Impedance Transformation on a Transmission Line

Ralf Schlatterbeck 2021-10-28 12:00

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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 50Ω 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:

\begin{equation*} \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)} \end{equation*}

In this formula \(Z_d\) is the impedance at distance \(d\) from the load, \(Z_l\) is the impedance at the load (e.g. at the antenna) and \(Z_0\) is the characteristic impedance of the transmission line and the radio, typically 50Ω, \(\gamma\) is the complex transmission coefficient which can be split into the real- and imaginary parts, where \(\alpha\) is the attenuation in Nepers/m and \(\beta\) is the phase constant, \(j\) is imaginary unit (often written as \(i\) but in electrical engineering it is most often denoted as \(j\)):

\begin{equation*} \gamma = \alpha + j \cdot \beta \end{equation*}

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 \(\alpha\) in the formula above is 0. We can express \(\beta\) in terms of frequency and, finally, the wavelength \(\lambda\):

\begin{equation*} \beta = \frac{2\pi f}{c * \mathbb{VF}} \end{equation*}

and:

\begin{equation*} \lambda = \frac{c}{f} \end{equation*}

The frequency \(f\) is given in Hz, \(c\) is speed of light and \(\mathbb{VF}\) is the velocity factor of the transmission line. When we express the distance from the load \(d\) in Chipman's formula as \(l_\lambda\) a multiple of of \(\lambda\), \(f\), \(c\), and \(\mathbb{VF}\) cancel and for the lossless case we get:

\begin{equation*} \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) } \end{equation*}

The complex reflection coefficient \(\rho\) is given as (e.g. in Chipman [1] 7.9 p.128):

\begin{equation*} \rho = \frac{Z - Z_0}{Z + Z_0} \end{equation*}

where \(Z\) 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 \(S\) for convenience:

\begin{equation*} S = \frac{1+|\rho|}{1-|\rho|} \end{equation*}

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 72Ω and various multiples of \(\lambda\) we get the same value for the standing wave ratio:

\(l_{\lambda}\)	\(Z_d\)	\(S\)
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 \(\rho\) 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 50Ω 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 72Ω impedance at the load to a 46.86-17.43jΩ 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 50Ω 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 50Ω resulting in a standing wave ratio of 1:1.

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