1. Active Low Pass FilterThe most common andeasily understood activefilter is the Active LowPass Filter. Its principle ofoperation and frequencyresponse is exactly thesame as those for thepreviously seen passivefilter, the only differencethis time is that it uses anop-amp for amplificationand gain control. Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 2

2. Active Low Pass FilterThis first-order low passactive filter, consistssimply of a passive RCfilter stage providing a lowfrequency path to theinput of a non-invertingoperational amplifier. Theamplifier is configured asa voltage-follower (Buffer)giving it a DC gain of one,Av = +1 or unity gain as opposed to the previous passive RC filter whichhas a DC gain of less than unity. The advantage of this configuration isthat the op-amps high input impedance prevents excessive loading onthe filters output while its low output impedance prevents the filterscut-off frequency point from being affected by changes in theimpedance of the load.Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 3 3. Active Low Pass Filter with AmplificationThe frequency response of the circuit will be the same as that for thepassive RC filter, except that the amplitude of the output is increased bythe pass band gain, AF of the amplifier. For a non-inverting amplifiercircuit, the magnitude of the voltage gain for the filter is given as afunction of the feedback resistor (R2) divided by its corresponding inputresistor (R1) value and is given as:Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 4 4. Active Low Pass Filter with AmplificationGain of a first-order low pass filter:Where: AF = the pass band gain of the filter, (1 + R2/R1) = the frequency of the input signal in Hertz, (Hz) c = the cut-off frequency in Hertz, (Hz)1. At very low frequencies, < c,2. At the cut-off frequency, = c,3. At very high frequencies, > c, Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 5 5. Active Low Pass Filter with AmplificationExample No1Design a non-inverting active low pass filter circuit that has a gain of tenat low frequencies, a high frequency cut-off or corner frequency of159Hz and an input impedance of 10K. Assume a value for resistor R1 of1k.Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 6 6. Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 7 7. Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 8 8. Second-order Low Pass Active FilterAs with the passive filter, a first-order low pass active filter can beconverted into a second-order low pass filter simply by using anadditional RC network in the input path. The frequency response ofthe second-order low pass filter is identical to that of the first-ordertype except that the stop band roll-off will be twice the first-orderfilters at 40dB/decade (12dB/octave). Therefore, the design stepsrequired of the second-order active low pass filter are the same. Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 9 9. Active High Pass FiltersThe basic operation of an Active High Pass Filter (HPF) is exactly thesame as that for its equivalent RC passive high pass filter circuit, exceptthis time the circuit has an operational amplifier or op-amp includedwithin its filter design providing amplification and gain control. Like theprevious active low pass filter circuit, the simplest form of an activehigh pass filter is to connect a standard inverting or non-invertingoperational amplifier to the basic RC high pass passive filter circuit asshown.Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 10 10. Active High Pass FiltersThis first-order high pass filter, consists simply of a passive filterfollowed by a non-inverting amplifier. The frequency response of thecircuit is the same as that of the passive filter, except that theamplitude of the signal is increased by the gain of the amplifier.For a non-inverting amplifier circuit, the magnitude of the voltage gainfor the filter is given as a function of the feedback resistor (R2) dividedby its corresponding input resistor (R1) value and is given as:Low frequenciesCut-offHigh frequenciesThomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 201211 11. Second-order High Pass Active FilterAs with the passive filter, a first-order high pass active filter can beconverted into a second-order high pass filter simply by using anadditional RC network in the input path. The frequency response ofthe second-order high pass filter is identical to that of the first-ordertype except that the stop band roll-off will be twice the first-orderfilters at 40dB/decade (12dB/octave). Therefore, the design stepsrequired of the second-order active high pass filter are the same.Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 12 12. Cascading Active High Pass FiltersHigher-order high pass filters, such as third, fourth, fifth, etc areformed simply by cascading together first and second-order filters. Forexample, a third order high pass filter is formed by cascading in seriesfirst and second order filters, a fourth-order high pass filter bycascading two second-order filters together and so on.Then an Active HighPass Filter with aneven order numberwill consist of onlysecond-order filters,while an odd ordernumber will startwith a first-orderfilterat thebeginning as shown.Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 13 13. Active Band Pass FilterThe Active Band Pass Filter is a frequency selective filter circuit used inelectronic systems to separate a signal at one particular frequency, or a range ofsignals that lie within a certain "band" of frequencies from signals at all otherfrequencies. This band or range of frequencies is set between two cut-off orcorner frequency points labelled the "lower frequency" (L) and the "higherfrequency" (H) while attenuating any signals outside of these two points.Simple Active Band Pass Filter can be easily made by cascading together a singleLow Pass Filter with a single High Pass Filter as shown. Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 14 14. Active Band Pass FilterThis cascading together of the individual low and high pass passive filtersproduces a low "Q-factor" type filter circuit which has a wide pass band.The higher corner point (H) as well as the lower corner frequency cut-off point(L) are calculated the same as before in the standard first-order low and highpass filter circuits. Obviously, a reasonable separation is required between thetwo cut-off points to prevent any interaction between the low pass and high passstages. The amplifier provides isolation between the two stages and defines theoverall voltage gain of the circuit. Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 15 15. Filter response Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 16 16. Fundamentals of Low-Pass FiltersThe most simple low-pass filter is the passive RC low-pass networkshown:For a steeper rolloff, n filter stages can be connected in series asshown in Figure 163. To avoid loading effects, op amps, operating asimpedance converters, separate the individual filter stages:Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 17 17. In comparison to the ideal low-pass, the RC low-pass lacks in the followingcharacteristics:The passband gain varies long before the corner frequency, fC, thusmplifying the upper passband frequencies less than the lower passband.The transition from the passband into the stopband is not sharp, but happensgradually, moving the actual 80-dB roll off by 1.5 octaves above fC.The phase response is not linear, thus increasing the amount of signaldistortion significantly.The gain and phase response of a low-pass filter can be optimized to satisfyone of the following three criteria:1) A maximum passband flatness,2) An immediate passband-to-stopband transition,3) A linear phase response.For that purpose, the transfer function must allow for complex poles andneeds to be of the following type:Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 18 18. The transfer function of a passive RC filter does not allow furtheroptimization, due to the lack of complex poles.The Butterworth coefficients, optimizing the passband for maximumflatnessThe Tschebyscheff coefficients, sharpening the transition frompassband into the StopbandThe Bessel coefficients, linearizing the phase response up to fC Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 19 19. Quality Factor QThe quality factor Q is an equivalent design parameter to the filterorder n. Instead of designing an nth order Tschebyscheff low-pass, theproblem can be expressed as designing a Tschebyscheff low-pass filterwith a certain Q. For band-pass filters, Q is defined as the ratio of themid frequency, fm, to the bandwidth at the two 3 dB points:For low-pass and high-pass filters, Q represents the pole quality and isdefined as: Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 20 20. Quality Factor QHigh Qs can be graphically presented as the distance between the 0-dBline and the peak point of the filters gain response. In addition, the ratio is defined as the pole quality. The higher the Q value, the more a filter inclines to instability. Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 21 21. In applications that use filters, the amplitude response is generallyof greater interest than the phase response. But in some applications,the phase response of the filter is important.It might be useful to visualize the active filter as two cascadedfilters. One is the ideal filter, embodying the transfer equation; theother is the amplifier used to build the filter.Filter design is a two-step process. First, the filter response ischosen; then, a circuit topology is selected to implement it. Thefilter response refers to the shape of the attenuation curve. Often,this is one of the classical responses such as Butterworth, Bessel, orsome form of Chebyshev. Although these response curves are usuallychosen to affect the amplitude response, they will also affect theshape of the phase response Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 22 22. Filter complexity is typically defined by the filter order, which isrelated to the number of energy storage elements (inductors andcapacitors). The order of the filter transfer functions denominatordefines the attenuation rate as frequency increases. The asymptoticfilter rolloff rate is 6n dB/octave or 20n dB/decade, where n is thenumber of poles. An octave is a doubling or halving of t he frequency;a decade is a tenfold increase or decrease of frequency. Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 23 23. Phase response Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 24 24. The center frequency can also be referred to as the cutoff frequency(the frequency at which the amplitude response of the single-pole, low-pass filter is down by 3 dBabout 30%). In terms of phase, the centerfrequency will be at the point at which the phase shift is 50% of itsultimate value of 90 (in this case).Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 25 25. Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 26 26. For the second-order, low-pass case, the transfer function has aphase shift that can be approximated by:The phase response of a 2-pole, high-pass filter can be approximatedby:Where is the damping ratio of the filter. It will determine thepeaking in the amplitude response and the sharpness of the phasetransition. It is the inverse of the Q of the circuit, which alsodetermines the steepness of the amplitude rolloff or phase shift.Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 27 27. Butterworth Low-Pass FiltersThe Butterworth low-pass filter provides maximum passband flatness.Therefore, a Butterworth low-pass is often used as anti-aliasing filterin data converter applications where precise signal levels are requiredacross the entire passband. Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 28 28. Tschebyscheff Low-Pass FiltersThe Tschebyscheff low-pass filters provide an even higher gain rolloffabove fC. However, as the Figure shows, the passband gain is notmonotone, but contains ripples of constant magnitude instead. For agiven filter order, the higher the passband ripples, the higher thefilters rolloff. Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 29 29. Bessel Low-Pass FiltersThe Bessel low-pass filtershave a linear phase responseover a wide frequency range,which results in a constantgroup delay in that frequencyrange.Bessel low-passfilters, therefore, provide anoptimumsquare-wavetransmissionbehavior.However, the passband gainof a Bessel low-pass filter isnot as flat as that of theButterworth low-pass, andthe transition from passbandto stopband is by far not assharp asthat of aTschebyscheff low-pass filterThomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 30 30. Second-Order SectionsA variety of circuit topologies exists for building second-ordersections. To be discussed here are the Sallen-Key and themultiplefeedback. They are the most common and are relevanttopologies.The general transfer function of a low-pass filter isThe filter coefficients ai and bi distinguish between Butterworth,Tschebyscheff, and Bessel filters. The coefficients for all three typesof filters are tabulated for second order filters:Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 31 31. Sallen-Key TopologyThe general Sallen-Key topology in Figure 1615 allows for separategain setting via A0 = 1+R4/R3. However, the unity-gain topology inthe Figure is usually applied in filter designs with high gain accuracy,unity gain, and low Qs (Q < 3).Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 32 32. Sallen-Key TopologyThe transfer function of the circuit is:For the unity-gain circuit (A0=1), the transfer function simplifies to:The general transfer function of a low-pass filter is:The coefficient comparison between this transferfunction and the general transfer function is:Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 33 33. Sallen-Key TopologyThe coefficient comparison between this transferfunction and the general transfer function is:Given C1 and C2, the resistor values for R1 and R2 are calculatedthrough:In order to obtain real values under the square root, C2 must satisfythe following condition:Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 34 34. Problem: Design a second-order Sallen-Key unity-gain Tschebyschefflow-pass filter with a corner frequency of fC = 3 kHz and a 3-dBpassband ripple. Supposse C1=22 nF.From the Coefficients Table obtain a1 and b1 for a second-orderfilter. a1 = 1.0650 and b1 = 1.9305.Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 35 35. Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 36 36. Multiple Feedback TopologyThe MFB topology is commonly used in filters that have high Qs andrequire a high gain.The transfer function of the circuit is:Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 37 37. Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 38 38. Higher-Order Low-Pass FiltersHigher-order low-pass filters are required to sharpen a desired filtercharacteristic. For that purpose, first-order and second-order filter stages areconnected in series, so that the product of the individual frequency responsesresults in the optimized frequency response of the overall filter. In order tosimplify the design of the partial filters, the coefficients ai and bi for each filtertype are listed in the coefficient tables.Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 39 39. Fifth-Order FilterThe task is to design a fifth-order unity-gain Butterworth low-passfilter with the corner frequency fC = 50 kHz.First the coefficients for a fifth-order Butterworth filter are obtained:Then dimension each partial filter by specifying the capacitor valuesand calculating the required resistor valuesThomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 40 40. First FilterFirst-Order Unity-Gain Low-Pass With C1 = 1nF, Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 41 41. Second FilterSecond-Order Unity-Gain Sallen-Key Low-Pass Filter With C1 = 820 pF,Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 42 42. Third FilterThe calculation of the third filter is identical to the calculation of thesecond filter, except that a2 and b2 are replaced by a3 and b3, thusresulting in different capacitor and resistor values.Specify C1 as 330 pF: The closest 10% value is 4.7 nF.With C1 = 330 pF and C2 = 4.7 nF, the values for R1 and R2 are:R1 = 1.45 k, with the closest 1% value being 1.47 kR2 = 4.51 k, with the closest 1% value being 4.53 kThomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 43 43. Higher-Order High-Pass FilterLikewise, as with the low-pass filters, higher-order high-pass filters are designedby cascading first-order and second-order filter stages. The filter coefficients arethe same ones used for the low-pass filter design, and are listed in thecoefficient tables.Third-Order High-Pass Filter with fC = 1 kHzThe task is to design a third-order unity-gain Bessel high-pass filter with thecorner frequency fC = 1 kHz. Consider C1 = 100nF Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 44 44. Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 45 45. Thomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 46 46. Bessel Coeficients Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 47 47. Butterworth CoeficientsThomas Kugelstadt . Op Amps for Everyone Chapter 16 Active Filter Design Techniques. Available online :http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 48 48. Tschebyscheff Coefficients for 3-dB Passband Ripple Thomas Kugelstadt . Op Amps for Everyone Chapter 16Active Filter Design Techniques. Available online : http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 49