Design and CAE Analysis of Yagi-Uda Antennas

Absolutely, they are still active. They are widely used for terrestrial digital broadcast receiving antennas, amateur radio in HF to UHF bands, and even for IoT sensors in the ISM band. Their greatest strength is achieving high gain and high directivity despite their simple structure.

The Yagi-Uda antenna is composed of three types of elements. The radiator (driven element) is the only element supplied with power, typically a half-wave dipole. A reflector is placed behind it, and one or more directors are placed in front. The reflector and directors operate parasitically through electromagnetic coupling—meaning they are not directly fed but are driven by induced currents from the radiator.

Exactly. The reflector is set slightly longer than the resonant length to have inductive reactance, causing the phase of the reradiated wave to shift in a direction that reinforces forward radiation. Conversely, directors are set shorter than the resonant length to have capacitive reactance, achieving similar forward reinforcement. The length and spacing of each element determine this phase difference, which is the key to the design.

First, think in terms of the wavelength $\lambda$. The wavelength in free space is:

where $c = 3 \times 10^8$ m/s (speed of light) and $f$ is the operating frequency. For example, in the 430 MHz band, $\lambda \approx 0.698$ m. Typical design values for each element are:

Gain increases with more directors, but diminishing returns occur. Adding 1-3 directors has a significant effect, but beyond 10, the gain increase per director drops to about 0.2-0.3 dB. Simultaneously, the antenna becomes longer, raising issues like wind load and mounting strength. Practically, a balanced design is around 5-15 elements for UHF bands and 3-7 elements for VHF bands.

The starting point is Maxwell's equations. Under the time-harmonic assumption ($e^{j\omega t}$):

Introducing the vector potential $\mathbf{A}$, the electric field becomes:

For wire antennas, using the thin-wire approximation, Pocklington's integral equation for the current distribution $I(z')$ on the conductor becomes fundamental:

Here, $k = 2\pi/\lambda = \omega\sqrt{\mu\varepsilon}$ is the wavenumber, and $G(z, z')$ is the free-space Green's function:

where $a$ is the conductor radius.

Roughly speaking, it's a kernel function representing "how much electromagnetic field a tiny current source at one location creates at another location." Imagine it like the ripple spread when you throw a stone into a pond. The influence weakens with distance, and the phase also rotates. In a Yagi antenna, all points on each element are coupled to each other via this Green's function, resulting in a system of integral equations.

A commonly used simplified gain estimation formula is this:

But this is only a guideline; in reality, element spacing, individual element lengths, and conductor diameter have a combined effect. Accurately, the far-field must be calculated as the product of the array factor and the element pattern. The far-field from an array of $N$ elements is:

Here, $I_n$ is the current amplitude and phase of each element (obtained via MoM), $f_n(\theta)$ is the radiation pattern of each individual element, and $\mathbf{r}_n$ is the position vector of the element.

For a properly designed 5-element Yagi with appropriate spacing and lengths, typically around 9-10 dBi. For 10 elements, about 12-13 dBi; for 15 elements, about 14-15 dBi. Optimization using MoM or FDTD often yields a 1-2 dB improvement over initial hand calculations. In practice, this "extra 1 dB" impacts the overall system link budget, making simulation optimization highly valuable.

The Front-to-Back ratio (F/B ratio) is the ratio, expressed in dB, of the radiated power in the main direction ($\theta=0°$) to that in the opposite direction ($\theta=180°$):

It can be controlled by adjusting the length and spacing of the reflector. Generally, a target of 15-25 dB is aimed for. For instance, in terrestrial digital reception, if there is another relay station behind the antenna, a low F/B ratio can pick up interference, causing block noise in the video.

Using two reflectors or a reflector grid (lattice reflector) improves the F/B ratio, but it significantly impacts input impedance, making balance difficult. Practically, aiming for an F/B ratio above 20 dB with a single reflector and fine-tuning its spacing if further needed is more cost-effective.

Good question. Due to mutual coupling with parasitic elements, the radiator's input impedance drops significantly from the 73Ω of a standalone half-wave dipole, typically to around 20-35Ω. Direct connection to 50Ω coaxial cable worsens VSWR. Commonly used countermeasures include:

• Folded dipole: Transforms impedance by approximately a factor of 4 (bringing $Z \approx 4 \times 20 = 80$Ω closer).
• Gamma match / Delta match: An adjustable rod is placed between the radiator and the boom for impedance transformation.
• Balun (balanced-to-unbalanced transformer): Converts between coaxial cable (unbalanced) and dipole (balanced).

In simulation, the quality of matching is evaluated using the S-parameter $S_{11}$ and VSWR (Voltage Standing Wave Ratio):

A common target is VSWR below 1.5 ($S_{11} < -14$ dB).