active filter - analog filter design

8/8/2019 Active Filter - Analog Filter Design

1/10

Sensors - February 2002 - Demystifying Analog Filter Design Page 1 of 12

http://www.sensorsmag.com/articles/0202/filter/main.shtml 10/03/2002

FEBRUARY 2002

Demystifying

Analog Filter DesignArmed with a little helpand a little mathyou can hack your wayfearlessly through the wild world of analog filter design and gain the

confidence that comes with doing it yourself.

Simon Bramble, Maxim Integrated Products

Its a jungle out there. A small tribe, hidden from view in the densewilderness, is much sought after by headhunters from thesurrounding plains. The tribe knows its threatened, because itsnumberskilled off by the accelerating advance of moderntechnologyare dwindling at an alarming rate. This is the tribe of theAnalog Engineers. The guru of Analog Engineers is the Analog FilterDesigner, who sits on the throne of the besieged kingdom andimparts wisdom while reminiscing of better days.

But youre desperate. Your boss told you to design a data acquisitionsystem, and that means you need an analog filterfast. You try abook on filter design. Alas! The countless pages of equations found

in such books can frighten small dogs and children, let aloneengineers. What to do?

Fear not! This article unravels the mystery of filter design so that youcan design continuous-time analog filters quickly and with aminimum of mathematics. The throne will soon be vacant.

The Theory of Analog Electronics

Analog electronics has two distinct sides: theory (e.g., using theequations of stability and phase-shift calculations), taught byacademic institutions, and practical considerations (e.g., avoiding

oscillation by tweaking the gain with a capacitor), familiar to mostengineers. Unfortunately, filter design is based firmly on long-established equations and tables of theoretical results. The theoreticalequations can prove arduous, so this article uses a minimum ofmatheither in translating the theoretical tables into practicalcomponent values or in deriving the response of a general-purposefilter.

Simple RC low-pass filters have the following transfer function:

SENSOR TECHNOLOGY AND DESIGN

- Select -

March 10, 2002

Industry NewsUpdated Twice Weekly

Xicor's New ArchitectureAddresses Growing MarketOmniVision OpensEuropean Office

Building AutomationSystems Buy Into Internet, ITIntelligent Instrumentation,Entivity Partner

News From Sensors

Order Now!


2/10



Cascading such filters complicates the response by giving rise toquadratic equations in the denominator of the transfer function. Thus,the denominator of the transfer function for any second-order low-

pass filter is as2 + bs + c. Substituting values for a, b, and cdetermines the filter response over frequency. Anyone whoremembers high school math will note that the preceding expressionequals zero for certain values ofs given by the equation:

At the values ofs for which this quadratic equation equals zero, thetransfer function theoretically has infinite gain. These values, which

establish the performance of each type of filter over frequency, areknown as thepoles of the quadratic equation. Poles usually occur aspairs, in the form of a complex number (a +jb) and its complexconjugate (a jb). The termjb is sometimes zero.

The thought of a transfer function with infinite gain may frightenyou, but in practice, it isnt a problem. The poles real part, a,indicates how the filter responds to transients; its imaginary part,jb,indicates the response over frequency. As long as this imaginary partis negative (as it must be), the system is stable. The followingdiscussion explains how to transfer the tables of poles found in many

textbooks into component values suitable for circuit design.

Filter Types

The most common filter responses are the Butterworth, Chebyshev,and Bessel types. Many other types are available, but 90% of allapplications can be solved with one of these three.

AButterworth filterensures a flat response in the pass band and anadequate rate of rolloff. A good all-round filter, the Butterworth issimple to understand and suitable for such applications as audioprocessing.

A Chebyshev filtergives a much steeper rolloff, but pass-band ripplemakes it unsuitable for audio systems. Its superior for applicationsin which the pass band includes only one frequency of interest (e.g.,the derivation of a sine wave from a square wave by filtering out theharmonics).

ABessel filtergives a constant propagation delay across the inputfrequency spectrum. Therefore, applying a square wave (consistingof a fundamental and many harmonics) to the input of a Bessel filter

(1)

(2)


3/10



yields an output square wave with no overshoot (i.e., all of thefrequencies are delayed by thesame amount). Other filters delaythe harmonics by differentamounts; the result is an overshooton the output waveform.

One other popular type, the

elliptical filter, is a much morecomplicated beast, and it will notbe discussed in this article. Similarto the Chebyshev response, it hasripple in the pass band and severerolloff at the expense of ripple inthe stop band.

Standard Filter Blocks

The generic filter structure (see

Figure 1A) lets you realize a high-pass or low-pass implementationby substituting capacitors orresistors in place of componentsG1G4. Considering the effect ofthese components on the op-ampfeedback network, you can easilyderive a low-pass filter by making G2/G4 into capacitors and G1/G3into resistors (see Figure 1B). The opposite yields a high-passimplementation (see Figure 1C).

The transfer function for the low-pass filter (see Figure 1B) is:

This equation is simpler with conductances. Replace the capacitorswith a conductance ofsCand the resistors with a conductance ofG.If this looks complicated, you can normalize the equation. Set theresistors to 1 or the capacitors to 1 F, and change the surroundingcomponents to fit the response. Thus, with all resistor values equal to

1 , the low-pass transfer function is:

This transfer function describes the response of a generic, second-order low-pass filter. You take the theoretical tables of poles (seeTables 17) that describe the three main filter responses and translatethem into real component values.

Figure 1. By substituting for G1-G4 in thegeneric filter block (A), you can implement alow-pass filter (B) or a high-pass filter (C).

(3)

(4)


4/10



The Design Process

To determine the filter type most appropriate for your application,use the preceding descriptions to select the pass-band performanceneeded. The simplest way to determine filter order is to design asecond-order filter stage and then cascade multiple versions of it asrequired. Check to see if the result gives the desired stop-band

rejection. Next, proceed with correct pole locations as shown in thetables at the end of the article. Once pole locations are established,the component values can soon be calculated.

First, transform each pole location into a quadratic expression similarto that in the denominator of our generic second-order filter. If aquadratic equation has poles of (a jb), then it has roots of (s a jb) and (s a +jb). When these roots are multiplied together, the

resulting quadratic expression is s2 2as + a2 +b2.

In the pole tables, a is always negative, so for convenience you

declare s2 + 2as + a2 +b2 and use the magnitude ofa regardless of itssign. To put this into practice, consider a fourth-order Butterworthfilter. The poles and the quadratic expression corresponding to eachpole location are as follows:

You can design a fourth-order Butterworth low-pass filter with this

information. Simply substitute values from the preceding quadraticexpressions into the denominator of Equation 4. Thus, C2C4 = 1 and2C4 = 1.8478 in the first filter, which implies that C4 = 0.9239FandC2 = 1.08F. For the second filter, C2C4 = 1 and 2C4 = 0.7654,implying that C4 = 0.3827Fand C2 = 2.61F. All resistors in bothfilters equal 1 . Cascading the two second-order filters yields afourth-order Butterworth response with rolloff frequency of 1 rad/s,but the component values are impossible to find. If the frequency orcomponent values just given are not suitable, read on.

It so happens that if you maintain the ratioof the reactances to the

resistors, the circuit response remains unchanged. You mighttherefore choose 1 k resistors. To ensure that the reactancesincrease in the same proportion as the resistances, divide thecapacitor values by 1000.

You still have the perfect Butterworth response, but unfortunately therolloff frequency is 1 rad/s. To change the circuits frequencyresponse, you must maintain the ratio of reactances to resistancesbut simply at a different frequency.

Poles (a jb)0.9239 j0.38270.3827 j0.9239

Quadratic Expression

s2 + 1.8478s + 1

s2 + 0.7654s + 1


5/10



For a rolloff of 1 kHz rather than 1 rad/s, the capacitor value must befurther reduced by a factor of 2 1000. Thus, the capacitorsreactance does not reach the original (normalized) value until thehigher frequency. The resulting fourth-order Butterworth low-passfilter with 1 kHz rolloff takes the form of Figure 2.

Using this technique, you can obtain any even-order filter responseby cascading second-order filters. Note, however, that a fourth-orderButterworth filter is not obtained simply by calculating thecomponents for a second-order filter and then cascading two suchstages. Instead, two second-order filters must be designed, each withdifferent pole locations. If the filter has an odd order, you can simplycascade second-order filters and add an RC network to gain the extrapole. For example, a fifth-order Chebyshev filter with 1 dB ripple hasthe following poles:

To ensure conformance with the generic filter described by Equation4 and to ensure that the last term equals unity, the first two quadraticshave been multiplied by a constant. Thus, in the first filter, C2C4 =2.488 and 2C4 = 1.127, which implies that C4 = 0.5635Fand C2 =4.41F. For the second filter, C2C4 = 1.08 and 2C4 = 0.187, whichimplies that C4 = 0.0935Fand C2 = 11.55F. Earlier, you saw that anRC circuit has a pole when 1 + sCR = 0: s = 1/CR. IfR = 1, then to

obtain the final pole at s = 0.28, you must set C= 3.57F.

Using 1 k resistors, you can normalize for a 1 kHz rollofffrequency (see Figure 3).

Figure 2. These two nonidentical second-order filter sections form a fourth-orderButterworth low-pass filter.

Poles Quadratic Expression0.2265 j0.5918

s2 + 0.453s + 0.402

2.488s2 + 1.127s + 1

0.08652 j0.9575s2 + 0.173s + 0.9241.08s2 + 0.187s + 1

0.2800 See text


6/10



Thus, designers can boldly go and design low-pass filters of anyorder at any frequency.

All of this theory also applies to the design of high-pass filters.Youve seen that a simple RC low-pass filter has the transferfunction of:

Similarly, a simple RC high-pass filter has the transfer function of:

Normalizing these functions to correspond with the normalized poletables give:

for low-pass and

for high-pass.

Note that the high-pass pole positions, s, can be obtained by invertingthe low-pass pole positions. Inserting those values into the high-pass

filter block ensures the correct frequency response. To obtain thetransfer function for the high-pass filter block, you need to go back tothe transfer function of the low-pass filter block. Thus, from:

you obtain the transfer function of the equivalent high-pass filter

Figure 3. A fifth-order, 1 dB ripple Chebyshev low-pass filter is constructed from two nonidentical

second-order sections and an output RC network.

(5)

(6)

(7)

(8)

(9)


7/10



block by interchanging capacitors and resistors:

Again, life is much simpler if capacitors are normalized instead ofresistors:

Equation 12 is the transfer function of the high-pass filter block. Thistime, you calculate resistor values instead of capacitor values. Giventhe general high-pass filter response, you can derive the high-passpole positions by inverting the low-pass pole positions andcontinuing as before. But inverting a complex-pole location is easier

said than done. For example, consider the fifth-order, 1 dB-rippleChebyshev filter discussed earlier. It has two pole positions at (0.2265 j0.5918).

The easiest way to invert a complex number is to multiply and divideby the complex conjugate, thereby obtaining a real number in thenumerator. You then find the reciprocal by inverting the fraction.

Thus:

gives:

and inverting gives:

(10)

(11)

(12)

(13)

(14)


8/10



You can convert the newly derived pole positions to thecorresponding quadratic expression and values calculated as before.The result is:

From Equation 12, you can calculate the first filter component valuesasR2R4 = 0.401 and 2R2 = 0.453, which implies thatR2 = 0.227andR4 = 1.77 . You can repeat the procedure for the other polelocations.

Because weve shown that s = 1/CR

, a simpler approach is to design

for a low-pass filter by using suitable low-pass poles and then treatevery pole in the filter as a single RC circuit. To invert each low-passpole to obtain the corresponding high-pass pole, simply invert thevalue ofCR. Once youve obtained the high-pass pole locations, youensure the correct frequency response by interposing the capacitorsand resistors.

You calculated a normalized capacitor value for the low-passimplementation, assuming thatR = 1 . Hence, the value ofCRequals the value ofC, and the reciprocal of the value ofCis the high-pass pole. Treating this pole as the new value ofR yields the

appropriate high-pass component value.

Considering again the fifth-order, 1 dB ripple Chebyshev low-passfilter, the calculated capacitor values are C4 = 0.5635Fand C2 =4.41F. To obtain the equivalent high-pass resistor values, invert thevalues ofC(to obtain high-pass pole locations) and treat these polesas the new normalized resistor values:R4 = 1.77 andR2 = 0.227.This approach provides the same results as does the more formalmethod mentioned earlier.

Thus, the circuit in Figure 3 can be converted to a high-pass filter

with 1 kHz rolloff by inverting the normalized capacitor values,interposing the resistors and capacitors, and scaling the valuesaccordingly. Earlier, we divided by 2 fR to normalize the low-passvalues. The scaling factor in this case is 2 fC, where Cis thecapacitor value andfis the frequency in hertz. The resulting circuit isshown in Figure 4.

(15)

Poles (a jb)0.564 j1.474

Quadratic Expression

s2 + 1.128s + 2.490

0.401s2 + 0.453s + 1


9/10



Conclusion

By using the methods described here, you can design low-pass andhigh-pass filters with response at any frequency. Band-pass andband-stop filters can also be implemented (with single op amps) byusing techniques similar to those shown, but those applications arebeyond the scope of this article. You can, however, implement band-pass and band-stop filters by cascading low-pass and high-passfilters.

This article promises to be your guide through the wilderness, yourdefense against the headhunters, and your key to the analog kingdom.

Figure 4. Transposing resistors and capacitors in the circuit in Figure 3 yields a fifth-order, 1 dBripple, Chebyshev high-pass filter.

TABLE 1

Butterworth Pole Locations

Order Real-a

Imaginary+/-jb

2 0.7071 0.7071

3 0.50001.0000

0.8660

4 0.92390.3827

0.38270.9239

5 0.80900.30901.0000

0.58780.9511

6 0.96590.70710.2588

0.25880.70710.9659

7 0.90100.62350.22251.0000

0.43390.78180.9749

8 0.98080.83150.55560.1951

0.19510.55560.83150.9808

9 0.93970.76600.50000.17371.0000

0.34200.64280.86600.9848

10 0.98770.8910

0.15640.4540

TABLE 2

Bessel Pole Locations

Order Real-a

Imaginary+/-jb

2 1.1030 0.6368

3 1.05091.3270

1.0025

4 1.35960.9877

0.40711.2476

5 1.38510.96061.5069

0.72011.4756

6 1.57351.38360.9318

0.32130.97271.6640

7 1.61301.37970.91041.6853

0.58961.19231.8375

8 1.76270.89551.37801.6419

0.27372.00441.39260.8253

9 1.80811.65321.36830.87881.8575

0.51261.03191.56852.1509


10/10


http://www sensorsmag com/articles/0202/filter/main shtml 10/03/2002

0.70710.45400.1564

0.70710.89100.9877

TABLE 3

0.01 dB Chebyshev Pole

LocationsOrder Real

-aImaginary

+/-jb

2 0.6743 0.7075

3 0.42330.8467

0.8663

4 0.67620.2801

0.38280.9241

5 0.51200.19560.6328

0.58790.9512

6 0.53350.39060.1430

0.25880.70720.9660

7 0.43930.30400.10850.4876

0.43390.78190.9750

8 0.42680.36180.24180.0849

0.19510.55560.83150.9808

9 0.36860.3005

0.19610.06810.3923

0.34200.6428

0.86610.9848

TABLE 4

0.1 dB Chebyshev Pole

LocationsOrder Real

-aImaginary

+/-jb

2 0.6104 0.7106

3 0.34900.6979

0.8684

4 0.21770.5257

0.92540.3833

5 0.38420.14680.4749

0.58840.9521

6 0.39160.28670.1049

0.25900.70770.9667

7 0.31780.22000.07850.3528

0.43410.78230.9755

8 0.30580.25920.17320.0608

0.19520.55580.83190.9812

9 0.26220.2137

0.13950.04850.2790

0.34210.6430

0.86630.9852

TABLE 5

0.25 dB Chebyshev PoleLocations

Order Real-a

Imaginary+/-jb

2 0.5621 0.7154

3 0.30620.6124

0.8712

4 0.45010.1865

0.38400.9272

5 0.32470.12400.4013

0.58920.9533

6 0.32840.24040.0880

0.25930.70830.9675

TABLE 6

0.5 dB Chebyshev PoleLocations

Order Real-a

Imaginary+/-jb

2 0.5129 0.7225

3 0.26830.5366

0.8753

4 0.38720.1605

0.38500.9297

5 0.27670.10570.3420

0.59020.9550

6 0.27840.20370.0746

0.25960.70910.9687

active filter - analog filter design

Documents