Q6Antenna And Wave Propagation

(a) Rhombic Antenna Design at 50 MHz, Elevation Angle 30 Degrees

Operating wavelength: at f=50 MHz:

A rhombic antenna consists of four long-wire radiating legs arranged in a horizontal diamond (rhombus) shape, terminated at the far end in a resistive load matched to the antenna's characteristic impedance (to absorb residual traveling-wave energy and suppress back-lobe reflection), with the whole structure mounted at a height above ground and oriented so that its main lobe of radiation is directed along the rhombus's long (major) axis, at a desired vertical elevation angle above the horizon determined jointly by the antenna's height above ground and the leg length/tilt angle geometry.

Design relationships: for a given desired vertical elevation angle of maximum radiation, Delta, the rhombic antenna's leg length (L, in wavelengths) and tilt angle (phi, the half-angle of the rhombus at the feed/termination apex, measured from the major axis) are jointly selected using standard rhombic antenna design charts (originally developed by Foster and subsequently refined by other authors), which are based on the requirement that the individual traveling-wave currents on each of the four legs reinforce constructively in the desired elevation direction. A widely used approximate design relationship (the 'alignment' design, optimizing gain for a given elevation angle) gives the required tilt angle approximately as phi=90-Delta (in degrees) for the simplest first-order alignment condition, and the corresponding leg length (in wavelengths) typically increases as the desired elevation angle decreases (lower desired elevation angles, corresponding to longer-range sky-wave communication paths, generally require longer rhombic antenna legs to achieve good gain and pattern alignment) — for a moderate elevation angle such as the 30 degrees specified in this problem, practical rhombic antenna designs typically use leg lengths in the range of about 2 to 6 wavelengths (here, approximately 12 m to 36 m at the 6 m operating wavelength), with the tilt angle typically in the range of 60-70 degrees for this elevation angle, the precise combination selected using the specific design chart/method the antenna is being engineered against, since different classical rhombic design references (Foster's original charts, the 'maximum gain' design, and the 'alignment' design) give somewhat different specific optimum leg-length/tilt-angle combinations for the same target elevation angle, each representing a different trade-off between achieved gain, side-lobe level, and antenna size.

Height above ground: in addition to the leg length and tilt angle, the antenna's height above ground, h, must also be selected consistent with the desired elevation angle, since ground reflection beneath a horizontal rhombic antenna causes constructive reinforcement of the desired elevation angle when h approximately equals lambda/(4sin(Delta)) for the lowest-order (most common) reinforcement condition — for Delta=30 degrees and lambda=6m, this gives an approximate design height of h=6/(4sin(30))=6/(4*0.5)=3 m as a representative first-order value, though practical installations often use a somewhat greater height (corresponding to a higher-order reinforcement condition) to keep the antenna's physical wire structure conveniently elevated above the ground and any nearby obstructions.

Termination and Impedance Matching Practice for Rhombic Antennas

A practical rhombic antenna is almost always operated as a terminated (non-resonant, traveling-wave) rather than resonant (standing-wave) structure, since terminating the far end apex in a non-inductive resistive load matched to the wire legs' own characteristic impedance (typically in the range of 600-800 ohms, using a suitable high-power, non-reactive resistor or a matched dissipative line) absorbs the residual wave energy that would otherwise reach the far apex and reflect back, causing an unwanted standing-wave (bidirectional, resonant-mode) pattern with degraded front-to-back ratio and gain. With correct termination, the current on each leg is a pure outgoing traveling wave (rather than a standing wave), producing the desired unidirectional, sharply forward-directed rhombic radiation pattern with good front-to-back ratio, at the cost of the terminating resistor dissipating a portion (typically around half) of the input power as heat, which is an accepted trade-off in exchange for the rhombic antenna's simple construction, wide operating bandwidth (since a traveling-wave structure is inherently far less frequency-sensitive than a resonant one), and ease of achieving high gain over a large HF frequency range from a single physical structure. The antenna's own characteristic (traveling-wave) impedance, and hence the required termination resistance, is set primarily by the leg wire's diameter and the rhombus's included tilt angle, and practical installations typically use a matching/feeding transformer or matched open-wire transmission line to couple the rhombic antenna's characteristic impedance to the standard 50-ohm or 600-ohm feeder line from the transmitter.

(b) Rectangular Microstrip Patch Antenna Design

Given: center frequency fr=10 GHz, dielectric constant epsilon_r=10.2, substrate height h=0.127 cm.

Step 1 - Patch width W: the standard design formula for microstrip patch width, chosen to give good radiation efficiency, is:

Step 2 - Effective dielectric constant: accounting for the fringing fields that extend partly into the substrate and partly into the air above it:

Step 3 - Length extension due to fringing: the effective patch length appears electrically longer than its physical length due to fringing fields at each radiating edge, requiring a length extension correction:

Step 4 - Effective length and physical patch length: the effective (electrical) patch length required for resonance at fr is:

and the actual physical patch length, after subtracting the fringing-field extension from both radiating edges:

Result: the calculated physical dimensions of the rectangular microstrip patch are approximately W=0.634 cm and L=0.426 cm, illustrating the standard four-step design procedure (width, effective dielectric constant, fringing length extension, and final physical length) universally used for microstrip patch antenna design at any target frequency and substrate combination — the comparatively high dielectric constant (10.2) specified in this problem results in a notably compact patch size (well under a centimeter in each dimension at 10 GHz), consistent with the general design principle that higher-permittivity substrates produce smaller (though typically narrower-bandwidth) microstrip patch antennas compared to lower-permittivity substrates at the same operating frequency.

Bandwidth and Feed-Point Considerations for Microstrip Patches

Microstrip patch antennas are inherently narrowband radiators, typically achieving only about 1-5% impedance bandwidth for a standard single-layer rectangular patch, since the patch behaves essentially as a resonant cavity (bounded by the radiating edges) with a high effective quality factor Q — the achievable bandwidth increases with substrate thickness h and decreases with dielectric constant epsilon_r, so a high-permittivity substrate such as the epsilon_r=10.2 used in this design, while producing a conveniently compact patch (as computed above), also tends to yield a narrower bandwidth than a lower-permittivity substrate of equal thickness would, an inherent trade-off between miniaturization and bandwidth that the designer must balance against the specific application's frequency-stability requirements. Bandwidth can be improved through techniques such as using a thicker substrate (at the cost of increased surface-wave loss and a bulkier structure), a lower-permittivity substrate (at the cost of a physically larger patch), or more advanced structures such as stacked/parasitic patches, U-slot patches, or aperture-coupled feeding.

Feed-point selection: the feed location on a microstrip patch critically determines the impedance match to the feeding transmission line, since the patch's input impedance is maximum (typically 100-300 ohms) at the radiating edge and decreases toward the patch center. The probe (coaxial) feed connects the inner conductor of a coaxial cable directly through the substrate to a point on the patch's underside, with the feed point selected inward from the edge along the patch's resonant length until the input impedance matches the feed line (commonly 50 ohms), offering easy impedance matching and a simple, low-spurious-radiation feed structure, though it introduces a small probe inductance (significant at higher frequencies) and requires a soldered mechanical connection. The inset feed instead uses a microstrip line feeding the patch directly from the edge but recessed (inset) a calculated distance into the patch along a narrow notch, exploiting the smooth impedance variation along the patch length to achieve the desired match without a coaxial through-substrate connection, giving a simpler, fully planar (single-layer, no-drilling) fabrication process well suited to printed-circuit and microwave integrated-circuit implementation, at the cost of a somewhat more involved feed-notch depth calculation and a modest amount of spurious radiation from the notch discontinuity itself.