Microwave baluns

A balun is a device which converts an unbalanced (non-symmetric) input, e.g. from coax, into a two (or possibly an even number other than two) port balanced output. See Munk (section in “Antennas and wave propagation” by Kraus et. al.) for a good account of available forms.

We will distinguish between “perfect” and “imperfect” baluns.¬† A perfect balun is one which provides a perfectly balanced output over its operational frequency range. On the other hand an imperfect balun operates over a frequency range only part of which provides a balanced output. The cutaway balun is an example of the latter sort, which operates over very large bandwidths with transmission down to near zero frequency. A Marchand balun is an example of the former sort.

For low frequency operation a transformer provides a near perfect balun, with a secondary winding magnetically coupled but otherwise isolated from the primary. However, transformers are limited in their technical capabilities to operation below the low GHz frequencies. For antenna applications, microwave baluns with near perfect performance are often required. High performance ultra wide band spiral and sinuous antennas are a case in point. There is considerable scope for innovative design but a question raises itself. “Are there any fundamental bandwidth limitations?”.

It turns out that for non-transformer perfect baluns there are indeed fundamental bandwidth limitations. Our paper  (balun_limits) is, as far as we know, the first on this topic. By way of example, suppose we consider such a perfect balun whose reflection coefficient at the unbalanced input is a rectangular distribution with frequency. Let the lower and upper angular frequencies be \omega_1 and \omega_2 and define the fractional bandwidth,

    \begin{equation*} \beta = \frac{\omega_2-\omega_1}{\sqrt{\omega_1\omega_2}} \end{equation*}

Let \Gamma_{db} be the reflection coefficient in dB at the unbalanced input and define,

    \begin{equation*} h=\frac{-\Gamma_{dB}}{20\log_{10}(e)} \approx -0.1151\,\Gamma_{dB} \end{equation*}

then the maximum attainable loss bandwidth product is given by,

    \begin{equation*} \beta h \approx \frac{\pi^2 Z_c}{2R_L} \end{equation*}

where Z_c is the characteristic impedance between the balanced output ports (odd mode impedance if the two output ports are referenced to ground) and R_L is the load impedance between the balanced output ports. This loss bandwidth product is attainable if the reverse reflection coefficient (i.e. the reflection coefficient at the balanced output ports looking towards the input) is minimum reflection phase and a design procedure is provided for this purpose.