Active Filter Design Using Operational Transconductance Amplifiers: Tu
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Geiger "Active Filter Design Using Operational Transconductance Amplifiers: Tutorial," IEEE Circuits Devices Magazine, Vol. pp.20-32, March 1985.
Active Filter Design Using Operational Transconductance Amplifiers: Tutorial
Randall Geiger Edgar
This question currently difficult answer reasons. First, there near void literature active filter structures employing alternative amplifier types. Second, evolution good integrated transresistance, transconductance, current amplifiers kept pace with that voltage amplifiers, although devices these alternate categories commercially available (e.g., transconductance amplifiers such 3080 13600 transresistance amplifiers such 3900) [l]-[5]. Comparisons some characteristics these amplifiers were recently discussed Brugger [40]. this paper, basic first- second-order structures using transconductance amplifier (often termed operational transconductance amplifier: OTA) discussed. shown that these structures offer improvements design simplicity programmability when compared based structures well reduced component count. Many basic based structures only OTAs capacitors and, hence, attractive integration. Component count these structures often very (e.g., second-order biquadratic filters constructed with OTAs capacitors) when compared VCVS designs. Convenient internal external voltage current control filter characteristics attainable with these designs. They attractive frequency referenced (e.g., master/slave) applications. Several groups have recently utilized OTAs continuous-time monolithic filter structures [28] [40]. From practical viewpoint, high-frequency performance discrete bipolar OTAs, such 3080, quite good. transconductance gain, varied over several decades adjusting external bias current, IABC. major limitation existing OTAs restricted differential input voltage swing required maintain linearity [5]. 3080, limited about maintain reasonable degree linearity. Although feedback structures which sensitivity filter parameters reduced goal based filter design) will discussed, major emphasis will placed upon those structures which standard filter parameters interest directly proportional OTA. Thus, will
Abstract Basic properties Operational Transconductance Amplifier (OTA) discussed. Applications voltage-controlled amplifiers, filters, impedances presented. versatile family voltage-controlled filter sections suitable systematic design requirements described. total number components used these circuits small, design equations voltage-control characteristics attractive. Limitations well practical considerations based filters using commercially available bipolar OTAs discussed. Applications OTAs continuous-time monolithic filters considered.
Introduction
conventional operational amplifier amp) used active device vast majority active filter literature. design purposes, assumption that ideal generally made, large amounts feedback used make filter gain essentially independent gain amp. host practical filter designs have evolved following this approach. also become apparent, however, that operational mplifier limitations preclude these filters high frequencies, attempts integrate these filters have been unsuccessful (with exception nondemanding applications), convenient voltage current control schemes externally adjusting filter characteristics exist. With realization that MOSFET inherently current transconductance amplifiers, respectively, following question naturally arises. improvements filter characteristics, performance, flexibility obtained using other basic types amplifiers (e.g., transconductance, current, transresistance) place voltage amplifier specifically operational amplifier) basic active device filter structure?
design parameter much resistors capacitors. Since transconductance gain assumed proportional external bias current, external control filter parameters bias current obtained. Most existing work based filter design approached problem either concentrating upon applying feedback make filter characteristics independent transconductance gain modifying existing structures inclusion some additional passive components OTAS. either case, circuits were typically component intense cumbersome tune. Some earlier works listed Refs. [6]-[16]. Some most practical circuits found manufacturer's application notes [3]-[5].
gm(V+
shown model, input output impedances model assume ideal values infinity. Current control transconductance gain directly obtained with control IABC- Since techniques abound creating current proportional given voltage, voltage control gain also attained through IABC input. Throughout this paper, when reference made either current voltage controllability based circuits' assumed attained control IABC.
Basic Building Blocks
Some basic building blocks introduced this section. brief discussion about these circuits follows. Voltage amplifiers using OTAs shown Fig. along with voltage gain output impedance expressions. basic inverting non-inverting configurations Figs. have voltage gain directly proportional which makes current (voltage) control gain IABC straightforward. Furthermore, observe that differential amplifier easily obtained using both input terminals Figs. major limitation these circuits relatively high output impedance. voltage buffer, such used Figs. often useful reducing output impedance.* Although gain characteristics these circuits ideally identical, performance circuits same. performance differences differences effects parasitics circuits. Specifically, parasitic output capacitance Fig. along with instrumentation parasitics, parallel resistor discrete component structures, thus causing roll-off frequency response circuits. circuit Fig. parasitic output capacitance connected across null port thus negligible effects when functions properly. Likewise, instrumentation parasitics will typically appear impedance output amp, thus have major effect performance. with conventional amplifier design using resistors amp's, amplifier bandwidth these structures warrants consideration. circuits Figs. major factor limiting
Fig.1 OTA. Symbol. Equivalent circuit ideal OTA.
Model
symbol used shown Fig. along with ideal small signal equivalent circuit. transconductance gain, assumed proportional* IABC. proportionality constant dependent upon temperature, device geometry, process [2]. hIABC
linear dependence bias current typically obtained bipolar OTAs configurations operating weak inversion. structures operating saturation region typically exhibit quadratic dependence IABC. output current given
Fig. Voltage amplifiers. Basic inverting. Basic noninverting. Feedback amplifier. Noninverting feedback amplifier. Buffered amplifier. Buffered VCVC feedback. amplifiers.
bandwidth generally finite gain bandwidth product amps. amps modeled popular single-pole roll-off model, A(s) GB/s, OTAs assumed ideal, follows that bandwidth circuits Fig. Fig. independent voltage gain amplifier. This contrasted bandwidths GB/K GB/1 basic single non-inverting inverting amplifiers gains respectively. Note that circuits Figs. differ only labeling terminals. circuits presented this paper, interchanging terminals will result only changing sign coefficient equation derived original circuit. Henceforth, will reader's responsibility determine when such interchange provides useful circuit. circuits Figs. feedback structures. circuit Fig. offers gains that continuously adjusted between positive negative values with parameter interchanging terminals OTA, very large gains obtained approaches approaches infinity). Gain nonlinearly related Control range reduced these structures when compared amplifiers Figs. components sized fitly, gain these structures made essentially independent conventional inverting non-inverting configurations) output impedances made reasonably small. amplifier Fig. attractive since contains passive components. Gain adjustment attained with either gm2. total adjustment range gain this structure double that attainable with single structures considered Figs. Furthermore, both OTAs same chip, variations with temperature gm's cancelled. Several standard controlled impedance elements shown Fig. along with input impedance expression. These controlled impedances used place passive counterparts (when applicable) active structures attain voltage control filter characteristics building blocks structures. circuit Fig. grounded Voltage Variable Resistor (VVR). circuit Fig. behaves floating VVR, provided matched. mismatch occurs, structure modeled with floating between terminals value gm1, along with voltage dependent current source value (gm1-gm2) driving node
Fig. Controlled impedance elements. Single-ended voltage variable resistor (VVR). Floating VVR. Scaled VVR. Voltage variable impedance inverter. Voltage variable floating impedance. Impedance multiplier. Super inductor. FDNR. circuit Fig. acts scaled VVR. Higher impedances possible than with simple structure Fig. expense additional resistors. voltage variable impedance inverter shown Fig. Note doubling adjustment range this circuit, with amplifier Fig. special interest case where this circuit loaded with capacitor. this case, synthetic inductor obtained. doubling adjustment range particularly attractive synthetic inductor since cutoff frequencies active filter structures generally involve inductor values raised power. making adjusting both simultaneously, first-order rather than quadratic control cutoff frequencies possible. floating impedance inverter shown Fig. Note that necessary match proper operation. circuit Fig. serves impedance multiplier. That Fig. behaves super inductor that Fig. FDNR.
First-Order Filter Structures
voltage variable integrator structure with differential input shown Fig. integrator serves basic building block many filter structures. different lossy integrators (first-order lowpass filters) shown Figs.
cutoff frequency lowpass filter Fig. given expression
(3a)
Linear adjustment f3dB with attainable with this circuit while maintaining unity frequency gain. structure Fig. fixed pole location adjustable gain with transconductance gain resistor this circuit replaced with controlled resistor Fig. circuit would have independently adjustable gain break frequency. highpass structure Fig. also cutoff frequency given
(3b)
Figure Integrator structures. Simple. Lossy. Adjustable. circuit Fig. loss that fixed product gain controllable circuit Fig. offers considerably more flexibility. pole frequency adjusted (interchanging input terminals actually allows pole enter right half plane), gain subsequently adjusted gm1. should noted that structure Fig. contains resistors obtained from circuit Fig. replacing resistor with controlled impedance Fig. Another lossy integrator without adjustable gain with adjustable pole location very simple structure shown Fig. When designing cascaded integrator-based filter structures, case that input impedance some stages infinite. that case, unity gain buffer would required coupling, since output impedances integrators Fig. nonzero. Note, however, that buffer needed cascade integrators Fig. since input impedance each circuit ideally infinite. First-order filters readily built using OTAS. Considerable flexibility controlling those specific filter characteristics that usually interest possible with these structures. Several first-order voltage-controlled filters shown Fig. functional plot transfer characteristics function transconductance gains shown Fig.
observed that characteristic networks lowpass highpass structures Figs. identical, thus they have same pole structures. They differ only where excitation applied. circuits Figs. shelving equalizers. response both circuits continuously changed from lowpass allpass highpass adjusting seen from Fig. basic difference circuits that former fixed pole adjustable zero, whereas circuit Fig. adjustable pole fixed zero. circuit Fig. additional flexibility obtained grounded resistor circuit Fig. replaced with controlled resistor Fig. circuit Fig. acts lowpass filter with highfrequency gain determined ratio. Both pole zero this circuit adjustable through parameter ratio held constant. This preserves shape transfer characteristics thus represents only frequency shift response, shown Fig. circuit Fig. utilizes additional offers considerable flexibility. either fixed, circuit behaves much like shelving equalizers discussed above. adjusted simultaneously, then fixed pole-zero ratio and, hence, shape preserving response possible. this case, circuit lowpass, allpass, highpass, depending upon ratio. terminals interchanged transconductance gains adjusted that =gm2, circuit behaves phase equalizer.
Fig. First-order voltage-controlled filters. Lowpass, fixed gain pole adjustable. Lowpass fixed pole, adjustable gain. Highpass, fixed high-frequency gain, adjustable pole. Shelving equalizer, fixed high-frequency gain, fixed pole, adjustable zero. Shelving equalizer, fixed high-frequency gain, fixed zero, adjustable pole. Lowpass filter adjustable pole zero, fixed ration. Shelving equalizer, independently adjustable pole zero. Lowpass highpass filter, adjustable zero pole, fixed ratio independent adjustment. Phase shifter, adjustable with
circuit Fig. also preserves shape transfer function, provided adjusted such manner that their ratio remains constant. this case, shape response determined gm1:gm2 Cl:C2 ratio. Depending upon these ratios, response either lowpass highpass nature, indicated Fig. gm2R circuit Fig. behaves phase equalizer, used adjust phase shift. monolithic applications, resistor replaced with third OTA, using configuration Fig.
changed, poles zeros approximating function must moved simultaneously with appropriate ratio constant-Q manner. group second-order voltage-controlled filter structures discussed this section. Circuits with constant-Q pole adjustment, circuits with constant bandwidth adjustment, circuits with independent pole zero adjustment presented. Some circuits with simultaneous constant-Q adjustment both poles zeros also presented along with general biquadratic structure. These structures have immediate applications voltage-controlled Butterworth, Chebyschev, Elliptic designs. simple second-order filter structure shown Fig. [17], [19]. This structure canonical sense that only four components needed obtain second-order transfer functions. output voltage, given expression
Second-Order Structures
Second-order filter structures find widespread applications directly design higher-order filters. Although emergence practical voltage current-controlled first-order filters amplifiers been slow, even fewer techniques exist design controlled second- higher-order structures. Switchedcapacitor techniques have been successfully used build voltage-controlled filter structures building voltagecontrolled oscillator using output required clock switching capacitors. Although useful some applications, these structures continuous time nature, have limited dynamic range, limited reasonably low-frequency applications. Concentration here will continuous-time voltage controlled structures. common requirement design voltage controlled filter structures that filter characteristics adjusted manner that essentially results frequency scaling. all-pole applications, such lowpass Butterworth Chebyschev case, well bandpass highpass versions these approximations, frequency scaling tantamount moving poles prescribed distance constant-Q manner. Those familiar with active filter structures will recall that pole movement second-order structures through adjustment single component always circular path (constant straight line (constant bandwidth) parallel imaginary axis -plane. challenges associated with constant-Q pole adjustment through simultaneous tuning more components should obvious. seemingly more difficult situation exists when considering design popular elliptic filters. maintain elliptic characteristics cutoff frequency
transfer function specific excitations listed Table. Note that lowpass, bandpass, highpass, notch versions this circuit behave adjustable circuits with fixed pole Q's. pole determined capacitor ratio, which accurately maintained monolithic designs. interesting note that zeros notch circuit also move constant-Q (i.e. along axis) manner with poles, adjusted. Occasionally, desirable have circuits which poles independently adjusted. circuits with these characteristics shown Fig. [18], [19], [24] Fig. [18], [24]. output voltages these circuits are, respectively,
Fig. Transfer characteristics first-order structures Fig. Circuit Fig. Circuit Fig. Circuit Fig. Circuit Fig. Circuit Fig. Circuit Fig. Circuit Fig. Circuit Fig. Circuit Fig.
Table
Transfer functions biquadratic structure Fig.
circuits Figs. also used implement lowpass, bandpass, highpass, notch transfer functions through proper selection inputs circuit Fig. circuit Fig. expressions poles circuit given
poles moved constant-Q manner fixed, adjusted; whereas movement constant manner attainable adjusted when remain constant. independent adjustment apparent. circuit Fig. expressions poles become
(10)
adjusted linearly with constant. Such movement often termed constant bandwidth movement. gm1, gm2, adjusted simultaneously, constant-Q pole movement possible. Adjusting (for 1/2) moves poles along vertical lines parallel axis s-plane. circuit Fig. output given
(11)
poles are, respectively,
Fig. Second-order filter structures.
(12)
(13)
Although transfer function similar that above, note that since coefficient term numerator equals that denominator, adjustment bandpass version this circuit with will result constant bandwidth, constant gain response. monolithic structures, prove useful replace resistor Fig. with structure Fig. Likewise, bandwidth adjustment with needed, desirable replace third shown Fig. with fixed resistor some applications. Phase equalizers also possible with structures shown Fig. example, interchanging terminals first OTAs Fig. setting making results second-order adjustable phase equalizer.
(14)
Fig. General biquadratic structure. This circuit applications higher-order voltagecontrolled elliptic filters. higher-order structures obtained cascading these second-order blocks, gm's would made equal adjusted simultaneously. Buffering between stages using standard unity gain buffer required prevent interstage loading. Modifications other circuits Fig. obtain adjustable features also possible. Although ratio zero location pole location controlled with C2/C3 ratio discrete designs, this pose some problems monolithic structures. convenient control pole-zero ratio insert voltage-controlled amplifier Fig. between points Fig. transconductance gain either these additional OTAs control variable. final second-order structure considered here general biquad Fig. output this circuit given
C1C2 sC1g sC1g
Fig. Elliptic Filter structure.
circuit Fig. both poles zeros that adjusted simultaneously constant-Q manner. circuit similar those shown Fig. with exception that capacitor previous circuits been split allow adjusting pole-zero ratio. transfer function circuit given
(15)
tuning algorithms that unwieldy. Alternatives these earlier designs using conventional operational amplifiers have proven much better.
Practical Considerations
Although circuits presented this point this paper practical with ideal operational transconductance amplifiers, existing discrete OTAs from ideal. mentioned introduction, major limiting factor with commercially available OTAs limited differential input voltage swing. Recent activity literature concentrated upon designing OTAs with improved input characteristics [27]-[28]. Significant improvements performance over what currently available with discrete OTAs have been demonstrated. alternative voltage attenuators buffers input existing OTAS. This technique often suggested manufacturer's application notes illustrated Fig. This technique used obtain reasonable signal swings with circuits discussed this point. Although such circuits useful, rather high price paid this modification. First, circuit requires many more components. Second, finite bandwidth amps will limit frequency response structures. Finally, attenuation input signal causes
Fig. Signal conditioner OTAs. potential tuning both poles zeros (when desired value should apparent. Although somewhat component intense, argued that there capability completely arbitrary location pair poles pair zeros adjustment transconductance gain OTA, then least degrees freedom and, hence, OTAs required. This circuit uses only more than minimum! capability various types pole and/or zero movement through simultaneous adjustment more transconductance gains should also apparent. Many other biquadratic structures, some which offer more flexibility expense additional complexity, also exist discussed here. Emphasis this section been placed entirely upon second-order structures which desired filter characteristics depend directly upon transconductance gain OTA. Very simple structures which filter characteristics adjustable through parameter resulted. stated introduction, readily controllable bias current over wide range values, thus making these circuits directly applicable voltagecontrolled applications. Several more recent works applications [18]-[24] have followed this approach. Most earlier works [7]-[16] circuits presented manufacturer's application notes [3]-[5] concentrated upon topologies which filter characteristics independent only mildly dependent upon transconductance gain. Most these structures very complicated, very component-intense, require
Fig. Macromodel bias current port bipolar OTA. serious loss dynamic range. From topological point view, some based structures inherently more susceptible differential voltage limitations than others. This parallels concern based active switched-capacitor
Fig. Schemes obtaining voltage control with OTA. thus, external control voltage,
shown Fig.
Fig. Single input-multiple output bias current
generator monolithic applications. Fig. Schemes simultaneous adjustment. structures that signal amplitudes output internal amps assume acceptable values. These considerations become more serious high high dynamic range applications. macro model bias current (IABC) input port typical bipolar shown Fig. This actually forms part internal current mirror that discussed later. Several schemes controlling current (IABC) and, Fig. attenuator. first circuit simplest very sensitive small changes approaches second circuit, control voltage referenced zero small sensitive mismatches between voltage transistor forward diode voltage drop. circuit Fig. 12c, control voltage also referenced ground dependent upon matching cancellation voltages across external forward biased
junctions. zener diode used maintain common mode voltage reasonable level. frequency response concern here since used only control path. should noted that amplifier bias current proportional schemes shown Fig. Since IABC typically adjusted over several decades, schemes will very sensitive small changes toward current IABC range. Logarithmic amplifiers often used control IABC with external control voltage wide adjustment range IABC effectively utilized. Many filter circuits discussed previous sections this paper require simultaneous adjustment matched gm's. Several schemes achieving this shown Fig. first circuit, easy adjust gm's trimming resistors fixed circuit quite sensitive slight differences voltage Fig. small values IABC. circuit Fig. again referenced ground essentially independent matching individual OTAs. scheme Fig. useful external single package current mirror with outputs available. discrete component version this mirror would practical. integrated circuit applications, amplifier bias currents several OTAs particularly easy match control. monolithic applications, simultaneous adjustment gain large number OTAs with single bias current easily attained using single input-multiple output current mirror such shown Fig. This structure actually replaces bias current mirrors each OTAS. transconductance gains ratioed, desired, correspondingly ratioing emitter areas width length ratio structures) outputs current mirror. With conventional operational amplifiers, slew rate, input impedance, output impedance, maximum output current essentially fixed design stage. OTAS, generally case that these parameters either proportional inversely proportional IABC. Thus adjusting IABC causes these parasitic parameters change accordingly. Although user should cognizant changes these parameters, problems they present manageable. output capacitance does cause concern output currents high frequencies. Much design conventional based circuits, designer must allow bias current path both input terminals OTA. Although amplifier Fig. serves effective attenuator, which will prove useful some applications, circuit useless since required input bias current will cause accumulation charge capacitors eventual saturation OTA. reader should cautioned that
more complicated circuits with same problem suggested literature [17]. Numerous nonlinear applications structures exist. Suffice that since amplifier bias current, IABC considered third signal input, simple multipliers, modulators, host other nonlinear circuits possible. reader referred application notes discussion some nonlinear applications. Some structures that only OTAs capacitors show promise monolithic applications bipolar processes. circuits should offer highfrequency continuous-time capabilities. Either external voltage-control internal reference circuit compensate process temperature variations will necessary make these circuits practical demanding applications. Finally, should noted that some filter structures presented earlier this paper have non-infinite input impedance, that output impedance generally quite high. Cascading such structures will require interstage buffer amplifiers, which will tend degrade bandwidth overall filter structures. Output buffers also generally required drive external loads.
Conclusions
group voltage-controlled circuits using basic active element have been presented. characteristics these circuits adjusted with externally accessible amplifier bias current. Most these circuits utilize very small number components. Applications include amplifiers, controlled impedances, filters. Higher-order continuous-time voltage-controlled filters such common Butterworth, Chebyschev, Elliptic types obtained. addition voltagecontrol characteristics, based circuits show promise high-frequency applications where conventional based circuits become bandwidth limited. major factor limiting performance based filters using commercially available OTAs severely limited differential input voltage capability inherent with conventional differential amplifier input stages. Recent research results suggested significant improvements input characteristics OTAs attained [27]-[28].
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Randall Geiger born Lexington, Nebraska, 1949. received B.S. degree electrical engineering M.S. degree mathematics from University Nebraska, Lincoln, 1972 1973, respectively. receiv Ph.D. degree electrical engineering from Colorado State University, Fort Collins, 1977. 1977, Geiger joined Faculty Department Electrical Engineering Texas University, College Station, currently holds rank Associate Professor. present research areas integrated circuit design active circuits. received Meril Reed Best Paper Award 1982 Midwest Symposium Circuits Systems, served Conference Chairman 1983 UGIM conference, currently serving Associate Editor IEEE Transactions Circuits Systems. Geiger member Kappa Sigma Epsilon, Sigma Tau; also Senior Member IEEE.
April 1967, joined Petroleum Institute Mexico, where associated with design instrumentation equipment until August 1967. worked Research Assistant Coordinated Science Laboratory University Illinois from September 1971 August 1973. 1974, held industrial postdoctoral position with Central Research Laboratories, Nippon Electric Company, Ltd., Kawasaki, Japan. From 1976 1983, Head Department Electronics Instituto Nacional Astrofisica, Optica (INAOE), Puebla, Mexico. Visiting Professor Department Electrical Engineering Texas University during academic years 1979-1980 1983-1984, where currently Professor. General Chairman 1983 26th Midwest Symposium Circuits Systems. coauthor book Switched-Capacitor Circuits (Van Nostrand-Reinhold, 1984). present interests areas active filter sign, solid-state circuits, computer-aided circuit design. Senior Member IEEE.
Edgar born Mexico City, Mexico, October 1944. received degree communications electronic engineering (professional degree) from National Polytechnic Institute Mexico, Mexico City; M.S.E.E. degree from Stanford University, Califomia; Ph.D. degree from University Illinois Champaign -Urbana, 1966, 1970, 1973, respectively. During graduate studies, awarded with fellowships from United Nations Educational, Scientific, Cultural Organization; Mexican Atomic Energy Commission; Consejo Nacional Ciencia Tecnologia Mexico. From January 1965 March 1967, worked with Mexican Atomic Energy Commission, Design Engineer.
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