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In order to achieve broadband matching a multi-section transformer can be designed with a large number of matching stages. As the number of stages is increased the impedance change between sections becomes smaller. In the limit of a transformer with an infinite number of stages the result is a smooth, continuous variation in impedance between feed line and load, represented in the figure by Z(x) where x is the distance along the matching section. For maximally wide passband response and a specified passband ripple the taper profile has an analytic form known as the Klopfenstein taper. This design problem will be analysed using a genetic algorithm optimisation approach and the solution compared to the analytic result.
The Klopfenstein taper is the optimal taper profile to achieve the minimum reflection coefficient over a specified passband in the shortest possible transition length. It is derived from a stepped Chebychev transformer in the limit of an infinite number of stages. The Chebychev filter characteristic has the maximum passband width for an accepted passband ripple.
The present design problem is to match a 50W transmission line to a 100W load using a tapered matching section. The results are shown in terms of non-dimensional variables, normalised to the length of the matching section L. The Klopfenstein taper was designed with a maximum allowable passband ripple in the reflection coefficient of 0.02, or -34dB.
The genetic algorithm calculation was set up to solve not a continuous taper but or a multi-section transformer with a large number of stages. To approximate a continuous taper, 20 matching sections are used. Each stage is represented as an 8 bit binary encoded impedance.
The objective function was designed to minimise the sum of the reflection coefficient at all frequencies within the passband. For a Klopfenstein taper the passband is specified as bL>1.13p, where b is the propagation constant for the transmission line.
The bandwidth used in the GA calculation was that of the Klopfenstein taper, allowing direct comparison of the designs, although in the GA case both a lower and upper limit were necessary to evaluate the design performance. A total bandwidth of 4 octaves was used with 40 sample frequencies at equal intervals throughout the band. The sum of reflection coefficients was weighted to give greater importance to the lower frequencies, being the more difficult to match.
The numerical value of Z as a function of normalised length along the taper is shown in in the following figure. An exponential taper design has been included for comparison.
The following figure shows the frequency domain characteristic of each of
these transformers. The GA design can be seen to be approaching the ideal
of the Klopfenstein taper with a flat passband characteristic. The exponential
taper achieves good high frequency performance at the expense of low frequency
performance. Note that the GA computation can be biased to provide whatever
passband characteristic is required by defining a suitable objective function
for the optimisation process.
