FOR an electronics engineer, the design

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1 ENINV IE 2007 Encuentro de Investigación en Ingeniería Eléctrica Zacatecas, Zac, Marzo 28 30, 2007 A Tool for the Design and Synthesis of Basic Active RC Filters Martha M. Mena A., Carlos A. Ochoa O. Electrical Eng. Dept., Autonomous University of Zacatecas Zacatecas ZAC TEL: +(492) , ext aguu16@gmail.com Abstract The availability of tools for the design and synthesis of RC active filters was not very common, until recently. In this work, a tool with such capabilities is presented. The developed tool calls for the filter specifications like cutoff and stopband frequencies with their corresponding gains, the type of approximation function, i.e., Butterworth, Chebyshev, Elliptic or Bessel, the type of frequency response: low-pass or high-pass, and then it delivers circuit schematics and the appropriate R and C values. A brief description of the calculations behind this tools is included. The expected contribution of this work is to help reducing considerably the time of the design and implementation process of these filters and, on the other hand, to develop home made tools to solve the synthesis problem. This tool is also the result of an educational exercise, a programming competition, proposed by one of the authors while he was teaching a course on Java programming. Resumen La disponibilidad de herramientas de diseño y síntesis de filtros activos RC no era muy común, hasta hace algún tiempo. En este trabajo se presenta una herramienta con tales prestaciones. La herramienta desarrollada pide las especificaciones del filtro como frecuencias de corte y de rechazo con sus respectivas ganancias, el tipo de función de aproximación, es decir, Butterworth, Chebyshev, Elíptico o Bessel, el tipo de respuesta en frecuencia: pasa-bajas o pasa-altas, y luego entrega diagramas esquemáticos con los valores de los componentes R y C. Se incluye una breve descripción de los cálculos que se encuentran detrás de esta herramienta. La contribución esperada en este trabajo es, por un lado, ayudar a reducir considerablemente el tiempo de diseño y construcción de estos filtros y por el otro, el desarrollar herramientas propias que den solución al problema de síntesis. Esta herramienta es también el resultado de un ejercicio educativo, un concurso de programación, propuesto por uno de los autores mientras impartía un curso de programación en Java. Index Terms Analog filter design, RC active filters, network synthesis. I. INTRODUCTION FOR an electronics engineer, the design and construction of filters is surely a common task, at least in part of the development of his career. In a previous work [1], a tool for the design and implementation of digital filters on a DSP was presented. Even

2 M. Mena, A. Ochoa: A tool for active RC filter synthesis 193 when today there exist many digital based systems, the need of analog filters will be very important for many reasons. One of these reasons is the fact that before a digital signal processing system is fed with a signal, the signal must be passed through an antialising filter in order to meet the sampling theorem; any frequency component above the Nyquist frequency has to be removed. There exist many other reasons, which will not be discussed in this work, but it is for sure that having a tool that helps in the calculation of the RC elements of an active filter is important. Through the students community, it is not unusual this need when they are working in their thesis projects or some times during the coursework in a final project. Albeit the importance of the filter design and synthesis (d&s) tools, it was not easy to find them for free or at a low cost. There are, certainly, some semiconductor companies that give for free some application notes describing the design process with some formulas or design tables [2]. Sometimes, a Windows c based program, in trial or demo version, can be available, but allowing only, in some cases, the design of a very low order filters [3]. FilterWiz Pro [4] offers a version without time restriction, but the resitor values are masked, making it useless, until a licence is purchased and the registration information is entered. In some cases, the demo version allows the design of defined filters only for testing purposes [5]. A web page with a list of many sources is Circuit Sage [6]. When a design table is available, the design process becomes a long and tedious task, and so prone to errors. Until recently, Texas Instruments has released a free version of a filter design package called FilterPro c [7], originally from Burr Brown. This work started before this application was available, so, the developed home made tool is presented in this work. The expected contribution is also the development of such tools, independently from foreign companies. II. THEORETICAL PRINCIPLES To show the fundation of this work and the developed tool, it is included in this report the theoretical fundamentals of the d&s process. Then, it is shown how the d&s is solved by the implementation of a computer program that integrates all the steps involved up to the calculation of the RC element values and the corresponding electronic circuit. So, the description of all the calculations involved is also a guide and explanation of the development of the tool. The results are presented with some examples, for Butterworth filters. A. First order active RC filters A basic first order low-pass active filter can be constructed as shown in the Figure 1. The transfer function can be obtained from the basic input output relationship for a inverter amplifier as H(s) = Z r Z e, (1) where Z r is the feedback impedance and Z e is the input impedance. Replacing the Z s by the circuit elements, the following transfer function is obtained [8] H(s) = R 2 1/(R 2 C) s + (1/R 2 C), (2) where s is the complex variable. Let K = R 2 and ω c = 1 R 2 C, then ω c H(s) = K. (3) s + ω c The first order RC active high-pass filter can be constructed as shown in the Figure 2 with

3 194 ENINV IE 2007 Encuentro de Investigación en IE, Marzo, 2007 C R Figure. 1. First order RC active low-pass filter. a transfer function given by H(s) = R 2 s s + (1/ C). (4) C - + R 2 Figure. 2. First order RC active high-pass filter. Let K = R 2, and ω c = 1 C then H(s) = K s. (5) s + ω c If ω c = 1 rad/s, R 2 = = 1Ω then K = 1 y C = 1 F, will be the R and C values for a normalized filter. B. Frequency scaling The R and C values found in the previous section are not typical and practical values for the construction of these filters (see the 1F capacitor). In addition, the cutoff frequency of 1 rad/s may not be the desired cutoff frequency of the filter. Even so, there is a good reason to have normalized filters and normalized R and C values. It is possible to change the cutoff frequency and the gain by means of very simple transformations, called frequency scaling and gain scaling, as follows: } R n = K m R magnitude, (6) C n = C/K m where R n is the new resistance value, C n is the new capacitor values, and K m is the magnitude scaling. } R n = R frequency, (7) C n = C/K f where R n is the new resistance value, C n is the new capacitor value, and K f is the frequency scaling, given by K f = ω n /ω 0 = 2 π f, where ω n is the new desired frequency, ω 0 is the normalized frequency, and f is the desired frequency in Hz. Combining both, magnitude and frequency scaling in a single operation: R n = K m R C n = C/(K m K f ) } magnitude&frequency. (8) Knowing K f, then K m can be determined by K m = C C n K f, (9) so it is necessary to give a value for C n. C. Low-pass design example If a normalized low-pass filter is going to be scaled to have a new cutoff frequency of 1 khz and a gain of -3 db at the cutoff frequency, then a capacitor value of C = 0.01 uf is proposed. We have initially the following values = R 2 = 1, C = 1 F, ω c = 1 rad/s, then

4 M. Mena, A. Ochoa: A tool for active RC filter synthesis 195 after the calculations we have: K f = 2π 1000 = , C n = 0.1 uf, K m = 1F 0.1uF = R C 1 R - + so n = R 2 n = K m R = (1Ω) = Ω and because ω c = 1/(R 2 C) and K = R 2 /, = R2. To prove the scaling operation using Matlab, the following code was used: R2=1591; R1=1591; C=0.1e-6; Gnum=(R2/R1)*(1/(R2*C)); Gden=[1 1/(R2*C)]; [H,w]=freqs(Gnum,Gden, s ); semilogx(w/(2*pi), 20*log10(abs(H))) grid and the frequency response is shown in Figure 3. Magnitude [db] db at 1 khz Frequency [Hz] Figure. 3. Frequency response of the non-normalized filter D. Second order low-pass RC active filters The electronic circuit of a second order lowpass filter, in a Sallen-Key configuration is shown in Figure 4. With R = = R 2, the transfer function C 2 Figure. 4. Second order Sallen-Key low-pass filter. will be given by H(s) = 1/(R 2 C 1 C 2 ) s 2 + (2/C 1 )s + (1/R 2 C 1 C 2 ), (10) where R = 1 Ω will be used. Once the transfer function of the building blocks are known, for first and second order filters, the R and C element values can be found to build non-normalized filters of any order. For order n > 2 several building blocks, connected in series, can be used. When the transfer function contains a polynomial of the form s n + b 1 s n b n, it has to be factored in second order sections. If n is an odd order, then it is completed with a first order section. Using the same principle of digital filter designs [9],[1] a table of R and C values for normalized filters is constructed for n = 1 to 8, for low-pass and high-pass filters, using Butterworth, Chebyshev, Bessel, and Elliptic function approximations. The construction of the table is done as follows: Let n = 4 for a low-pass Butterworth filter. Using a normalized second order Sallen-Key filter we have ω c = 1, R = 1, and the C 1 y C 2 values will be found. Starting with a fourth order Butterworth polynomial in fac-

5 196 ENINV IE 2007 Encuentro de Investigación en IE, Marzo, 2007 tored form 1 B 4 (s) = (s s + 1)(s s + 1), (11) so, two second order sections will be necessary. For any second order polynomial (s 2 + b 1 s + b 2 ), and comparing with (10), the following relations arise: 1 C 1 C 2 = b 2 2 C 1 = b 1. (12) Applying these relations to the second order section we get C 1 (1) = 2.614F, and C 2 (1) = 0.328F, where C i (j) stands for the C i element of the j section. For the second order section we get C 1 (2) = F, and C 2 (2) = F. These calculations have to be repeated for each n value, including the needed number of first order and second order sections, as well as for the low-pass and high-pass configurations. In addition, all these calculations have to be repeated for Chebyshev, Bessel, and Elliptic function approximations. It can be concluded that a lot of work has to be done, just to obtain the R and C values for normalized filters. The frequency and gain scaling is another phase of the work. A tool capable of performing all these calculations will be useful and will save a lot of work. So, the proposed tool requires only the (non normalized) filter specifications and, once the order of the filter is calculated, it will go through a look-up table to get the R and C normalized values, then after applying frequency and gain scaling, the R n and C n values will be shown together with the circuit configuration. One difference with the FilterPro c package is the calculation of the filter order n. In the developed tool, this calculation is done for Butterworth and Chebyshev filters. For Bessel and Elliptic filters, like FilterPro c, the user has to enter the desired order. Another capability of the tool is the generation of a SPICE file or netlist, which will be useful for simulation. Finally, unlike FilterPro c, the tool runs on different operating systems, like Windows and Linux. D.1 SPICE simulation The previous results can be verified, before the circuit implementation, using a circuit simulation program. One of these simulation programs is SPICE (Simulation Program with Integrated Circuits Emphasis) from Berkley University, available for Linux or Windows c and several commercial distributions. In this work, the Micro-Cap 7.2.4, Evaluation version was used [10]. Figure 5 shows the circuit diagram of a high-pass normalized filter with the corresponding R and C element values, and the Figure 6 shows the obtained frequency response. It is worth to say that SPICE simulation is different to the Matlab simulation since in the former, many physical properties of the circuit elements are modeled, like the operational amplifiers, resistors and capacitors, including noise and temperature affects. III. RESULTS Figure 7 shows the results for a high-pass Butterworth filter design. The main window of the program calls for the selection of the type of filter approximation (Butterworth and Chebyshev shown only), and the filter specifications (in Spanish): cutoff frequency (F. de corte(w1)) in Hz, passband gain (Atenuación(k1)), a proposed capacitor value (Capacitor Propuesto), stop-

6 M. Mena, A. Ochoa: A tool for active RC filter synthesis 197 Figure. 5. Electronic circuit for a normalized high-pass filter. Figure. 7. Application window and Butterworth filter design selected. Figure. 6. Frequency response from the normalized high-pass filter simulation. band frequency (F. de Rechazo(w2)), and the corresponding gain (Atenuación(k2)). After entering the values shown in the window, the solution can be obtained, selecting the buttom (Solución Propuesta). The solution shows the order of the filter that meets the filter specifications, and the values for the RC elements. Figure 8 shows the proposed circuit implementation with the RC values found by the program. For the solution one stage is necessary. The single pole section, upper diagram, is shown without values, that means it is not needed. A second order section is shown below the single pole section, with the RC values for the implementation. As a final step of the design process, a simu- Figure. 8. Result showing the proposed filter circuit and the RC values. lation was made with the RC values found by this tool. Figure 9 shows the modified circuit with the calculated R n and C n values after de-normalization, for a high-pass design with cutoff frequency of 1 khz. Figure 10 shows the frequency response obtained in the simulation. For each case the results agree with a good approximation to the expected values, that is, -3dB at 1 rad/s (0.159 Hz), and -3 db at 1kHz for the non-normalized case. IV. CONCLUSIONS The design and synthesis of active RC filters can be a long and tedious task which involves many calculation steps before the R n and C n values can be obtained. In this work, a set of tables for normalized RC active filters with low-pass and high-pass frequency responses, and Butterworth, Chebyshev, Bessel, and El-

7 198 ENINV IE 2007 Encuentro de Investigación en IE, Marzo, 2007 Figure. 9. Electronic circuit of the non-normalized high-pass filter. Figure. 10. Frequency response of the non-normalized filter from simulation. 10 MHz Butterworth Filter Using the Operational Amplifier THS4001, Application Report, SLOA032, Texas Instruments (October 1999). [3] AktivFilter 2.3 Demo Edition, available at: Updated , (2007). [4] Filter Wiz PRO active filter designer version 4, available at: /filter_wiz_files/fwpro.htm, (2006). [5] OMICRON FilterDesign, available at: /news/index.html, (2002) [6] Circuit sage, available at: Copyright Last update on 08/22/2005. [7] FilterPro(TM) Filter Design Program, Texas Instruments Incorporated, TI Blvd. Dallas Texas 75243, (2006). [8] Active Filters, Electronics Club, available in: (May 2006). [9] G. Miramontes de León, Procesamiento Digital de Señales: Introducción con Teoría y Práctica, Coord. de Investigación y Postgrado, Universidad Autónoma de Zacatecas, ISBN , pp , (2005). [10] MicroCap Evaluation Version, Spectrum Software, 1021 S Wolfe Road, Sunnyvale CA 94086, ( ). liptic function approximations were done. It is worth to say that these tables are not available over the Internet or at least they are not easy to find. With the tool presented in this work, the task is reduced to enter the filter specifications, hiding all the design steps, which are time consuming and can lead to calculation errors. Usually, this type of applications are developed by foreign companies; so this work is also an attempt to develop home made filter design software. References [1] G. Miramontes de León, FDI: Herramienta para Diseño e Implantación de Filtros Digitales para la Familia DSP56000, Encuentro de Investigación en Ingeniería Eléctrica ENINVIE-2004 Marzo 4-5 Zacatecas, Zac. pp , (2004). [2] Dirk Gehrke and Andreas Hahn MSLP/AAP,