Q1Antenna And Wave Propagation

(a) Ideal Dipole and Short Dipole Antenna

Ideal (infinitesimal) dipole: an ideal dipole, also called a Hertzian dipole or infinitesimal dipole, is a purely theoretical, mathematically idealized current element of length dl that is vanishingly small compared to the operating wavelength (dl much less than lambda, in the limiting sense dl approaching zero), carrying a current of uniform (constant) magnitude and phase along its entire length. Because no physically realizable antenna can actually maintain a perfectly uniform current all the way to its open (unconnected) ends — the current must necessarily fall to zero at a free end, since charge cannot accumulate indefinitely there — the ideal dipole is not itself a practically constructible antenna, but instead serves as the fundamental mathematical building block from which the radiation field of any actual, more complex antenna current distribution can be computed, by treating the actual antenna as a continuous series of infinitesimal ideal-dipole current elements and superposing (integrating) their individual radiated fields. The ideal dipole produces the canonical elementary radiation pattern (a sin(theta) far-field variation, i.e., a figure-eight-shaped, omnidirectional-in-azimuth doughnut pattern with nulls along the dipole's own axis), which underlies the derivation of the radiation characteristics of essentially every practical linear-wire antenna.

Short dipole: a short dipole is a physically realizable, finite-length linear (wire) antenna whose total length is small compared to a wavelength (conventionally, length L less than approximately 0.1 lambda to 0.2 lambda), fed at its center. Unlike the ideal dipole, a short dipole's current cannot be uniform along its length, since the current must necessarily taper to zero at each of its two open (unconnected) end tips — for a short dipole, this current distribution is well-approximated as triangular, i.e., varying linearly from a maximum value at the center feed point down to exactly zero at each end. Because the average current along a short dipole (with its triangular taper) is exactly half of its peak (feed-point) current, a short dipole's radiated field, radiation resistance, and radiated power are all reduced by a factor of one-half (or one-quarter, for power, since power depends on the square of the effective current) compared to a hypothetical ideal dipole of the same physical length carrying the same peak current — this triangular-current correction is essential for accurately predicting the practical radiation resistance (and hence radiation efficiency) of real short/small dipole antennas, such as those used in electrically compact antenna designs where the physical size is deliberately kept much smaller than the operating wavelength.

(b) Difference Between Directivity and Gain

Directivity (D0): the ratio of the radiation intensity in a given direction (usually the direction of maximum radiation) to the average radiation intensity over all directions, D0=4piUmax/Prad, a quantity that depends purely on the shape of the antenna's radiation pattern and is entirely independent of any ohmic (conductor or dielectric) losses within the antenna structure — directivity describes how well an antenna concentrates whatever power it actually radiates into a preferred direction, without regard to how much of the input power supplied to the antenna terminals actually gets radiated in the first place.

Gain (G0): the ratio of the radiation intensity in a given direction to the radiation intensity that would result if the total power accepted (input) at the antenna terminals were radiated isotropically, G0=4piU/Pin — gain, unlike directivity, explicitly accounts for the antenna's ohmic losses, since it is referenced to input power Pin rather than radiated power Prad. Gain and directivity are related through the antenna radiation efficiency, e_cd=Prad/Pin=Rr/(Rr+Rloss):

Are they the same in any case? Yes — directivity and gain are numerically equal precisely when the antenna's radiation efficiency e_cd equals 1 (100%), i.e., when the antenna is lossless (Rloss=0, so that all power delivered to the antenna terminals is actually radiated, Prad=Pin). This is the case for an idealized, perfectly conducting antenna with no ohmic (I^2*R) heating losses in its conductors or any associated dielectric material. For any real antenna with non-zero loss resistance, gain is always strictly less than directivity (G0<D0), since some fraction of the input power is inevitably dissipated as heat rather than radiated; well-designed, electrically large antennas (such as full-size half-wave dipoles and larger arrays/reflectors made of good-conductivity metal) typically have efficiency very close to unity and hence gain very close to directivity, whereas electrically small antennas (such as short dipoles and small loops) can have significantly reduced efficiency (because their inherently small radiation resistance becomes comparable to, or even smaller than, the unavoidable ohmic loss resistance), causing their gain to fall noticeably below their directivity.

(c) Proof: Radiated Power of Quarter-Wave Monopole, Pr=36.5*Ieff^2

Setup using image theory: a quarter-wave (lambda/4) monopole antenna is mounted vertically on top of an infinite, perfectly conducting ground plane, fed at its base between the monopole and the ground plane. By the classical method of images, a perfectly conducting ground plane can be replaced (for purposes of field calculation above the ground plane) by removing the ground plane entirely and instead placing an image of the monopole's own current distribution in the region below where the ground plane used to be, with the image current directed such that it exactly reproduces the boundary condition of zero tangential electric field at the (now-removed) ground plane's original location. For a vertical monopole fed at its base, this image is itself another vertical current element of the same length, oriented so that the image current flows in the same direction (upward) as the actual monopole current when both are referenced to the same absolute vertical direction — the combination of the quarter-wave monopole plus its quarter-wave image current exactly reconstructs the current distribution of a complete half-wave dipole (total length lambda/4 + lambda/4 = lambda/2), fed at its center.

Field and power relationship via image theory: because the monopole-plus-image combination produces exactly the same current distribution as a free-space half-wave dipole, the electric and magnetic far-field pattern produced by the quarter-wave monopole (for the upper half-space, z>0, above the ground plane) is identical in functional form and magnitude to the upper-half-space field of an equivalent half-wave dipole carrying the same feed-point current. However, a crucial distinction arises for total radiated power: the actual physical monopole-over-ground system only occupies (and only radiates energy into) the upper half-space (z>0), since the conducting ground plane blocks/prohibits any field or power flow into the lower half-space (z<0) — whereas the equivalent free-space half-wave dipole radiates its power into the entire full sphere (both upper and lower half-spaces, z>0 and z<0 together). Since the field intensity (and hence the Poynting vector / power density) in the upper half-space is identical for both the monopole-over-ground system and the equivalent half-wave dipole (both carrying the same feed current), but the monopole-over-ground system's radiated power is obtained by integrating this same power density only over the upper hemisphere (solid angle 2pi steradians) rather than the full sphere (4pi steradians) that applies to the free dipole, the quarter-wave monopole's total radiated power is exactly half that of the equivalent half-wave dipole, for the same feed-point current:

Radiation resistance of half-wave dipole: the well-known, rigorously derived radiation resistance of a free-space half-wave dipole (obtained via the standard induced EMF method or direct integration of its radiated power) is Rr,dipole=73 ohms (more precisely, about 73.1 ohms), so that its radiated power in terms of its feed-point effective (RMS) current Ieff is:

Radiation resistance and power of quarter-wave monopole: applying the half-power relationship derived above from image theory (the monopole radiates exactly half the power of the equivalent dipole, for the same feed current):

Result: this directly establishes the required result, Pr=36.5*Ieff^2, and correspondingly shows that the quarter-wave monopole's own radiation resistance, Rr,monopole=Pr/Ieff^2=36.5 ohms, is exactly half the 73-ohm radiation resistance of the equivalent free-space half-wave dipole — a direct and elegant consequence of image theory, since the monopole-over-ground system produces the identical upper-half-space field of the corresponding half-wave dipole while only radiating into (and hence only depositing power into) half of the total solid angle that the free dipole radiates into, exactly halving the total radiated power for the same feed-point current.