07.Circuits and Systems - Analog and Digital Signal Processing

1121 Pages·1999·31.46 MB·English

by  John G. Webster (Editor)

07• Circuits and Systems: Analog and Digital Signal Processing
07• Circuits and Systems: Analog and Digital Signal Processing

Active Filters Abstract | Full Text: PDF (248K)
Adaptive Filters Abstract | Full Text: PDF (108K)
All-Pass Filters Abstract | Full Text: PDF (1662K)
Analog Computer Circuits Abstract | Full Text: PDF (237K)
Analog Processing Circuits Abstract | Full Text: PDF (306K)
Analog-to-Digital Conversion Abstract | Full Text: PDF (314K)
Asynchronous Circuits Abstract | Full Text: PDF (146K)
Asynchronous Sequential Logic Abstract | Full Text: PDF (192K)
Attenuators Abstract | Full Text: PDF (2793K)
Bandpass Filters Abstract | Full Text: PDF (114K)
Band-Stop Filters Abstract | Full Text: PDF (88K)
Bipolar and Mos Logic Circuits Abstract | Full Text: PDF (152K)
Bootstrap Circuits Abstract | Full Text: PDF (86K)
Bridge Circuits Abstract | Full Text: PDF (323K)
Butterworth Filters Abstract | Full Text: PDF (123K)
Cascade Networks Abstract | Full Text: PDF (452K)
Cellular Arrays Abstract | Full Text: PDF (528K)
Chebyshev Filters Abstract | Full Text: PDF (162K)
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07• Circuits and Systems: Analog and Digital Signal Processing
Circuit Stability Abstract | Full Text: PDF (188K)
Circuit Tuning Abstract | Full Text: PDF (243K)
Combinational Circuits Abstract | Full Text: PDF (159K)
Comparator Circuits Abstract | Full Text: PDF (416K)
Counting Circuits Abstract | Full Text: PDF (84K)
Current Conveyors Abstract | Full Text: PDF (158K)
Data Acquisition and Conversion Abstract | Full Text: PDF (226K)
DC Amplifiers Abstract | Full Text: PDF (203K)
Delay Circuits Abstract | Full Text: PDF (217K)
Differential Amplifiers Abstract | Full Text: PDF (258K)
Differentiating Circuits Abstract | Full Text: PDF (113K)
Digital Filters Abstract | Full Text: PDF (711K)
Digital Filter Synthesis Abstract | Full Text: PDF (128K)
Digital-To-Analog Conversion Abstract | Full Text: PDF (290K)
Diode-Transistor Logic Abstract | Full Text: PDF (145K)
Discrete Event Systems Abstract | Full Text: PDF (330K)
Discrete Time Filters Abstract | Full Text: PDF (220K)
Distributed Amplifiers Abstract | Full Text: PDF (243K)
Elliptic Filters Abstract | Full Text: PDF (143K)
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07• Circuits and Systems: Analog and Digital Signal Processing
Feedback Amplifiers Abstract | Full Text: PDF (1023K)
Fir Filters, Design Abstract | Full Text: PDF (243K)
Frequency Synthesizers Abstract | Full Text: PDF (111K)
Harmonic Oscillators, Circuits Abstract | Full Text: PDF (197K)
Hysteresis in Circuits Abstract | Full Text: PDF (183K)
IIR Filters Abstract | Full Text: PDF (591K)
Integrated Circuits Abstract | Full Text: PDF (278K)
Integrating Circuits Abstract | Full Text: PDF (104K)
Intermediate-Frequency Amplifiers Abstract | Full Text: PDF (733K)
Ladder Filters Abstract | Full Text: PDF (194K)
Lattice Filters Abstract | Full Text: PDF (145K)
Logarithmic Amplifiers Abstract | Full Text: PDF (377K)
Logic Design Abstract | Full Text: PDF (160K)
Low-Pass Filters Abstract | Full Text: PDF (110K)
Majority Logic Abstract | Full Text: PDF (544K)
Matched Filters Abstract | Full Text: PDF (195K)
Mixer Circuits Abstract | Full Text: PDF (333K)
Multipliers, Analog Abstract | Full Text: PDF (351K)
Multipliers, Analog CMOS Abstract | Full Text: PDF (174K)
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07• Circuits and Systems: Analog and Digital Signal Processing
Multivibrators Abstract | Full Text: PDF (178K)
Network Parameters Abstract | Full Text: PDF (575K)
Noise Generators Abstract | Full Text: PDF (668K)
Nonlinear Filters Abstract | Full Text: PDF (78K)
Overvoltage Protection Abstract | Full Text: PDF (144K)
Phase-Locked Loops, Applications Abstract | Full Text: PDF (399K)
Phase Shifters Abstract | Full Text: PDF (297K)
Power System On-Line Transient Stability Assessment Abstract | Full Text: PDF (819K)
Preamplifiers Abstract | Full Text: PDF (324K)
Programmable Filters Abstract | Full Text: PDF (308K)
Pulse-Shaping Circuits Abstract | Full Text: PDF (294K)
Ramp Generator Abstract | Full Text: PDF (125K)
Rectifying Circuits Abstract | Full Text: PDF (195K)
Relaxation Oscillators and Networks Abstract | Full Text: PDF (695K)
Sample-and-Hold Circuits Abstract | Full Text: PDF (236K)
Sequential Circuits Abstract | Full Text: PDF (136K)
Smoothing Circuits Abstract | Full Text: PDF (243K)
Summing Circuits Abstract | Full Text: PDF (209K)
Switched Capacitor Networks Abstract | Full Text: PDF (247K)
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07• Circuits and Systems: Analog and Digital Signal Processing
Threshold Logic Abstract | Full Text: PDF (223K)
Time-Varying Filters Abstract | Full Text: PDF (502K)
Translinear Circuits Abstract | Full Text: PDF (291K)
Tunnel Diode Circuits Abstract | Full Text: PDF (180K)
UHF Receivers Abstract | Full Text: PDF (160K)
Variable-Frequency Oscillators Abstract | Full Text: PDF (604K)
Voltage References Abstract | Full Text: PDF (180K)
Wideband Amplifiers Abstract | Full Text: PDF (106K)
Wiener Filters Abstract | Full Text: PDF (179K)
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F. William Stephenson1 l Print this page phrases
1Virginia Polytechnic Institute and State University, Blacksburg,
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Copyright © 1999 by John Wiley & Sons, Inc. All rights Search
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DOI: 10.1002/047134608X.W2201 Acronym Finder
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Article Online Posting Date: December 27, 1999
Abstract | Full Text: HTML PDF (248K)
Abstract
The sections in this article are
Second-Order Structures
Higher-Order Realizations
Integrated Filters
Summary
About Wiley InterScience | About Wiley | Privacy | Terms & Conditions
Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.
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210 ACTIVEFILTERS
ACTIVE FILTERS quality performance and low cost resulted in a fundamental
changeindesignphilosophy.
An electrical filter may be defined as ‘‘a transducer for sepa- Designers, previously cost-constrained to single-amplifier
rating waves on the basis of their frequencies’’ (1). There are second-order sections, were now able to consider multiampli-
numerous everyday uses for such devices ranging from the fier sections whose performance and multipurpose functions
filter that allows one to select a particular radio station, to made commercial production a viable proposition. In particu-
the circuit that detects brainwaves, to resonant cavities that lar,thestatevariabletopology(5)formedthebasisforauni-
operate at microwave frequencies. Indeed, filters are needed versal filter yielding all basic filtering functions from a sin-
for operation across the electromagnetic spectrum. Further- glestructure.
more,theyarerequiredtoperformfrequencyselectiontosat- Inductor replacement and direct simulation techniques
isfy various specialized approximating functions, not neces- such as the leapfrog approach (6) offered low-sensitivity ana-
sarily confined to the conventional low-pass, bandpass, high- logs of classical LC filters. The difficulty in tuning these de-
pass,andband-stopforms. vices was simplified enormously by the introduction of com-
However,thepurposeofthisarticleistofocusonapartic- puter-controlledlasertrimmingusinghybridmicroelectronics
ular category of filter, the active filter, whose evolution over technology. Indeed, by the mid-1970s, sophisticated fifth-or-
thepast 40yearshasbeen heavilyinfluencedby advancesin derellipticcharacteristicfilterswereinlarge-scaleproduction
microelectronic circuit fabrication. The earliest active filters withintheBellSystem(7).
were motivated by the need to overcome significant limita- Thus, over a period of 20 years (1954–1974), active filter
tionsofinductor–capacitor(LC)passivefilters,namely: designershadcometorelyuponarelativelysmallnumberof
basic building blocks to form second-order sections, or were
1. In the audio band, inductors are bulky and prone to basing higher-order designs on analogs of LC structures. Al-
pickup. though many realizations used discrete components, larger-
scaleproductionofthickandthinfilmhybridmicroelectronic
2. Resistor–capacitor (RC) filter structures offer a limited
structureswasquitecommon.
range of responses and are subject to substantial pass-
The advent of switched-capacitor filters in 1979 (8) over-
bandattenuation.
came the need to laser trim resistors and yielded the first
fully integrated active filters. While truly a sampled-data
By contrast, active RC structures can realize (theoreti- technique,theuseofsufficientlyhighclockfrequenciesmeant
cally)losslessfiltercharacteristicsinminiaturizedform.Pas- that active filters could be used up to 100 kHz, far higher
siveandactivefilterpropertiesaresummarizedinTable1. thanbyconventionalanalogtechniques.Subsequentdevelop-
A disadvantage of the active filter is its need for a power mentshaveledtometal-oxidesemiconductorfield-effecttran-
supply and the incorporation of one or more active elements, sistor-capacitor (MOSFET-C) and operational transconduc-
usually operational amplifiers. As a result, highly selective tance amplifier-capacitor (OTA-C) filters (9) which yield
filtersneedcarefuldesignsoastoavoidinstability.However, authentic analog performance at frequencies exceeding 1
asactivefilterdesignhasmatured,asmallnumberofhighly MHz.
reliable topologies have evolved that provide solid perfor- Thefollowingsectionswillconcentrateonafewfundamen-
manceacrossavarietyoffabricationtechnologies. tal filter design techniques that form the basis for modern
The earliest active filters used discrete components and active filter design. The Sallen and Key, multiple loop feed-
were based upon direct synthesis of RC sections with appro- back,andstatevariablestructureshavestoodthetestoftime
priatelyembeddedactivedevicessuchasthenegativeimped- andhaveproventobeaseffectiveindiscretecomponentreal-
ance converter (2). Second-order sections were then cascaded izationsastheyhaveinMOSFET-Cstructures.Theyallform
toformhigherorderstructures. higher-order filters when cascaded with similar sections. Fi-
Subsequently, a catalog of building blocks was developed nally, the leapfrog design and direct replacement techniques
by Sallen and Key (3), which led to a much broader interest are discussed as examples of direct higher-order filter syn-
in active filters. This was due in no small part to removal of thesis.
theneedforclassicalsynthesisexpertise.
However, widespread use of active filters was still inhib-
itedbyconcernsoversensitivity,particularlywhencompared SECOND-ORDERSTRUCTURES
tothepassbandperformanceofpassivefilters.Thiswasover-
come by the simulation of the floating inductor (4) and the ThefundamentalbuildingblocksforactiveRCfiltersaresec-
widespread availability of operational amplifiers whose high- ond-orderstructureswhichcanreadilybecascadedtorealize
higher-order approximating functions described by the gen-
eralvoltagetransferfunction:
(cid:24) (cid:25)
ω
Table1. ComparisonofActiveandPassiveFilterProperties s2+ zs+ω2
V Q z
AudioBandFilters o =H· (cid:24) z (cid:25) (1)
V ω
LC ActiveRC i s2+ ps+ω2
Q p
p
Bulky Small
Lossy(lowQ) Lossless(highQ) where (cid:5), Q and (cid:5), Q refer to the zero and pole frequency
Stable(absolutely) Stabilitydependsupondesign z z p p
and Q, respectively. All-pole functions (low-pass, bandpass,
Transmissionloss Capableoftransmissiongain
high-pass) occur when only one of the numerator terms (s0,
J.Webster(ed.),WileyEncyclopediaofElectricalandElectronicsEngineering.Copyright#1999JohnWiley&Sons,Inc.
ACTIVEFILTERS 211
G
2
3 G C
1 2
K
1 RC 2
K
network
V C G V
1 1 3 3
V V
1 3
Figure 2. Sallen and Key second-order bandpass filter using posi-
Figure1. SallenandKeystructureconsistingofacontrolledsource tive-gaincontrolledsource.
andanRCnetwork.AppropriatechoiceoftheRCnetworkyieldsall
basicformsofsecond-ordertransferfunctions.
Table2illustratesfourtopologiesandtheresultingvoltage
transfer function when that RC structure is used in the cir-
s1, or s2) is present. A notch occurs when the s1 term disap- cuit of Fig. 1. Thus, it is seen that low-pass and high-pass
pearsinthenumerator.Wewillnotdiscussthemoregeneral sections can be achieved with a positive-gain controlled
case (10) that arises when all numerator terms are present source,whereasthetwobandpasssectionsrequireanegative-
simultaneously. gaincontrolledsource(11).
Thetopologiesthatfollowaresuitablefordesignusingdis- Although not included in the original catalog, the five-ele-
crete, hybrid, or fully monolithic fabrication. Furthermore, mentbandpasscircuitofFig.2,whichutilizesapositive-gain
they have stood the test of time for ease of design, tuning controlled source, is now generally incorporated under the
simplicity,andrelativelylowcost. SallenandKeybanner.Inthiscase:
SallenandKey KG
s 1
V C
Sallenand Keyoriginally proposed(3)a familyof all-polefil- 3 = (cid:24) 1 (cid:25) (3)
tersbaseduponthecircuitshowninFig.1,forwhich V1 s2+ G2(1−K)+(G1+G3)+G3 s+G3(G1+G2)
C C C C C
1 1 1 1 2
V −Ky
3 = 21 (2)
V y +Ky Design is most commonly restricted to the positive-gain
1 22 23
controlled source realizations, despite the inherent positive
ByappropriatechoiceofthepassiveRCnetwork,itispossible feedback used to enhance Q. In general, if realizations are
p
to realize all forms of basic filter. However, because the cre- restrictedtoQ (cid:18)10,theadvantagesofthelowercomponent
p
ationofaband-stop(notch)sectionrequirestheuseoftwinT spread in this design outweigh the stability considerations.
networks, which are inherently difficult to tune, we confine Design is relatively straightforward and proceeds by coeffi-
ourdiscussiontotherealizationofall-polefunctions. cientmatching.
Example1. Forourexample,wedesignasecond-orderlow-
Table 2. Sallen and Key Realizations pass Chebyshev filter having 0.5 dB passband ripple and a
passband gain of 20 dB. From the standard tables (12–15),
Circuit
a thenormalizedtransferfunctionis
No. RC Structure Voltage Transfer Function
1 1 3 C1 2 (cid:3) (cid:2)KCG11CG22(cid:5) (cid:4) VV31 = s2+1.42H6s+1.516 (4)
G1 G2 C2 s2 + s G2+G1 + G2—KG2 +G1G2
C2 C1 C2 C1C2 Apassbandgainof20dBisequivalenttoanabsolutegainof
10, so that H (cid:5) 15.16. By matching the low-pass expression
3
G1 Ks2 fromTable2andthecoefficientsofEq.(4),weobtain
(cid:3) (cid:4)
2 1 2 s2 + s G2+G2+G1 (1 — K)+G1G2
C1 C2 G2 C2 C1 C1 C1C2 KG1G2 =15.16 (5a)
C C
1 2
3 G1 s(cid:2)KCG22(cid:5) GC1CG2 =1.516 (5b)
3 1 C1 G2 C22 s2 + s(cid:3)GC22+G1 C+1 G2(cid:4) + (1 — K)GC11CG22 GC2 + G1C+G2 − K1CG22 =1.426 (5c)
2 1 2
3 C1 s(cid:2)(1 —KG K1)C(cid:5)1 Thus, K (cid:5) 10 [from Eqs. (5a) and (5c)]. The remaining two
4 1 2 (cid:3) (cid:4) equations contain four unknowns, indicating freedom of
G1 C2 G2 s2 + s G1+G2+G2 + G1G2 choice for two of the elements. Such freedom of choice is a
(1 — K) C1 C1 C2 (1 — K)C1C2
characteristic of the coefficient-matching technique. For con-
aK > 0 for circuits 1 and 2, K < 0 for circuits 3 and 4. venience, set C (cid:5) C (cid:5) IF, since this is highly desirable in
1 2
212 ACTIVEFILTERS
Table 3. MFB Structure and Voltage Transfer Functions
Filter Type Network Voltage Transfer Function
3
1 neRtwCork 2 +– (a) Low-pass G1 GCG423 C+—5 s2C2C5 + sC5(G—1 G+1 GG33 + G4) + G3G4
V V
1 3
C4 G5
Figure3. Multiplefeedback(MFB)structureconsistingofanopera- (b) High-pass C1 GC23 +— s2C3C4 + sG5(—C1s 2+C C1C3 3+ C4) + G2G5
tional amplifier and an RC network. Appropriate choice of the RC
networkyieldsallbasicformsofsecond-ordertransferfunctions.
C4 G5
manypracticalrealizations.Asaresult, (c) Bandpass G1 GC23 +— s2C3C4 + sG5(C—3 s+G C1C4)3 + G5(G1 + G2)
G G =1.516 (6a)
1 2
G −8G =1.426 (6b)
1 2
TheonlysolutionyieldingpositivevaluesisG (cid:5)4.268Sand
1
G (cid:5) 0.355 S. Impedance and frequency denormalization can
2 stable, negtive-feedback circuit. Specific realizations of the
beapplied,dependinguponspecificdesignrequirements.
all-polefunctionsareshowninTable3.
Realizationof thebasic low-passSallen andKey filterhas
As for Sallen and Key sections, design proceeds by coeffi-
been widely discussed in the literature (11). Popular alterna-
cient matching.A widelyused design setis illustratedin Ta-
tivestotheC (cid:5)C (cid:5)IFusedaboveareasfollows:
1 2 ble 4, for which both bandpass and high-pass circuits use
equal-valued capacitors. No such solution is possible for the
1. SettingC1(cid:5)C2(cid:5)C,G1(cid:5)G3(cid:5)G,andK(cid:5)3(cid:3)(1/Q) low-pass circuit, though an equal-valued resistor pair is pos-
2. SettingK (cid:5)1 (therebyeliminating twogain-setting re- sible.
sistors),G (cid:5)nG andC (cid:5)mC Although highly stable, the MFB structure has a pole Q
1 3 1 2
3. Setting K (cid:5) 2 (for equal-valued gain-setting resistors), dependent upon the square root of component ratios. Thus,
C1(cid:5)C2(cid:5)CandG3(cid:5)Q2G1 foraQpofn,themaximumcomponentspreadwillbepropor-
tional ton2. As aresult, the MFBarrangement isbest suited
tomodestvaluesofQ,typicallynotgreaterthan10.
MultipleFeedbackStructure p
The multiple feedback (MFB) structure (16) is derived from
the general feedback configuration of Fig. 3, in which the ac- ModifiedMultiple-LoopFeedbackStructure
tiveelementisanidealoperationalamplifier.
In positive-feedback topologies such as the Sallen and Key,
The most common realization of the RC network is shown
Q is enhanced by subtracting a term from the damping (s1)
inFig.4,whichyieldstheMFBtransferfunction p
coefficient in the denominator. By contrast, in negative-feed-
V −Y Y back topologies such as the MFB, high values of Qp are ob-
3 = 1 3 (7) tainedattheexpenseoflargespreadsinelementvalues.The
V Y (Y +Y +Y +Y )+Y Y
1 5 1 2 3 4 3 4 two techniques are combined in the modified multiple-loop
Thebasicall-polefunctionscanberealizedbysingle-element
replacement of the admittances Y (cid:1) Y, yielding a highly
1 5
Table4. ElementValuesforMFBRealizations(Histhe
3 NumeratorConstantinEachCase)
ElementValue
Bandpass High-pass Low-pass
1 Y1 Y4 Y3 Y52 G1(cid:5)H C1(cid:5)H G1(cid:5)(cid:5)Hp
G2(cid:5)2(cid:5)pQp(cid:3)H G2(cid:5)(cid:5)p(2(cid:1)H)Qp C2(cid:5)Qp(2(cid:5)(cid:5)2p2(cid:1)H)
p
Y2 C3(cid:5)1 C3(cid:5)1 G3(cid:5)(cid:5)p
C (cid:5)C C (cid:5)C G (cid:5)G
4 3 4 3 4 3
(cid:5) (cid:5) (cid:5)2
Figure 4. Three-terminal, double-ladder structure for use in MFB G5(cid:5)2Qp G5(cid:5)Q(2(cid:1)p H) C5(cid:5)Q(2(cid:5)2p(cid:1)H)
p p p p
sections.
ACTIVEFILTERS 213
versatility and ease of tuning. The advent of the operational
C amplifier eliminated earlier cost concerns, and the ability to
4
G
5 realize relatively high-Q sections remains an attractive con-
G
1
sideration. However, it is the abilityof the circuit to yield all
–
+ basic forms of second-order sections by appropriate choice of
C
3 output terminal that has made it so popular for commercial
V V manufacture(19).Customfiltersarereadilyfabricatedbyap-
i o
G Gb propriateinterconnection ofterminals, yieldingthe universal
a
filterterminologyofseveralvendors.Inparticular,highlyre-
liablenotchfiltersarepossiblethroughtheadditionofasum-
mingamplifiertothebasicthree-amplifierarray.
The circuit shown in Fig. 7 is an example of a state-vari-
Figure5. Modifiedmultiple-loopfeedback(MMFB)structuredueto
Deliyannis which yields a second-order bandpass function. By judi- ablesectionandcanberecognizedasananalogcomputerre-
cious use of positive feedback, this circuit reduces the large compo- alization of a second-order differential equation. It is more
nent spreads which are characteristic of the MFB structure while commonly referred to as the Huelsman-Kerwin-Newcomb
yielding greater stability margin than the Sallen and Key ar- (HKN)filter(5).Inthefrequencydomainitiscapableofyield-
rangement. ing a variety of voltage transfer functions, according to the
particular output connections used. Assuming ideal opera-
tionalamplifiers,thespecifictransferfunctionsareasfollows:
feedback(MMFB)circuit(17)ofFig.5,forwhich
1. Thelow-passresponsewith
V −sC G (1+k)
o = 3 1 (8) (cid:11) (cid:12)(cid:26)
Vi s2C3C4+s{G5(C3+C4)−kC3G1}+G1G5 V1 = R2[R3+R10] D(s) (9a)
V R [R +R ]
where k (cid:5) G /G, and the Q-enhancement term signifies the i 3 1 2
b a
presenceofpositivefeedback.
Design of this bandpass circuit proceeds by coefficient 2. Thebandpassresponsewith
matching, although the reader is advised to adopt the step- (cid:11) (cid:12)(cid:26)
V R [R +R ]
by-stepproceduredevelopedbyHuelsman(11). 2 =−R C s 2 3 10 D(s) (9b)
A generalization of the MMFB circuit, yielding a fully bi- Vi 9 2 R3[R1+R2]
quadratic transfer ratio has been developed by Friend et al.
(18), as shown in Fig. 6. This arrangement was used exten-
3. Thehigh-passresponsewith
sivelyintheBellSystemwherethebenefitsofcomputer-con-
(cid:26)
trolled (deterministic) laser trimming techniques and large- V R (R +R )
scale manufacture were utilized. Although this resulted in V3 = R2(R3 +R10)C1C2R8R9s2 D(s) (9c)
quite exacting realizations using tantalum thin-film technol- i 3 1 2
ogy,thestructureislesssuitedtodiscretecomponentrealiza-
where
tions. An ordered design process based upon coefficient
(cid:24) (cid:25)
matchingispresentedelsewherebyHuelsman(15). R (R +R ) R
D(s)=C C R R s2+ 1 3 10 C R s+ 10
1 2 8 9 R (R +R ) 2 9 R
StateVariableStructure 3 1 2 3
Based upon analog computer design techniques, the state Ageneralbiquadraticfunctionmaybeobtainedbycombin-
variable (SV) structure (5) assumed popularity because of its ing the various outputs via a summing network, as shown in
Fig.8.Thecompositevoltagetransferfunctionthenbecomes
C2 Vo =R2(R10+R3)R5(R6+R7)
V (R +R )R (R +R )R
i 1 2 3 4 (cid:11)5 7 (cid:12) 
R2  C C R R s2+ [R4+R5]R6 R C s+ R4  (10)
R4 C1 · 1 2 8 9 (cid:11) R5[R6+R7](cid:12) 9 2 R5
– C C R R s2+ R1[R3+R10] C R s+ R10
R6 1 2 8 9 R [R +R ] 2 9 R
3 1 2 3
+
R
c Nowconsiderthedesignofalow-passresponse
V R V
i b o H
R5 R7 RD T(s)= s2+(ω s/Q )+ω2 (11)
p p p
ItisclearfromEqs.9(a)and(10)thatthereisconsiderable
Figure6. TheFriendbiquadwhichgeneralizestheMMFBstructure flexibility in the design since there are nine passive compo-
ofFig.5soastoyieldbiquadraticfilters. nentsandonlythreespecifiedvariablesinEq.(11).