Filtering in Signal Conditioning Modules, SCMs

Transcription

1 AN Dataforth Corporatio Page of 5 DID YOU KNOW? Pierre Simo, Marquis de Laplace (749-87) the famous Frech mathematicia ad astroomer was bor i Normady, Frace. His parets were humble farmers uable to provide him with a educatio; fortuately, wealthy eighbors recogized his talets ad helped fiace his educatio. Laplace became a professor of mathematics i Paris at the age of 0. As a political opportuist, he chaged his political positios as eeded to support his career, which prospered through three Frech revolutioary regimes. Napoleo made him a cout ad Kig Louis XVIII made him a marquis. The mathematical trasformatio, kow as the Laplace Trasform is but oe of his may outstadig accomplishmets. This trasformatio allows aalyzig complex time depedet behavior with simple algebraic equatios. The Laplace Trasform is extesively used for aalyzig time depedet electric circuits; iterestigly, Laplace developed this trasformatio more tha 50 years before the world had AC circuits to aalyze! Filterig i Sigal Coditioig Modules, SCMs Preamble Sigal coditioig modules, SCMs, used for measurig process cotrol variables such as temperature, pressure, strai, positio, speed, level, etc. are always subject to exterally iduced oise sigals. Electrically ad magetically iduced oise voltages/currets are ievitable. Field sesors with output voltages i the millivolt rage are certaily degraded by iduced oise levels o the order of volts. Cosequetly, sigal coditioig modules must provide filterig to elimiate iduced oise compoets. frequecy ad semilog plots of phasor phases i degrees versus frequecy. Figure illustrates Bode magitude plots of three fudametal RC filter types; curve # sigle sectio RC low-pass (LP) filter, curve # sigle sectio high pass (HP) filter, ad curve # two sectio badpass filter. Hudreds of articles ad text books have bee writte o filters that provide a multitude of frequecy characteristics such as the Bessel, Butterworth, Chebyshev, Cauer, etc. I-depth reviews of this iformatio are beyod the scope ad itet of this documet. The objective of this applicatio ote is a brief review of filter fudametals primarily focused o the amplitude respose of low-pass (LP) filters eeded i idustrial sigal coditioig modules. Brief Review of Some Fudametals Filter topologies are characterized by their trasfer fuctios, which are frequecy depedet ratios of output voltage (Vout) to iput voltage (Vi) expressed as ratios of polyomials as show. For example, Vout(s) N s b *s b *s... b = =, s= jπf. Vi(s) D s a *s a *s... a More about this math later. Filter trasfer fuctio behavior is geerally characterized by Bode plots, which are semilog plots of phasor amplitudes i decibels {0*log (Vout/Vi)} versus Figure Passive Filter Output Resposes: Iput is Volt Curve # is LP, Curve # is HP, Curve # is BP Low-Pass (LP) filters allow trasmissio of oly a rage of low frequecies from DC to a higher cutoff frequecy. See Figure, circuit # ad Figure, curve #. This type of filter is ideal for elimiatig high frequecy iduced oise o low frequecy sigal data. Low-Pass filterig is required i most idustrial SCMs to elimiate iduced 60

2 AN Dataforth Corporatio Page of 5 cycle harmoics ad radomly iduced high frequecy trasiet oise. High-Pass (HP) filters allow trasmissio of a rage of high frequecies from a lower cutoff frequecy to a upper limit of circuit compoet performace. See Figure, circuit # ad Figure, curve #. These type filters are ideal for elimiatig iduced low frequecy oise o high frequecy sigal data. Badpass (BP) filters allow trasmissio of a rage of frequecies betwee a lower ad upper cutoff limit. The characteristics of a simple BP filter are show i Figure, curve # ad ca be simulated by cascadig the HP ad LP circuits i Figure. These type filters are ideal for sigal selectio withi a give frequecy rage. Coversely, badpass filter trasfer fuctios ca be rearraged to fuctio as otch filters, which elimiate frequecies betwee a lower ad upper cutoff limit. i C out i R Figure Sigle Sectio RC Passive Filters # is High Pass (HP): # is Low-pass (LP) Basic Filter Characteristics Traditioal filter trasfer fuctios are implemeted with either passive or active circuit topology. Passive filter circuit topology is cofigured with idividual resistors, capacitors, ad iductors; whereas, the typical active filter circuit uses operatioal amplifiers with resistors ad capacitors i various feedback arragemets. Uique filter trasfer fuctio characteristics are implemeted with active filters by selectig resistor ad capacitor values i the feedback topology ad by adjustig the filter s amplifier gai. Some basic SCM filterig characteristics are illustrated i the followig list. R C out cutoff frequecies. Bode plots locate these cutoff frequecies decibels below the flat mid rage; hece, the term -db poits. Ideal filters have o respose below their lower -db cutoff, o respose above their upper -db cutoff, ad very steep respose slopes approachig these -db poits. The term badwidth defies the frequecy rage betwee -db cutoff frequecies. For low-pass filters, the badwidth is from DC to the -db frequecy. See Figure, Curve #. Note : Aalysis o LP filters shows that cutoff behavior ear -db frequecies is determied by a mathematical combiatio of the idividual roots (called poles ) of the filter trasfer fuctio deomiator polyomial, D(s). These roots are fuctios of the time costats withi a filter circuit ad have the uits of Hz or radias/sec. The umber of roots (order of deomiator polyomial) is equal to the umber of circuit time costats. For example, a RC low-pass filter cotaiig four capacitors has four RC time costats ad a fourth order deomiator polyomial with four roots (poles). The mathematical combiatios of deomiator roots ad the frequecy spread betwee them determie the actual -db frequecy. For istace, if all the idividual roots differ by a factor of 0 the the lowest root (largest time costat) essetially is the -db poit. O the other had if a filter has equal roots, the the -db poit is { f * ( )} /. For situatios betwee these two simple extremes, the math becomes complicated ad computer simulatios with bech testig are more effective ways to determie -db poits. Figures ad 6 illustrate the effects multiple poles have o the -db frequecy. Note : Cotiued aalysis of LP filter trasfer fuctios reveals that the umber of roots (or umber of filter time costats) cotrols the slope of a filter s respose as it approaches the -db poit. Beyod the cutoff frequecy, each pole cotributes a amplitude decrease of 0 db per frequecy decade to the filter respose. For example, a 4- pole filter respose beyod the -db poit has a limitig slope of -80 db per decade. Badpass: This filter trasfer fuctio applies equally to all frequecies withi the rage (bad) of desired filterig with o amplitude variatios withi i the filter s desired badpass (i.e. a flat respose with o ripples). Frequecy compoets outside the badpass are sharply atteuated. See Figure, Curve #. Sharp Respose at Cutoff Frequecies: Frequecies where the filter s power respose has dropped 50% or / (0.707) of the desired output voltage are defied as

3 AN Dataforth Corporatio Page of 5 Figure ad Figure 6 illustrate the effect of multiple poles (RC sectios) for a low-pass filter. Figure Low-pass (LP) RC Passive Filter # is Sigle RC Sectio: # is 4 Idetical RC Sectios Over shoot: Sesors ofte output step fuctios or ear istataeous trasitios betwee sigal levels. SCM filters should ot allow excessive overshoot or rigig i resposes to these step fuctios. RC time costats ad filter gai determie active filter overshoot. Figure 4 shows the effects of modifyig active filter time costats ad gai. Filter Respose Calculatios Maual aalysis of filter trasfer fuctios is laborious ad best implemeted with special computer applicatio programs; oetheless, may saliet filter characteristics are illumiated by maual aalysis of simple filter topologies usig Steimetz s phasor cocepts ad Laplace trasformatios. As a example cosider the simple RC sectio passive LP filter topology show i Figure 5. The fudametal time depedet behavioral loop equatios for this circuit are; Vi(t) = R*Ι(t) + * Ι(t) dt - * Ι(t) dt C C 0 = - * Ι(t) dt + + * Ι(t) dt + R*I(t) C C C Vout(t) = * Ι(t) dt C Trasfer fuctio (Vout/Vi) derivatios from these equatios require cosiderable effort. Usig phasor defiitios, Laplace trasformatios, ad steady-state frequecy depedet siusoidal voltages, reduce this effort to simple algebraic equatios where Laplace trasformatio allows s to represets the complex frequecy term jω. The LP filter trasfer fuctio time depedet equatios for Figure 5 ow become; Vout(s) Vi(s) = N(s) D(s)... s CRCR +scr+r+r C + or rearragig CRC R ( R+R ) s +s + + CR C( RR ) RCR C Eq. Eq. Note: D(s) i both these equatios is a quadric deomiator of the form ( a*s + b*s + c ) with two roots. Figure 4 Filter Respose to Iput Voltage Pulse # is Iput: # is Maximum Flat: # is Flat-Rigig Vi R I C I R C Vout Figure 5 Simple Passive Bad-Pass Filter Topology

4 AN Dataforth Corporatio Page 4 of 5 As a example, choose Figure 5 compoet values to be; R=0kΩ, C=0.5µF, R=56kΩ, ad C= 0.00µF. A computer simulatio program that sweeps the iput siusoidal voltage from oe hertz to te megahertz geerates the results show below i Figure 6. 4 Figure 6 Computer Simulatio of LPCircuits i Figure 5 # Sigle RC Sectio: 5.9Hz # Sigle RC Sectio: 84Hz # RC-RC Combied: 5.9Hz #4 Idetical RC Sectios: 5.69Hz Further examiatio of this example, illumiates some importat fudametal low-pass filter characteristics. Each separate low-pass RC sectio has a cutoff frequecy (respose dow by -db) related to their idividual RC time costat. Recall that for a sigle RC LP sectio the cutoff frequecy f is / (πrc) Hz. See Figure 6, curve # ad curve # Low-pass filters with idetical RC sectios have a much lower cut off frequecy tha the idividual RC sectio. See curve #4 i Figure 6. Each sigle low-pass RC sectio cotributes a 0 db chage i voltage respose for each factor of 0 chage i frequecy (0dB per decade). Recall that a 0 db chage represets a factor of 0 chage i voltage ( i.e. a voltage chage from to 0. is a 0 db chage). As RC sectios with ear equal time costats are added, the cutoff frequecy decreases ad, most importatly, the respose slope icreases. For example, oe RC sectio gives 0 db / decade whereas five RC sectios give 00dB/ decade slope. SCM filters eed low cutoff frequecies with steep rates of respose to esure effective rejectio of oise beyod the LP cutoff. The quadric deomiators D(s) i Eq. ad Eq. have idetical roots; 5.7 Hz ad.8794 khz. The deomiator roots of these RC filter trasfer fuctios are determied by circuit RC time costats ad are defied as poles, which cotrol the filter s cutoff frequecy. The above example illustrates a fudametal filter characteristic; amely, low-pass RC filter cutoff frequecies are domiated by the smallest pole. See curve # i Figure 6 Low-pass RC filter trasfer fuctios, which are arraged as show i Eq, always have deomiators with b, the coefficiet of the first power of s equal to the sum of the filters ope circuit time costats ad the filter cutoff frequecy ca be estimated by /(πb). I Figure 5, the sum of ope circuit time costats is [RC + C*(R+R)] = 0.076mS ad frequecy associated with this time costat is 5.8 Hz. Recall that; frequecy is related to a time costat T by the expressio [f=/(*π*t)] ad that a ope circuit time costat is the product of each capacitor ad the effective resistor this capacitor sees with all other capacitors ope. Ope circuit time costats are referred to by the memoic OCTs. The umerators N(s) of valid filter trasfer fuctios, N(s)/D(s), force the filter respose to zero ad are defied as filter zeros. Low-pass SCM filters have o zeros Filter trasfer fuctios have frequecy depedet phase shifts ad time delays, both of which are extremely crucial i high frequecy commuicatio applicatios. Substitutig s = jπf i Eq ad illustrates frequecy depedet phase agles. Phase characteristics of SCMs i idustrial data acquisitio systems are ot geerally a issue but may be crucial i high-speed cotrol loops. Elimiatio of oise ad maitaiig sigal itegrity are the pacig issues for idustrial SCM applicatios. Active Filters Figure 7 illustrates a popular cost effective active filter circuit topology. Circuits of this type offer umerous advatages over the typical passive circuit topologies. Vi Z Z Z Z4 Z5 Gai Vout Figure 7 Example Geeralized Active Filter

5 AN Dataforth Corporatio Page 5 of 5 The trasfer fuctio for Figure 7 is; Vout N(s) = =... Vi D(s) Gai Z 4+Z Z Z4 ZZ4 Z Gai Z5 Z Z5 ZZ5 Z Eq. 4 Hz -0 db per decade Figure 6 filter topology ca implemet either low-pass, high pass, or bad pass characteristics by selectig R ad C combiatios for Z,Z, Z, Z4, ad Z5. A typical low-pass active filter cofiguratio has; Z=R, Z4=R, Z=/sC, Z5=/sC, ad Z=ope. The trasfer fuctio for this topology is show below i Eq 4. Vout N(s) =... arraged as... Vi D(s) Gai [ ] s (R C R C )+s C (R +R )+C R (-Gai) + Eq.4 Active LP filter trasfer fuctios arraged i the same form as Eq. ad Eq. 4 exhibit characteristics similar to passive filters. For example, the roots of D(s) are poles ad the smallest pole domiates the LP cutoff frequecy. I additio, the b coefficiet of the s term i D(s) is the sum of the filter s ope circuit time costats (OTC) ad the filter cutoff frequecy is closely approximated by / (πb). Moreover, a sigificatly large OTC domiates the LP filter -db frequecy. Figure 8 Dataforth SCM Low-pass 6-Pole Filter 4 Hz with -0 db per decade fall-off Figure 9 is the block diagram of Dataforth s DIN-Rail sigal coditioig geeral purpose aalog iput module, DSCA4. This module represets Dataforth s 4-way isolatio scheme with a multipole filter partitioed to provide ati-alaisig filterig o the field side of the sigal isolatio barrier followed by multipole filterig o the system side of the isolatio barrier. Readers are ecouraged to visit Dataforth s web site ad explore their complete lie of sigal coditioig modules. See Ref. Active filters have a advatage over passive filters sice their behavior is electroically adjustable be adjustig by the filter gai, as illustrated i Eq. 4. For example, active filter phase agles ca be tailored to fit special applicatios i commuicatio ad cotrol loops. Dataforth SCM Filters Dataforth sigal coditioig products all have exactig filters icorporated i their desigs. These filters are tailored to idividual SCM applicatios. Dataforth SCMs have multi-pole low-pass filters desiged for low frequecy ad widebad applicatios. Figure 8 illustrates the typical respose of a Dataforth SCM 6-pole low-pass filter with a 4 Hz -db frequecy ad a roll-off of 0 db per frequecy decade. Note that Hz sigals are atteuated by over 00 db, a factor of 00,000. This represets a very effective LP filter for elimiatig iduced idustrial oise. Figure 9 DSCA4, DIN Rail Aalog Iput Module, 4-Way Isolatio, Multi-Pole LP, Ati-aliasig Filter Dataforth Refereces. Dataforth Corp.,