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Log‑Periodic Dipole Array Calculator

Serge Stroobandt, ON4AA

Michael McCue

Copyright 2014–2017, licensed under Creative Commons BY-NC-SA

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Overview

HF wire antenna rendering of a log‑periodic dipole array. Drawing by: Michael McCue ©2017

HF wire antenna rendering of a log‑periodic dipole array.
Drawing by: Michael McCue ©2017

Free-space directivity

LPDA
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
Free-space directivity of a log-periodic dipole array as a function of its taper \tau and spacing \sigma for Z_{0f}=100\,\Omega and \frac{\ell_i}{⌀_N}=125. Source: Hutira et al.7

Free-space directivity of a log-periodic dipole array as a function of its taper \(\tau\) and spacing \(\sigma\) for \(Z_{0f}=100\,\Omega\) and \(\frac{\ell_i}{⌀_N}=125\). Source: Hutira et al.7

Input

Input
lowest frequency* \(f_1\) MHz
highest frequency* \(f_n\) MHz
diameter of the shortest element N \(⌀_N\)  
characteristic input impedance \(Z_{0i}\)
taper \(\tau\) 0.800.98
relative spacing \(\sigma\) 0.03\(\sigma_{opt}\)

Input guidelines:

Resulting design

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:

Copy & Paste

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.

License: GNU GPL version 3
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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Last update: Monday, October 9, 2017.