New Proposed Algorithms for nth Order Butterworth Active Filter Computer-Aided Design

Transcription

1 New Proposed Algorithms for nth Order Butterworth Active Filter Computer-Aided Design Information and Communication Department, Al-Khwarizmi Engineering College Baghdad University,Jaderia,Baghdad, Iraq. doi: 0.456/ijact.vol.issue3.7 Abstract This paper describes new proposed algorithms for constructing the transfer function of nth order Butterworth LPF using the idea of the cascade combination of active filters. Computer-aided design is used with new algorithms to achieve the procedure of constructing the transfer function and give the practical results of the nth order Butterworth active LPF. C++ program is used for designing procedure by linear programming. The proposed algorithms are very fast, flexible, and exact. The program allows users to design different frequency, components, and order of Butterworth active LPF with high flexibility. Keywords: Programming algorithms, Active filter design, Frequency-dependent component.. Introduction This paper introduces new proposed algorithms for nth order Butterworth LPF using the idea of the cascade combination of active filters. This wor is to simplify active filter design by writing a program which can be used even without any bacground about filter design and to get high flexibility of components selection for getting the most economical design. Analog electronic filters are present in just about every piece of electronic equipment. There are the obvious types of equipment, such as radios, televisions, and stereo systems. Test equipment such as spectrum analyzers and signal generators also need filters []. Detection of a wanted signal may be impossible if unwanted signals and noise are not removed sufficiently by filtering. Electronic filters allow some signals to pass, but stop others. To be more precise, filters allow some signal frequencies applied at their input terminals to pass through to their output terminals with little or no reduction in signal level []. Winder [] has written simple programs for filter design lie Super FILTER and Filter Master. He has written Active_F as an active filter design program []. Chen [] has provided a lot of information in filters with significant additions in the areas of computer-aided design of active filters. Yue Wu [3] has introduced the design of active filters for RF receivers. He has also introduced the design of active bandpass filters which are based on CMOS active inductors [4].. Active Filters A filter is a device which passes signals of certain frequencies and rejects or attenuates those of other frequencies. Passive filters are constructed with inductors, capacitors, and resistors, but for certain frequency ranges inductors, because of their size and practical performance limitations, are undesirable. Consequently there has been, for some years, a trend toward replacing inductors by active devices which simulate the effect of inductors [,5]. This trend has accelerated with the advances in IC integration which have made the active devices available at prices competitive with, and in many cases cheaper than, those of inductors. The active device we use to construct active filters is the operational amplifier [6]. 70

2 International journal of Advancements in Computing Technology Volume, Number 3, August, General Circuits and Equations A low-pass filter is a device which passes signals of low frequencies and attenuates those of high frequencies. There are many numbers of ways of designing LPFs using active devices. In Fig.(), the op-amp, together with resistors R3 and R4 constitutes a voltage controlled voltage source (VCVS) and the overall networ is second order LPF which has the following transfer function [7]: V ( s) V ( s) = s K + as + b () Figure. Second order active LPF In Fig.(), the networ is first order LPF which has the following transfer function[8]: (where R3=R4) V s B = ( ) V ( s) s + B () Figure. First order active LPF An analysis of Fig.() shows that it achieves Eq.(): (where R3=R4) (where R3=R4) An analysis of Fig.() shows that it achieves Eq.() with: K = R R C C (3) a = (4) R C b = R R C C (5) B RC = (6) 7

3 Higher order filters may be obtained by cascading two or more second-order filters. To implement odd nth order, we can add a first order stage. 4. Low-Pass Butterworth Filters A filter which approximates the ideal low-pass filter with a relatively flat pass band characteristic is Butterworth filter. Its amplitude response [5] is: H ( jw) = (7) + ( w / w c ) n Where n is the order of the filter. It may be seen that the filter is improved as n increases. The Butterworth filter has the advantage of maximally flat response in the pass band. The poles of the nth order Butterworth filter can be calculated [9] by: For =,,,n p ( ) π ( ) π sin + j cos n n = (8) The poles of nth order are complex conjugates only if the n is odd there is an additional real pole equal to s= -. The complex conjugate poles can be calculated using the equation []: For =,,, n/ 5. Design Principles p p * = s ( ) π + (sin ) s + n The nth order Butterworth active filter can be implemented using multistage. The transfer function of each is multiplied by the other. The representation of the stages by second order for each and the final stage is first order if the order n is an odd number. For example if the designed filter is 7 th order then the stages will be: (9) Figure 3. Stages of 7 th order LPF The calculation of the transfer function of each stage depends on the calculated conjugate poles from Eq. (9). 6. Realization of the Prototype Filter The designed filter for the calculated poles is prototype filter with cut-off frequency rad./sec., therefore the realization steps must be performed to get the real cut- off frequency and real filter components. The impedance scaling can be made without changing the transfer function or the cut-off frequency value by multiplying by the same factor each of resistors and capacitive reactances [0]. Therefore the transformation is by multiplying each resistance by a factor () and dividing the capacitance by the same factor (). In the other hand to get frequency transformation from rad./sec. to required cut-off frequency (f), we must divide each capacitance by ( ( π f ) [,]. 7

4 International journal of Advancements in Computing Technology Volume, Number 3, August, Frequency Transformation In this paper we have only discussed the design of low pass (LP) filters. There are many other types of filters which find use in signal processing. Once the low-pass prototype filter has been designed, the other filters (High-pass, band-pass, and band-stop) can be obtained by a simple frequency transformation [6,9]. 8. Design Algorithms Flowchart () shows the algorithms of nth order active filter which gives flexibility to choose the resistors or capacitors as well as other parameters of filter. The algorithms are optimized to give highest possible accuracy and flexibility for choosing the order and components. 9. Design Example Results Example results of the program (in C++ Language) which follows the algorithms of Flowchart () are as follows: For third order with freq.=00hz and with selected capacitance C=nF The first stage: R=59.5 Ω, R= 59.5 Ω The second stage : R=59.5 Ω Fig.( 4) shows above third order filter Figure 4. Third order LPF For sixth order with freq.=00hz and with selected capacitance C=nF The first stage: R= Ω, R=838 Ω The second stage: R= 5 Ω, R= 5 Ω The third stage: R= 83.8 Ω,R= Ω Fig.(5) shows above sixth order filter Figure 5. Sixth order LPF For tenth order with freq.=00hz and with selected capacitance C=nF The first stage:r=5087 Ω, R= Ω The second stage: R=753 Ω, R=445 Ω The third stage: R=5 Ω, R= 50 Ω The fourth stage: R=893. Ω, R= 836 Ω The fifth stage: R= 8057 Ω, R =343.9 Ω Fig.(6) shows above tenth order filter 73

5 Figure 6. Tenth order Start Input n, f Input C Is the choice for resistance? Input R.=. =(-)*pi/n.a= sin().=. =(-)*pi/n.a= sin() R=/(.pi.f.a.C) R=/(4.pi.pi.f.f.R.C.C) C=/(.pi.f.a.R) R=/(4.pi.pi.f.f.R.C.C) of the th stage. of the th stage. Is < n/?.=+.=+ Is < n/? Is n/= n/? Is n/= n/? R=/(.pi.f.C) of (+)th stage. Stop R=R, C = /(.pi.f.r) of (+)th stage. LPF Flowchart. Algorithms of Active Filter Design For third order with freq.=00hz and with selected resistance R=0 Ω The first stage: R=0 Ω, C= 0.59nF The second stage : R=0 Ω, C= 0.59nF Fig.( 7) shows above third order filter 74

6 International journal of Advancements in Computing Technology Volume, Number 3, August, 00 Figure 7. Third order LPF For sixth order with freq.=00hz and with selected resistance R=0 Ω The first stage: R=679.5 Ω,C=0.307nF The second stage: R= 0 Ω, C= 0.5nF The third stage: R= 3730 Ω,C=8.384pF Fig.(8) shows above sixth order filter Figure 8. Sixth order LPF For tenth order with freq.=00hz and with selected resistance R=0 Ω The first stage:r=978.8 Ω, C= nF The second stage: R=844 Ω, C=0.753nF The third stage: R=0 Ω, C= 0.54nF The fourth stage: R=3755 Ω, C= 89.3pF The fifth stage: R= 3900 Ω, C =80.569pF Fig.(9) shows above tenth order filter 0. Conclusions Figure 9. Tenth order LPF The program is very simple to use and does not require bacground in filtering principles in order to run it. The performance of each designed filter was evaluated using Electronic Worbench 9, which resulted that the design is exact and convenient for any particular application. The main outputs from this program are the components of each stage of the filter.. References [] Steve Winder, Analog and Digital Filter Design, Newens, 00. [] Wai-Kai Chen, Passive, Active, and Digital Filters ( The Circuits and Filters Handboo, 3 rd Edition),Taylor and Francis Group, 009. [3] Yue Wu, Active Filter Design for RF Receivers, KTH lic thesis, Stocholm, 000. [4] Yue Wu, Xiaohui Ding, Mohammed Ismail, and Haan Olsson, RF Band pass Filter Design Based on CMOS Active Inductors, IEEE Transactions on Circuit and Systems, vol.50, 003, pp

7 [5] Rolf Schaumann, Design of Analog Filters, Oxford University Press, 00. [6] Mohammed Shuaib Ghausi, Modern Filter Design Active RC and Switched Capacitor, SciTech Publishing, 003. [7] Yue Wu, Active Filter Design for RF Receivers, Circuits and Systems, II: IEEE Transactions on Analog and Digital Signal Processing, Volume 50, Issue, Dec. 003, Pages: [8] Don A. Meador, Analog Signal Processing with Laplace Transforms and Active Filter Design, Thomson Learning, 00. [9] Arthur B. Williams, Electronic Filter Design Handboo, McGraw-Hill Publisher, 995. [0] Yannis P., Integrated Continuous-Time Filter Design, IEEE Journal of Solid-State Circuits, Volume 9, Issue 3, Mar. 994, Pages: [] Bram Nauta, Analog CMOS Filters for Very High Frequencies, Khuwer Academic Publishers,