a49!3!4 jurisic08 low sensitivity single amplifier active rc allpole filters using tables

8/2/2019 A49!3!4 Jurisic08 Low Sensitivity Single Amplifier Active RC Allpole Filters Using Tables

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ISSN 00051144ATKAAF 49(34), 159173 (2008)

Dra`en Juri{i}, George S. Moschytz, Neven Mijat

Low-Sensitivity, Single-Amplifier, Active-RCAllpole Filters

Using Tables

UDK 621.372.57IFAC 4.3.2

Original scientific paper

A single-amplifier, active-RC filters design procedure for common filter types, such as Butterworth andChebyshev, using tables with normalized filter component values, is presented. The considered filters consistofRC ladder network in operational amplifiers positive feedback path. Tables with normalized component

values having equal capacitors and equal resistors were presented by some authors [1, 2]. In this paper, wepresented new tables for designing filters with optimized sensitivity to passive circuit components. A consid-erable improvement in sensitivity is achieved using the design technique called impedance tapering. Thepresented filters are up to the 6th-order. The sensitivity problem generally limits the use of higher than 6th--order single-amplifier filters. Low-pass and high-pass filters design is presented.

Key words: active-RC filters, minimal-RC filters, low power, low sensitivity, normalized components

1 INTRODUCTION

Passive filters are usually designed using hand-books with filter tables [35]. They are typicallyapplied for the design of passive ladder-RLC fil-ters. Besides filter transfer function parameters formany common filter types, handbooks usually con-tain tables with normalized filter component val-ues, as well [3, 4].

Active filters are commonly designed usingclosed-form design equations and/or filter-design

programs [6, 7]. In a majority of applications theyare realized as cascade structures of second-order(biquads) and/or third-order (bitriplets) sec-tions filters, and explicit formulas for calculation

of filter components, as well as, step by step de-sign procedures are available.

However, the high-order, single-amplifier, non--cascade filter designs have very often complicateddesign procedures and since there exists no explic-it formulas, numerical calculations have to be per-formed. Those filters are referred to as canonicalor minimal-RCbecause they have minimum num-

ber of components. In this paper we present filtertables with components for designing single-ampli-fier, active-RC filters up to the 6th-order, forButterworth and Chebyshev approximation types.

Based on the preliminary results from previous

paper [8], extended (final) results including practi-cal design examples as guidance for engineers are

presented. The need for design tables with normal-ized filter component values in the form of a hand-

book is approved. Compared to the filter design ta-bles for some common active-RC filters given in[1, 2], the tables in this paper are produced usingan optimization procedure for the low sensitivityto passive component tolerances.

In active-RC filter design, the most importantparameters for evaluation of a filters section qual-ity are: simple realizability, repeatability, a possi-

bility of straightforward procedure of parametercalculation, small number of components, low

power consumption, low noise performance and themost often low filters magnitude sensitivities to

passive component tolerances and/or active gainvariations. The filters, designed in this paper, poss-es minimum passive sensitivities for a given topol-ogy. The sensitivity is reduced using the designtechnique, called impedance tapering [9]. All cal-culations are performed numerically, including thecalculation of filters components, as well as, opti-mizations for filters low sensitivity. Schoefflerssensitivity measure is used as a basis for compari-son of sensitivity to component tolerances of thevarious filters. Monte Carlo runs are performed asa double check to the Butterworth and Chebyshev

filter examples.

AUTOMATIKA 49(2008) 34, 159173 159


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sented in [1] (and repeated in [2] p. 252). Tablespresented there, are calculated for some typical am-plifier gains ( = 2.0 and 2.2) and are not optimizedfor sensitivity.

2.1 Optimization of Sensitivity

In this paper we calculate component values ofthe filters that are optimized for minimum sensi-tivity to component tolerances. Component valuesare obtained numerically and shown in Tables 12.The low sensitivity performance is achieved byimpedance tapering design method (more pre-cisely by capacitive tapering), first introduced in[9], and applied to the LP filter structure in Fig. 1.At the same time a value of the so-called designfrequency 0=(R1C1)

1 [9] is optimized iterative-

ly to converge to the value which provides mini-mum sensitivity filter. Therefore, in Tables 12 weobtain various optimal values forR1 and the corre-sponding values of the gain . (Suppose we havechosen C1=1, thus for optimal 0, an optimal valueofR1=(0C1)

1 readily follows.) Optimal values forR1 are calculated using the procedure shown in Fig.2, which is implemented with program Mathemat-ica [11].

It should be stressed that the capacitive-taperingfactorC for higher-order filters is smaller, thanthat for filters of lower-ordern. For example, if we

try to find solution of the 6th-order Butterworth fil-ter with largerC (for example C3), it is not

possible, because the Newtons method does notconverge. Furthermore, larger capacitive taperingfactorC is not permitted since the last capacitorCn (e.g. C6=C1/C

5) becomes too small and com-parable to the parasitic capacitance of the circuit.Luckily, in high-order filters, even the small taper-ing factor satisfies enough our needs in degree ofdesensitization (e.g. C=2.0 is good enough for thefilter ordern =6).

In the sequel, we briefly explain the block dia-

gram shown in Fig. 2. The procedure in Fig. 2 usesNewtons iterative method to calculate the compo-nents of the high-order (i.e. 4th-, 5th- and 6th-order)allpole low-pass filters. In addition, the filters arecapacitively tapered and numerically optimized forlow sensitivity. Optimization of 2nd- and 3rd-orderfilters follows the same steps in the block diagramshown in Fig. 2, but theirs procedures are simplerin that the calculation of filter components can be

performed analytically.

The input data are the values of Butterworth orChebyshev coefficients ai (i =0,, n1), chosen

values ofC, and C1. Because of capacitive taper-ing, the procedure calculates Ck=C1/Ck1 (k=2,,

160 AUTOMATIKA 49(2008) 34, 159173

Low-Sensitivity, Single-Amplifer, Active-RCAllpole Filters Using Tables Dra`en Juri{i, George S. Moschytz, Neven Mijat

2 SINGLE-AMPLIFIER, MINIMAL-RC,

ACTIVE-RC FILTER

Consider an nth-order, allpole, single-amplifier,low-pass filter circuit with positive feedback pre-sented in Fig. 1. The filter belongs to the type ofclass-4 or Sallen and Key [2, 10]. The voltagetransfer function T(s) =V2(s) /V1(s) of the filter inFig. 1 is given by

(1)

where the pass-band gain K is given by

(2)KR

R

F

G

= = + 1

T sN s

D s

Ka

s a s a s a s an nn

ii

i

( ) =( )

( )=

=+ + + + + +

0

11

0

The coefficients ai (i = 0, , n1) as functions ofcomponents Rk and Ck (k= 1,, n) are presentedin Appendix I for the canonical filters up to the6th-order. There are many different ways how thosecoefficients can be calculated. In this paper, in Ap-

pendix I, we present one recursive way which iseasily programmed in any program that can, be-sides numeric, perform symbolic calculations (e.g.Mathematica, Matlab or MathCad).

Knowing coefficients ai in the transfer functions(1) denominator polynomial, we can develop a setof nonlinear equations by equating each of the co-efficients in the polynomial to the coefficient val-ues of the appropriate Butterworth or Chebyshev

polynomials. Butterworth and Chebyshev polyno-mials are readily obtained from tables [3, 6] or can

be calculated using closed-form equations. We can,finally, numerically solve a set of nonlinear equa-tions and calculate passive (normalized) compo-nents Rk, Ck (k= 1,, n) and to build the filtercircuit in Fig. 1.

The results of such computer calculation forequal capacitors and equal resistors cases are pre-

Fig. 1 General nth-order single-amplifier low-pass filter


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n). In the first cycle the initial value of the resistorR1 has to be defined. The resistorR1 is the designparameter to be adjusted. We choose C1=1, then byvarying the value ofR1 we indeed vary the valueof the design frequency 0=(R1C1)

1. With the

value ofR1 we solve the system of non-linear equa-tions for the vectorR2,, Rn and . To achieve asolution, we start with vector of random values for

R20,, Rn

0 and 0. Random initial resistors valuesRk

0 are within the interval and gain has a value in the range . If the properstarting vector is chosen, the Newtons methodwill, with prescribed accuracy, converge in severalsteps to the solution, i.e. to the vectorR2,, Rnand . If the method fails to converge, we must tryanother random starting vector. If the convergenceis achieved but we have solution with negative re-sistor values or gain less than unity, then we,

again, choose another random starting vector. Weperform random starting vectors for maximum

1000 times. This process of finding solution isknown as random search. If we choose startingvectors for Newtons iterative solving method byapplying some rule, which tries to find all possiblesolutions, we have exhaustive search (brute forcealgorithm). In solving some problems, if there existseveral local minimums, exhaustive search tries tofind global minimum, but only for a particular typeof problems.

If we do not find all real and positive compo-nent values R2,, Rn and gain 1, we proceedwith another value of resistorR1. Finally, when wefind realizable elements we proceed to the sensi-tivity optimization. To accomplish it we calculatemulti-parametrical statistical measure M, which isdefined in [12], with

(3)

where S2() is Schoefflers sensitivity function de-fined by

(4)

Function SFxi in (4) represents relative sensitivityof the function F (it is the transfer function mag-nitude, i.e. F=|T(j)| to the parameterx

i

(thereareN= 2n + 2 passive filter componentsxi in an nth--order filter). S2() in (3) is a function of frequen-cy, while M is a number instead of a function, andcan be used as a goal of optimizing process. Mrep-resents area under the function S2(), with bordersof integration from 1 to 2. Disadvantage is thedependence of numberM on the selected bordersof integration 11 and 2. Therefore, during wholeoptimization process, we choose the same pair of

1 and 2. The whole above procedure is repeatedwith new value forR1, until minimum value ofMis found.

For the 2nd-order filters we do not perform nu-merical optimization. Instead we apply capacitivetapering factors C in Tables 12 that must obeythe following constraint

(5)

The value max in (5) represents an upper boundof resistive tapering factorC. For equal resistors

C pq

R

R

R

R

=+

maks2

2

1

2

2

1

1

S SxF

i

N

i2

2

1

( ) = ( )=d

M S= ( ) 21

2

d

AUTOMATIKA 49(2008) 34, 159173 161

Dra en Juri{i, George S. Moschytz, Neven Mijat Low-Sensitivity, Single-Amplifer, Active-RCAllpole Filters Using Tables

Fig. 2 Block-diagram for solving capacitively-tapered 4th-,5th- and 6th-order filters and optimising design frequency 0

for minimum sensitivity


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(i.e. R1 =R2), we have minimum sensitivity of the2nd-order filters and another form of the constraint(5) given by

(6)

If we use maximum desensitization with C= 4q2p

in (6) we obtain in Tables 12 low-Q realizationof 2nd-order filters (having unity gain, i.e. =1)[6]. Furthermore, it is apparent from tables that for

the 3rd-order filter we have the values ofR2 andR3 close to each other. This corresponds to the con-clusions for the 3rd-order low-pass circuits withminimum sensitivity derived in [9]. For higher-than-third order filters no such rules exist for theminimum sensitivity design; instead the numericaloptimization should be carried out.

2.2 Sensitivity Analysis

In this section we compare sensitivities of newlydesigned capacitively-tapered filters to the filters

having equal capacitors and equal resistors.A sensitivity analysis was performed to the log-

arithmic gain function ()=20log|T(j)|/dB, as-suming the relative changes of the resistors and ca-

pacitors to be uncorrelated random variables, witha zero-mean Gaussian distribution and 1% standarddeviation. The standard deviation () /dB (whichis related to the Schoefflers sensitivities) of thevariation of the logarithmic gain =8.68588|T(j)|/|T(j)|/dB, with respect to the passiveelements, is calculated for the optimized elementvalues for Butterworth filters given in Table 1, as

well as for the equal capacitors and equal resistorsfilters designed using tables in [1, 2]. For all those

C R R pq =

=max 1 24 2

filters the standard deviations curves () are pre-sented in Fig. 3. Observing Fig. 3 we conclude thatthe capacitively impedance tapered filters haveminimum sensitivities to component tolerances ofthe circuits for all filter orders. Very close resultsare achieved with filter circuits, having equal re-sistors. One possible explanation is in the slightlytapered capacitor values, which can be seen inTable 1. The worst sensitivity performances havefilters with equal capacitors. Although it is usually

not practical to mass produce discrete componentactive-RC filters having unequal capacitors, in thecase of IC design various (tapered) capacitor val-ues are acceptable. The sensitivity curves in Fig. 3are repeated in Fig. 4, sorted by different designs.

Same investigations performed for Chebyshevfilter approximations lead to the same conclusions,

but they are not presented here.

It is well known that Butterworth filter comparedto a Chebyshev filter of equal order has lower poleQs. Since the sensitivities are related to the magni-

tudes of filter Q-factors, from the sensitivity point,a Butterworth filter is always preferable to aChebyshev filter and a low-ripple Chebyshev filteris always preferable to a Chebyshev filter withhigher ripple. Preferable filters with regard to lowsensitivity have lowest possible pole Qs. Further-more, filters with high order n have both larger

pole Qs and larger number of components thanlow-order filters and therefore much higher sensi-tivities. The reason for high sensitivities to compo-nent variations of high-order canonical filters is ob-vious from filter coefficients, presented as func-

tions of components, in Appendix I. Qualitativelythis can be explained by the fact that in a canonic

162 AUTOMATIKA 49(2008) 34, 159173


Table 1 Normalized components of min. sensitivity LP filters using capacitive tapering: Butterworth approximation

Table 2 Normalized components of min. sensitivity LP filters using capacitive tapering: 0.5 dB Chebyshev approximation

n C C1 C2 C3 C4 C5 C6 R1 R2 R3 R4 R5 R6

2

34

5

6

2

33

2.5

2

1

11

1

1

0.5

0.33330.3333

0.4

0.5

0.11110.1111

0.16

0.25

0.0370

0.064

0.125

0.0256

0.0625 0.03125

1.41421

1.090.7

2.29

0.675

1.41421

6.012557.07694

2.26474

8.63542

4.1198318.1004

8.21287

5.19291

8.13008

26.8796

8.97413

8.32969

23.3141 5.17416

1.0

1.142311.34647

1.5333

1.74047

n C C1 C2 C3 C4 C5 C6 R1 R2 R3 R4 R5 R6

2

3

4

5

6

2.9841

3

3

2.5

2

1

1

1

1

1

0.3351

0.3333

0.3333

0.4

0.5

0.1111

0.1111

0.16

0.25

0.0370

0.064

0.125

0.0256

0.0625 0.03125

1.40289

1.71

1.31

3.96

1.8

1.40289

6.5827

9.61917

4.86466

14.2659

3.35148

19.9327

11.4959

9.77697

7.65695

29.8471

11.7128

8.06382

23.8853 5.17416

1.0

1.31082

1.4989

1.66703

1.84611


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realization all the components interact with eachother, and a change in any component will be mag-nified through its repeated occurrence in all coeffi-cients. The larger order n of the filter, the morecomplicated and larger are the filter coefficientsand therefore the larger sensitivities. In cascade re-alization, in contrast, the change in a componentwill affect only a localized segment, i.e. only one

biquad in a cascade. Therefore, the cascade designhas lower sensitivities. But it has more amplifiers(consumes more power) and passive components(resistors) than single-amplifier filters.

As stated above, in the design of active-RC fil-ters, very important quality of the filter is its low

power consumption. Thus, for low-pass filters ofreasonably low order (for example n6), the useof single-amplifier filters, as in Fig. 1 is advanta-geous over the cascaded 2nd- and 3rd-order Sallenand Key sections [10], although the latter havesmaller sensitivities to component tolerances of thecircuit. Another advantage of canonical, single-am-

plifier filters, which are presented here, is that theyhave only n capacitors and n resistors as well, andtheir sensitivities are improved applying imped-ance tapering technique. Finally, advantages ofsingle-amplifier filters lie also in the fact that thereis no need for signal level optimization procedure

which must be performed in multiple amplifierconfigurations.

AUTOMATIKA 49(2008) 34, 159173 163


Fig. 3 Schoefflers sensitivity of normalized Butterworth low-pass filter circuits (up to 6th-order), with components given inTable 1 and in tables presented in [1] and [2] p. 252

Fig. 4 Schoefflers sensitivity of normalized Butterworth LP filter circuits in Fig. 3, sorted by type of impedance tapering


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An anti-aliasing low-pass filter has to satisfy thespecifications imposed by the tolerance schemeshown in Fig. 5. The requirements are the maxi-mum pass-band attenuation ofAp = 0.5dB for the

frequencies up to thefp= 20 kHz, and the minimumstop-band attenuation ofAs = 10 dB for the frequen-cies above fs = 32 kHz. The filter should have aunity gain in the pass band (K=1).

The specifications in Fig. 5 can readily be con-verted to the normalized LP specifications havingnormalized cut-off frequency Fs =fs /fp (or s==s/p, where s=2 Fs ). In that case we write1 instead fp and we write Fs instead fs in Fig. 5.In all subsequent design equations the ratio fs /fp isthen replaced by the single value Fs.

Butterworth transfer function magnitude has thefollowing form

3 TRANSFORMATIONS

3.1 LP-LP Transformation

Low-pass to low-pass (LP-LP) frequency trans-formation (i.e. denormalization) defined by

(7)

can be performed by new elements calculations.From normalized element values in Tables 12 wecalculate denormalized element values by applying

(8)

where index n denotes normalized components val-ues in tables. It is an advantage of normalized com-

ponent values in that they can be used as a tem-plate to directly construct the filter with any cut--off frequency. We do not consider inductances be-cause we deal with active-RC filters. Denormaliza-tion constants 0 is in [rad/s] and R0 in .

3.2 LP-HP Transformation

Low-pass to high-pass (LP-HP) frequency trans-

formation defined by

(9)

can be performed by new elements calculations.From normalized LP filter components in Tables12, HP filter components can be readily calculat-ed using (RC-CR transformation)

(10)

Note, that while applying LP-HP transformation(9) [i.e. while calculating HP filter componentsusing (10)]; capacitive taperingby a factorC atLP filters, transforms itself into resistive taperingfactor at HP filters. The value of gain remainsthe same as in the LP prototypes.

4 DESIGN OF LOWPASS FILTERS

Suppose we build mixed-signal-processing cir-

cuitry on the chip for the hearing aids. It consistsof continuous-time to discrete-time converter (C/D)

RR

CC R RHP

nLPHP nLP= = ( )

00 0

1,

ss

0

R R R CC

RLP nLP LP

nLP= =00 0

,

ss

0

which prepares the signal for the digital signal pro-cessing circuitry. Before C/D converter we have to

prepare voice signal using an integrated low-passfilter to suppress high frequency components and

eliminate aliasing. On the other hand the low-passfilter has to be as simple as possible (therefore werealize it using an active-RC filter), must have low

power consumption (it has a single amplifier), andmust be selective (it is a high order filter).

Because of relatively high filter order it musthave acceptably small sensitivity to component tol-erance to be realizable. For those reasons we de-cided to use the nth-order allpole low-pass filtercircuit presented in Fig. 1. Because, this filter israther complicated to calculate (especially for theorders higher than 3rd, where there are no closed

form equations) the Tables 12 with componentvalues shows to be of enormous help. In the de-sign process we will first decide which type of fil-ter (Butterworth or Chebyshev) best suites ourneeds.

164 AUTOMATIKA 49(2008) 34, 159173


Fig. 5 The attenuation specifications for the LP filter


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(11)

The filter order n for the Butterworth filter canbe readily calculated

(12)

We choose next larger integer value for n. Thecut-off frequency 0 is given by

(13)

where the radian frequency p=2 fp. The cut-offfrequency 0 is the frequency on which the filtersmagnitude has the 3 dB attenuation. Note that forthe calculation of the Butterworth cut-off frequen-cy 0, we use the lower frequency p, i.e. the pass-

band edge in specifications, because the suppress-ing of the high frequency components in the anti-aliasing filter is then guaranteed. The Table 1 pro-vides a set of necessary component values for thedesigning required Butterworth filter.

Chebyshev transfer function magnitude has thefollowing form

(14)

where the constant 0 < 1 describes the pass-band ripple and Tn() an n

th-order Chebyshev poly-nomial. The constant readily follows from mini-mum pass-band attenuation (i.e. pass-band ripple)

Ap given in dB

(15)

The filter order n for the Chebyshev filter canbe calculated using

= eAp

10 1

ALossn

( ) = +

dB20 1

0

2

log

A TLoss n

( )[ ] = +

dB 20 1 2 2

0

log

0

102

10 1

=

p

An

p

n

A

A

s

p

1

2

10 1

10 1

10

10

log

llogf

f

s

p

AUTOMATIKA 49(2008) 34, 159173 165


(16)

The cut-off frequency 0 in (14) is the highestfrequency on which the filters attenuation reachesthe maximum value in the pass-band (i.e. Ap). Inthe Chebyshev case we choose the cut-off frequen-cy to be

0=p (17)

where the radian frequency p=2 fp. Note thesimilarity between (16) and the Butterworth formu-la (12). The Table 2 provides a set of necessarycomponent values for the designing requiredChebyshev filter.

If the cut-off frequency in the Chebyshev filtersdesign is required to be the frequency at which themagnitude reaches 3dB value (not the pass-bandripple valueAp) then we have to calculate the valueof0 in (14) another way using

(18)

where

(19)

The value is given by (15). In this case wealso use normalized component values of the LPlow-sensitivity Chebyshev filter in Table 2. Thiscase will not be considered in this paper, insteadwe will design Chebyshev filters having cut-off fre-

quency at the pass-band ripple value Ap /dB.

4.1 Example

Using equations (11) to (19) we can readily cal-culate the filter order n and the cut-off frequency

0 for the design of the Butterworth or Chebyshevfilters. Indeed those calculations are the only thatwe need. The filter design then proceeds using ta-

bles and is straightforward.

As descriptive examples we present the designsof filters that satisfy our specifications in Fig. 5: i)

Butterworth, ii) Chebyshev with pass-band rippleRp = 0.2 dB and iii) Chebyshev with Rp = 0.5 dB.

n f

f

A

A

s

p

s

p

Arch

Arch

10 1

10 1

10

10

=

3dB Cosh1

nArCosh

1

0 3=

p

dB


8/15

Using (12) and (16) we calculate required filterorder n. Thus we have for the three possible real-izations the following filter orders and cut-off fre-quencies: i) Butterworth n = 5, 0 =155084 rad/s;

ii) Chebyshev n = 4, 0 =125664 rad/s; and iii)Chebyshev n = 3, 0 =125664 rad/s. Note that theordern of the Chebyshev filters is smaller than theorder of the Butterworth filter. Also note that theChebyshev filter with higher ripple requires lowerfilter order. Consequently, in what follows we real-ize the Butterworth filter and the Chebyshev filterwith 0.5 dB ripple.

Using normalized predefined component valuesfrom above design tables, we proceed with the de-normalization procedure. To calculate filter compo-nents we need to choose denormalizing resistorR0

value and the denormalization frequency 0.Generally speaking 0 and R0 are free constants.But, in our case, the frequency 0 is already knownand follow from filter specifications (e.g. it is

0 =155084 rad/s for the Butterworth filter exam-ple). The resistor value R0 could be chosen to ob-tain, for example, desired total capacitance valueCTOT. The latter constraint arises because we in-tend to realize the filter circuit on the chip, thuswe are limited with maximum allowable capaci-tance still realizable on the chip.

Suppose that we want to obtain a total capaci-

tance CTOT= 300 pF, we have to calculate denor-malization resistance R0. For the filters with ta-

pered capacitors by a factorC of the order n thetotal capacitance equals

(20)

For the 5th-order Butterworth filter from Table 1and using (20) we obtain normalized valueCTOT= C1.1.6496 and therefore maximum value for

capacitor C1 should be C1=300pF/1.6496==181.862 pF. Starting from this we calculate mini-mum denormalization resistance R0

(21)

We can choose for example reference resistorR0 =36 k thus we have R0R0min. We calculatereference capacitor value C0=1/(0R0)=179.114 pF

and according to (8) the circuit capacitor is thenC=C0.CnLP and the circuit resistor is R =R0 .RnLP,

RC

00 1

1 1

155084 181 86

35 4561

min.

.

= =

=

= pF

k

C C C C TOT ii

n

Ci

i

nC

n

C

= = = =

=

1

10

1

1 1

1

1

166 AUTOMATIKA 49(2008) 34, 159173


where CnLP and RnLP represent normalized valuesfrom Tables 12. The fifth capacitorC5 has a valueof C5=C0 . C5nLP=4.5853 pF. This is very smallvalue and it is dangerously near to the order of the

parasitic capacitance in the circuit. We chooseRG=10 k and calculate

(22)

Figure 6a shows the normalized network withthe reference values 0 = 155084 rad/s and

R0 =36k (all normalized values follow directlyfrom the 5th row in Table 1). Figure 6b shows thedenormalized network with resistor values in kand capacitors in pF. A simple first-order check forthe correctness of these results is to verify that

a0=(R1R2R3R4R5C1C2C3C4C5)1

.Now we can continue with the next example of

the 3rd-order Cheby-0.5 dB filter. Using the samedesign steps as above we choose R0 =39k. This

provides that the total capacitance is less than 300pF. The reference capacitor value is C0=1/(0R0) == 204.044 pF.

We choose RG =10 k and calculate RF= 3.1082k. Note that the third capacitorC3 has a value ofC3 = 22.67 pF, which is, in this case, quite largerthan the parasitic capacitance in the integrated cir-

cuit.

R RF G= ( ) = 1 5 333. k

Fig. 6 Butterworth LP filters (n=5). a) Normalizedelements. b) Denormalized elements


9/15

Figure 7a shows the normalized network withthe reference values 0 =125664 rad/s and R0=39

k, whereas Figure 7b shows the denormalizednetwork.

In what follows for both denormalized filtersshown in Fig. 6b and Fig. 7b, magnitudes areshown in Fig. 8 to demonstrate their functionalityand the satisfaction of filter specifications. Noticethat magnitudes in Fig. 8 have unity pass-band gain

(K=1, i.e. 0 dB). The realization of the desiredpass-band gain value K will be presented in thesection 4.2, below. As a double check of filter sen-sitivities, Monte Carlo runs (using PSPICE simula-tion) are performed and shown in Fig. 9.

Note that Chebyshev filter although has 0.5 dBripple in the pass-band and larger pole Q factors,

shows lower sensitivity than the other realization,a higher order Butterworth filter. This is becauselower filter order has substantially lower sensitivi-ty, especially when filters are realized using singleopamp circuit as in Fig. 1. Therefore our choicewill be the Chebyshev filter realization.

Recall that the filters presented in design tablesabove are capacitively tapered and optimized forlow sensitivity.

4.2 The Gain Factor K

The DC forward gain K of the filter transferfunction will generally not coincide with the am-

AUTOMATIKA 49(2008) 34, 159173 167


Fig. 7 Chebyshev-05 dB LP filters (n=3). a) Normalizedelements. b) Denormalized elements

Fig. 9 MC runs (a) Butterworth (n=5) and (b) Chebyshev0.5 dB (n=3) filters

Fig. 8 Butterworth and Chebyshev LP filters transferfunction magnitudes


10/15

5 DESIGN OF HIGHPASS FILTERS

In this section we present the design of high--pass (HP) filters starting from given specificationsand using the same tables as in the low-pass filter(LP) case presented above. The filters are designedin the same straightforward way and final filter cir-cuits are optimal in the sense that they have mini-mum sensitivity to passive components variationsof the circuit, and low power consumption.

Consider an example of HP filter which satisfiesthe specifications shown in Fig. 11. The specifica-tion requires the maximum pass-band attenuation of

Ap = 0.5 dB for the frequencies above fp =32kHzand the minimum stop-band attenuation of

As =10 dB for the frequencies below fs =20kHz.

The filter should have a unity gain in the pass band(K=1).

plifier gain , required to obtain filter transfer func-tion parameters. The gain factorK may be speci-fied by the filter designer, but the amplifier gain isdetermined by the values for given in design ta-

bles. Recall that all values in above tables includ-ing are obtained using optimization procedure forlow sensitivity filter. Thus, the value of cannot

be freely chosen; it depends on the design equa-tions for the filter, whereas the overall DC filtergain K may very likely be required to have a dif-ferent value. Fortunately, there are various schemesfor the decoupling ofK and [9], one of whichwill be presented in what follows. In terms of ourfilter, this implies that

(23)

If the desired value of is less than unity, i.e. >K (and because 1), then a resistive voltagedivider can be inserted at the input of the network,as shown in Fig. 10. The gain decoupling was ap-

plied to the 3rd-order Chebyshev filter with 0.5 dBshown in Fig. 7b. In this case input resistorR1 wassubstituted by voltage divider (by factor) con-sisting of resistors R1 and R1. We have

(24)

i.e. R1=R1/ and R1=R1/(1). Since is, in thiscase, less than unity, R1 is always positive. In ourexample to realize K= 1 we have =K/ ==0.7629, and R1=R1/=87.419 k; R1=281.251k.

=K

=

+

=

+

R

R R

RR R

R R

1

1 11

1 1

1 1

and

The input of the LP filter with K and decou-pled is shown in Fig. 10 when K will not be shown in this paper andis presented in [9].

168 AUTOMATIKA 49(2008) 34, 159173


Fig. 10 Final Chebyshev LP filter circuit with unity gain inpass-bad (K=1). (Rs in k and Cs in pF)

Fig. 11 The attenuation specifications for the HP filter

When we include specifications values into (12)and (16) we readily calculate required filter ordern. Recall that the HP specifications can be trans-formed to the normalized LP specifications withthe normalized cut-off frequency Fs =fp /fs. In bothLP and HP filter examples we obtain the same or-ders n because both specifications have the samerequirements, i.e. we have the same frequency Fs.

In what follows we are going to design a HP fil-ter of the 3rd-order with Chebyshev approximation0.5 dB ripple, which is dual to the filter circuit inFig. 7b.

To design HP filter we should calculate capaci-tors values as reciprocals of resistors values inTables 12 and vice versa, resistors values are re-ciprocals of capacitors values (RC-CR transforma-tion), which is accomplished by (10). In (10) thedenormalization of components values is perform-ed, as well. The denormalization is performed to

the cut-off frequency fp =32 kHz or 0=2fp =201061.9 rad/s. To properly select the denor-


11/15

Note that in (26)RnHP=1/CnLPand CnHP=1/RnLP.Fig. 12a shows the normalized network with the

0=201061.9 rad/s and R0=18k, whereas the de-normalized network is shown in Fig. 12b. It is ob-

vious from Fig. 12a that the circuit is resistivelytapered by the factor 3 (as in the LP case, wherethe capacitors are scaled by the factorC= 3). Theresistors RG and RF in the amplifier feedback real-ize the same gain as in the LP case.

To realize the gain K= 1 we substitute an inputcapacitor C1 by voltage divider (by factor) con-sisting of capacitors C1 and C1. We have

(27)

i.e. C1=C1/(1) and C1=C1/. In our example wehave =K/=0.7629, thus the input capacitor C1

=

+ = +

C

C CC C C1

1 11 1 1and

AUTOMATIKA 49(2008) 34, 159173 169


malization resistance R0 we must be able to realizecapacitors on the chip.

Therefore we calculate (with CTOT= 300 pF)

(25)

We choose reference resistor value R0 =18 kand this provides reference capacitor value C0==1/(0R0)=276.311 pF. The circuit capacitor isthen C= C0 . CnHP and the circuit resistor is R ==R0 .RnHP, where index n denotes normalized val-

ues. Thus we have the following values for com-ponents in HP filter

(26)

RR

CR R

R R

CR

HPnLp

nHP

HP HP

HPnLp

10

10 1

2 3

11

18

54 162

1

= = =

= =

=

k

k k

; ;

0 00 1

2 3

161 6

41 98 82 44

RC C

C C

nHP

HP HP

= =

= =

.

. .

pF;

pF; pF

RC

C

C

in

TOT0

0 1 0

1

1 03508

201062 30017160

min

.

= = =

=

=

pF

Fig. 13 Chebyshev HP filter transfer function magnitude

Fig. 14 MC runs of Chebyshev HP filterFig. 12 Chebyshev HP filters. a) Normalized elements. b)

Denormalized elements (Rs in k and Cs in pF)


12/15

is split into C1=681.45 pF and C1=211.81 pF (C1is connected to the input signal generator). Corres-

ponding magnitudes and MC runs are shown inFig. 13 and Fig. 14, respectively. The obtained HP

filter circuit in Fig. 12b satisfies the specificationsin Fig. 11.

Comparing MC runs in Fig. 14 (for the HP 0.5dB Chebyshev filter) to those in Fig. 9b for its dualcounterpart LP filter (Fig. 7b), both satisfying thesame LP prototype specifications, we can concludethat both of these filters have identical sensitivi-ties. This is because they are designed starting fromthe same optimized component values in the sec-ond row (for n = 3) in Table 2.

6 CONCLUSIONS

A procedure for the design of allpole low-sensi-tivity, low-power, active-RC filters using tableswith predefined normalized filter component val-ues for some common filter types (Butterworth andChebyshev 0.5 dB) is presented. The filter usesonly one operational amplifier, and a minimumnumber of passive components. The amplifier it-self ensures realization of conjugate-complex filter

poles, and low output impedance. For reasons re-lated to the filter topology, the application of thecapacitive impedance tapering has improved thesensitivity of the low-pass filters magnitude tocomponent tolerances [9]. The proposed design isuniversal and straightforward by using design ta-

bles; thus there is no need for numeric calculations.It can be extended to the design of single-amplifi-er, low-sensitivity high-pass filters, as well. Beca-use, the high-pass filters are dual to the low-passfilters, resistive tapering is applied to reduce thesensitivity of the high-pass filter.

Furthermore, the reduction in power and com-ponent count achieved with the single-amplifier LP

filters is obtained at a price: a cascade of imped-ance-tapered second-order and/or third-order sec-tions has lower sensitivity than impedance-taperedsingle-amplifier filter. Thus the decision on whichway to go is typically one of tradeoffs: low powerand component count versus low sensitivity. In ourexample, of hearing aid circuit realization, we pre-ferred former solution having low power and lowcomponent count and acceptably low sensitivity.

APPENDIX I: COEFFICIENTS OF T(s)

Consider the nth-order allpole low-pass filter cir-cuit with positive feedback presented in Fig. A.1.

Note the descending notation ofRn, Cn to R1, C1from the driving source to the amplifier input. Thisreverse order notation is convenient to develop re-cursive formulas for determining transfer functioncoefficients d

1to d

nin eq. (A.2) of the nth-order

polynomial Dn(s) as functions of resistors Ri, ca-pacitors Ci and gain .

Recursive formulas follow from characteristicsof the continuants. The continuants are used tosolve ladder-networks, and they can be calculatedrecursively. The filter in Fig. A.1 has a ladder net-work in the amplifiers positive feedback loop withgain , where gain = 1+RF/RG represents the gainin the class-4 filter circuit. The transfer function ofthe nth-order filter as presented in the Fig. A.1 hasthe form given by

(A.1)

As shown in [1] and [2] p. 252, the nth-order de-nominator polynomial in transfer function (A.1),i.e.

(A.2)

can be calculated from polynomials of n 1 and

n2 order:

(A.3)

and

(A.4)

using

(A.5)

where 1j = 0, forj1; and 1j = 1, forj=1 where1 j n. Note that b0=c0=d0=1. Note also that,for the start of the recursive process polynomials

D0=1 and D1=R1C1s+1.

To perform the symbolic calculations (to calcu-late coefficients in terms of components) we usesymbolic calculation program Mathematica. At the

end of recursive process we change to the ascend-ing notation, i.e. we substitute RnR1, CnC1,

( ) = +=D s d sn j jj

n

11

T sD sn

( ) = ( )

d c bR

Rc

c R C R C

j j jn

nj

j n n j

n

n n

= ( ) + +

+ ( ) +

1

1 11 1

2

( ) = +=

D s b sn j jj

n

21

2

1

( ) = +=

D s c sn j jj

n

11

1

1

170 AUTOMATIKA 49(2008) 34, 159173



13/15

Rn1R2, Cn1C2,, R1Rn, C1Cn, resultingin the filter circuit with notation shown in Fig. 1.Consequently, we perform multiplication of nomi-nator and denominator by the same factor

(A.6a)

(A.6b)

We obtain the form of the transfer function ofthe nth-order filter given by (note the unity coeffi-cient with the highest, i.e. nth power ofs)

(A.7)

The transfer function (A.7) has the same formas the transfer function (1). In what follows we

present the transfer functions coefficients up toreasonable (6th) order obtained by the recursive

procedure shown above. Those coefficients corre-spond to the filter circuit with ascending notationshown in Fig. 1, and are used in this paper for all

numerical calculations carried out by Mathematica.It can be seen that increasing the filter order n, co-efficients become even more complicated, andtherefore the canonical high-order filters sensitivi-ty rapidly increases.

2nd Order

a1= (C1 R1 + C2 R1 + C2 R2 C1 R1 ) / (C1C2 R1 R2)

a0= 1 / (C1 C2 R1 R2)

3rd Order

a2= (C1 C2 R1 R2 + C1 C3 R1 R2 + C1 C3 R1

R3 + C2 C3 R1 R3 + C2 C3 R2 R3 C1 C2R1 R2 ) / (C1 C2 C3 R1 R2 R3)

ad

dj nj

j

n

= , 0

N sN s

d a D sD s

dn n( ) =

( )= ( ) =

( ) 0 ,

T sN s

D sa

s a s a s a s an nn

ii

( ) =( )

( )

=

=+ + + + + +

0

11

1 0

a1= (C1 R1 + C2 R1 + C3 R1 + C2 R2 + C3 R2+ C3 R3 C2 R1 C2 R2 ) / (C1 C2 C3R1 R2 R3)

a0= 1 / (C1 C2 C3 R1 R2 R3)

4th Ordera3= (C2 C3 C4 (R1 + R2) R3 R4 + C1 R1 (C3 C4

(R2 + R3) R4 + C2 R2 (C4 (R3 + R4) + C3(R3 R3 )))) / (C1 C2 C3 C4 R1 R2 R3 R4)

a2= (C1 C2 R1 R2 + C4 (C3 (R1 + R2 + R3) R4+ C2 (R1 + R2) (R3 + R4) + C1 R1 (R2 + R3+ R4)) C3 (C2 (R1 + R2) R3 + C1 R1 (R2+ R3)) (1 + )) / (C1 C2 C3 C4 R1 R2 R3R4)

a1= (C1 R1 + C2 R1 + C2 R2 + C4 (R1 + R2 +R3 + R4) C3 (R1 + R2 + R3) (1 + ) C1R1 ) / (C1 C2 C3 C4 R1 R2 R3 R4)

a0= 1 / (C1 C2 C3 C4 R1 R2 R3 R4)

5th Order

a4= (C2 C3 C4 C5 (R1 + R2) R3 R4 R5 + C1 R1(C3 C4 C5 (R2 + R3) R4 R5 + C2 R2 (C4 C5(R3 + R4) R5 + C3 R3 (C5 (R4 + R5) + C4(R4 R4 ))))) / (C1 C2 C3 C4 C5 R1 R2 R3R4 R5)

a3= (C1 C2 C3 R1 R2 R3 + C5 (C4 (C3 (R1 + R2+ R3) R4 + C2 (R1 + R2) (R3 + R4) + C1 R1(R2 + R3 + R4)) R5 + C3 (C2 (R1 + R2) R3+ C1 R1 (R2 + R3)) (R4 + R5) + C1 C2 R1

R2 (R3 + R4 + R5)) C4 (C2 C3 (R1 + R2)R3 R4 + C1 R1 (C3 (R2 + R3) R4 + C2 R2(R3 + R4))) (1 + )) / (C1 C2 C3 C4 C5 R1R2 R3 R4 R5)

a2= (C1 C2 R1 R2 + C1 C3 R1 R2 + C1 C3 R1R3 + C2 C3 R1 R3 + C2 C3 R2 R3 + C5 (C4(R1 + R2 + R3 + R4) R5 + C3 (R1 + R2 +R3) (R4 + R5) + C2 R2 (R3 + R4 + R5) + R1(C1 R2 + (C1 + C2) (R3 + R4 + R5))) C4(C3 (R1 + R2 + R3) R4 + C2 (R1 + R2) (R3+ R4) + C1 R1 (R2 + R3 + R4)) (1 + ) C1 C2 R1 R2 ) / (C1 C2 C3 C4 C5 R1 R2R3 R4 R5)

a1= (C1 R1 + C2 R1 + C3 R1 + C2 R2 + C3 R2+ C3 R3 + C5 (R1 + R2 + R3 + R4 + R5) C4 (R1 + R2 + R3 + R4) (1 + ) C2 (R1+ R2) ) / (C1 C2 C3 C4 C5 R1 R2 R3 R4R5)

a0= 1 / (C1 C2 C3 C4 C5 R1 R2 R3 R4 R5)

6th Order

a5= (C2 C3 C4 C5 C6 (R1 + R2) R3 R4 R5 R6 +C1 R1 (C3 C4 C5 C6 (R2 + R3) R4 R5 R6 +C2 R2 (C4 C5 C6 (R3 + R4) R5 R6 + C3 R3(C5 C6 (R4 + R5) R6 + C4 R4 (C6 (R5 + R6)

+ C5 (R5 R5 )))))) / (C1 C2 C3 C4 C5 C6R1 R2 R3 R4 R5 R6)

AUTOMATIKA 49(2008) 34, 159173 171


Fig. A1 General nth-order single-amplifier low-pass filterwith reverse notation


14/15

a4= (C1 C2 C3 C4 R1 R2 R3 R4 + C6 (C5 (C4(C3 (R1 + R2 + R3) R4 + C2 (R1 + R2) (R3+ R4) + C1 R1 (R2 + R3 + R4)) R5 + C3 (C2(R1 + R2) R3 + C1 R1 (R2 + R3)) (R4 + R5)+ C1 C2 R1 R2 (R3 + R4 + R5)) R6 + C4(C2 C3 (R1 + R2) R3 R4 + C1 R1 (C3 (R2 +R3) R4 + C2 R2 (R3 + R4))) (R5 + R6) + C1C2 C3 R1 R2 R3 (R4 + R5 + R6)) C5 (C2C3 C4 (R1 + R2) R3 R4 R5 + C1 R1 (C3 C4(R2 + R3) R4 R5 + C2 R2 (C4 (R3 + R4) R5+ C3 R3 (R4 + R5)))) (1 + )) / (C1 C2 C3C4 C5 C6 R1 R2 R3 R4 R5 R6)

a3= (C1 C2 C3 R1 R2 R3 + C1 C2 C4 R1 R2 R3+ C1 C2 C4 R1 R2 R4 + C1 C3 C4 R1 R2R4 + C1 C3 C4 R1 R3 R4 + C2 C3 C4 R1R3 R4 + C2 C3 C4 R2 R3 R4 + C6 (C5 (C4(R1 + R2 + R3 + R4) R5 + C3 (R1 + R2 +

R3) (R4 + R5) + C2 R2 (R3 + R4 + R5) + R1(C1 R2 + (C1 + C2) (R3 + R4 + R5))) R6 +C4 (C3 (R1 + R2 + R3) R4 + C2 (R1 + R2)(R3 + R4) + C1 R1 (R2 + R3 + R4)) (R5 +R6) + C2 C3 R2 R3 (R4 + R5 + R6) + R1(C3 (C2 R3 + C1 (R2 + R3)) (R4 + R5 + R6)+ C1 C2 R2 (R3 + R4 + R5 + R6))) C5 (C4(C3 (R1 + R2 + R3) R4 + C2 (R1 + R2) (R3+ R4) + C1 R1 (R2 + R3 + R4)) R5 + C3 (C2(R1 + R2) R3 + C1 R1 (R2 + R3)) (R4 + R5)+ C1 C2 R1 R2 (R3 + R4 + R5)) (1 + ) C1 C2 C3 R1 R2 R3 ) / (C1 C2 C3 C4 C5

C6 R1 R2 R3 R4 R5 R6)a2= (C1 C2 R1 R2 + C1 C3 R1 R2 + C1 C4 R1

R2 + C1 C3 R1 R3 + C2 C3 R1 R3 + C1 C4R1 R3 + C2 C4 R1 R3 + C2 C3 R2 R3 + C2C4 R2 R3 + C1 C4 R1 R4 + C2 C4 R1 R4 +C3 C4 R1 R4 + C2 C4 R2 R4 + C3 C4 R2R4 + C3 C4 R3 R4 + C6 (C2 R2 R3 + C2 R2R4 + C3 R2 R4 + C3 R3 R4 + C2 R2 R5 +C3 R2 R5 + C3 R3 R5 + C2 R2 R6 + C3 R2R6 + C3 R3 R6 + C5 (R1 + R2 + R3 + R4 +R5) R6 + C4 (R1 + R2 + R3 + R4) (R5 + R6)+ R1 (C1 R2 + C3 (R4 + R5 + R6) + (C1 +

C2) (R3 + R4 + R5 + R6))) C5 (C4 (R1 +R2 + R3 + R4) R5 + C3 (R1 + R2 + R3) (R4+ R5) + C2 R2 (R3 + R4 + R5) + R1 (C1 R2+ (C1 + C2) (R3 + R4 + R5))) (1 + ) C3(C2 (R1 + R2) R3 + C1 R1 (R2 + R3)) ) /(C1 C2 C3 C4 C5 C6 R1 R2 R3 R4 R5 R6)

a1= (C1 R1 + C2 R1 + C3 R1 + C4 R1 + C2 R2+ C3 R2 + C4 R2 + C3 R3 + C4 R3 + C4 R4

+ C6 (R1 + R2 + R3 + R4 + R5 + R6) C5(R1 + R2 + R3 + R4 + R5) (1 + ) (C1R1 + C3 (R1 + R2 + R3)) ) / (C1 C2 C3 C4C5 C6 R1 R2 R3 R4 R5 R6)

a0= 1 / (C1 C2 C3 C4 C5 C6 R1 R2 R3 R4 R5

R6)

REFERENCES

[1] R. S. Aikens and W. J. Kerwin, Single amplifier, min-imal RC, Butterworth, Thompson and Chebyshevfilters to 6th order, Proc. of Intern. Filt. Symposium,Santa Monica, California, April 1972, 8182.

[2] G. S. Moschytz, Linear integrated networks: design(Bell Laboratories Series, New York: Van NostradReinhold Co., 1975).

[3] A. I. Zverev, Handbook of Filter Synthesis, JohnWiley and Sons, New York 1967.

[4] R. Saal, Handbook of Filter Design, AEG-Tele-funken, Berlin, 1979.

[5] E. Christian and E. Eisenmann, Filter Design Tablesand Graphs, John Wiley and Sons, New York, 1966.

[6] G. S. Moschytz and P. Horn, Active Filter DesignHandbook, John Wiley and Sons, Chichester, 1981.(IBM Program Disk: ISBN 0471-915 43 2)

[7] M. Biey and A. Premoli, Tables for Active FilterDesign, Artec House, Massachusetts, 1985.

[8] D. Juri{i}, G. S. Moschytz and N. Mijat, SingleAmplifier, Active-RC, Butterworth, and ChebyshevFilters Using Impedance Tapering Proc. of theSPPRA 2002, (Crete, Greece), June 2528. 2002, pp.461465.

[9] G. S. Moschytz, Low-Sensitivity, Low-Power,Active-RC Allpole Filters Using ImpedanceTapering, IEEE Trans. on Circuits and Systems, vol.CAS-46(8), Aug. 1999, 10091026.

[10] R. P. Sallen and E. L. Key, A practical Method ofDesigning RC Active Filters, IRE Trans. on CircuitTheory, vol. CT-2, 1955, 7885.

[11] S. Wolfram, The MATHEMATICA Book, (WolframMedia, IL-USA and Cambridge University Press,Cambridge, UK, 1999).

[12] K. R. Laker and M. S. Ghausi, Statistical Multipa-rameter Sensitivity A Valuable Measure for CAD,Proc. of the IEEE Int. Symp. on Circuits and Systems,ISCAS 1975, April 1975., 333336.

172 AUTOMATIKA 49(2008) 34, 159173


Projektiranje aktivnih RC filtara niske osjetljivosti s jednim poja~alom pomo}u tablica. Prikazan jepostupak projektiranja aktivnih RC filtara s jednim poja~alom pomo}u tablica s normaliziranim filtarskimkomponentama za poznate tipove aproksimacija kao {to su Butterworth i Chebyshev. Filtri su gra|eni od RC

ljestvi~aste mre`e u pozitivnoj povratnoj vezi poja~ala. Tablice s normaliziranim vrijednostima elemenata sjednakim kapacitetima i s jednakim otporima su ve} objavljene od strane raznih autora [1, 2]. U ovom radu,


15/15

AUTOMATIKA 49(2008) 34, 159173 173


AUTHORS ADDRESSES

Dra`en Juri{i}Faculty of Electrical Engineering and Computing,Unska 3, HR-10000, Zagreb, Croatia

George S. MoschytzSwiss Federal Institute of Technology, (ETHZ-ISI)Sternwartstr. 7, CH-8092, Zrich, Switzerland

Neven Mijat

Faculty of Electrical Engineering and Computing,Unska 3, HR-10000, Zagreb, Croatia

Received: 2008-06-29Acceptved: 2008-10-15

prikazujemo nove tablice za projektiranje filtara koji imaju optimalnu osjetljivost na varijacije pasivnih kom-ponenata. Zna~ajno smanjenje osjetljivosti posti`e se posebnom tehnikom projektiranja pomo}u skaliranjaimpedancija. Prikazani filtri su do uklju~uju}i {estog reda. Problem osjetljivosti ne dopu{ta da red promatra-nih filtara bude ve}i od {estog reda. Prikazano je projektiranje niskopropusnih i visokopropusnih tipova fil-tara.

Klju~ne rije~i: aktivniRC filtri, minimalni RC filtri, mala potro{nja, niska osjetljivost, normalizirane vrijed-nosti komponenata

a49!3!4 jurisic08 low sensitivity single amplifier active rc allpole filters using tables

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