hamwaves.com
;

Log‑Periodic Dipole Array Calculator

Serge Stroobandt, ON4AA

Michael McCue, W7YZT

Copyright 2014–2020, licensed under Creative Commons BY-NC-SA

  1. Home
  2. LPDA Calculator

Overview

HF wire antenna rendering of a log‑periodic dipole array.
Drawing by: Michael McCue, W7YZT ©2017–2018

Detail of the end‑fed feed of a VHF log‑periodic dipole array. The feed coax is routed through the inside of the right boom. Only the center conductor of the coaxial cable is connected to the left boom by means of a rivet. The dipole elements are threaded on the outside. Polymer standoff insulators between the parallel booms are also visible. Use my parallel square transmission line calculator to determine the correct boom separation distance, given \(Z_\text{c,feed}\).

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 \(\sigma\).15 Decreasing \(\sigma\) will decrease the boom length \(L\). Decreasing \(\tau\) will decrease both the boom length \(L\) and the number of elements \(N\). Due to space and resource constraints, amateur radio log-periodic antennas are often 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 \(Z_\text{c,feed}=100\,\Omega\) and \(\frac{\ell_i}{⌀_N}=125\). Source: Hutira et al.7

Input

Input
lowest frequency* \(f_1\) MHz
highest frequency* \(f_\text{n}\) MHz
diameter of the shortest element N \(⌀_N\)  
characteristic input impedance \(Z_\text{c,in}\)
taper \(\tau\) 0.800.98
relative spacing \(\sigma\) 0.03\(\sigma_\text{opt}\)

Input guidelines:

Resulting design

Resulting design
relative operating bandwidth \(B\)
optimal spacing \(\sigma_\text{opt}\)
cotangent of the apex half-angle \(\cot{\alpha}\)
relative bandwidth of the active region \(B_\text{ar}\)
relative bandwidth of the structure \(B_\text{S}\)
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 \(Z_\text{term}\) \(\ell_{Z_\text{term}}\)
average characteristic impedance of the shortest element N \(Z_{\text{c,}N}\)
required characteristic impedance of the feeder connecting the elements \(Z_\text{c,feed}\)

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.

\[B = \frac{f_\text{n}}{f_1}\]

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

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

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

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

\[B_\text{S} = B\cdot B_\text{ar}\]

\[N = 1+\frac{\ln B_\text{S}}{\ln\frac{1}{\tau}} \qquad \begin{cases} \{N\} \gt 0.3 \rightarrow \lfloor N\rceil = \lceil N\rceil\\ \{N\} \le 0.3 \rightarrow \lfloor N\rceil = \lfloor N\rfloor \end{cases}\]

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

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

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

\[\ell_{Z_\text{term}} = \frac{\lambda_1}{8}\]

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

\[Z_\text{c,feed} = \frac{Z_\text{c,in}^2}{8\,\sigma'Z_{\text{c,}N}} + Z_\text{c,in} \sqrt{\left( \frac{Z_\text{c,in}}{8\,\sigma'Z_{\text{c,}N}} \right)^2 + 1}\]

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 3. 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.
R. Carrel. The design of log-periodic dipole antennas. In: IRE International Convention Record. Vol 9.; 1961:61-75. doi:10.1109/IRECON.1961.1151016
2.
W.M. Cheong, R.W.P. King. Log-periodic dipole antenna. Radio Science. 1967;2:1315-1325.
3.
G. De Vito, Giovanni B. Stracca. 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.
G. De Vito, Giovanni B. Stracca. 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.
P.C. Butson, G.T. Thompson. 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. http://www.arrl.org/shop/Antennas/
7.
Frantisek Hutira, Jan Bezek, Vladimir Bilik. Design and investigation of a log-periodic antenna for DCS, PCS and UMTS mobile communications bands. http://hamwaves.com/lpda/doc/hutira.pdf
8.
Raymond H. DuHamel, James P. Scherer. Frequency-independent antennas. In: Richard C. Johnson, ed. Antenna Engineering Handbook. 3rd ed. McGraw-Hill, Inc.; 1993:35-53.
5
Creative Commons Licence
This work is licensed under a Creative Commons Attribution‑NonCommercial‑ShareAlike 4.0 International License.
Other licensing available on request.
GNU GPL v3
Unless otherwise stated, all originally authored software on this site is licensed under the terms of GNU GPL version 3.
cookie
This static web site has no backend database.
Hence, no personal data is collected and GDPR compliance is met.
Moreover, this domain does not set any first party cookies.

All Google ads shown on this web site are, irrespective of your location,
restricted in data processing to meet compliance with the CCPA and GDPR.
However, Google AdSense may set third party cookies for traffic analysis and
use JavaScript to obtain a unique set of browser data.
Your browser can be configured to block third party cookies.
Furthermore, installing an ad blocker like EFF's Privacy Badger
will block the JavaScript of ads.
Google's ad policies can be found here.
This page employs a Python Bottle server‑side script.
This page includes an open-source client-side script, written in Python and
transcoded by Brython to make it run as secure JavaScript in the browser.
Static XHTML generated from Markdown by Pandoc and
the GNU/Linux make, sed and gpp commands.
LaTeXmath markup rendered with MathJax.
BibTeX references are best read with JabRef.
Unattended CSS typesetting with Prince.
This work is published at https://hamwaves.com/lpda/en/.
profile for Serge Stroobandt on Stack Exchange, a network of free, community-driven Q&A sites
GnuPG
Use my OpenPGP public key to encrypt messages for:

echo c2VyZ2VAc3Ryb29iYW5kdC5jb20K |base64 -d
Last update: Wednesday, March 24, 2021.