Microstrip Filter Optimisation Using Masks
Introduction
In this article, the FEKO optimisation mask feature will be used to optimise the response of a microstrip filter. General familiarity with FEKO optimisation options is assumed. The optimisation mask feature allows for the comparison of an output data array obtained from one simulation result, with a predefined array, in the evaluation of the fitness of an optimisation step.
Start by considering the following generic microstrip line structure (this is a parametric generalisation of a model available in [1]):


Figure 1: Parametric geometry of the microstrip structure, with example CADFEKO mesh. 
The objective is to optimise this structure, to automatically realise specified filtering characteristics (i.e. transfer functions). The optimisation variables will be the set {l21; l22; l23}. The feed lines have 50 Ohm characteristic impedance and are driven by microstrip ports.
The following optimisation features will be employed:
 Masks: the FEKO optimiser allows the specification of masks with respect to which a given output series of values can be optimised. In this case, the behaviour of the transfer coefficient (S_{21}) will be optimized as a function of frequency.

Weighted, combined, optimisation goals: the FEKO optimiser allows the user to combine multiple, specified goals, such that all are targeted simultaneously. The combination action allows weighting of the individual goals before combination. The combination procedure can be an averaging step, or a minimisation/maximisation action on the set of weighted goals. With regards to realising filter characteristics, this will clearly be an important feature, as the transfer coefficient needs to be either maximised or minimised depending on the frequency band.
LowPass Realisation
Consider optimising the transfer coefficient in the band 4 – 6 GHz, with a lowpass characteristic. A set of 11 linearly spaced frequency points are defined. The linear interpolation of S_{21} at these points is the function that will be optimised. Note that continuous, adaptive frequency interpolation cannot be used together with optimisation masks, at present. The masks specified in FEKO are shown in Figure 2.
Figure 2: Masks for lowpass filter realisation. 
Note the following important points regarding mask definitions:
 Optimising with respect to any given mask implies that the difference between the output function (as defined above) and the function obtained by linearly interpolating the mask data, will be optimised.
 The number of mask data points do not need to correspond with the number of output points (as the optimiser will use a piecewise linear fitting on the mask array to determine the values for comparison at the correct points). For instance, in the present case 11 output points are specified, but only 4 mask data points. However, the mask should always span the whole range under consideration. For example, in the present case it means that points at both 4 and 6 GHz must form part of the mask definitions.
The output function is required to be less than the first mask, and greater than the second. These two subgoals are weighed by 10 and 1 (the cutoff characteristic is considered the more important of the two) and combined by way of set maximisation to form the final goal. The optimisation method (algorithm) is set to Automatic and the convergence accuracy set to High. The resulting optimisation required 68 iterations to converge, with the optimum parameter set found as follows:
Parameter  Value 

l21  7.657237107e+00 mm 
l22  5.551954736e+00 mm 
l23  7.869152734e+00 mm 
Sparameter results for the final structure, as calculated with adaptive frequency interpolation, are shown in Figure 3. Observe the lowpass behaviour as specified. Note that the masks are nearly, but not exactly satisfied. This is due to the use of a combined goal, which forces the optimiser to a compromise. Also, with optimisation one must realise that it might not always be possible to achieve a goal exactly within the specified parameter space.
Figure 3: Sparameters of the optimised lowpass filter. (a) 4 – 6 GHz range used for optimisation. (b) 1 – 10 GHz range. 
BandReject Realisation
Again consider optimising the transfer coefficient in the band 4 – 6 GHz, but now towards a bandreject characteristic. Again 11 linearly spaced frequency points are specified for the analysis. The masks specified in FEKO are shown in Figure 4.
Figure 4: Masks for bandreject filter realisation. 
As before, the output function is required to be less than the first mask, and greater than the second. In this case the two subgoals are weighed by 100 and 1 and combined by way of set maximisation to form the final goal. The optimisation method is set to Automatic and the convergence accuracy set to High. The resulting optimisation required 84 iterations to converge, with the optimum parameter set found as follows:
Parameter  Value 

l21  3.591053158e+00 mm 
l22  4.532049193e+00 mm 
l23  9.483669760e+00 mm 
Sparameter results for the final structure, as calculated with adaptive frequency interpolation, are shown in Figure 5. Observe the bandreject behaviour as specified.
Figure 5: Sparameters of the optimised bandreject filter. (a) 4 – 6 GHz range used for optimisation. (b) 1 – 10 GHz range. 
Conclusion
Here, S_{21} as a function of frequency constituted the array to which the masks were applied, but other data sets may also be targeted, such as functions of angle (e.g. a radiation pattern) or functions of position (e.g. a set of near field points).
References
[1] 
A.C. Polycarpou, P.A. Tirkas, and C.A. Balanis, "The FiniteElement Method for Modeling Circuits and Interconnects for Electronic Packaging," IEEE Trans. on Microwave Theory and Techniques, Vol. 45, No. 10, October 1997. 