Element spacing is arguably the most critical geometric parameter in a Log periodic antenna design, directly dictating its bandwidth, gain, impedance matching, and beam characteristics. It’s not a single value but a carefully controlled progression that defines the antenna’s fundamental log-periodic nature. Getting the spacing wrong can turn a high-performance antenna into a mediocre one, as it governs the electromagnetic coupling between adjacent elements, which is the engine of the antenna’s operation.
The Science of the Scaling Factor (τ) and Spacing Factor (σ)
To talk about spacing, we have to introduce the two key design constants. The scaling factor, tau (τ), determines how much each successive element is scaled down from the previous one. If the longest element has a length L1, the next is L2 = τ * L1, and so on. The spacing factor, sigma (σ), defines the distance between corresponding points on adjacent elements relative to the length of the larger element. Specifically, the distance from element n to element n+1 is given by d = (σ * L_n) / 2. These two factors are deeply intertwined. Their product, τσ, is particularly important as it relates to the antenna’s gain. A common design choice is to hold the ratio τ/σ constant to maintain consistent performance across the band.
The table below shows how different combinations of τ and σ influence the antenna’s key characteristics.
| Scaling Factor (τ) | Spacing Factor (σ) | Typical τ/σ | Primary Impact on Performance |
|---|---|---|---|
| High (e.g., 0.95) | Low (e.g., 0.04) | > 20 | Very gradual scaling. Results in a larger number of elements for a given bandwidth, leading to higher gain and smoother impedance response, but a physically longer and heavier antenna. |
| Moderate (e.g., 0.88) | Moderate (e.g., 0.08) | ~11 | A common balanced design. Offers a good compromise between gain, bandwidth, and physical size. Provides stable performance for most applications. |
| Low (e.g., 0.78) | High (e.g., 0.16) | < 5 | Rapid scaling. Creates a shorter boom with fewer elements for the same bandwidth, but at the cost of lower gain and potentially poorer impedance matching, especially at the low-frequency end. |
Direct Impact on Bandwidth and Frequency Response
The spacing between elements is what allows the antenna to be truly frequency-independent. The “active region”—the set of three or four elements that are approximately half a wavelength long at the operating frequency—moves along the boom as the frequency changes. Closer spacing (a higher σ value with a moderate τ) means the transition of the active region from one set of elements to the next is smoother. This results in a more consistent input impedance and a lower Standing Wave Ratio (SWR) across the entire bandwidth. Think of it like a staircase with very small steps versus large jumps; the smaller steps provide a smoother climb. However, if the spacing is too tight, the mutual coupling between elements becomes excessively strong, which can distort the radiation pattern and actually reduce the bandwidth by causing undesirable resonances.
The Gain and Directivity Trade-Off
Gain is directly proportional to the number of elements that are effectively contributing to the radiation at any given frequency. A smaller τ (faster scaling) and a smaller σ (tighter spacing) both lead to a shorter boom with fewer elements for a given frequency range. Fewer elements mean less directive capability and therefore lower gain. Conversely, to maximize gain, you want a larger τ (slow scaling) and a larger σ (wider spacing), which packs more elements into the structure. But there’s a ceiling. Beyond a certain point, increasing the spacing too much (a very high σ) causes the active region to become less defined. The antenna starts to behave less like a coordinated array and more like a collection of individual dipoles, leading to sidelobes and a squinting main beam (where the direction of maximum radiation shifts with frequency). The optimal gain for a standard log-periodic dipole array (LPDA) is typically achieved when τ is around 0.88 and σ is around 0.14, giving a τ/σ ratio near 6.3.
Impedance Matching and SWR
The magic of a log-periodic antenna’s consistent 50-ohm or 75-ohm input impedance isn’t magic at all—it’s a direct result of the element spacing and the feed system. The alternating connection of the elements to the two sides of the feed line (often a coaxial line inside a square boom) creates a phasing effect. The precise spacing ensures that the electromagnetic fields from the “active” elements induce currents in the smaller, non-resonant elements ahead of them (towards the feed point) in such a way that the reflected impedance seen at the feed point remains nearly constant. If the spacing is incorrect, this delicate balance is disrupted. Too wide a spacing, and the coupling is too weak, leading to high SWR peaks between the natural resonant frequencies of the individual elements. Too close a spacing, and the strong coupling can create a capacitive or inductive loading effect that detunes the active region, also degrading the SWR.
Beamwidth and Front-to-Back Ratio
Element spacing has a pronounced effect on the antenna’s radiation pattern. Tighter spacing generally results in a wider E-plane (the plane parallel to the elements) beamwidth. This can be desirable for applications like television reception where the signal may be coming from multiple directions. Wider spacing narrows the beamwidth, increasing directivity. The H-plane (the plane along the boom) beamwidth is primarily controlled by the size of the active region, which itself is a function of τ and σ. Furthermore, a key metric for directive antennas is the front-to-back ratio (F/B ratio)—the ratio of power radiated in the desired forward direction to the power radiated in the opposite direction. A well-designed spacing progression ensures that the elements behind the active region act as reflectors, while those in front act as directors, much like in a Yagi-Uda antenna. Poor spacing can cause these parasitic elements to be improperly tuned, significantly reducing the F/B ratio and allowing more noise or interference to enter from the rear.
Practical Design Considerations and Performance Data
Let’s look at a concrete example. Suppose we’re designing an LPDA to cover 100 MHz to 1 GHz, a 10:1 bandwidth ratio. The longest element will be roughly half a wavelength at 100 MHz (~1.5 meters).
- Design A (High Gain): τ = 0.92, σ = 0.10. This design would have approximately 22 elements. The boom length would be calculated to be around 4.2 meters. Expected gain would be relatively flat, around 8.5 dBi ± 0.5 dB across the band. The SWR would be excellent, staying below 1.5:1.
- Design B (Compact): τ = 0.82, σ = 0.16. This design would have only about 12 elements. The boom length would be a much more manageable 1.8 meters. However, the gain would suffer, especially at lower frequencies, varying from about 5 dBi at 100 MHz to 7.5 dBi at 1 GHz. The SWR would likely show peaks above 2.0:1 at certain frequencies within the band.
The choice between Design A and Design B hinges entirely on the application’s priorities: maximum performance versus size and weight constraints. In real-world engineering, these parameters are first calculated and then refined using sophisticated electromagnetic simulation software (like NEC-based simulators or HFSS) to model mutual coupling effects with extreme accuracy before a prototype is ever built.
The Relationship Between Spacing, Boom Length, and Element Length
It’s crucial to understand that element spacing (d) is not independent of element length (L). They are linked by the spacing factor σ, via the formula d_{n, n+1} = σ * L_n. This means that as the elements get shorter towards the front of the antenna, the spacing between them automatically decreases. This progressive shortening of both length and spacing is the fundamental principle that makes the structure periodic on a logarithmic scale. Therefore, you cannot simply change the spacing without recalculating the entire element length progression, or vice-versa. The boom length is then simply the sum of all the inter-element spacings. A smaller σ leads to a shorter boom for the same number of elements, but again, with potential performance trade-offs.