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Band pass filtering techniques have been a challenging task due to requirement of keeping Quality factor, gain and mid-frequency of the filter independent of each other. Other most important aspect is ...

Band pass filtering techniques have been a challenging task due to requirement of keeping Quality factor, gain and mid-frequency of the filter independent of each other. Other most important aspect is keeping the filter stable, keeping mid-frequency immune to circuit component tolerances and to achieve the mid-frequency at the accurate value. The Biquad family of topologies are typically used for a stabilized filtering application, however the design requirements on Biquad topology for low frequency application is still a challenge. The requirements become more stringent for bandwidth curve, roll-off curve and preciseness of frequency filtering as we move down to low frequency applications where the shift in few Hz would cause great frequency errors.. This paper evaluate the performance and design improvements for the Biquad filter topologies for designing a extremely stable and precise narrow band-pass filter at low frequency.

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- 1. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 20051 Copyrights © 2005 Raman K. AttriDesign of stable Narrow Band-Pass Filter using Multi-Stage Biquad TopologyRAMAN K. ATTRIGlobal Scientific and Technical Consultant (Instrumentation)Rkattri@rediffmail.comAbstract - Band pass filtering techniques have been achallenging task due to requirement of keeping Qualityfactor, gain and mid-frequency of the filter independent ofeach other. Other most important aspect is keeping the filterstable, keeping mid-frequency immune to circuit componenttolerances and to achieve the mid-frequency at the accuratevalue. The Biquad family of topologies are typically used fora stabilized filtering application, however the designrequirements on Biquad topology for low frequencyapplication is still a challenge. The requirements becomemore stringent for bandwidth curve, roll-off curve andpreciseness of frequency filtering as we move down to lowfrequency applications where the shift in few Hz wouldcause great frequency errors.. This paper evaluate theperformance and design improvements for the Biquad filtertopologies for designing a extremely stable and precisenarrow band-pass filter at low frequency.Keywords: Band-Pass filter, Biquad Topology,State-Variable Filter topology, Narrow BPF, Multi-stage filters1 Introduction:Band-pass filter design has nevertheless been achallenge in view of many interrelated dependenciesin the circuit parameters. In Band-pass filter, qualityFactor (Q) and Gain of the filter are generally interrelated and thus do not give the independent control.Always there have to be some design tradeoffs. Incase of narrow band-pass filter the circuit stabilityposes difficult requirements. Generally the Narrowband pass filtering action is achieved by increasingthe Q value of the normal Band-pass filter. Thehigher Q value creates circuit instability, oscillationsand makes the circuit very sensitive to the circuitcomponent tolerances. Certain application requiresextremely stable narrow band-pass filter with veryhigh Q value with nominal gain. Such filters areused in the devices used to detect a particularfrequency accurately. The filter performance verymuch depends on the filter topology chosen. Biquad,Akerberg and other multi-amplifier filter topologiesare best bet for clean band-pass filter (Aghashani,2000). However, there are practical challenges inmaking a Narrow band pass filter circuit aroundBiquad topologies especially when this filter is usedfor low frequency applications where precision offiltering selected frequency is critical. The requiredQ value and Gain and accuracy needed in centrefrequency determine the practical challenges that wemay encounter.In this paper, a performance evaluation of Biquadtopology of filter design is presented and a designapproach is evolved using a multi-stage Biquad filterto design a stable narrow band-pass filter to ensurethe accuracy within 1% of the centre frequency withhigh Q value and precisely narrow bandwidth.2 Design RequirementsThe current application required to detect a tone of577 Hz with + 5 Hz precision and for this we neededto build an extremely narrow band pass filter whichpeaks exactly at the desired mid frequency. Insummary, following are the design requirements:-o Centre frequency: 577Hzo Bandwidth less than 20Hz (+ 10 Hz) to ensureattenuation of nearest power harmonicso Accuracy & stability of mid-frequency: + 1 % (+6 Hz max)o Roll-off of minimum 12 db per decade on eachside of the centre frequencyo High overall Q value (Q>25)o Independent adjustment of Q without affectingcentre frequencyo Independent adjustment of gain withoutaffecting Qo Gain should not be very high (G=5)o Single supply (+5V) operationo Power supply Harmonics Rejection(540Hz,550Hz, 600Hz)o Use not more than 2 discrete IC chips (fewer thebetter)
- 2. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 20052 Copyrights © 2005 Raman K. Attri577Hz is low frequency, where a 1% error in centrefrequency means a shift of 6 Hz on either side, thusdefeating the design purpose. The circuit wasrequired to filter out all the rest of the frequencies bysubstantial attenuation. It is worth mentioning that innarrow band-pass filter, the bandwidth parameterjust indicates the total span of 3db down points onboth sides of the mid frequency curve and do notindicates if the mid-frequency occurred at thedesired value or not. Since the Q value is ratio ofmid-frequency with the band-width, a particularvalue of the Q will impose restrictions on achievingcalculated bandwidth. Thus desired Q value can beachieved by limiting the bandwidth parameter, butdo the circuit peaks exactly at the said mid-frequency? This is the key performance parameter ofsuch circuits. Another challenge in such designs isthat can we keep the mid-frequency stable andinsensitive to component tolerance values?The requirements 6 & 7 are very stringent. Thisseems so easy, but in actual practice most of thefilters exhibit strong Q and Gain relationship thatchanging one will either change other parameter orwill shift the centre frequency of the filter.The requirement of having high Q poses anotherproblem of choosing a right amplifier. With High Q,the amplifier gain bandwidth product GBW can beeasily reached, even with the gain 20dB. At least40dB of headroom should be allowed above thecentre frequency peak (Kugelstadt, 2002).Operational amplifier slew rate should also besufficient to allow the waveform at centre frequencyto swing to the amplitude required.When working with high Qs one has to be verycareful with layout and component selection. This isbecause high Q circuits have a tendency to exhibitinstability with slight component mismatch. Theyalso are more likely to oscillate due to this instability(Lacanette, 1991).The bandwidth of 20Hz was selected to ensure thepower supply harmonics to filter out along withother undesired frequencies. The nearest harmonicsof 60Hz power harmonics is 540Hz and 600Hz andthat of 50Hz (US Version) is 550Hz and 600Hz.Maximum 10 Hz bandwidth can be allowed oneither side of the mid-frequency to ensure more than20 db attenuation to 540Hz and 600Hz frequencies.This requirement of filtering the harmonics alsoneeded a steeper roll-off from 3db points. The roll-off requirement needed a minimum 2ndorder filter.The response type does not matter for the band-passfilter, since it is narrow band. However the filtershaving low pass and high pass stages cascaded to getthe band-pass function may be Chevyshev or elliptic(mind the ripples in pass band & attenuation band)for steeper roll-off (Lacanette, 1991).This seemingly easy task itself is so critical becausecircuit ability to accurately peak at desired frequencywith an accuracy of + 5 Hz with a narrow band-width of 20Hz on this peak mid-frequency and toreject the rest of the frequency determined theproject success.3 Selection of topologySelecting the right topology for the filter is the keyto the overall design efforts. A simple survey oninternet would reveal that there are so manytopologies for designing the filters. Only thoseoptions have been reviewed and analyzed whichcould be driven by single supply. We did a range ofexperiments, simulation and test to rule out some ofthe topologies and in order to seek better control.Following topologies were analyzed in detail. Theresults and observations are provided briefly here.a) Sallen-key topology uses 1 amplifier andappears very attractive since it uses only oneamplifier and a few passive components (Karki,1999). However, Q and Am can not be adjustedindependently because both are dependent uponthe inner gain G (Maxim 2002).b) Multiple Feedback (MFB) topology uses 1amplifier and is very versatile, low cost, andeasy to implement and allows to adjust Q, Am,and f mindependently (Kugelstadt, 2002).However, MFB particularly is very sensitive tovariation in attenuation resistor, but not to othercomponent variations and very precise resistorsand capacitors are needed to make narrow bandpass filter with MFB topology (Elliot, 2000)c) Deliyannis filter with 1 amplifier is just theMFB modified filter with attenuator resistormissing [3]. The Daliyaanis is supposed to bebetter in terms of the variation due to componenttolerances. However, the circuit performance is
- 3. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 20053 Copyrights © 2005 Raman K. Attrinot as good as MFB and not very much suitableto our application.d) Active Twin-T circuit has the advantage that thequality factor Q can be varied via the inner gainG without modifying the mid frequency f m.However, Q and Am cannot be adjustedindependently (Kugelstadt, 2002).e) Fliege filter has lowest component count out ofthe 2 amplifier topologies offers an excellentoption of controlling the Q value by singleresistor R. The big trouble with this circuitturned out to be its sensitivity to componenttolerances. Even the small tolerances shifted themid-frequency by large amount.The most desirable situation is of course toimplement a filter with a single op amp thusreducing the cost, but controlling all the designparameters requires intensive calculations and non-standard value of components. As a designer, wewould like to have filter which can work withordinary components without severely degrading theperformance. An additional quad amplifier just cost$0.30 whereas high quality 2% PPS capacitor itselfcosts around $0.20 each whereas a 1% metal filmresistor costs around $0.02 each. So MFB or Twin-Tfilter requiring two capacitors each requiring highquality 2-3 capacitors and similar number of resistorwould cost more than a multiple amplifiersconfiguration which are stable enough to work withnormal components.As a design target, a complete control over the filtercorner / center frequency, the gain of the filter circuitand Q of band-pass filters is required. More controlusually means more op amps, which may beacceptable in designs that will not be produced inlarge volumes, or that may be subject to severalchanges before the design is finalized (Carter, 2000).There are number of possible topologies which use 3or 4 amplifiers such as: State variable, Biquad, Tow-Thomas Biquad, Akerberg-Mossberg Biquad, KHNTopology, Berka-Herpy topology and Michael-Bhattacharya topology. We resorted to theoreticalmethod of ascertaining the comparison of thesetopologies and selected Biquad topology for designimplementation.Biquad topology showed very stable performance,however to meet the stringent design requirements,the design required further investigations on designoptimization to meet roll-off criteria, gainrequirements and band-width requirements. In thenext section, we will discuss the design techniquerequired to implement Biquad topology assuccessful narrow band-pass filter.4. Design modifications for stable Biquadtopology4.1 Design analysis of state- variable band-pass filterState variable Biquad is the first topology weanalyzed in multiple amplifier family. Figure [1]shows the basic architecture of the 3 amplifier statevariable Biquad. It comprises a summing nodefollowed by two integrators. This architecture isquite versatile in that it gives a high-pass, band-passand low-pass output, but it also allows independentcontrol of f mand the Q (Maxim, 2002). This is athree to four amplifier topology. The fourthamplifier is only required for notch filters. It is alsovery easy to tune, and it is easy to change the styleof low pass and high pass, and easy to change the Qof the band pass and notch. Unfortunately, it is notas nice a topology as Akerberg-Mossberg. The sameresistor is used for gain and style of filter / Q,limiting control of the filter. It use more number ofamplifiers, but integrate LP, HP and BP filters in oneand any transfer functions can be realized bycombining the outputs. The configuration can workvery well with single supply (Carter, 2000).Fig [1]: Three amplifier state-variable Biquad with singlesupplyThe circuit can be integrated with single value of thecapacitor and resistors by choosing
- 4. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 20054 Copyrights © 2005 Raman K. AttriR1=R2=R3=R4=R5=R6=R and C1=C2=C in such away that Resistance, R is given by equation (1):CR f mπ2/1= (1)Where C= Capacitance, f m= Mid-frequencyR7 controls the gain since mid-frequency is given bythe equation (2) [2]:RRAm/7=− (2)Where Am= peak Mid-frequency gainFor better performance value of R7 is chosen in sucha manner that R7= 3.Q.R, so roughly G= 3.Q.Higher values of R7 ensure high Q while lowervalues of R7 will give low Q value. R7 have anotherimportant impact on the type of HP and LP filter. IfR7> R/2, the LP and HP filters type turns out to beChebyshev (Carter, 2000). For steeper roll off in BPfilters, R7 is chosen to be greater than R/2. It isworth mentioning that BP output is coming due tosuperimposition of HP and LP filters, thus the styleof these LP and HP filters will control the steepnessin the roll off in BP filter too (Tobin, 1998). Theresponse due to superimposition of LP and HP filteris shown in the figure [2].Fig [2]: Band-pass filter response of Biquad due tosuperimposition of LP and HP response (Courtesy Tobin,1998)The dependency of Q and gain could be eliminatedby slight design modification. By adding a 4thamplifier, independent control of the Q and gain isrealized as shown in Fig [3]. The state variable isideal for high Q circuits. Qs of 500 or more areeasily attainable with proper filter design (Maxim,2002). In addition to high Q, state variable also givesvery high pass output. Another feature is that Q andf mare independent of each other (Tobin, 1998).Fig [3]: State variable filter with independent control ofQ. Adding a 4thamplifier in feedback path ensures Qindependent of gain.The frequency response is shown in Fig [4]. Theonly issue State variable Biquad has is that Q andGain are controlled by same resistance R7, whichinhibited setting high Q and A value. However, flattop of the frequency curve indicates that that thefilter performance is going to be very stable in band-pass mode. This inherent stability in the pass-bandled us to explore this design option further for theapplication under question.Independent the control of Q as well as gain is thusrealized and also stability of mid-frequency is alsotaken care of by State-variable topology. And unlikethe single amplifier architectures, the open loop gain(3Q) need only be slightly higher than the filtersoutput gain (Q), and the low-pass gain is Q, whichreduces the requirements on the op amps GBW(Maxim, 2002).
- 5. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 20055 Copyrights © 2005 Raman K. AttriFig [4]: State variable BPF response without independentQ control. The filter shows a wide and almost flat pass-band with steep roll-off on the sides.Of the topologies we experimented, the state-variable is the least sensitive to componentvariations. It also showed another unique attribute:as the frequency f mchanges, the Q and percentagebandwidth remains constant. That is, as you shiftf min the frequency domain, the Q value remainsthe same, but the bandwidth of the filter decreases asyou increase f mand increases as you decrease f m.Maxim (2002) defines the percentage bandwidth asin equation (3) as under:( ) ( ) %100/ ×+−= ffff LuLuBandwidthPercentage (3)Wheref u= upper 3dB bandwidth pointf L= lower 3dB bandwidth pointf m= f uf LThis topology uses 4 amplifiers which have thenegative effect of in power sensitive applicationsdue to power supply rejection ratio. However,effects of adding another dual amplifier muchoutweigh the cost of special passive components, soit is a viable topology (Maxim, 2002).4.2 Design analysis of Biquad band-pass filterBased on the analysis and results of State-variabletopology, we analyzed Biquad circuit also whichessentially is same as state-variable topology withthe difference that it comprises an integratorfollowed by another integrator and then inverterrefer to Fig [5]. Some circuits present it withintegrator and inverter reversed, which makes nodifference to response. However, NP output isavailable at first integrator output.Fig [5]: Biquad BPF with Single Supply. The basicstructure consists of 3 amplifiers cascaded together withtwo integrators followed by an inverter and output fedback to input.Just a small change in the circuit organization interms of individual blocks provides a circuit thatbehaves different than the state variable filter. Thebig difference is that for a Biquad, as f mchanges,the bandwidth stays constant, but the Q valuechanges. Thus Q factor and f mare not independent(Tobin, 1998). If we change f min the frequencydomain, as f mincreases, the Q value increases andas f mdecreases, the Q value also decreases. Otherthan this difference, the Biquad behaves like thestate variable. It allows very high Q values, it can beconfigured in a 3 or 4 amplifier configuration, and ittoo is less sensitive to external component variations(Maxim, 2002).For implementation with simpler values of resistors,we choose R1=R2=R5=R and C1=C2=CR5 and R6 are not critical, they can be equalresistors. As mentioned designer can chose any
- 6. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 20056 Copyrights © 2005 Raman K. Attrivalue, as a guideline, R6 is chosen in such a mannerthat (Carter, 2000):R6 = 0.707 R (4)Keep R3=R4 for unity gain, otherwise the absolutegain at mid frequency will be given by equation (5):-3/4 RRAm= (5)The component selection addressed all the threeconcerns discussed earlier for successful Narrowband filter design as under:-a) Independent Mid-frequency ControlThe value of resistor R and capacitor C determinethe mid frequency. Both are related as in equation(6):RCf mπ2/1= (6)Where R= Resistance, C= Capacitance, f m= Mid-frequencyIt is worth noting that R has effect on f mbut it doesnot have much effect on Q. For 577Hz mid –frequency R=274K (nearest E96 resistor tocalculated value of 276K) and C=1nf.b) Independent Control of Q:The Q of the circuit depends upon the value of R4,which need to be high. Low R4 means low Q andhigh R4 value gives high Q. We chose R4=10M forreally good Q value. This simple one resistor Qcontrol makes it such a simple to use topology. Therelative ratio of R4 and R determine the Q value ofthe Q as in equation (7):-Q= R4 / R (7)Here it is to be asserted that change in R will not benot very much and once fixed will never be changed.So f mand Q can be taken independent of eachother with R mainly controlling f mand R4controlling the Q value (Carter, 2000).c) Independent Gain ControlAs mentioned earlier, mid-frequency gain is givenby equation (5) which is ratio of Resistor R4 and R3.Generally gain is not provided in the Biquad, thegain is controlled mainly by R3. We keep R4=R3=10M to provide unity gain to integrator, andR5=R6=10K for unity gain of the gain stage. Theactual gain needed was given by an additional gainamplifier, which allowed separate gain controlindependent of Q value.From implementation aspect we found that the 3amplifier implementation can be done with the helpof one quad amplifier. The circuit is found quiteimmune to external component variations. It isworthy to mention that a Biquad has only 2 criticalcapacitors and 3 resistors inside the Biquad loop.Instead of 2% PPS capacitors NPO 5% capacitorscould be used easily. This actually saved good coston overall design and driven the design evaluation totest the circuit.Even though 3 and 4 amplifier circuits draw morepower, and usually require more design time and aremore costly, but in terms of overall cost of active aswell as passive components viz a viz performance,Biquad proved to be most economical.The measured frequency response of one stageBiquad is shown in figure [6]. The response givesbandwidth =18 Hz, f m= 580 (with 274K standardresistor, exact 577Hz can be achieved by using aseries 2.2 K resistor), Q= 32.Fig [6]: Biquad Stage 1 frequency response withamplitude in dB plotted as function of frequency. The -3dB bandwidth is 18 Hz. The centre frequency peak isseen at A sharp peak seen at 577 + 3 Hz.Mid-frequency was quite stable at the desiredfrequency with desired accuracy. The circuit did notshow any oscillations too.
- 7. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 20057 Copyrights © 2005 Raman K. Attri4.3 Designing of a multi-stage Biquad filterThe disadvantage with Biquad as we observed thatits roll off may not be steeper enough, now thesteeper roll-off could be achieved with the help ofcascading two Biquad BPF together; thisarrangement gives really very good steep roll-off,much better than other topologies and remarkablestability of mid-frequency, immunity to variation tothe external components and relaxed tolerancerestrictions on the components. A gain stage mayalso be inserted in between stage-1 and Stage-2 ofthe Biquad as shown in figure [7] [Kugelstadt,2002).Fig [7]: Cascading of Biquad Stages to achieve bettersteep roll-off and independent gain controlFrom a performance point of view, the cascadinggives much better results in terms of higher Gain(20dB), high Q value (apprx. 52 ), narrowerBandwidth B (11 Hz) and sharper roll-off (24db/decade) as shown in figure [8]. Since the Q curvebecome sharper due to extra roll-off and by theunique combination of superimposition of LP andHP signals, the circuit is bound to be stable even atvery high Q value.Fig [8]: Biquad Stage 2 frequency response plotted as function of frequency. The peak frequency occurs at 580 Hz andbandwidth is further reduced to 11 Hz with steeper roll-off curveThe complete two stage Biquad circuit is shown in Fig [9].
- 8. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 20058 Copyrights © 2005 Raman K. AttriFig [9]: Highly stabilized narrow band-pass filter designed as two-stage Biquad with an independent gain controlamplifier. The circuit is designed around LMV324 amplifier series with centre frequency of 577Hz and bandwidth of 20Hz.The Monte Carlo Simulation of the circuit suggestedthat maximum variation of Mid-frequency us from571 Hz to 582 Hz, a + 6 Hz variation withcomponent tolerance, well within the range. TheSimulation results are shown in Fig [10]. Furthergood temperature stability of the mid-frequency isalso obtained.Fig [10]: Monte Carlo Simulation of Biquad double stagecircuit. The circuit shows very stable variations forcomponent tolerances. The simulation is plotted for boththe Biquad stages which contain 7 amplifiers andnumerous resistors and capacitors. In spite of thecomponent tolerance, the band-width and the mid-frequency does not get changed much.5. Guidelines for component selection tolimit sensitivity & mid-frequency shiftsTheoretically, any values of R and C that satisfy theequations may be used, but practical considerationscall for component selection guidelines to befollowed. Given a specific corner frequency, thevalues of C and R are inversely proportional—as Cis made larger, R becomes smaller and vice versa.Deviations from nominal values of the passivecomponents of course have influence on thefrequency response of the filter. These deviationsmay be caused by component tolerances or due tothe fact, that under normal circumstances the idealvalues are not available. As a rough estimation it ispossible to say: The lower the stage order is, thelower the influence of deviations on the frequencyresponse is. Higher stage orders have a higherquality factor Q and deviations of R and C impingeon the resulting frequency response roughlyproportional to the Q-factor. We conducted theMonte Carlo simulation of desired parameter overall the tolerance ranges of the components involved.5.1 Selection guidelines for capacitorsCapacitors are the real accuracy controlling andvariation controlling components, thus it requiregreater attention in their selection particularly for
- 9. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 20059 Copyrights © 2005 Raman K. Attrinarrow band-pass filter circuits. Various types ofcapacitors used in filter design are shown in theTable [2]. To minimize the variations of f mand Q,NPO ceramic capacitors are used for high-performance filters (Kugelstadt, 2002). Thesecapacitors hold their nominal value over a widetemperature and voltage range. The varioustemperature characteristics of ceramic capacitors areidentified by a three-symbol code such as: COG,X7R, Z5U, and Y5V. We used NPI COG typeceramic capicitors to achieve highly precise results.Their nominal values range from 0.5 pF toapproximately 47 nF with initial tolerances from +0.25 pF for smaller values and up to +1% for highervalues. Their capacitance drift over temperature istypically 30ppm/oC (Kegelstadt, 2002)Table [2]: Various types of capacitors suitable for filterdesign [1]Type Temp. Coeff.ppm/degCCommentsNPOceramic0 +/- 30 most popular foractive filtersFilm: MPC 0 +/- 50 metallizedpolycarbonateFilm:Polystyrene-120 larger than MPC;melts at low temp.Mica -200 ..+ 200 larger, costlier thanNPOOther type of capacitors which could also be usedfor these applications are: X7R is OK in a pinch.X7R-type ceramic capacitors range from 100 pF to2.2 uF with an initial tolerance of +1% and acapacitance drift over temperature of +15%. Forhigher values, tantalum electrolytic capacitorsshould be used (Kugelstadt, 2002). For suchapplications, avoid Z5U and other low qualitydielectrics. In critical applications, even higherquality dielectrics like polyester, polycarbonate,mylar, etc., may be required. Other precisioncapacitors are silver mica, metalized polycarbonate,and for high temperatures, polypropylene orpolystyrene. Predictable negative temperaturecoefficient of polystyrene capacitors can be used toadvantage with metal film or cermet film resistors tominimize passive sensitivity (Kugelstadt, 2002).We faced an issue with capacitive values. To makean accurate filter, it is necessary to measure theindividual capacitor values, and to calculate theresistors accordingly. However, for productiondesign, we need to resort to standard values. Thecapacitor range is chosen depending upon the mid-frequency range. A simple guideline is enumeratedin the table [3]. Since capacitor values are not asfinely subdivided as resistor values, the capacitorvalues need to be defined prior to selecting resistors.Capacitor values can range from 1 nF to several uF.The lower limit avoids coming too close to parasiticcapacitances (Kugelstadt, 2002). As a designguideline, avoid values less than 100 pF.Table [3]: Mid frequency vs recommended capacitorvalues [1]f m(Mid-Frequency) Capacity in pFfrom to from to10 100 100000 470000100 500 22000 100000500 1000 6800 390001000 5000 2700 100005000 10000 1000 330010000 50000 560 1500100000 500000 330 1000The selection of the tolerance of the capacitorsdepends on the filter sensitivity and on the filterperformance. The important filter parameters toconsider while selecting capacitor tolerance is: thecorner frequency, f mand Q. For example, when Qchanges by 2% due to a 5% change in thecapacitance value, then the sensitivity of Q tocapacity changes is expressed in equation (8) as(Kugelstadt, 2002):s(Q/C) = 2% / 5%= 0.4 %/ % (8)where ‘s’ represents the sensitivity. Although 0.4%/% is a small difference from the ideal parameter,in the case of higher-order filters, the combination ofsmall Q and f mdifferences in each partial filter cansignificantly modify the overall filter response fromits intended characteristic. This effect is highlightedin Figures [11] which shows how an intended
- 10. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 200510 Copyrights © 2005 Raman K. Attrieighth-order Butterworth low-pass can turn into alow-pass with Tschebyscheff characteristic mainlydue to capacitance changes from the partial filters(Kugelstadt, 2002). The difference between idealand real response peaks with 0.35 dB atapproximately 30 kHz, which is equivalent to anenormous 4.1% gain error, can be seen.Fig [11]: Deviation from ideal response due to change inCapacitor (Courtesy Texas Instruments, Kugelstadt,2002)As a general rule, we used 1% tolerancecomponents. 1%, 50V, NPO, SMD, ceramic caps instandard E12 series values are available fromvarious sources. Capacitors with only 5% tolerancesshould be avoided in critical tuned circuits.5.2 Selection guidelines for resistorsA more general rule is that any resistor value in theop amp RC network should at least ten times theoutput resistance of the op amp and less than one-tenth the input resistance of the op amp.Resistor values should stay within the range of 1 kohms to 100 k ohms [1]. The lower limit avoidsexcessive current draw from the op amp output,which is particularly important for single supply opamps in power-sensitive applications. Thoseamplifiers have typical output currents of between 1mA and 5 mA. At a supply voltage of 5 V, thiscurrent translates to a minimum of 1k resistor. Theupper limit of 100 k ohm is to avoid excessiveresistor noise (Kugelstadt, 2002). We used surfacemount components for high heat dissipation andprecision characteristics.Secondly resistors come in various packaging withdifferent precision. For filter applications, weselected 1% resistor tolerances from E96 series.Various types of resistors suitable for good filterdesign is given in Table [4]. Metal film resistors arepreference for filter networks. Carbon film resistorsmay also be used because their negative temperaturecoefficient can be used to advantage to minimize thepassive sensitivity of a circuit (if the capacitors havea positive temperature coefficient).Table [4]: Suitability of Various resistors [1]Type Temp.Coeff.ppm/degCStandardTolerances%CommentsMetalfilm-25 -->100 1 low cost; mostwidely usedCermetfilm200 0.5,1 larger, costlierthan metal filmCarbonfilm-200-->-500 2,5,10,20 low cost; negativetemp. coeff.5.3 Selection guidelines for operationalamplifierThe most important op amp parameter for properfilter functionality is the unity-gain bandwidth. Ingeneral, the open-loop gain (AOL) should be 100times (40 dB above) the peak gain (Q) of a filtersection to allow a maximum gain error of 1%. Theconcept is shown in Fig [12].
- 11. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 200511 Copyrights © 2005 Raman K. AttriFig [12]: Q value relationship with amplifier’s Gain-Bandwidth Product. Open loop gain of amplifier shouldbe 40dB above the peak gain (Q) at a given frequency.(Courtesy KugelStadt, 2002)Besides good DC performance, low noise, and lowsignal distortion, another important parameter thatdetermines the speed of an op amp is the slew rate(Kugelstadt, 2002). For adequate full powerresponse, the slew rate must be greater than thevalue given in the equation (9):fV mPPSR π= (9)Where V PP= peak-to-peak voltage, f m= mid-frequencyWe can use the minimum, typical and maximumUnity GBW values listed in datasheet from whichthe GBW tolerance can be estimated [1] by equation(10) below:Tolerance (in %) = (TYP – MIN) / TYP x 100(10)Where TYP = Typical Gain bandwidth productMIN = Minimum gain bandwidth productThe range of op amp tolerances is wide; from around15% to about 50%, with the mean approximately30%. The GBW temperature coefficient must beestimated from a graph of (normalized) unity gainbandwidth versus free air temperature. This graph isnot always provided. Typical coefficient valuesmight be 1000 to 7000 ppm/degC. Some of theLMV series amplifiers from National or TI and TLVseries amplifiers from TI are good candidates forsingle supply extremely narrow band pass filterapplications. We used LMV series from seriesamplifier for the current application.To avoid auto-oscillations connect two 100 nFcapacitors between ground and power pins (+V, -V)of op amp IC as shown in Fig [13].Fig [13]: Power supply Decoupling to group with a largecapacitor connected between power supply and ground toavoid oscillations [1]In summary, component selection, especially thepassive components, plays a big role in definingfilter mid-point accuracy, temperature characteristicsand tolerances. In addition to design modifications,the appropriate selection of components leads to adesign conforming to stringent bandwidth, stability,temperature, Q and centre frequency accuracyrequirements.6. ConclusionBiquad multi-stage topology proved very efficient interms of mid-frequency stability, high Q factor,independent gain and Q values, high roll-off andrejection of power supply harmonics. Whereas theother 2- amplifiers topologies required some trade-offs and has inherent dependencies of Q, f mandGain, the Biquad proved to most efficient in terms ofstability of the circuit and immunity to the externalcomponents. It is worth noting that multi-stageBiquad filter topology we used has total 7 amplifiersin its circuit and numerous passive components. Inspite of all this the Biquad is not that much sensitiveto external component variations. This particular factmakes it a very stable topology for the currentapplication. In case of broad band-pas filter onestage of Biquad is suffice. But for narrow band-passfilter two stages may be needed to be cascaded. Intotal the circuit performed as per the requirements.
- 12. R. Attri Instrumentation Design Series (Electronics), Paper No. 4, September 200512 Copyrights © 2005 Raman K. AttriReferences1. Aghashani, A. (2000), State Variable Topology (Second-Order Active Filters Based on the Two-Integrator-LoopTopology), Filter Design Web Assisted Course EE175, SanJose State University, http://www.engr.sjsu.edu/filter/(Accessed 1 Sept 2005)2. Carter, B. (2000), A single Supply op-amp circuitcollection, Texas Instruments Application ReportSLOA058. Available athttp://courses.cit.cornell.edu/bionb440/datasheets/SingleSupply.pdf3. Elliott, R. (2000), Multiple Feedback Bandpass filter,Elliott Sound Products available athttp://sound.westhost.com (Accessed 1 July 2009)4. Karki, J. (1999), Analysis of the Sallen-key Architecture,Texas Instruments Application Report SLOA024A availableathttp://www.dei.unipd.it/~pel/Elettronica_Analogica/Sallen_Key_architecture_TI.pdf (Accessed 1 July 2009)5. Kugelstadt, T. (2002), Chapter 16: Active Filter Designtechniques in Mancini, R. (Ed.) Op-amps for every one,Texas Instruments Design Reference SLOD006B. Pp 16-1to 16-63. Available athttp://focus.ti.com/lit/an/slod006b/slod006b.pdf (Accessed1 July 2009)6. Lacanette, K. (1991) A basic Introduction to filters-Active,Passive and Switched-capacitor, National SemiconductorApplication Note 779. Available athttp://www.national.com/an/AN/AN-779.pdf (Accessed 1July 2009)7. Maxim Inc. (2002), A beginners Guide to filter topologies,Maxim Application Note 1762 available athttp://www.maxim-ic.com/appnotes.cfm/an_pk/1762(Accessed 1 July 2009)8. Tobin, P. (1998), Chapter 7 & 8: Biquad Circuit & StateVariable Toplogy, Electric Circuit Theory Notes, DublinInst of Technology, Pp 68-71 available athttp://www.electronics.dit.ie/staff/ptobin/waed3notes.htm(Accessed 1 July 2009)Web References[1] Electronics Circuit Collection – Second order Band-pass Filter Design Topologies, available online athttp://www.circuitsarchive.org/index.php/Notes_on_Filters (accessed 1 July 2009)[2] Texas Instruments On-line Filter design Guide: SingleSupply Analog Expert, Available at http://www-k.ext.ti.com/SRVS/Data/ti/KnowledgeBases/analog/document/faqs/ssexpert.htm ,Further Readings• Bies, U. (2005), Design and Dimensioning of ActiveFilters, available athttp://www.beis.de/Elektronik/Filter/ActiveLPFilter.html• Carter, B. (2001), Filter design in thirty second, TexasInstruments Application Report SLOA093 Available athttp://focus.ti.com/lit/an/sloa093/sloa093.pdf (Accessed 1July 2009)• Green, S. (2003), Design Notes for 2-pole filter design withdifferential inputs, Cirrus Logic Application Note AN48available at www.cirrus.com (Accessed 1 July 2009)• Maxim Inc (2001), A Filter Design Primer, MaximApplication Note 733. Available athttp://microblog.routed.net/wp-content/uploads/2006/08/an733.pdf (Accessed 1 July 2009)• Maxim Inc (2002), Analog Filter Design Demystified,Maxim Application Note AN 1795. Available athttp://www.maxim-ic.com/appnotes.cfm/an_pk/1795/(Accessed 1 July 2009)Author Details:Author is Global Learning and Training Consultantspecializing in the area of performance technology.His research and technical experience spans over16 years of project management, productdevelopment and scientific research at leadingMNC corporations. He holds MBA in OperationsManagement, Executive MBA, Master degree inTechnology and Bachelor degree in Technologywith specialization in Electronics andCommunication Engineering. He has earnednumerous international certification awards -Certified Management Consultant (MSI USA/ MRAUSA), Certified Six Sigma Black Belt (ER USA),Certified Quality Director (ACI USA), Certified Engineering Manager (SMEUSA), Certified Project Director (IAPPM USA), to name a few. In addition tothis, he has 60+ educational qualifications, credentials and certifications inhis name. His interests are in scientific product development, technicaltraining, management consulting and performance technology.Contact: +44 20 7979 1979E-mail: rkattri@rediffmail.comWebsite: http://sites.google.com/site/ramankumarattriLinkedIn: http://www.linkedin.com/in/rkattri/Copyright InformationWorking paper Copyrights © 2005 Raman K. Attri. Paper canbe cited with appropriate references and credits to author.Copying and reproduction without permission is not allowed.

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