Why is the impedance of a log periodic antenna relatively constant over its bandwidth?

At its core, the log periodic antenna maintains a relatively constant impedance across its wide operating bandwidth because of its unique self-scaling design. Unlike a simple dipole whose dimensions are fixed for a single frequency, the log periodic antenna is essentially an array of dipoles of varying lengths, each one tuned to a different frequency. The key is that the electrical relationships between these elements—their lengths, spacings, and widths—are kept constant. This is known as a constant scaling factor, or tau (τ). As the frequency of operation changes, the active region—the part of the antenna that is effectively radiating and receiving—simply shifts along the structure to a set of elements that are the correct size for that new frequency. Because the electrical environment around this active region is always geometrically similar, the impedance it “sees” remains largely unchanged. It’s like having a series of perfectly tuned antennas in one package, with a smooth handoff from one to the next.

To really grasp this, we need to break down the geometry. A typical Log periodic antenna consists of a series of dipole elements mounted on a boom, with the longest element at the back for the lowest frequency and the shortest at the front for the highest frequency. The defining parameters are the scaling factor (τ) and the relative spacing constant (σ). The scaling factor τ is the ratio of the lengths and spacings of successive elements. If one dipole is 1 meter long and the next is 0.9 meters long, τ is 0.9. The spacing between them will also be scaled by this same factor. The relative spacing constant σ is derived from τ and the angle of the antenna’s triangular outline (α). This meticulous scaling ensures that the antenna is electrically identical at all frequencies it’s designed for; only the physical scale changes.

The magic happens in what engineers call the “active region.” At any given frequency, only a small group of elements—typically two or three—are actively resonating and doing most of the work. Elements much longer than the resonant wavelength act as passive reflectors, while elements much shorter act as directors, similar to a Yagi-Uda antenna. However, as the frequency shifts, this active region moves along the boom. For a higher frequency, it shifts toward the shorter, more closely spaced elements at the front. For a lower frequency, it moves toward the longer, wider-spaced elements at the rear. Crucially, because the geometry scales perfectly, the electromagnetic coupling between the active dipole and its immediate neighbors (which are now acting as reflector and director) is identical regardless of where on the structure the active region is located. This consistent electromagnetic environment is the primary reason for the stable impedance.

Let’s look at some concrete numbers to illustrate this scaling. The table below shows a simplified design for a log periodic dipole array (LPDA) with a scaling factor of τ = 0.95 and a design ratio σ = 0.05. The longest element (Element 1) is set for 100 MHz.

As you can see, both the lengths and the spacings decrease by the same 0.95 factor. This means that at 100 MHz, the active region is around Element 1, and the spacing to Element 2 is a specific electrical distance. At 280 MHz, the active region is around Element 20, and the spacing to Element 19 is the exact same electrical distance in wavelengths. This geometric similarity directly translates to similar current distributions and phase relationships on the elements, which in turn produces a consistent feed-point impedance, typically designed to be close to a standard 50 or 75 ohms.

The feed system also plays a critical role in impedance stability. The elements are not connected in parallel; instead, they are fed 180 degrees out of phase by alternating the connection of the twin-line transmission line that runs along the boom. This phasing is crucial for achieving the desired end-fire radiation pattern (beam off the front of the antenna). More importantly for impedance, this feed method helps isolate the non-resonant elements. Since the feed line is transposed between each element, the impedance transformation effects of the long transmission line are minimized. The feed point at the apex (the front of the antenna) effectively only “sees” the impedance of the active region, which, as we’ve established, is constant due to scaling. This prevents the high impedance variations that would occur if all elements were simply connected in parallel.

Comparing the log periodic to other wideband antennas highlights its advantage. A discone antenna, for example, also offers wide bandwidth but its impedance can be more sensitive to the precise geometry of the disc and cone, and it has an omnidirectional pattern. A biconical antenna has a relatively constant impedance but is often physically larger for comparable gains at lower frequencies. The log periodic’s combination of a directional pattern, high gain, and stable impedance is unique. The trade-off is complexity; it has many more parts than a simple dipole or discone. The impedance isn’t perfectly flat—there are always minor ripples—but it is remarkably stable over bandwidths that can exceed 10:1 (e.g., 50 MHz to 500 MHz). This performance is why LPDAs are industry standards for applications like TV reception, EMC testing, and wideband communications systems where a reliable, predictable connection to the transmitter or receiver is non-negotiable.

Finally, the material and construction choices influence how well the theoretical impedance stability is achieved in practice. The use of low-loss conductors like aluminum for the elements and boom minimizes resistive losses that would distort the impedance. The rigidity of the structure is also important; if elements bend or sag, the precise spacing and coupling between them change, which can introduce impedance variations and pattern distortion. High-quality manufacturing ensures that the designed scaling factor is maintained accurately from the longest to the shortest element, which is essential for achieving the predicted smooth impedance response across the entire band. This attention to physical detail is what separates a well-performing log periodic antenna from one that merely looks the part.