# Parallel Circular Conductor Transmission Line Calculator

## Serge Y. Stroobandt, ON4AA

### Copyright 2015–2017, licensed under Creative Commons BY-NC-SA

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# Introduction

This calculator is a tool for designing balanced transmission lines with a specific desired characteristic impedance \(Z_c\) and made of parallel circular conductors of a given diameter \(d\). Round open wire ladder line, window line, and twin-line are all balanced transmission lines which are frequently encountered as the feed line of severely mismatched multi-band wire antennas, especially the G5RV antenna. The conductors being **massive or hollow does not affect the characteristic impedance.**

# Construction

Here is an excellent construction suggestion by Leon Salden, VK3VGA. He devised a **ladder line spreader** built from durable and readily available materials; a black cable tie and black polyethylene (PE) **irrigation extension tube.** For smaller separation distances, **black LED spacers** can also be used.

# Formulas

Following formula can be derived for the characteristic impedance of a parallel wire transmission line:^{1}

\[Z_c = \frac{Z_0}{\pi\sqrt{\epsilon_r}}\;\text{acosh}\left(\frac{D}{d}\right)\]

The characteristic impedance of free space is exactly:

\[Z_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} = \mu_0\cdot c_0 \approx 376.73\,\Omega\]

where:

\(c_0 = 299\,792\,458\,\frac{\text{m}}{\text{s}}\): the speed of light in free space

\(\mu_0 = 4\pi\cdot10^{-7}\frac{\text{H}}{\text{m}}\): the free space permeability

\(\epsilon_0 = \frac{1}{\mu_0 c_0^2}\): the absolute permittivity of free space

\(Z_0\): the characteristic impedance of free space

Rearranging and solving (1) for \(D\):

\[D = d\cdot\cosh{\left(\pi\frac{Z_c}{Z_0}\sqrt{\epsilon_r}\right)}\]

\[s = D - d\]

where:

\(D\): the centre to centre distance

\(d\): the diameter of the circular conductors

\(Z_c\): the desired characteristic impedance of the transmission line

\(Z_0\): the characteristic impedance of free space

\(\epsilon_r\): the relative dielectric constant of the surrounding medium (1.00054 for air)

\(s\): the space between the circular conductors

# 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: `zc.circular.py`

# References

1. Robert A. Chipman. *Theory and Problems of Transmission Lines*. McGraw-Hill Book Company; 1968.

This work is licensed under a Creative Commons Attribution‑NonCommercial‑ShareAlike 4.0 International License.

Other licensing available on request.

Unless otherwise stated, all originally authored software on this site is licensed under the GNU GPL version 3.

transcoded by to make it run as secure JavaScript in the browser.