Design of low-pass active filters using MATLAB

International Journal of Electrical Engineering Education, Jul 1998 by Joaquim, Marcelo Basilio

Abstract In this paper a program developed in MATLAB programming language to aid in the design of all-pole active filters using operational amplifiers is presented. The filter structure chosen was the Sallen-Key that furnishes a filter with low sensitivity to element tolerances and it is easy to implement in hardware.

INTRODUCTION

Active filters using operational amplifiers are largely utilized and vital in modern electronics, but their design is tedious due to the great number of calculations involved, or the use of curves and tables. Thus, the best way to design filters is to use digital computer programs. In this paper we describe a program developed in MATLAB for the windows programming language to aid in the design of filters. The program has been developed for applications in the teaching course `Applications of Linear Integrated Circuits' which is in the curriculum of the Electronic Engineering degree at the School of Engineering of Sao Carlos. Another objective reached with this program is the introduction of the students to a high level programming language, necessary nowadays for the modern engineer. The program described here was developed to design an all-pole low-pass active filter, with approximation to Butterworth or Chebyshev characteristics. The structure chosen was the Sallen-Key (voltagecontrolled voltage source) which furnishes a filter with low sensitivity to element tolerances and is easy to implement in hardware.

THE LOW-PASS SALLEN-KEY ACTIVE FILTER

Active filters are composed of circuits containing operational amplifiers, resistors and capacitors. Many structures for active filters have been developed1-3. One of the most popular structures is the voltage-controlled voltage source circuit (or Sallen-Key) with equal resistors in the feedback network. This kind of structure has been widely used because it is easy to implement and because it furnishes stable filters with low sensitivity to circuit element tolerances.

Filter synthesis directly from polynomials offers an elegant solution to filter design. However, it may involve laborious computations to determine circuit element values. A number of design methods have been developed to simplify the computations by the use of curves, tables and step-by-step calculations. In general the design procedure involves:

* the choice of the response characteristic,

* the determination of the order of the filter using curves,

* the determination of the normalized filter elements using tables,

* the removal of the normalization of the filters' elements.

Further on, if the filter order is greater than three, the procedure above has to be repeated at least once. Consequently, to find the values of the filter capacitors is somewhat laborious and tedious. Thus, the best way to design this kind of filter is to accomplish this using programming languages.

THE DESIGN FILTER PROGRAM

A program to design the filters has been implemented in the MATLAB 4.2cl for Windows programming language. It uses graphic facilities and built-in functions for the design of filters. The built-in functions buttord, chebord, butter and chebyl are used to determine the order and filter coefficients. In addition, there are the root function which is used to determine the pole locations, and functions like freqs, plot and semilogx, lets the graphic visualization of the magnitude and pole locations. MATLAB also has a powerful graphical user interface (GUI), in particular the uicontrol function, which helps the user to manipulate menus, buttons, lists and fields and create friendly windows for running a program. All these functions were used easily with success to make the design of active filters program.

The window shown in Fig. 2 illustrates the program interface for designing low-pass active filters. On the top of the window are plotted the two structures of the filters given in Fig. 1. In the center of the window we have the graphical output to show the magnitude of the transfer functions and pole locations. And on the bottom there is a text window where the program shows the filters' parameters, i.e., the number of sections to be used in cascade, the filter order and the value of the capacitors C^sub 1^, C^sub 2^ and C^sub 3^ in nanoFarads for each section. If the section has order two then the last capacitor (C^sub 3^) is set to 0. On the right side of the window there is implemented a menu of options of the program with some default values. This menu lets the user interact with the program to modify the value of the resistors and design parameters such as passband and stopband frequencies with their respective maximum and minimum attenuations. The user can choose between the two family characteristics shown previously. For the Chebyshev filter it is possible to obtain ripples of 0.1, 0.5, 1 and 3 dB and the program selects automatically one of these ripples from the design data.

For the selection of the capacitor we preferred to create a data file (table.mat) with all normalized values of the capacitors1 to reduce the computational charge of the program. The final values are obtained by means of the frequency transformations given by equations (7), (8) and (9).