 |
| |
 |
The objective of this design exercise is to use genetic algorithm (GA) methods to evaluate optimum values of Z1, Z2 and Z3 for the three stage transformer. In this design example the characteristic of the transformer is chosen to be maximally flat at the design frequency of the transformer. An analytic solution is available to provide this frequency domain characteristic, that of the binomial transformer. This is used to compare with the GA result
An impedance transformer is used to match mismatched impedances, for example a transmission line terminated with a mismatched load, or a transition from one transmission line geometry to another (with associated change in impedance).
The quarter wave transformer is a section of line that can be used to modify the impedance presented to the feed line by the load. At the design frequency the transformer section is a quarter wavelength long and the intrinsic impedance of this line section is chosen to cause the wave impedance at the transition to match the feed line impedance, giving a perfect match.
Away from the design frequency the transformer degrades significantly and he passband for a single stage transformer is narrow. The bandwidth of the transformer can be increased by using more quarter wave sections sequentially along the transmission line. The frequency domain characteristic of the resulting transformer is determined by the impedance values of each stage, shown in the above figure as Z1 to Z3 for a three stage design.
Various frequency domain characteristics can be achieved using varying impedance values. For example the maximum passband width is provided by the Chebyshev transformer, although this is achieved at the expense of ripple within the passband. The binomial transformer provides a characteristic that is maximally flat at the design frequency. This latter characteristic will be used as an objective function for a genetic algorithm optimisation of the three stage transformer.
In this design example a transmission line of 50W is matched to a load of 100W using a three stage transformer.
The genetic algorithm optimisation computation was designed using the standard operators of selection, crossover and mutation. The specific encoding for the problem involved a binary encoding of the three transformer stage impedance ratios, each stored at 12 bit resolution, resulting in a 36 bit chromosome. The population size was 1000 and the initial designs were generated randomly.
The GA was set up with an objective function shown in Equation (1).
(1)
where r
= reflection coefficient
f = frequency
The coefficients a and b were chosen to give equal emphasis to reduction in reflection coefficient and reflection coefficient gradient at the design frequency.
The GA calculation was run for 100 generations although convergence was achieved within 40 generation. The convergence history is is shown in the following plot.
The GA designed transformer impedance ratio values, Z1 to Z3, were 1.0908, 1.4146 and 1.8339 respectively (referenced to the 50W line). These values compare to an three stage binomial transformer where the analytically derived impedance rations are 1.0907, 1.4142 and 1.8337 respectively. The frequency domain characteristic for both transformers is shown below.
It can be concluded that the GA optimisation route, taking less that a few seconds computing time in this case, is an effective optimisation tool for this class of problem producing a design close to the analytic optimum. Although this example is trivial it demonstrates the utility of the GA. The real benefit of the GA is that it may be used to optimise designs to provide a characteristic for which no analytic solution is readily available.