# Log‑Periodic Dipole Array Calculator

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# Free-space directivity

The free-space directivity of a log-periodic dipole array (LPDA) is a function of its taper $$\tau$$ and its chosen spacing $$\sigma$$.1–5 Decreasing $$\sigma$$ will decrease the boom length $$L$$. Decreasing $$\tau$$ will decrease both the boom length $$L$$ and the number of elements $$N$$. Because of constrained resources, amateur radio log-periodic antennas often are limited to values of $$\tau$$ between 0.88 and 0.95, with values of $$\sigma$$ between 0.03 and 0.06.6

# Input

 lowest frequency* $$f_1$$ MHz highest frequency* $$f_n$$ MHz diameter of the shortest element N† $$⌀_N$$ mm  inch characteristic input impedance‡ $$Z_{0i}$$ Ω taper $$\tau$$ 0.800.98 relative spacing $$\sigma$$ 0.03$$\sigma_{opt}$$

#### Input guidelines:

• * Choose a slightly broader frequency range to compensate for construction inaccuracies.
• The characteristic input impedance $$Z_{0i}$$ of an LPDA is to a large extent determined by the dimensions of the shortest element. In a true LPDA, element diameters should scale proportionally to element length. However, keeping element diameters the same throughout an LPDA turns out to be pretty forgiving in many cases.
• Every so often, HF LPDAs are designed with a characteristic input impedance of 200 Ω; a 4÷1 current balun transforms the impedance to 50 Ω. On VHF and higher frequencies, LPDAs are normally designed for a direct connection to a 50 or 75 Ω characteristic input impedance.

# Resulting design

 relative operating bandwidth $$B$$ optimal spacing $$\sigma_{opt}$$ cotangent of the apex half-angle $$\cot{\alpha}$$ relative bandwidth of the active region $$B_{ar}$$ relative bandwidth of the structure $$B_S$$ number of elements, rounded to an integer6 $$\lfloor N\rceil$$ boom length $$L$$ length of dipole element $$i$$ $$\ell_i$$ distance between element centres $$i$$ and $$i+1$$ $$d_{i,i+1}$$ length of the terminating stub* $$Z_t$$ $$\ell_{Z_t}$$ average characteristic impedance of the shortest element N $$Z_{0N}$$ required characteristic impedance of the feeder connecting the elements† $$Z_{0f}$$

#### Design notes:

• * The terminating stub $$Z_t$$ is only required when the characteristic impedance of the feeder $$Z_{0f}$$ is low. This is typical of VHF and UHF arrays.6,8 A terminating stub improves the front‑to‑back ratio at «weak spots» in the frequency domain.6,7 Moreover, a stub prevents static charge build‑up on the connecting transmission line.6 Hutira7 prescribes a stub length of λ1/4; i.e. double the length given by Cebik6 and this calculator. Model or experiment with different stub lengths to select the most useful radiation pattern.6
• The feeder connecting the elements is a parallel conductor transmission line. The separation distance between the parallel conductors determines the characteristic impedance $$Z_{0f}$$ of this transmission line, which is different from the input impedance $$Z_{0i}$$ of the antenna. The separation distance can be calculated using either my parallel square or circular conductor transmission line calculator.

# Formulas

This LPDA calculator is mainly based on the design procedure as described by L. B. Cebik, W4RNL (SK) in the 21st edition of The ARRL Antenna Handbook.6 The calculator was successfully tested against the examples listed in this reference. Unlike the book, this calculator employs the velocity of light $$c$$ at full precision, resulting in slightly shorter, but more precise lengths. Furthermore, the formula for computing the boom length $$L$$ has been improved by not including the distance to the virtual apex $$2\alpha$$ of the antenna.

$B = \frac{f_n}{f_1}$

$\tau\equiv\frac{\ell_i}{\ell_{i-1}} \qquad 0.8 \leq \tau \leq 0.98$

$\sigma\equiv\frac{d_{1,2}}{\lambda_1} \qquad \sigma_{opt} = 0.243\:\tau - 0.051 \qquad 0.03 \leq \sigma \leq \sigma_{opt}$

$\cot\alpha = \frac{4\,\sigma}{1 - \tau}$

$B_{ar} = 1.1 + 7.7\,\left(1 - \tau\right)^2\cot\alpha$

$B_S = B\cdot B_{ar}$

$N = 1+\frac{\ln B_S}{\ln\frac{1}{\tau}} \qquad \begin{cases} \{N\} \gt 0.3 \rightarrow \lfloor N\rceil = \lceil N\rceil\\ \{N\} \le 0.3 \rightarrow \lfloor N\rceil = \lfloor N\rfloor \end{cases}$

$c \equiv 299\,792\,458\,\frac{m}{s}$

$\ell_1 = \frac{\lambda_1}{2}=\frac{c}{2\,f_1} \qquad \ell_i = \tau \cdot \ell_{i-1} \qquad \ell_{tot} = \sum\limits_{i=1}^{n}\ell_i$

$d_{i,i+1} = \frac{\ell_i-\ell_{i-1}}{2}\cot\alpha \qquad L = \sum\limits_{i=1}^{n-1} d_{i,i+1}$

$\ell_{Z_t} = \frac{\lambda_1}{8}$

$Z_{0N} = 120 \left[ \ln \left( \frac{\ell_N}{⌀_N} \right) - 2.25 \right] \qquad \sigma'\equiv\frac{\sigma}{\sqrt{\tau}}$

$Z_{0f} = \frac{Z_{0i}^2}{8\,\sigma'Z_{0N}} + Z_{0i} \sqrt{\left( \frac{Z_{0i}}{8\,\sigma'Z_{0N}} \right)^2 + 1}$

# Brython source code

Here is the Brython code of this calculator. Brython code is not intended for running stand alone, even though it looks almost identical to Python. Brython code runs on the client side in the browser, where it is transcoded to secure Javascript.

Download: lpda.py

# References

1. Carrel R. The design of log-periodic dipole antennas. In: IRE International Convention Record.Vol 9.; 1961:61-75. doi:10.1109/IRECON.1961.1151016.

2. Cheong WM, King RWP. Log-periodic dipole antenna. Radio Science. 1967;2:1315-1325.

3. De Vito G, Stracca GB. Comments on the design of log-periodic dipole antennas. IEEE Transactions on Antennas and Propagation. 1973;21(3):303-308. doi:10.1109/TAP.1973.1140476.

4. De Vito G, Stracca GB. Further comments on the design of log-periodic dipole antennas. IEEE Transactions on Antennas and Propagation. 1974;22(5):714-718. doi:10.1109/TAP.1974.1140881.

5. Butson P, Thompson G. A note on the calculation of the gain of log-periodic dipole antennas. IEEE Transactions on Antennas and Propagation. 1976;24(1):105-106. doi:10.1109/TAP.1976.1141278.

6. Cebik LB, W4RNL (SK). Log periodic arrays. In: Straw RD, N6BV, ed. The ARRL Antenna Book. 21st ed. The American Radio Relay League, Inc.; 2007:10.1-10.28. Available at: http://www.arrl.org/shop/Antennas/.

7. Hutira F, Bezek J, Bilik V. Design and investigation of a log-periodic antenna for DCS, PCS and UMTS mobile communications bands. Available at: http://hamwaves.com/lpda/doc/hutira.pdf.

8. DuHamel RH, Scherer JP. Frequency-independent antennas. In: Johnson RC, ed. Antenna Engineering Handbook. 3rd ed. McGraw-Hill, Inc.; 1993:35-53.

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