Q6Antenna And Wave Propagation

(a) Principle of Operation and Properties of Yagi-Uda Antenna

Principle of operation: the Yagi-Uda antenna is a linear end-fire array consisting of a single driven element (typically a folded or ordinary half-wave dipole, directly connected to and excited by the transmission line feed), together with one or more parasitic elements — a single reflector element positioned behind the driven element (on the side opposite the desired direction of maximum radiation) and one or more director elements positioned in front of the driven element (on the side of desired maximum radiation) — none of which are directly connected to the feed line. The parasitic elements operate purely through electromagnetic mutual coupling with the driven element: the driven element's own radiated near-field induces currents in each nearby parasitic element, and the magnitude and phase of each induced parasitic current is determined by that element's own length (relative to resonant half-wavelength) and its spacing from the driven element. The reflector element is made slightly longer than resonant length (giving it an inductive reactance), causing its induced current to lag in phase in a manner that produces constructive reinforcement of the array's radiated field toward the director side and destructive cancellation toward the reflector's own side; conversely, the director elements are made slightly shorter than resonant length (giving a capacitive reactance), producing an induced current phase relationship that further reinforces radiation in the forward (director) direction as the wave progresses along the successive directors. The cumulative effect of this correctly-phased mutual coupling across the reflector, driven element, and one or more directors produces the Yagi-Uda antenna's characteristic unidirectional end-fire radiation pattern, achieving moderate gain from an antenna structure using only a single actively-fed element.

Properties: Directivity and Bandwidth

Directivity with number of directors: the Yagi-Uda antenna's overall directivity/gain increases as additional director elements are added to the array (extending the effective end-fire aperture length along the boom), since each additional properly-phased director contributes further constructive reinforcement to the forward-directed beam, narrowing the overall beamwidth and increasing the on-axis gain — a simple three-element Yagi (one reflector, driven element, one director) typically achieves a gain of only about 7-9 dBi, while practical designs with 7 to 12 (or more) director elements can achieve gains in the range of roughly 10-15 dBi or somewhat higher, though the incremental directivity gained per additional director progressively diminishes as more directors are added (a classic case of diminishing returns), and beyond a certain practical boom length, adding further directors yields little additional gain while continuing to increase the antenna's physical size, weight, and wind-loading.

Bandwidth: the Yagi-Uda antenna's operating bandwidth (the frequency range over which its gain, pattern, and input impedance remain acceptable) is relatively narrow, typically only a few percent of its center design frequency, because the parasitic reflector and director elements' entire mode of operation depends on each element being close to a specific resonant length relative to the operating wavelength — since each parasitic element's induced current phase (essential for producing the correct constructive/destructive interference pattern) is highly sensitive to how far the operating frequency has shifted from that element's own individual resonant frequency, even a modest frequency shift away from the design frequency substantially degrades the careful phase relationships between all the elements, causing the antenna's gain, front-to-back ratio, and impedance match to all deteriorate relatively quickly outside a fairly narrow frequency band — this narrow bandwidth is widely recognized as one of the principal practical limitations of the Yagi-Uda antenna design, in contrast to genuinely broadband antenna types such as log-periodic dipole arrays.

Practical design trade-offs: the reflector-to-driven-element spacing (typically around 0.15-0.25 lambda) and director-to-driven-element and inter-director spacings (typically around 0.1-0.3 lambda, often progressively increased for successive directors in longer designs) both strongly affect the achieved gain, input impedance, and front-to-back ratio, meaning practical Yagi-Uda design is typically carried out using established empirical design tables or numerical optimization (method-of-moments antenna simulation software) rather than simple closed-form formulas, since the mutual coupling between every pair of elements in the array (not just between the driven element and each parasitic element individually) contributes to the overall current distribution and hence the final radiation pattern. The driven element is also very commonly implemented as a folded dipole rather than an ordinary half-wave dipole specifically because the parasitic elements' loading effect tends to pull the driven element's own input impedance down to a value lower than a free-standing half-wave dipole's 73 ohms, and the folded dipole's inherent fourfold impedance step-up (discussed in the corresponding non-OR part of this question) conveniently compensates for this loading effect, helping restore a practical, easily-matched feed-point impedance.

(b) Design Procedure of Rectangular Patch Antenna with Example

The standard rectangular microstrip patch antenna design procedure follows four sequential steps, using the specified resonant (center) frequency fr, substrate dielectric constant epsilon_r, and substrate height h.

Example given: fr=2.4 GHz, epsilon_r=4.4 (standard FR4 substrate), h=0.16 cm — chosen here to illustrate the design procedure at the common 2.4 GHz WiFi/Bluetooth/ISM band frequency on the widely available, low-cost FR4 substrate material, as a distinct illustrative example from the 10 GHz/epsilon_r=10.2 example already covered in this paper's companion 2021 examination answer.

Step 1 - Patch width W:

Step 2 - Effective dielectric constant:

Step 3 - Length extension due to fringing:

Step 4 - Effective length and physical patch length:

Result: the calculated physical dimensions of the 2.4 GHz FR4 rectangular microstrip patch are approximately W=3.81 cm and L=2.97 cm, an appreciably larger patch than the earlier 10 GHz/epsilon_r=10.2 example (as expected, since both the lower operating frequency and the lower substrate dielectric constant each independently favor a physically larger patch).

Applications of Rectangular Microstrip Patch Antennas

Microstrip patch antennas are extremely widely used in modern compact, low-profile wireless devices, owing to their light weight, thin planar profile, and ease of low-cost fabrication using standard printed-circuit-board processes: WiFi and Bluetooth antennas, operating in the 2.4 GHz and 5 GHz ISM bands as designed in the above example, integrated directly onto device circuit boards or access-point enclosures; GPS receiver antennas, typically circularly-polarized patch designs operating at the GPS L1 (1575.42 MHz) frequency, valued for their compact size and ability to be conformally mounted; RFID tag and reader antennas, where the patch's low profile and low manufacturing cost suit high-volume tag production; and phased array elements, where large numbers of individual microstrip patches are combined into planar phased-array panels for radar, satellite communication, and 5G mmWave beamforming systems, taking advantage of the patch's small size, planar compatibility with array feed networks, and ease of fabrication in large uniform arrays.