COMPONENT VALUE SELECTION FOR ACTIVE FILTERS USING GENETIC ALGORITHMS

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1 COMPONENT VALUE SELECTION FOR ACTIVE FILTERS USING GENETIC ALGORITHMS David H. Horrocks and Mark C. Spittle, School of Electrical, Electronic and Systems Engineering, University of Wales, College of Cardiff, Cardiff, CF2 1XH, Great Britain. Abstract In the realisation of analogue electronic circuits it is common practise, because of costs, to specify discrete passive component values from a range of manufactured preferred values. The original design typically results in values that do not all coincide with preferred value and conventionally the designer selects the nearest preferred value thus causing a design deviation. In general a better set of preferred values will exist. However this set will be in a solution space of all component-value combinations that is normally huge. We show that Algorithms can be successfully used to search this space. The application chosen is a second order state variable active filter. Fully-discrete and semiintegrated forms are considered. The designs produced are much superior to those achieved using the conventional method. 1 INTRODUCTION. Since the early days of electronics, components such as resistors and capacitors have been produced in a range of 'preferred' values. Here, discrete components are manufactured to have a fixed number of nominal values in each decade, over a wide decade range, the values being approximately evenly spread on a logarithmic scale. Typical is the well known 'twelve-series', for which the preferred values are..., 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82, To reduce costs, components for discrete element circuits are typically selected from either this, or another, preferred value range wherever possible. Despite the dominance of integrated circuits, discrete components are still in widespread use for analogue circuits, for example active filter circuits. Usually, analogue circuit design procedures result in component values that do not all exactly coincide with preferred values. The performance therefore deviates from the ideal when the nearest preferred value is selected. This deviation can be held to within an acceptable amount by selecting from a more closely spaced series, by connecting pairs of components in series or parallel, or by specifying special values. However all of these methods carry penalties. In analogue circuits, it is common that there are more components than there are significant response parameters. Classical design methods usually reduce the consequent number of degrees of freedom by, for example, choosing certain components to be equal, thus enabling direct design formulae to be obtained. This simplification in the design procedure means that combinations of preferred component values could be excluded that would have resulted in the design being closer to the ideal. To perform an exhaustive search on all possible combinations of preferred values, in order to obtain an optimised design, is often not feasible. For example, in the eight-component circuit considered in this paper, if components are selected from the twelve-series over a four-decade range then the search space contains about points, which is beyond the range that could be computer-searched in a reasonable time.

2 An alternative method of determining discrete component values is therefore required. It is the aim of this paper to show that the application of Algorithms (GAs) to the problem of preferred value selection can result in very much improved component value assignments. Various genetic techniques are presented and compared with a classical design approach. The circuit example chosen is the state variable second order active filter. However, the approach can be similarly applied to other analogue electronic circuits. Two forms of the state variable active filter are considered. The first is the conventional op amp version in which all the resistor and capacitor passive components are selected from a series of preferred values. The second is the semi-integrated form as exemplified by the AF100 from National Semiconductors [1], which is typical of several of this type. In this form, the op amps and some of the passive components are integrated and the remaining discrete components are attached externally. In the examples considered, the twelve-series of preferred values is assumed for the discrete resistors and capacitors. 2 THE STATE VARIABLE ACTIVE FILTER The state variable filter is illustrated in Figure 1, and is well described in the literature, for example [2]. The low pass output is assumed here to be the desired output. The response of a second order lowpass circuit is completely specified by the passband gain, H, the passband cut-off frequency ω 0 = 2π f 0, and the selectivity factor, Q. For the state variable circuit these quantities are given, in terms of the passive component values, by the following expressions. H R R R 2 ( ) = R ( R + R ), ω = F H G R I R K J F H G 1 C C R R I K J, and Q R R R 3( ) = R ( R + R ) C R R C R R (1). R4 C1 C2 R2 R5 R6 input R1 R2 lowpass output Figure 1. State Variable Filter With Low-Pass Frequency Response The specification chosen here is ω 0 = π = rad/sec, and Q = 2 = The passband gain, H, is not very critical in most applications since it can be readily compensated for by other cascaded analogue circuits. In the conventional design procedure, H is fixed at some value; however for the Algorithm method described below it is unconstrained. As a result of choosing preferred values for the components, the cut-off frequency and selectivity will deviate from the specification by ω and Q respectively. It is important that these deviations be as low as possible, especially where the circuit is one of a number of sections that in cascade make up a high order filter in which the response is more sensitive to errors. The error criterion adopted here is

3 ω Error = a + a 1 ω 2 0 Q Q (2), where a 1 = 0.5 and a 2 = 0.5 is assumed. Other values for these constants would be chosen where the acceptable design tolerances for cut-off frequency and selectivity are unequal. 3 CONVENTIONAL DESIGN PROCEDURE For the fully discrete circuit there are eight component values (six resistors and two capacitors) whose values are to be chosen to give the specified values for ω 0 and Q, and also H where this is important. The choice of component values is made tractable by introducing further constraints. The method that is conventionally proposed is to arbitrarily choose to make both capacitors equal to C and all resistors except R 2 equal to R. Equations (1) then reduce to the following. and ω 0 = 1 RC (3), R1 = R3 = R4 = R5 = R6 = R ; R2 = ( 2Q 1) R and C1 = C2 = C (4) The procedure is to first choose a pair of values for R and C to satisfy (3); then to calculate the circuit components using equations (4). The resulting passband gain is H=1. For an exact design, a sensible way is to first choose a preferred value for R in the middle of the range since it results in five of the resistors also having preferred values. The values for the remaining R 2, C 1 and C 2 then follow from (3) and (4), and in general have to be special non-preferred values if an exact design is to be obtained. For a design using preferred values only these exact values can be rounded to the nearest preferred values. A simple improvement to this procedure is to repeat it for all possible preferred values for R and to select the design having the lowest error. It is only necessary to do this for the twelve values in one decade since the errors produced repeat for each decade because of the constraints introduced. The result of doing this for this example is shown in the second column, 'Nearest Preferred', of Table 1. The design error obtained is %. For the semi-integrated circuit AF100, components R 1, R 5, and R 6 are connected externally. The other five passive components are integrated in the device and have fixed values. In the conventional design the user specifies the three performance parameters, H, ω 0 and Q and solves for the three external resistor values by means of (1). 4 COMPONENT SELECTION USING GENETIC ALGORITHMS 4.1 Algorithms Algorithms are optimisation procedures based on ideas borrowed from natural selection and evolution in nature. Detailed descriptions of GAs are to be found in literature; books on the subject include those by Goldberg [3] and Davis [4]. Research in the subject is very active and is expanding. A good source of the latest work are the Proceedings of the International Conference on Algorithms and their Applications, which has been held every two years since There is by now a variety of GAs, However the principal steps in most are as follows.

4 1. The representation of the set of variable of a potential solution to the problem as 'gene', which is often in the form of a bit string. 2. The definition of a 'fitness function' that quantifies how good any potential solution is from the contents of its gene. 3. The establishment, that is 'seeding', of an initial population of genes. 4. The creation of the next population using breeding operators such as the random selection of 'parent' genes for breeding according to their finesses, and the generation of 'children' genes using cross-over and mutation operations. 5 The repeated generation of new populations, and on convergence the best solution is output. 4.2 Implementation For the application considered here the basic form of GA was found to perform satisfactorily, and it was not necessary to implement a more elaborate scheme. The fully discrete circuit problem was represented by having eight locations in each gene - one location for each passive component in the circuit. Each of these locations was allocated a group of bits to identify a particular preferred value for that component. The number of bits allocated depends on the representation method. For the 'One-Bit' method, one bit is allocated which allows a choice of from two preferred values that are nearest to the exact design values that emerged from the conventional design procedure described above. The number of bits is progressively increased in the 'Two-Bit', 'Three-Bit' and 'Four-Bit' methods. In the latter method the four bits allow sixteen preferred values spread on either side of the exact values to be offered for selection by the GA. The purpose of these methods is to explore the effect of increasing the range of the search. In the 'Full Evolution' method eight groups of six bits are used to specify any preferred value over a four-decade range. Two of these bits signify the decade in the range 10 3 to 10 6 ohms for the resistors and 10-9 to 10-6 farads for the capacitors. Values outside these ranges were judged to lead to unwanted practical effects such as stray capacitance effects or large signal currents. However the decade range can be readily modified by changing the number of bits. The remaining four bits are used to signify any of the twelve preferred values in the decade range. For the semi-integrated circuit the 'Full Evolution' method has been applied using the same gene data structure as for the fully discrete circuit, but three rather than eight groups of six bits to specify the three externally connected components. The fitness function was defined as the simple reciprocal of error as defined by equation (2). This produces the desired characteristic that fitness is larger for better circuits with lower errors. and also causes increased fitness separation for groups of solutions that are near the optimum. The population size was taken to be fifty. From the population 'Parent' genes were selected with a probability that uniformly increases according to the fitness ranking of the genes. 'Child' genes were generated using random single-point cross-over applied with a probably of 0.3, and mutation was applied to each bit in the gene with a probability of No formal criterion was used to detect convergence; populations were generated repeatedly until the best fitness appeared to have settled at a relatively stable value. 4.3 Results The results obtained with these GAs are shown in table 1. Columns three to seven apply to the fully discrete circuit. A very large reduction in design error is evident. Even the highly restricted One-Bit Algorithm produces a ten-fold improvement over the conventional 'Nearest Preferred' method.

5 For the full-evolution method, the design error is extremely low at 0.001%. In practical circuits this low figure is likely to be masked by manufacturing tolerance errors in the components themselves even in the most stringently specified cases. The last column shows the results for the semi-integrated AF100 circuit. This has five fixed internal components, indicated by parenthesis. The remaining three parameters are available for choice by the GA. Because this is more restrictive than the fully discrete circuit, the error obtained is larger. Even so, the figure achieved is good considering the large separation between the preferred values of about 8%. In the genetic evolution of these designs, the passband gain, H, was unconstrained. The resulting passband gains, taken to be H=1 in the conventional design, range from to in the GA designs. These values would be acceptable for many applications. In the AF100 an extra op amp is included which can be used for adjusting the pass band gain. However it is conceivable that unacceptable values could emerge. A strategy to avoid this could be to add a term to the fitness function that depends on the resulting H. This term would be set to zero if H was in the acceptable range and set to a large negative value if not. Nearest Preferred One-Bit Two-Bit Three-Bit Four-Bit Full Evolution National AF100 Q ω R R (100000) R (100000) R (10000) R R C (1 10-9) C (1 10-9) Error (%) Conclusions Algorithms have been applied to component selection in an analogue active circuit. Significant reductions in design error can be achieved compared with the conventional approach. The more freedom of component choice allowed to the Algorithm generally results in a lower design error being achieved. It is expected that these methods are applicable to other types of analogue circuits and of greater complexity. The reduction in design error is large enough to consider restricting the search to minimise the number of different preferred values in any one design, thereby reducing inventory costs. An attractive feature of GAs is the ease with which mixed criteria can be included with the fitness measure, such monetary costs of components and power supply loading. Our work is continuing on these and related topics. References Table 1. Component Values and Performance Figures for Various Design Procedures. [1] National Semiconductor Corp., Data Aquisition Data Book, National Semiconductors Corp., Santa Clara, CA, USU, pp , (1993).

6 [2] P. Bowron and F. W. Stephenson, Active Filters for Communications and Instrumentation, McGraw-Hill, (1979). [3] D. E. Goldberg, Algorithms in Search, Optimisation, and Machine Learning, Addison- Wesley, (1989). [4] L. Davis, Ed., Handbook of Algorithms, Van Nostrand Reinhold, (1991).