FOR YEARS, PEOPLE WORKING ON CIRCUITtheory have been trying to do-in the induc-tor. This is especially true in the audio andsub-audio regions where inductors are in-herently big, expensive, difficult to adjust,and subject to fields and hum pickup. Aftera lot of false trys and some rather poorways of going about this, a batch of solid,reliable methods now exist that can do thejob. Almost all these methods use low cost,readily available operational amplifiers. Inmost of the methods, the energy storage ofan inductor is simulated by taking energyfrom a power supply and delivering it atthe right point and in the right amount in acircuit to simulate exactly the bahavior ofan inductor.

Actually most methods don't workdirectly on replacing inductors. Instead,they look at the whole picture and attemptto come up with a functionally equivalentcircuit that does exactly the same thing thatthe original one did, but internally does itin a wildly different way. These function-ally equivalent circuits are often called ac-tive filters, and an active filter is simply anycircuit that uses at least one operationalamplifier or its equivalent to simulate ex-actly a circuit that normally would need atleast one inductor to get the same result.

A filter itself is any frequency selectivenetwork. Three popular styles are the low-pass filter that passes only low frequenciesand stops higher ones; the bandpass filterthat passes only a few or a range of medianfrequencies; and a high-pass filter that al-lows only high frequencies to reach its out-put. A rumble filter on a turntable is ahigh-pass filter. The tuning on an AM radiois a bandpass filter, and the treble cut con-trol on a hi-fi is a form of low-pass filter.

A comparisonBefore we go into the nuts and bolts

details of how to build your own active fil-ter, let's compare a simple active low-passfilter with an equivalent low-pass passiveone (see Fig. 1). And, if we wanted to, we

PASSIVE

FIG. 1-LOW-PASS FILTERS. The passive L-Ctype is simpler, the active is more versatile.

How ActiveFilters Work

Here are full details on how to build filters withop-amps instead of inductors. This stable and reliable

method works for practically all audio andsub-audio low-pass, bandpass, and high-pass designs

tance decreases as we increase frequency,shunting more and more signal to ground.In the active filter, we essentially have twocascaded R-C sections at very high fre-quencies that also shunts the signal toground. The problem is that if we left theamplifier out of the circuit, the responsewould droop very sloppily and very badlyaround the cutoff frequency.

IDCO

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-15

-21

-24

-27

0.707 VOLTAGEOR -3dB AT

CUTOFF

X4

could select the L-C ratio of the passive fil-ter or the ratio of Cl and C2 in the activefilter to get a response that looks like Fig.2.

The way we get the response differsfor the two circuits, but the result is thesame. In the passive filter, the inductive re-actance increases and the capacitive reac-

42 RADIO-ELECTRONICS • NOVEMBER 1973

X0.5 X1 X2CUTOFF FREQUENCY

FIG. 2-RESPONSE OF LOW-PASS FILTER,Slope shown is 12 dB per octave.

What the op amp does is use circuitfeedback via Cl. It takes energy from thesupply and introduces it in the middle ofthe R-C network to simulate exactly thesame effect as energy storage in the induc-tor. Thus no R-C network by itself can everhope to be as good as an L-C one, but anR-C network with some energy feedbackcontrolled by an op amp is another story,and you can replace virtually any L-C net-work with a group of op amps, resistors,and capacitors. In fact, there's even thingsyou can do actively that you can't withconventional circuits. Gain for instance.

We picked this particular response be-cause it has the maximum possible flatnessin the passband. It is called a Butterworthfilter. If we try to steepen the responsewithout adding any more parts, we'd get ahump in the passband, and the size of thehump would decide the initial but not theultimate rate of falloff. Filters with humpsare called Chebycheff filters if the humpsare in the passband and Elliptical filters ifthe humps are both in the passband andthe stopband. We could also make the re-sponse less flat and more gradual. Thiswould improve the pulse response andovershoot at the expense of tilt in thepassband and a more gradual rolloff. The

by DON LANCASTER

best of these is called a Bessell filter. Theterm that controls the shape of the filternear the cutoff frequency, but not at verylow or very high frequencies is called thedamping of the filter. The damping is con-trolled by the L/C ratio in the passive filterand the Cl to C2 ratio in the active filter,or by holding Cl and C2 constant andchanging the ratio of R3 and R4.

We picked the Butterworth here be-cause it is the most popular and the easiestto use. We'll stick with Butterworth filtersall the way through this story. Other typesare just as easy to build. All you have todo is move the damping and cutoff fre-quencies around a bit.

If we wanted something steeper than a12-dB-per-octave rolloff, we'd have to addmore parts. Two inductors and a capacitorwould give you a 18-dB-per-octave filter,and two inductors and two capacitorswould give you a 24-dB-per-octave rolloff,and so on. We call this the order of the fil-ter. Second-, third-, and fourth-order filtershave rolloff rates of 12, 18, and 24-dB-per-octave, and are the most popular normallyused. Normally it takes one op amp for asecond or third-order filter and two for afourth.

Note that the damping of the filtercontrols the response near cutoff, partic-ularly the flatness in the passband, the timedelay and overshoot, and the initial rate offalloff. The order of the filter controls theultimate or asymptotic rate of falloff for thefilter.

Why go active?The operational amplifier serves as a

gain block with a very high input imped-ance and a very low output impedance. Itsessential function is to provide for energyfeedback to simulate the effect of energystorage in an inductor. Two nice benefitsare the ability to drive any load and to usehigher impedance (and almost alwayscheaper) components. So what are the ben-efits of an active filter? What do we gainand what do we lose when we go active?

The first and obvious thing we lose isthe inductor, along with its cost, size, diffi-culty of adjustment, and sensitivity to humand other magnetic fields. Note also thatthe passive filter has a load resistor. Thevalue of this resistor is critical, for if youchange it, the relative effects of the reac-tance changes of the inductor and capacitor

change and the response shape or the cut-off frequency may change. This is not trueof the op amp active filter, for the op ampcan drive most any reasonable load withoutchanging the filter's response. We can varythe load from an open circuit down to any-thing the op-amp can reasonably drivewithout changing the response.

The input to the op amp is a very highimpedance. This means you can use high-impedance resistors and capacitors for agiven response at a given frequency. Thebenefits here are obvious. A high-imped-ance resistor costs the same as a low-ohmsone, but a high-impedance capacitor ismuch smaller, and much cheaper.

Passive filters are inherently lossy, andthe best we could expect to hope for wouldbe slightly less than unity gain. With opamps and active filter designs, you some-tunes can design for any circuit gain youwant. Those we're going to show you havegains above unity.

Another big benefit is tuning. Largevariable capacitors are nonexistent, whilelarge variable inductors are expensive and apain to adjust. On the active side, we haveresistors Rl and R2 and surely changingthem will change the response. For thisparticular circuit, we have to change both atonce to change frequency without hurtingthe damping and response shape. This iseasy to do with a dual pot, and we can eas-ily get at least a 10:1 range. Even for slighttuning adjustments, the resistors are easy tochange to get exactly the response youneed. Because of this, active filters are gen-erally more tuneable and easier to adjustthan passive ones.

A final benefit is a bit subtle, but veryimportant when we want a fancier higherorder filter with faster cutoff slopes. We cancascade active filter blocks without any in-teraction, since they are free from fieldsand mutual inductance and since they gen-erally have a high input impedance and alow output impedance. Cascadeability is avery big benefit. You normally can't simplycascade identical stages, for what was a -3-dB point becomes a -6 and so on. Whatyou do is take the math expression for thehigher order filter you want and factor itinto second-order terms, and then buildeach second-order term separately. Gener-ally, the individual block responses will beless damped and appear peaked when com-pared to the final result.

Disadvantages and problemsIf active filters are so good, why

doesn't everybody use them? First and fore-most, it's because very few people under-stand or appreciate what they are and whatthey can do. But, over and above this, thereare some limitations and disadvantages totheir use. Let's take a closer look.

Obviously, we need some supply powerand the noise characteristics of the op ampcan effect very low-level signals. More im-portant is the high-frequency limitations ofthe operational amplifier. As you increaseoperating frequency, an op amps open-loopgain decreases and its phase characteristicschange so that you are limited to the upperfrequency you can handle with a given op-erational amplifier.

For amplifiers like the 741 style or itsdual and quad combinations, a reasonableupper frequency limit for active filters isbetween 20 and 50 kHz for low-pass andhigh-pass versions, and between 2 and 5

10K 5.6K

FIG. 3-ACTIVE FILTERS HAVE GAIN. Pas-sive types are lossy circuits.

kHz for bandpass designs. If you go to ahigher performance internally compensatedamplifier such as the National LM318, youcan work up into the hundreds of kilohertz.Finally, if you go to really exotic op ampsyou can work higher, and even microwaveactive filter structures have been built.Thus, we have an upper audio limit for ac-tive filters built with the cheapest availableop amps and a fractional megahertz limitfor op amps in the $3 to $5 class.

Bandpass filters need more gain forresonance and thus are generally limited tolower frequencies. One way around theproblem is to distribute the problem amongtwo or more op amps so that each only hasto provide some of the gain.

The low-frequency limit is anotherstory. It's decided mostly by how much youwant to pay for big capacitors and howhigh you're willing to let impedances get.With FET op amps, this can be a bunch,and operation down below 0.1 Hertz is cer-tainly possible. Thus active filters are idealfor such sub audio work as brain wave re-search, seismology, geophysics, and fieldslike this.

One limitation, and the big one, iscalled the sensitivity problem. You have toask how the individual components in theactive filter are going to change the re-sponse if they are out of tolerance or driftwith time. For instance, if a particular pa-rameter such as a gain or a capacitancevalue happens to have a sensitivity of 0.5,the result is a 5% change in cutoff fre-quency or damping for a 10% change incomponent value. On the other hand, if a1% variation makes a 50% change in some-thing you've got problems. This is clearlyungood. When picking a way to build ac-tive filters, you have to be aware of thesensitivity problems and how to use them.The method we'll be showing you in aminute is very well behaved at fixed lowgains and for lower to moderate Q band-pass designs.

A final limitation is one of method.There are about a dozen good and provenways to design active filters. These all varywith their ease of understanding and whatthey can and cannot do. Some can't handleall three basic responses. Some allow singleresistor tuning; others allow separate tuningof bandpass gain, center frequency, and Q.Some are well behaved at certain gains, butat others are too highly sensitive or actuallyunstable. You have to pick a method thatworks for you, is reliable, behaves well, and

"VSR

I 10K-*—•vA/v—'wv—•—*

100K

?te.016.0016

O1-10 10-100 100- 1 kHz- '

1kHz 10kHz (

FIG. 4—AN ADJUSTABLE LOW-PASS FILTER covering from 1 Hz to 100 kHz in five frequencyranges. Switched capacitors select the bands, ganged potentiometers do the variable tuning.

NOVEMBER 1973 • RADIO-ELECTRONICS 43

does what you want it to. The one we'll beshowing you is very easy to understand,stable and forgiving of component varia-tions for fixed low gains, and useable in thebandpass case for low to moderate Q's. Itusually takes two resistors simultaneouslyadjusted to tune, and in the bandpass case,you cannot separately set the Q, gain, andcenter frequency without a major change incomponents.

The method is called the Sallen-Key orVoltage Controlled Voltage Source (VCVS)method, and first appeared in the March1955 IRE Transactions on Circuit Theory.Other popular filter methods are called theIntegrator Lag, the Biquadratic Section, theMultiple Feedback, and the State Variable.Another type of active filter uses the gyratoror impedance converter but these generallytake a bunch of parts and have a high-im-pedance output.

Building your ownSo, now we should know why we'd

want to use active filters, and where to goto get complete design details, let's concen-trate on how to actually build one. Here's asecond-order Butterworth low-pass filterwith a cutoff frequency of 1 kHz and again of 1.6 (see Fig. 3).

The response is identical to the curvein Fig. 2 with f = 1 kHz, 2 f = 2 kHz,and so on. As with any low-pass active fil-ter, there must be a low dc impedance toground at the source. Thus your source hasto be less than 10,000 ohms and must pro-vide a route to ground for the op amp'sbias current. Again, the response is But-terworth, giving us the flattest possiblepassband, and an attenuation of -3 dB or0.707 amplitude at the cutoff frequency,and smoothly falls off at -12 decibels peroctave. This means that in the stopband asyou double frequency, you get only onequarter the amplitude, and so on.

The above circuit looks deceptivelysimple and it is except, that a "magic" gainof 1.6 has been used that lets you use equalresistors, equal capacitors, and still have thedesired shape. Change anything from theabove, and the mathematics behave wildly.The circuit is forgiving of component varia-tions and 5% components should be morethan adequate for practically all uses.

To change frequency (in Fig. 1) youchange Rl and R2 to identical val-ues, or you simultaneously change Cl andC2 to new values. Raising R lowers the op-erating frequency. Raising C lowers the op-erating frequency. Thus, a 5000-ohm valueinstead of 10,000 ohms puts you at a 2 kHzcutoff frequency, and so on. A 0.032 /nF ca-pacitor value puts you at 500 hertz and soon. If you change one capacitor, you mustchange the other. Similarly if you changeone resistor, you must change the other, orthe response shape will also change.

It's easy to see how we can use a dualpot to tune 10:1 and switch capacitors toget decade ranges. Fig. 4 shows a circuitthat covers any cutoff frequency you wantfrom 1 hertz to 100 kHz:

The pot rotation will generally be non-linear since the frequency varies inverselywith pot rotation and resistance value. Oneway to linearize the pot is to use a dual au-dio log pot, with a normal taper if the dialis on the pot shaft and a reverse taper ifthe dial is on the panel.

If you just want one frequency differ-

44 RADIO-ELECTRONICS

10K 5.6K

FIG. 5-IN HIGH-PASS FILTERS, the R and Cshunt and series elements are transposed.

ent from 1 kHz, just calculate the capacitorvalue you need and change the capacitors,or change the resistors. It's simply the ratioof the capacitors equals the ratio of the fre-quencies and vice versa for the resistors.

High-pass designsThe high-pass filter is a snap—you in-

side the circuit out and by a network prin-ciple called duality you're done. Fig. 5 is a1-kHz Butterworth, second-order high-passcircuit.

We can now see another big advantageto the "magic" gain value of 1.6—this cir-cuit lets us switch from highpass to lowpasswith a 4pdt switch without any change ofcomponent values.

Steeper skirtsWe can cascade two low-pass second-

order sections to get a fourth-order But-terworth with a 24-dB-per-octave cutoff.We can't use identical sections, but we canmake everything identical except for R4(the feedback resistor) on each section.Finding the right R4 takes a lot of math,but here's the final circuit (see Fig. 6). Ithas an overall gain of 2.5:

The response is twice as good as be-fore on a decibel scale. The passband istwice as flat and still drops only to -3 dB atthe cutoff frequency of 1 kHz. The attenua-tion drops at 24-dB-per-octave, meaningthat every doubling of frequency gives youonly one-sixteenth the power and so on.

The higher performance circuit issomewhat harder to tune, since you simul-taneously have to change four capacitors or

10K 1.6K 10K 11K

FIG. 6-TWO LOW-PASS SECTIONS IN CASCADE produce a fourth-order Butterworth filter witha rolloff slope of 24 dB per octave. The circuit's overall gain is about 8 dB.

OP AMP GAIN MUST GREATLY EXCEED8Q2 AT OPERATING FREQUENCY

CIRCUIT GAIN = 2Q

LM3I8GAIN = 16

eOUT

741GAIN = 16

10K

- OP AMP GAIN MUST GREATLY EXCEED2Q AT OPERATING FREQUENCY

CIRCUIT GAIN = 2Q

FIG. 7-ACTIVE BANDPASS CIRCUITS re-quire high-gain operational amplifiers.

You simply interchange the resistorsand capacitors on the input and you nowhave a highpass circuit. Again, if youchange frequency, change both resistors orboth capacitors to identical new values, orelse the response shape will also change.

FIG. 8— CIRCUIT CONSTANTS for bandpasscircuit where Q Is 8 and frequency is 1 kHz.

four pots. Quadriphonic audio pots are aneat way to handle the tuning and they arereasonably available. Highpass to lowpassswitching can be handled by a 8-pole-double-throw switch or two ganged 4-pole-double-throw pushbuttons, arranged so oneis up when the other is down and viceversa.

Other orders and shapes of active filter(continued on page 71)

NOVEMBER 1973

SERVICING RECORD CHANGERS(continued from page 53)

wheel. Replace the retaining C clipand any trim you might have had toremove to gain access to the clip.

18. Refer to Fig. 2 and apply asingle drop of lubricant to the over-arm shaft assembly. Work the arm upand down to insure proper lubrication.

OPTIONAL STEPS19. Place 4 dowels, one under

each corner to support the changerslightly off the bench. Connect theuniversal power cord to the changerand plug into an ac outlet. Place astrobe disc on the turntable and testfor correct speed. The pattern shouldappear stationary.

20. Thoroughly clean the changerusing a liquid spray cleaner for metalsurfaces, and spray wax for thewooden surfaces.

21. Replace the changer in theunit and performance test. If anyproblems are noted consult troublechart (Table II).

The complete changer just de-scribed requires about 45-minutes.While these simple overhaul tech-niques will solve about 80% of yourrecord-changer problems, Table IIgives you additional hints for servicingproblems. R-E

ACTIVE FILTERS(continued from page 44)

are just as easy to do.

Bandpass designsBandpass filters are generally much

harder to design and more subtle to use.About all we have room for here is to showyou two circuits that will do the job (seeFig. 7). They're shown for any Q at a cen-ter frequency of 1 kHz. And here in Fig. 8are the same circuits for a Q = 8: (1 kHz)

The two-amplifier job requires far lessstable gain and works better for higher Q'sand higher frequencies. Either circuit givesyou the equivalent of a single series "pole"or tuned RLC circuit. This circuit, like itspassive counterpart has a nasty feature thatyou must allow for. Its response starts fall-ing off very steeply either side of resonance,but for very low or very high frequencies, it

FIG. 9-BANDPASS RESPONSE CURVE mea-sured at 3-dB point depends on circuit Q.

falls off at a more gradual rate of six deci-bels per octave. The response shape lookslike the diagram in Fig. 9.

Normally, you cascade several poles toget the desired bandpass response. If weput the poles on top of one another, we geta very sharp response that is not very flatin the passband. We can control the re-sponse shape by staggering the poles in fre-quency and by altering their Q. Spreadingthe poles flattens out the passband, until fi-nally you get a dip in the middle if you gotoo far. Another more formal way to designis to build a lowpass filter that does the jobyou want and then use a math processcalled transformation to get the desiredbandpass shape. When you only need twopoles, the simplest thing is to sit down witha breadboard and experiment with the Qand staggering for the response you need(the circuit moves around just like the low-pass and bandpass ones do by simulta-neously changing capacitors or resistors);this is also a trivial problem for any com-puter that speaks BASIC, but the math is abear otherwise. That's about all the detailson bandpass design we have room for here.If enough readers are interested, we canput together another story with completedesign curves for the two-pole bandpass de-signs in some other issue. R-E

BUILD IN ACTIVE FILTERNext month in Radio-Electron-

ics Don Lancaster presents com-plete details on how to build an ac-tive filter to meet your own needs.Don't miss it.

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... Just ask Detroitand the E.PA

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