How to calculate the dimensions for a log periodic antenna based on desired frequency?

Understanding the Core Design Principles

Calculating the dimensions for a log periodic antenna boils down to a systematic application of geometric scaling laws governed by a key parameter called the scaling factor. The entire structure is a series of dipole elements where each subsequent element is a scaled version of the previous one. The longest dipole determines the lowest operating frequency, and the shortest dipole determines the highest. The fundamental parameters you need to define are the desired frequency range (F_low and F_high), the scaling factor (τ), and the relative spacing factor (σ). Once these are set, the rest of the dimensions cascade from these values. The design is not based on a single resonant frequency but on a band of frequencies, making it distinct from a simple Yagi-Uda antenna.

The Essential Design Parameters and Their Interplay

Before you can calculate a single length, you must lock in three critical, interrelated values. These form the foundation of your design and directly impact performance metrics like gain, bandwidth, and impedance.

Scaling Factor (τ – Tau): This is the ratio of the lengths and spacings of successive elements. It dictates how quickly the antenna transitions from the longest to the shortest element. A value closer to 1 (e.g., 0.95) results in more elements, smoother performance, and higher gain but a larger physical structure. A lower value (e.g., 0.85) creates a more compact antenna with fewer elements but potentially lower gain and less smooth impedance characteristics across the band. The typical practical range is 0.80 ≤ τ ≤ 0.98.

Spacing Factor (σ – Sigma): This defines the relative spacing between adjacent elements relative to the length of the longer element in the pair. It influences the coupling between elements, which is crucial for the active region to propagate correctly. A common optimal value, derived from research, is around σ = 0.15, but it can vary. The two factors are related by the angle of the antenna (α) through the formula: α = arctan( (1-τ) / (4σ) ).

Design Frequency Ratio: Your desired bandwidth (F_high / F_low) determines how many elements you’ll need. The antenna must have elements long enough to resonate at the lowest frequency and short enough for the highest frequency, with a sufficient number of scaled elements in between to maintain performance.

Step-by-Step Dimensional Calculation

Let’s walk through a concrete example. Suppose we need an antenna to cover the VHF TV band from 175 MHz (F_low) to 230 MHz (F_high).

Step 1: Determine the Number of Elements (N)
First, calculate the bandwidth needed: B = F_high / F_low = 230 / 175 ≈ 1.31.
We choose τ = 0.92 and σ = 0.15. The number of elements can be estimated by the formula involving the log of the bandwidth and τ. A simpler iterative approach is to calculate the length of successive elements until the shortest element’s half-wavelength is less than or equal to the half-wavelength of F_high.

Step 2: Calculate the Length of the Longest Element (L1)
The longest dipole is approximately a half-wavelength at F_low. The formula for a half-wavelength dipole is L (in meters) = 150 / F (in MHz), accounting for the end effect (velocity factor, typically around 0.95 for tubes).
Theoretical half-wavelength: 150 / 175 MHz = 0.857 meters.
With a velocity factor (k) of 0.95: L1 = (150 / 175) * 0.95 = 0.814 meters. This is the total length; each arm of the dipole would be half of this: 0.407 meters.

Step 3: Calculate the Length of the Shortest Element (Ln)
Similarly, the shortest element must be a half-wavelength at F_high.
Theoretical: 150 / 230 MHz = 0.652 meters.
With velocity factor: Ln = (150 / 230) * 0.95 = 0.620 meters (total length).

Step 4: Calculate the Lengths of All Intermediate Elements
Using the scaling factor, each element is τ times the length of the previous one.
L2 = τ * L1 = 0.92 * 0.814m = 0.749m
L3 = τ * L2 = 0.92 * 0.749m = 0.689m
… and so on, until the calculated length is less than or equal to Ln (0.620m). In this case, we would end up with around 6 or 7 elements.

Step 5: Calculate the Spacing Between Elements (d)
The spacing between element 1 and element 2 is determined by the spacing factor and the length of element 1.
d_(1 to 2) = 2 * σ * L1 = 2 * 0.15 * 0.814m ≈ 0.244 meters.
The spacing between element 2 and element 3 is: d_(2 to 3) = τ * d_(1 to 2) = 0.92 * 0.244m ≈ 0.224 meters.
This scaling continues for all subsequent spacings.

Boom Length and Apex Angle

The total boom length is simply the sum of all the spacings. From our table, that’s 0.244 + 0.224 + 0.206 + 0.190 = 0.864 meters. The apex angle (α) can be calculated from τ and σ: α = arctan( (1 – 0.92) / (4 * 0.15) ) = arctan(0.08 / 0.6) = arctan(0.1333) ≈ 7.6 degrees. This defines the triangular shape of the antenna array when you draw lines through the ends of the elements.

Feeding the Antenna: The Transmission Line

A critical and often overlooked part of the calculation is the feed system. The elements are not connected directly; instead, they are attached to a twin-line transmission line that runs along the boom. The unique feature is that the connections alternate between the two feed lines for successive elements. This phase reversal is what creates the traveling wave effect that allows the “active region” to move along the structure with frequency. The characteristic impedance of this feeder, often 50-100 ohms, must be matched to your coaxial cable feed (typically 50 ohms) using a balun at the feed point (the shortest element end).

Material Considerations and Real-World Adjustments

These formulas provide a theoretical starting point. In practice, you must account for the diameter of the elements. Thicker elements have a lower length-to-diameter ratio, which affects the resonant length (hence the velocity factor k). The initial calculation is a prototype. Fine-tuning, especially for optimal VSWR, often involves slightly adjusting the lengths and spacings of the first and last few elements through simulation software or empirical testing. For a robust and professionally engineered solution, consulting with a specialist manufacturer like the one found at this Log periodic antenna resource can save significant time and ensure performance meets specifications, especially for commercial or critical applications.

Advanced Considerations: Optimization and Trade-offs

Beyond the basic calculations, optimizing a log periodic design involves trade-offs. Increasing τ (closer to 1) and decreasing σ (closer to 0.1) generally increases gain but also increases boom length and the number of elements, raising cost and weight. The choice of materials for the boom and elements affects weight, wind loading, and corrosion resistance. For very wide bandwidths (e.g., 10:1 or more), the design may require multiple stacked log periodic arrays or a more complex feeding network to maintain consistent performance across the entire range. Computer-aided engineering (CAE) tools like NEC-based simulators are indispensable for analyzing these interactions and predicting far-field radiation patterns, front-to-back ratio, and impedance matching before any metal is cut.