadband antennas, bandwidth is often expressed as a ratio of upper to lower frequency (e.g., 2:1 bandwidth), while for narrowband antennas it is more commonly expressed as a percentage of the center frequency. Antenna bandwidth requirements vary greatly by application, with simple resonant antennas (like half-wave dipoles) typically offering only a few percent fractional bandwidth, while specifically broadband-designed antennas (log-periodic, biconical, spiral) can achieve bandwidth ratios of 10:1 or considerably more.

The physical root cause of an antenna's limited bandwidth is directly tied to its stored reactive energy relative to its radiated energy — a resonant antenna behaves, near its design frequency, much like a high-Q tuned circuit, and just as a high-Q tuned circuit exhibits a narrow bandwidth around its resonant frequency, an antenna whose structure stores a large amount of reactive near-field energy relative to the power it actually radiates will likewise show its input impedance and radiation performance change rapidly with frequency, yielding a narrow bandwidth. Electrically small antennas (whose physical size is much less than a wavelength) are especially bandwidth-limited for exactly this reason, since their small size forces a large fraction of the total field energy to remain reactive and non-radiating close to the antenna, while only a comparatively small fraction of the total stored energy is actually radiated away — this fundamental physical constraint, formalized in antenna theory as the Chu-Wheeler limit, states that the product of an electrically small antenna's achievable bandwidth and its achievable efficiency/gain is bounded above by a value set purely by the ratio of its physical size to the operating wavelength, meaning a designer can trade off bandwidth against size (or against efficiency) but cannot simultaneously make an antenna both very small and very broadband without incurring some other significant performance penalty. In practical antenna engineering, techniques used to broaden bandwidth beyond what a simple thin-wire resonant structure would achieve include increasing the effective conductor thickness or using a cage/fat-dipole structure (which lowers the effective Q of the resonant structure), adding matching or loading networks specifically designed to maintain an acceptable impedance match over a wider frequency span, and using inherently broadband geometries such as biconical, log-periodic, or spiral antenna structures whose self-similar or continuously-scaled geometry supports a naturally wideband, frequency-independent radiation and impedance behavior.

(b) Numerical: Maximum Effective Aperture and Directivity of Short Dipole at 450 MHz

Given: operating frequency f=450 MHz, antenna type = short dipole.

Step 1 - Wavelength: the free-space operating wavelength is calculated from the standard relation between wavelength, speed of light, and frequency:

Step 2 - Directivity of short dipole: the directivity of a short (infinitesimal/Hertzian, or triangular-current short) dipole is a standard, well-known result derived from its characteristic sin(theta) far-field radiation pattern (the same fundamental elementary-dipole pattern discussed in the earlier part of this paper), independent of the specific operating frequency or physical length (so long as the dipole remains electrically short):

This standard D0=1.5 result follows from integrating the short dipole's radiation intensity U(theta) proportional to sin^2(theta) over the full sphere to obtain Prad, then dividing 4piUmax by this Prad — the calculation yields exactly D0=1.5 regardless of the specific dipole length (provided it remains short/electrically small) or operating frequency, since the underlying sin(theta) field pattern shape itself does not change with these parameters.

Step 3 - Maximum effective aperture: the maximum effective aperture is related to directivity through the universal antenna relation:

Substituting D0=1.5 and lambda=0.6667 m:

Result: the short dipole operating at f=450 MHz has wavelength lambda=0.6667 m, directivity D0=1.5 (1.761 dB), and maximum effective aperture Aem=0.053052 m^2, illustrating that although a short dipole's physical size is small compared to a wavelength, its effective aperture (the effective 'capture area' it presents to an incident wave for maximum power transfer to a matched load) is a wavelength-scaled quantity determined entirely by its directivity, following the same universal Aem=D0lambda^2/(4pi) relation that applies to any antenna type regardless of physical size.