.. title: Impedance Transformation on a Transmission Line
.. slug: 28
.. date: 2021-10-28 12:00
.. tags: documentation,english,hardware,howto,open source
.. description:
.. wp-status: publish
.. has_math: true
.. |--| unicode:: U+2013 .. en dash
.. |ohm| unicode:: U+02126 .. Omega
:trim:
.. |_| unicode:: U+00A0 .. Non-breaking space
:trim:
As a ham-radio operator one is confronted with the challenge to connect
an antenna to a radio via a transmission line. Now often the impedance
of the antenna is different from the impedance used by the radio and the
transmission-line between antenna and radio. The impedance of radio and
a typical coax transmission line is usually 50 |Ohm| |_| in ham radio
applications. The impedance of the antenna varies and depends on the
technology and environmental factors.
A transmission line transforms the impedance of the antenna depending on
how long it is. A well-known formula for this is given in Chipman [1]_
p.134 formula 7.15:
.. math::
\frac{Z_d}{Z_0} =
\frac{e^{\gamma d}(Z_l/Z_0 + 1) + e^{-\gamma d}(Z_l/Z_0 - 1)}
{e^{\gamma d}(Z_l/Z_0 + 1) - e^{-\gamma d}(Z_l/Z_0 - 1)}
In this formula :math:`Z_d` is the impedance at distance :math:`d` from
the load, :math:`Z_l` is the impedance at the load (e.g. at the antenna)
and :math:`Z_0` is the characteristic impedance of the transmission line
and the radio, typically 50 |Ohm|, :math:`\gamma` is the complex
transmission coefficient which can be split into the real- and imaginary
parts, where :math:`\alpha` is the attenuation in Nepers/m and
:math:`\beta` is the phase constant, :math:`j` is imaginary unit (often
written as :math:`i` but in electrical engineering it is most often
denoted as :math:`j`):
.. math::
\gamma = \alpha + j \cdot \beta
It is sometimes claimed that a cable can improve the standing-wave ratio
at the radio because it transforms the impedance of the cable. This is
only true if the cable has a characteristic impedance *different* from
the input/output impedance of the radio (if we asume that the cable is
lossless which holds in many cases for short cables) which I'm going to
show in the following.
In the lossless case :math:`\alpha` in the formula above is 0. We can
express :math:`\beta` in terms of frequency and, finally, the wavelength
:math:`\lambda`:
.. math::
\beta = \frac{2\pi f}{c * \mathbb{VF}}
and:
.. math::
\lambda = \frac{c}{f}
The frequency :math:`f` is given in Hz, :math:`c` is speed of light and
:math:`\mathbb{VF}` is the velocity factor of the transmission line.
When we express the distance from the load :math:`d` in Chipman's
formula as :math:`l_\lambda` a multiple of of :math:`\lambda`, :math:`f`,
:math:`c`, and :math:`\mathbb{VF}` cancel and for the lossless case we get:
.. math::
\frac{Z_d}{Z_0} =
\frac{ e^{ 2\pi l_\lambda j}(Z_l/Z_0 + 1)
+ e^{-2\pi l_\lambda j}(Z_l/Z_0 - 1)
}
{ e^{ 2\pi l_\lambda j}(Z_l/Z_0 + 1)
- e^{-2\pi l_\lambda j}(Z_l/Z_0 - 1)
}
The complex reflection coefficient :math:`\rho` is given as (e.g. in
Chipman [1]_ 7.9 p.128):
.. math::
\rho = \frac{Z - Z_0}{Z + Z_0}
where :math:`Z` is the impedance at the point on the line we want to
know the reflection coefficient. From this we can compute the voltage
standing wave ratio (see e.g. Chipman [1]_ 8.21 p.165) which I'm calling
here :math:`S` for convenience:
.. math::
S = \frac{1+|\rho|}{1-|\rho|}
A simple `Python`_ program (python is very convenient for computations
like this because it supports complex numbers out of the box and is
free) for computing all this might look like::
from math import e, pi
def impedance (z_load, l_lambda, z0 = 50.0):
zl = z_load / z0
ex = 2j * pi * l_lambda
lp = e ** (ex) * (zl + 1)
lm = e ** (-ex) * (zl - 1)
return z0 * (lp + lm) / (lp - lm)
def vswr (z, z0 = 50.0):
absrho = abs ((z - z0) / (z + z0))
return (1 + absrho) / (1 - absrho)
If we compute several points for this, e.g., with an antenna that has
72 |Ohm| |_| and various multiples of :math:`\lambda` we get the same value
for the standing wave ratio:
+---------------------+--------------+-----------+
| :math:`l_{\lambda}` | :math:`Z_d` | :math:`S` |
+=====================+==============+===========+
| 0 | 72+0j | 1.44 |
+---------------------+--------------+-----------+
| 1/8 | 46.85-17.46j | 1.44 |
+---------------------+--------------+-----------+
| 3/8 | 46.85+17.46j | 1.44 |
+---------------------+--------------+-----------+
| 1/4 | 34.72-1.59j | 1.44 |
+---------------------+--------------+-----------+
| 1/2 | 72+0j | 1.44 |
+---------------------+--------------+-----------+
To prove that the standing wave ratio is *always* the same, no matter
how long the transmission line is, is left as an exercise to the reader,
a hint: It is enough to prove that the absolute value of :math:`\rho`
stays the same.
Now the more interesting case is when we take cable losses into account.
I've written a piece of software that can model a coax line from
manufacturer data, an idea that was published long ago by Frank Witt,
AI1H [2]_. The implementation is part of my open source
`antenna-optimizer`_ project
and features a command-line utility called ``coaxmodel``. With it you
can compute the input impedance (and standing wave ratio) for a real
cable. The implementation already contains models of some cables but it
is easy to add more (see at the end of ``coaxmodel.py``). In addition it
allows you to compute a stub match: Adding a parallel piece of
transmission line in parallel to the feed line at a certain distance
from the load (this piece of line is called a stub) will transform the
impedance in a way that the generator (the transceiver) will see a
50 |Ohm| |_| match. It would compute for the example given above (only a
subset of the output is given here, try it yourself)::
% coaxmodel -z 72 -f 435e6 -l .057 match
0.06 m at 435.00 MHz with 100 W applied
Load impedance 72.000 +0.000j Ω
Input impedance 46.857 -17.433j Ω
VSWR at load 1.440
VSWR at input 1.439
Inductive stub with open circuit at end:
Stub attached 0.06246 m from load
Stub length 0.20224 m
Resulting impedance 50.00 -0.00j
This tells us that a 5.7cm transmission line will transform the
72 |Ohm| |_| impedance at the load to a 46.86-17.43j |Ohm| |_| impedance
at the input end of the feed line. It is also visible that the
standing wave ratio at the input has improved very slightly due to
losses in the cable and that at this length the difference is
negligible.
Attaching a 20.2cm piece of 50 |Ohm| |_| wire with an open circuit at the
end in parallel to the feed wire at a distance of 6.2cm from the load
will transform the impedance to 50 |Ohm| |_| resulting in a standing wave
ratio of 1:1.
.. _`antenna-optimizer`:
https://github.com/schlatterbeck/antenna-optimizer
.. _`Python`: https://www.python.org
.. [1] Robert A. Chipman. Theory and Problems of Transmission Lines.
Schaums Outline. McGraw-Hill, 1968.
.. [2] Frank Witt. Transmission line properties from manufacturer’s
data. In Straw [3]_, pages 179–183.
.. [3] R. Dean Straw, editor. The ARRL Antenna Compendium, volume 6.
American Radio Relay League (ARRL), 1999.