Yagi vs log-periodic; LP design

Yagi vs log-periodic antenna

Working principle of log-periodic antenna

An LPDA is a set of dipoles of lengths L₁ < L₂ < … < Lₙ placed along a boom inside a triangle of half apex angle α and fed by a crossed (transposed) two-wire line from the short end.

            apex (feed)
             /|\  α
            / | \
       L1  --+--      shortest (fmax)
          ---+---
         ----+----    active region
        -----+-----
       ------+------  longest (fmin)
     <- beam toward apex (short end)

• Dimensions grow geometrically: τ = Lₙ/Lₙ₊₁ = Rₙ/Rₙ₊₁ = dₙ/dₙ₊₁ < 1, where Rₙ is the distance of dipole n from the apex and dₙ the spacing.
• At any frequency, only the 3–5 dipoles near half-wave length (the active region) carry large current and radiate. Longer dipoles behind act like reflectors; shorter ones in front act like directors, so the beam points toward the apex.
• Changing frequency only moves the active region along the array. Because the structure repeats itself when frequency is multiplied by 1/τ, the impedance and pattern repeat periodically with log f, hence the name. If the variation within one period is small, the antenna is effectively frequency-independent.
• The feeder is transposed between adjacent dipoles to give the 180° phase shift needed for end-fire toward the short end.

Design steps (Carrel method)

1. Choose directivity; from Carrel's chart pick the optimum scale factor τ (0.8–0.98) and spacing factor σ = dₙ/(2Lₙ).
2. Half apex angle: α = tan⁻¹[(1 − τ)/(4σ)].
3. Bandwidth of active region: Bₐᵣ = 1.1 + 7.7(1 − τ)² cot α.
4. Bandwidth of structure: Bₛ = B × Bₐᵣ, where B = f_max/f_min.
5. Number of elements: N = 1 + ln(Bₛ)/ln(1/τ).
6. Longest element: L_max = λ_max/2 (at f_min); then Lₙ₊₁ = τLₙ for the rest.
7. Boom length: L = (λ_max/4)(1 − 1/Bₛ) cot α.
8. Spacings dₙ = 2σLₙ; choose feeder impedance to match the source (typically 50–75 Ω via a balun).