11111111111111111111

by Rufus

TURNER

11111111111 LCRufus CircuitsP. Turner, Ph. D. by

Howard4300 WEST 62ND W. ST. Sams INDIANAPOLIS, & INDIANACo., 46268Inc. USA Copyright © 1980 by Howard W. Sams & Co., Inc. Technologically and historically, the familiar combination Preface FIRSTIndianapolis, PRINTINGEDITION Indiana -1980 46268 faringselectiveof inductance afieldof radio unitin manyapparatus, of and electronics. areas capacitance, oftheThis electronics. LC Originallybook circuit the describes LC has delegated circuit, found a number isapplication to the the ofbasic tun- practical LC circuits and writtenphotocopying,transmittedreproduced,All rights permission reserved. bystored recording,any inmeans,from No a retrievalpart theor electronic, otherwise,ofpublisher. this system, book mechanical,without No orshall patent be torsof resistance,offersthem. and enoughcapacitors.A sufficient since background that amount property theory of space is to inherent promotehas been in the devotedpractical understanding also induc- to assumedforthisprecautioninformationliability errorsbook, is for ortheassumed has contained omissions.damages publisher been with taken herein.resulting assumesNeither respect in theWhile fromisnoto preparation any theresponsibility every the useliability use of ofthe mayplanationsposes.dent, find technician, A certain minimum being parts andpreferred of of experimenter,mathematicsAlthough it useful, where theif feasible-andisonly materialmore employed-physical for advanced reference is addressed frequent readers pur- ex- il-to the electronics stu- theLibraryInternational information of Congress Standard contained Catalog Book herein. Number:Card Number: 0-672-21694-9 79-57616 virtuoso.lustrative examples demonstrateI hope that the this necessary book willcalculations. serve both the novice and the Printed in the United States of America. RUFUS P. TURNER 2.7 Broadband Tuning 50 2.122.112.102.92.8 DCSymmetricalWave -Tuned Traps Circuits Circuits SelfRange -Resonance Coverage 5453515653 2.132.14 Wavemeters Varactor Frequency Multiplier CHAPTER 3 5759 Contents FILTERS 3.43.33.23.1 PowerWaveFilterBasic Sections Filters-Supply Filtering Filters Properties of L and C 72646260 60 FUNDAMENTAL THEORY 1.1 CHAPTER 1 7 7 CHAPTER 4 1.51.41.31.2 NatureNatureThe ACof ofCapacitance Cycle InductiveInductanceResistance -Rate Reactance of Change 20191511 BRIDGES AND OTHER MEASURING DEVICES 4.24.34.1 MaxwellHayAnderson Bridge Bridge Bridge 787776 . 76 1.91.81.71.6 FigureResonanceCombinedNature of Merit,CapacitiveReactance Q inReactance LC Circuits 2827262624 4.74.64.54.4 ResonantBridgedResonanceOwen Bridge -T Circuit BridgesNull Network as Measuring Device 84828079 1.141.131.121.111.10 TimeInductive PracticalPureNature Constant L Couplingandof L Practical andC in C Combination in InductorCombination 32313028 APPENDIX A 1.151.16 Oscillations Range of Application in LC Circuit of LC Circuits CHAPTER 2 3635 ANGULAR VELOCITY (w) APPENDIX B 86 TUNED CIRCUITS 2.32.22.1 ResonantParallelSeries -Resonant -Resonant -Circuit Circuit Constants Circuit 424037 37 REACTANCE OF INDUCTORS AT 1000 Hz . . 87 2.62.52.4 CoupledCircuitSelectivity Q Resonant Circuits 464543 REACTANCE OF CAPACITORS APPENDIX C . . 88 APPENDIX D RC TIME CONSTANTS APPENDIX E 90 RL TIME CONSTANTS APPENDIX F 91 CHAPTER 1 RESONANT FREQUENCY OF LC COMBINATIONS APPENDIX G . . 93 Fundamental Theory CONVERSION FACTORS . . 95 readertheunderstandingare essentialmathematics(LC) has circuits. that to abackground. angeneral of These understanding electronics,This familiarity are chapter specific and of withdigests itemsit electricalis assumed thoserequiring theoryparts that of the basic electronics that inductance -capacitance for their and Asvector1.1 theTHE isvectorrotating ACE., CYCLE-RATEthe moves atmaximum constant inFig. a counterclockwise 1-lA OFvaluevelocity. CHANGE depicts attained The a magnitudeby sinusoidaldirection the ac voltage.from ofac thisvoltage its in terms of a completestartingextendingtaneouswhich(Although pointincreases rotation.voltage, from at the from thee,If vectorhorizontalisthe tip initialdepicted figure of is the zerorotating isaxis, byvector drawn to the 360°it at lengthgeneratesto constantto (27rthe scale, of horizontalradians) the angularan halfangle in chordvelocity,each 0 the instan- axis. theatvoltagerate; length see is Tableof zero this 1-1.)at half 0°,From chordsince all, Fig. heredoes and 1-1A, the not is half maximumchange it chord at has ata constant90°, no length since here the half is easily seen that the instantaneous chord has zero,its maximum(7r/2 increases radians), length. to the returns maximum Thus, to instantaneouszero positive at 180° (7r voltage radians), e startsincreases at value (+Emax) at 90° to 7 90° instantaneousunit radius voltage(i.e., E. at = any1), itpoint, is equal therefore to sin is 0.: The value of 180° 360°0° (A) Rotating vector. versus angle. This curveFig. is1-1B identical shows withthe familiar that of plotthe ofsine instantaneous voltage e = Emaxsin 0 (1-1) sinusoidal).function from trigonometry,IllustrativeSin(90°) 60° =value Example: 0.866. hence of From162.6 The the Equation volts.115-V term Calculate power sine 1-1, e -linewave = the162.6(0.866) voltage instantaneous (or has = a 140.8maximum value V. at 60°. 180 270 360 27, RADIANSDEGREES (B) Voltage curve. thebutinstantaneous1-1.) thecurve rateThis at of pointscan voltagechange be ofshown also continuouslyinterestA close is graphically changing. andexamination observingchanging (See by drawingColumn of during the Fig. slope 1-13tangentsthe in ofcycle,showsTable the to that not only is the theandthe vectorreturns maximum has to zerotraced negative at out360° value a (27rcomplete (-Erna.) radians). at 270°At this latter Fig. 1-1. Development of sine wave. cycle, and a new cycle (37r/2 radians), point, passingrateisattangent. (45°)noPoint of change change reveals throughP2Thus, (90°) at thus in aall its moderate Fig.has atmaximumis this zerono1-1B, tiltpoint. atrate the whatever90° points, of tiltIn and change, the of 270°and tangentsineand is whereas whenshows-wave maximum ab theat cycle,that tangentPoint cycle therewhen the P1 is cd age,ofbegins line is e,proportionalwith and the accordingly continued to the the sinerotation value of angle of the 0, vector. instantaneousand for The a circle of Table 1-1. Voltage Change and Rate Change length volt- 360°).ofvoltagethe the cycle angular around is passing rotationthe point throughThis at of that caninterest its pointbe zero shown divided equals points mathematically bythe (0°, the slope 180°,increment ; and: (1) The increment of thus, (degrees)Rotation 0-10 Voltage Change* (V) Voltage Change (V/degree) Rate of valueslopeingly(Point of= are AE/00.AO, P2), zero, so AE Fromthe whereas =slope 0, Fig. and isat 1-1B, steepthe 180° slope it atAE isthis andeasilyis pointvery rate seen largeandof change thatthe for changeat a accord-90°small 40-5030-4020-3010-20 0.643-0.7660.5-0.6430.0342-0.50.174-0.3420-0.174 0.01230.01430.01580.01680.0174 isofat is1,athatchange point great.so point. the of is(2) rate zerointerest The From of at cosine change those calculus,is proportional ofpoints. is 90° maximum the or The rate of to cosine270° ofthe at change those cosineis of zero, 0° pointsof ofand soa the sineofthe where angle360° ratewave = 1 V80-9070-8060-7050-60 0.985-10.940-0.9850.866-0.9400.766-0.866 0.00150.00450.00740.01 radianssinethe cycle function per crosses second is 1). the (where zeroAt a linef givenis in(the hertz, frequency maximum and 7T f,value = the 3.1416). voltageof the Theco- vector describes 27rf 8 betweenw.expression (Appendix 1 27rfHz A andis gives often 100 valuesrepresented MHz.) of The co byat maximum lowercasea number rateofGreek frequencies of omegachange : 9 followsin voltage from theis equal previous to 2711E. discussion (also that written this rate coE.), of change and it ifgrees, desired but : these time units are easily converted into an angle, themustinterest:rate rate occur of of change change at zero mayby-voltage the be foundcosine points by ofin multiplying thethe cycle. angle At theat every themaximum point point, of Rate of Change = 2irfEmaxcos 0 = coEmaxcos 0 (1-2) where, f is the frequency inis hertz,the angle in radians, 0 = 2irft = wt (1-3) For15 example, V, the rate in a of500 change -Hz cycle at 65° having (where a maximum cos 0 = 0.42262)voltage of is 125 Hz wt is is thean -f. time in seconds,is 3.1416, V) 0.004 0.005 0.006 0.007 0.008 directly,To convert change 0 to degrees,Equation multiply 1-3 to : by 57.296. To obtain degrees 0 = 360 ft (1-4) O 0.001 0.002 0.003 t (Seconds) ratetimeNoteFig. of instants, 1-2B changethe rate shows here andof change thegives(271-fErna0 relationship exhibitedsines andis 1570.8 betweenin cosines this V/s.cycle. selected of From these The Equation anglesmaximum angles. and t (A)0 Typical cycle. of21-2, change (3.1416) the rateat 1250.001 of (2)change s ( -1) at= 1570.80.004 second ( -1) = -1570.8(180°, 7T V/s radians) is (45°, 7r/4 radians) is 2 (3.1416)125 (2) ; the rate (seconds) 0.001 0 (degrees) 45 0 (radians) 0 sin 00.7071 0 cos 0 1 7r/20.7071 radians) = 1110.7 is 2(3.1416)125 V/s ; Itand should the (2)0 rate be = clearof 1570.8 change by xnow 0at = 0.002 that0. the s (90°, rate of change in voltage 0.0020.0050.0040.003 22590180135 3.92703.14162.35621.57080.7854 00.70711 -0.7071-1 00.7071 taneousprecedingmagnitudeat a selected voltage fromexample, point theat theinstantaneous forin 0.001 the instance, ac -second cycle voltage it isis point seenmarkedly at that (45°)that point. the different instan- In the in is (from 0.0060.007 270315 5.49784.7124 -1-0.7071-0.7071 -0.7071 10.70710 ofEquation change of 1-1) voltage 2 x sin at 45°that = point 2(0.7071) (from = the 1.414 foregoing V; but the rate para- 0.008 360 (B)Fig. 1-2. Illustrative cycle. Numerical relationship. 6.2832 0 voltagetionsinceofgraph) change and a isclearcycle of 1110.7 in LC theunderstandingapplies circuits acvolts cycle equally per Whatdemands must second. ofhaswell beinductor thisbeen to masteredThe accomprehension. said currentvarious and in and capacitorthis andaspects remembered, section the currentof aboutrate ac voltage and the opera- gree.change47,1242 (3.1416)500(where voltsis 2(3.1416)500 cos per (15)0.422620 degree.=1), theFig. (15)0At rate 1-2A 90°= of 19,915= (where change47,124illlstrates volts cos isx 0 2(3.1416)500 aper0 = specific= 0 degree.0), volts the example.per rate At (15)1 de- 0°of = This is one cycle 1.2voltage,cycle. NATURE It andis necessary theOF symbolsRESISTANCE only I andto substitute i for E and the e inword the discussion.current for horizontalof10 125 -Hz axisac voltage is divided having into a secondsmaximum of valuetime, insteadof 2 volts. of deThe - this point, since internal resistanceIt is necessary is unavoidable to introduce in inductors a discussion of resistance at 11 circuits.and capacitors and can influence the performance of LC filament-such as a-type vacuum lamp-is tube, sometimes transistor, employed semiconductor instead diode, of a re- or frictionflow of encounteredan electric current. byResistance flowing It is water.somewhat (R or Resistancer) is analogous the simple is directly to opposition the offered to the thissistor zero per -phasese. -shiftPure feature; resistance in both introduces the wave no plot phase shift. Fig. 1-3 illustrates , (Fig. rent,proportional as shown to by voltage Ohm's lawand :is inversely proportional to cur- R = I (1-5) propertiesherentpossibleage1-3B) are and inductance in to steptheare attain vector usually,at allin and practice,points.diagram butcapacitance, not Because (Fig.all always, resistors 1-3C) butpure so these possessminuteresistance extraneous thatsome is they im- in- current and volt- where,willaAn resistance applied produce voltage R; a likewise,voltage E thusIER isisdrop will athethe current currentvoltageEresistanceforce = IR Ia inthroughcurrentacross amperes.volts,in ohms, Ithe a= resistanceE/R resistance. through R can be ignored. submultiplessistors.howeverWhile all Resistance tiny,conductors of theresistance ohmis measured ofConventional (See electricity is Table primarily in ohms1-2) exhibit (ohmic) . andthe some in propertyresistors multiples resistance, areof and re-made principally from (A) Circuit. (B) Wave pattern. voltage,ohmic)resistivityfromsuitable carbon andmetals compositions, thermistors, or ingraphite; the form mixtures, whose and of resistancewire,from and oxides.certainstrip, depends ribbon, Nonlinear controllable upon or (nonfilm; tem- - resistors, whose resistance depends upon applied Resistance is a dissipative property : that is, resistors(C) Vector con- diagram. Fig. 1-3. Phase relationship: resistance. areintegratedsistors,pounds,perature, readily there suchare circuitsobtainable, madeare as integratedcomplex from(ICs) and semiconductors. Variable, oxides.all resistors resistors In as which addition wellare or availablefrom areas fixed,toprocessed suitable discrete inresistors a com-wide into re- and sizes. For latterfrictionway,causessume represents electrical power.gives heat torise Currentbe energy togenerated heat isthat in converted bodiesisin lostthe resistor, forrubbed into some heat justtogether. intended energy, as mechanical In andelec- this the (I) flowing through resistance (R) specialrange of purposes, resistance the and internal power resistance ratings, shapes, of some other device Unit Table 1-2. Units of Resistance Number of Ohms Symbol tricalexpressed work. asThe : electrical loss is power dissipation and P = PR = E2R may be (1-6) MilliohmMicrohm 1 x 10-310-6 p,f/mfg where, GigohmMegohmKilohmOhm 1 x 10°10' nGftMOkf) REIP isis the thethe currentpowervoltageresistance loss throughacross in ohms.watts, the the resistance resistance in in volts, amperes, 12 Teraohm 1 x 10" Ttrt inThe relation sine -wave to current pattern and shown voltage. in Fig. 1-4 shows resistor power 13 When operated within their ratings, good -grade conven- Unit Table 1-3. Units of Inductance tionaldesignedExceptionsresult resistors of variationsto are be undergo voltage in -dependentveryvoltage,sensitive, little temperature, changeand resistors thermistors, in (vdr's) resistance or frequency. , whichwhich as area MillihenryHenry Number of Henrys 0.0011 Symbol mHH designed to be temperature sensitive. 1.3 NATURE OF INDUCTANCE PicohenryNanohenryMicrohenry 1 xX 10-7210-910-6 pHttHnH VOLTAGECURRENT AXISAND Fig. 1-4. Power in resistor. or by a coil of wire (inductor)Inductance which (L) retardsis the property the buildup exhibited of by a conductor When resistors are connected in series (see Fig. 1-5A), the ducedinertia,whencurrent aby and voltagewhen the is magnetic caused a isvoltage removed. by fieldInductance the is surroundingapplied,counter It is sometimes is emf ormeasured the("back inductor.decay called voltage")in henrysof electrical current pro- (H) ; but since this is a (see total resistance of the combination is : Rt R1 + R2 + R3 + . . . + (1-7) inTablelarge(also a straight unit,1-3). called Whilesubmultiples wire,coils). inductance it A simple of isthe practicalpresent henry in areinductor any often conductor, consistsused even of a is primarily a property of inductors Whenequivalent resistors resistance are connected of the combination in parallel is : 1 (Fig. 1-5B) , the eternumbercylindricalcoil ofwhose the of coil turns form (d) (N),are and: wound the its length inductance in a ofsingle the (L)coillayer depends(1), on anda nonmetallic, theupon diam- the canIf only be simplifiedtwo resistors to R are = (R1R2)connected / (R1 in +parallel, R2) . the equation Req = R1 1 _t_ R2 1 R3 1 + R 1 If n resistors, (1-8) where, dL is the diameterinductance of in the microhenrys, winding in inches, L = 0.2 d2 N23d + 91 (1-9) RI R2 R3 R, TheN1 isis theinductancethe length number of formula ofthe turns. winding becomes in inches, somewhat different when each having the same value R, are connected in series, the total(A) Resistors in series. Fig. 1-5. Basic resistor circuits. (B) Resistors in parallel. forability--,a-ofturns)Theonthe a agiveninductor eachcorecore andincreasesinductance).of the ismagnetic the woundcore multilayer the material, in inductancematerialInductors several winding the layers fewer are(the (increasedmanufactured orhigher the when turns the it number neededisperme- wound in a of (iron, steel, ferrite, etc.) . lenthavingresistance14 resistance the ofsame the of value combinationthe combination R are connected is R, is = Re, nR. in =parallel, If R/n. n resistors, the equiva- each areresistancewide available range ratings ofin fixedinductance, and and in numerous variable current, types. shapes voltage, and sizes. and internalInductors - 15 components.Forductance,It isexample, inherent and Evenby in so theirother do short, the naturecomponentsInductance leads straight transformer of capacitors, wires andis also devices exhibit windings found resistors, than inductance, in haveplacesinductors. and in-other other than in inductors. (A) Circuit. -combinefrequencies,lowquencies.which-though frequencies with other the tiny-canis duefactors-suchwire Allto resistance the beinductors resistancedeleterious as core to have determine lossesof internaltheat very wire.and the resistance skin highAt total highereffect fre- re- (R) which at de and rapidlyfinalwhichsistanceBut current theacts increased of current inan isseries inductor. determined doesor with decreased. Whennot In the areach byhighinductance, a thede Any this-quality voltageresistance valuesuch is inductor, heldchangeisat appliedonce,of to the a is resistance,norminimum. winding. opposedto can an inductor,it be the resulting E applicationcurrentbyduced the counter doesby ofthe not the emf,surrounding evenvoltage. which begin Thus, is magnetic toopposite flowcurrent untilfield. in lags polarity aThis voltageshort means andtime (voltage is thatafter pro- the 90 thisintroduceleads1-6B) 90° current) and-lagging a 90° the in phasevector an phase inductive shift diagram Becausefeature. in an circuit. (Fig. ac Inof circuit. theboth 1-6C), action the Fig. current wave just 1-6 described, illustratesplotand voltage(Fig. pure inductance would (B) Wave pattern. Fig. 1-6. Phase relationship: inductance. (C) Vector diagram. thezeropassingchangeshowsare outac when cyclethat of inthrough stepvoltage themaximum is ratebyat zero) maximum). 90° isof maximum changeinductor at; all points. inFor current (thatvoltage a Examination clarification, is, flows iswhen zero when the (that of see acthe Fig. is. cycleSection rate when1-6B ofis and, conversely, inductor current is ternalinductor,thatpower).resistor of resistance the power,Note the current onlythat and of the powerthewave Fig. frequency inductor. 1-11, or loss the theis voltage ofthat wave the associated power patternwave. wave Infor with a capacitorispractical doublethe in- propertiesinherentis 1.1impossible for aresistance arediscussion tousually attain and ofsmall. in rate practice,capacitance ofNevertheless, change. all inductors andBecause inherent these purepossess extraneous resistance inductance some 360° E AND I turnedoneresultsducing (internalhalf fromtofull -cycle the 90° energy aclosses) phase (whengenerator being shift.preventsPure the stored duringfieldinductance, a is inpractical theexpanding) the next unlikemagnetic halfinductor resistance, and -cycle field being from during(when consumes re- intro- no power. This /720°P andinductorthe16 field voltage is power collapsing) (compare (for an withideal Fig.inductor) 1-4, thein relation wave pattern to current for . The wave pattern in Fig. 1-7 depicts Fig. 1-7. Power in ideal inductor. flowingmaterialHowever,resulttional inductorsthroughof will variationsthe decrease inductance a undergowinding in whenWhen voltage, ofsimultaneously a highcoiloperated frequency, wound values within onwith of a directcore antheir alternating of currentratings, magnetic good -grade very little change in inductance as a or temperature. conven- thektwoconnectionvice = turnsfirst 1)inductors versa that coil of of(the allthe induces two arethe other,effect so welllines tightlya M differs-spacedcounter of is force maximum,coupled from inductorsemf from intothat(coefficient oneand theinresulting whenseries).inductor second ofthe Whenfrom couplingcoil linktwo and thethein-all aretancesaturablecurrent spaced in saturate this reactor) far way. enough the is core.designed Whenapart One that inductorsto particular provide their magnetic are dctype -controlledconnected of inductorfields induc-doin (theseriesnot (Fig. 1-8A) and ductors1.4interacts NATURE are with completely OF the INDUCTIVE other) decoupled , M isREACTANCE zero. (that is, no flux from one Wheninteract, inductors the total are inductance connected of in theparallel combination and is : Lt = L1 + L2 + L3 + . are spaced far (1-10) age,and currentI is the forresulting a pure current,inductance.Fig. 1-9 and shows Here,E. isthe E, relationship is the applied between volt- sinusoidal voltage the counter emf. This equivalentenough apart inductance that their of themagnetic combination fields is do : not interact, the Le - 1 1 1 1 1 cordinglyincounter Fig. 1-9-the voltage decreases two opposes are asThe outthe the counteroffrequency applied phase voltage withvoltage, of theeach increases appliedsince-as other. andvoltage is seen the inductor current ac- eachcanIf be onlyhaving simplified two the inductors same to valueL = are (L1L2) L, connected / (L1 in+ L2). If n inductors, L1 L2 are connected in series, the total L3 parallel, the Lequation thisternatingforeasvoltageincreases, the oppositionoffers inductance increases current. andfrequency vice is increases,andByequal versa. -dependentmeansthe to inductor Atada. and of a calculus,Thisgivenvice opposition current versa.opposition frequency, it therefore canAn to inductorthebe is the shown flowtermed decreases counter there-of that al-in- arepointalenthavinginductance so is close inductance strictly the physicallysameof termedthe value ofcombination the self-inductance.that The L,combination are theirproperty connected is fields L, = whichis nL. Whenoverlap in If parallel, has two (thatbeen inductors the is,under equiv- the discussion L = L/n. n inductors, each up to this where,ductive reactance, is measuredLXi, is is the the ininductance inductive ohms, and inreactance is henrys, given byin ohms,: XL = = 2/rfL (1-12) inductancecoils are coupled), (M)-comes a second into play. variety Interaction of inductance-mutual LI L2 L3 Ln occurs because ircof is is the 27f, frequency in hertz,is 3.1416. (A) Inductors in series. pressedThe relations in a form between often current,called Ohm's voltage, law andfor ac:reactance are ex- XL = E, E = IXL, and I = -ELX (1-13) T (B) Inductors in parallel. where, XLEI is is in in ohms,amperes, volts. 18 Fig. 1-8. Basic inductor circuits. flowing in an inductiveFrom reactance Equation produces 1-13, it a is voltage seen that drop an alternating current 19 E, (APPLIED VOLTAGE) E, (COUNTER VOLTAGE) E, where, C = 4.45 t kA (1-14) 90° tA is kCis the is isthe the thethickness area dielectriccapacitance of one of constant,the plate in dielectric picofarads, in in inches. square inches, I (INDUCTORCURRENT) 90° andtype. are Both offered are manufactured in a wideCapacitors range in fixedof may capacitance beand of variable the twoand varieties-plate voltage type or multiple -plate (A) WaveFig. pattern. 1-9. Inductor current/voltage relationship. (B) Vector diagram. E, andhigheroil,ratings, thickness the shapes, dielectric for aand given sizes.constant, capacitance. Numerous the smaller In dielectrics addition theceramic, required to (air, discrete mica, area glass, plastic, etc.) are employed ; paper, the likevoltageIXL capacitive ; and, leads because thereactance current of Inductivethe (Sectionby phase 90°. reactance of1.6), inductive is the is aproperty reactance, major factor that this in all LC circuits. It, timestube,ternalintocapacitors, semiconductoris capacitanceintegrated employed there circuits insteadare of diodeintegratedsome of(ICs) oraother capacitor rectifier, capacitors device-such or whichtransistor-some- as are a vacuum processed . For special purposes, the in- per se. halvinginductance,imparts the frequency frequency doubling sensitivity halves the frequency the toreactance, these doubles circuits. and sothe Foron. reactance, Simi-a given Unit Table 1-4. Units of Capacitance Number of Farads 1 Symbol F toingandthelarly, 1000 reactance,toso for tenon. H. ainductancesAppendix given doubling frequency, B spacedgives the inductancethe inhalving 1000 decade -Hz the doublesrelationship reactance inductance, the correspond- reactance,from halves 1µH PicofaradNanofaradMicrofaradFarad 1 Xx1 10-2 X 10-12 pFnFtic quantityIt1.5 is NATURE somewhat of fluid. OF Capacitanceanalogous CAPACITANCECapacitance to is the measured ability (C or c) ofin is faradsa the tank ability (F)to store; tobut store a an electric charge. acrossbetweenremaincapacitorcomes the charged thecharged capactor. retains plates. because indefinitely. the WhenIf charge, the Whenenergy capacitorthe Whenthe avoltagethen dc original voltageis anwere stored isexternal disconnected,voltage perfect,is in applied the circuit Eelectric it appearing wouldto is a the con-field capacitor, the latter be- solididenticalofbetweengenerallysince capacitors. insulant,this any metalisare an two used Aextremelyor plates nearby simpleliquid (see or Table conductors,insulant), practicalfilmslarge 1-4). unit,separated capacitorand While submultiplesit is its primarilyby capacitance aconsists dielectric of a theproperty of (C)existsfarad (air,two is tweencanpacitorrentthroughnected flowflows the across in the through plates. theinto circuit theopposite the chargedthe capacitorand capacitor, directionits capacitor, voltage to chargebecause to falls dischargethe it,tocapacitor of andzero. the it. outdielectric Thus, No discharges of currentthe a cur- ca-be- portional20dielectricdirectly proportional toconstant the thickness (k) to of the the(t) area ofdielectric, the (A) dielectric of andone isplate : inversely and to the pro- quantityproportional of charge to the (Q) voltageWhen that the anda voltagecapacitor to the capacitance(E) receives is applied is (C) directly to : a given capacitor, the 21 Q = CE (1-15) Examination of Fig. 1-10B shows that maximum capacitor where, WhenECQ is thethe voltagecapacitance quantityvoltage in offirst volts. chargein isfarads, applied, in coulombs, a heavy current flows cussionIiscapacitorcurrent = zero C(i.e., (de/dt). (i.e., flowsof when current rate when when In ofthe thischange. isthe ac thezero connection,acvoltage rate voltage whenBecause of cyclechange the cycle pure seerate is in atisSection capacitanceof voltagezero)at change maximum). ; 1.1and, is in maximum foris conversely,voltage impos- aHere, dis- pacitorchargingcurrentincreases.intoThis the current voltage. ceasescapacitor voltage. When diminishesand When andthe Thus,the capacitor the ancapacitor the acvoltagewith voltage capacitor time becomes voltageacross whileis applied current the then fullythe capacitor capacitoristo leadscharged, equala capacitor, theis to voltagelow. theca- 90°losses)areresistancesible usuallyphase to preventsattain shift. and tiny. in inductance, practice, aNevertheless, practical all butcapacitorscapacitor inherentthese extraneous frompossess resistance introducing some properties (internal inherent full thecycle,charging other and half currentdischarging -cycle flows ; current into theflows capacitor out of the during capacitor one duringhalf - so an alternating current flows in the 360° E AND I 720° P thistroduceout1-10B)circuit, of 90° phase a-leading and90°but-again-not in thephase a capacitor vectorphase shift diagram Becausefeature. throughin circuit. an ac (Fig.In currentcircuit.thePure both capacitor.1-10C), capacitance theFig.leads wave 1-10current voltage plotillustrateswould and (Fig.in in- volt-a capacitor, the two are age are 90° out of step, with current leading at all points. .0-- I --I. (A) Circuit. thetance), electric consumes field during no power.Pure charge capacitance, This is returnedis because unlike to energythe resistanceac generator stored in(but like pure induc- Fig. 1-11. Power in ideal capacitor. ternalcapacitor,1-7,withcapacitorduring theFig. resistance discharge. wave 1-4,powerthe onlythe pattern ofin Thewave powertherelation sine capacitor,for pattern loss -wave inductorto currentis for thatandpattern resistor power). associated thisand in isvoltage Fig. power,small In 1-11 witha inpractical(compare and a depictsthe high- Fig. in- 90° areExceptionsationalgrade result designed capacitor.capacitors of variationsare to voltagebe undergo voltage inWhen -variable voltage, sensitive,very operating little capacitors temperature, and change withincompensating (varactors), in theircapacitanceor frequency. ratings, capaci- which as good -grade conven- 22 (B) Wave pattern. Fig. 1-10. Phase relationship: capacitance. (C) Vector diagram. equivalenttors, which capacitance are designed Whenof tothe be combinationcapacitors temperature are is sensitive.:connected in series (Fig. 1-12A), the 23 E, (APPLIED VOLTAGE) Ceq 1 -r 1 1 1 (1-16) E, (COUNTER VOLTAGE) Ec ,C2 - - - - I (CAPACITOR totalWhen capacitance capacitors ofare the connected combination in parallel is +C3+.. : (Fig. 1-12B), the / / / / / ,,/, ./ \ \ y CURRENT) 90° Ifcapacitance Ifn capacitors,only two of capacitors eachthe combination having are connectedthe same is C value= in (C1C2)/ C, (C1 + C2). Ct = Cl + C2 + C3 + series,+ Cn the equivalent are connected (1-17) 90° C/n.*series,Ifin nparallel, capacitors, the equivalent the eachtotal havingcapacitance the same of the value combinationcombination C, is C = are connected in Ct = nC. (A) Wave pattern. Fig. 1-13. Capacitor current/voltage relationship. (B) Vector diagram.E, Cl C2 C3If- C, (- - - Cl' C2 C3 c wC.developed This opposition that this opposition is termed capacitive is equal to reactance, the reciprocal is mea- of (A) Capacitors in series. Fig. 1-12. Basic capacitor circuits. (B) Capacitors in parallel. III where,sured in ohms, and is given by : X` a)C 1 = 27rfC 1 (1-18) voltage.pliedvoltage1.6 voltage, NATUREand The current latter I isOF the foris CAPACITIVE the resultinga Fig.pure voltage 1-13 capacitance. current, REACTANCEdropshows across andthe Here, relationshipE, the Eithe capacitor is counter the between the sinusoidal ap- The relations between irwfCX,current, isis is the 3.1416.2irf,the the frequency capacitancevoltage,capacitive and in reactance inhertz, reactance farads, in ohms,are ex- accordinglyoutthecarryingand ofappliedis phaseanalogous alternating increases voltage,with toeach the current.assince-as other. The thecounter frequencycounter The is emfseen counter developedvoltage inof Fig. the voltage, applied1-13-thedecreases by E( voltage two and the capacitor current an inductor opposes are where,pressed in a form often X,called is in Ohm'sohms, law for ac: X, = T , E = IX,, and I = X, (1-19) flowtorcreasesvoltageincreases, therefore of as alternatingdecreases theand offerscapacitance vice versa.andcurrent.frequency the increases,For capacitorBy a -dependent meansgiven and frequency, currentof vice calculus, opposition versa. therefore the Ait countercapaci- to thein- can be andflowingFrom because Equation in a capacitiveof the 1-19, phase EI reactanceit is is isof in in seen capacitiveamperes, volts. produces that an reactance, alternating a voltage this drop currentvoltage IX, ; 24Inc.,written Indianapolis, by William Indiana F. *Mullin For 46268. additional and published information by Howard W. Sams & on capacitance, see abc's of Capacitors Co., usedlags the on currentac. Like by inductive 90°.Capacitive reactance, reactance it is the isproperty a major that factor in all LC circuits 25 halvesquarteringSimilarly,capacitance,imparts the reactance,frequencythe for frequency doubling a given and sensitivity sofrequency,quadruples the on. frequency Appendix to these thedoubling reactance, halvesC circuits. the the capacitanceandFor reactance, a given gives the 1000 -Hz so on. where, f is in hertz, f' - 27r V LC 1 (1-21) 1.7reactance5000 COMBINED i.tF. for 126 REACTANCE common -stock IN capacitorsLC CIRCUITS between 5 pF and IllustrativecircuitITCL isis containingin3.1416. Example: henrys,farads, 1 CalculatemH and 250 the pF.resonant frequency, in kHz, of a other.andtion capacitive(series Because L and reactance, of C, these orWhen parallel phasebeing inductance Loppositerelationships-illustrated and and in capacitance are operated in combina-C), inductive reactance sign, oppose each by 1/1/(6.2832FromHere, (3.1416 Equation1 mH V2.5x =10-6) 0.001 1-21 x 10-13)= 318,309H,f, = and 1/(6.2832 = 2501/(6.2832 Hz pF= 318.3 V0.001= 2.5 x kHz.5 x x10-10x 10-7) 2.5 x F. 10-10)= where,circuitFigs. 1-6 is the and difference 1-10, Xrespectively-the XL,between and X,the are two totalin the reactance same units in (ohms, the kilohms, Xt = XL - XC : (1-20) meg- thissistance,1.9 FIGURE resistance they OF can is MERIT, low.functionSince The 0 efficiently ratiopractical of reactance inductors as reactors to and onlyresistance capacitors when have inherent re- FromoftorIllustrative this are Equation combination. operated Example: 1-12, in series XLA 16 = at wL-henry 400 =ohms, 40,212hertz. inductor etc.).Calculate ohms; and and 1-microfarad the from total Equation 1- reactance capaci- ForactsXL/Rfiguretherefore thefor = of inductortheXe/R, merit,is mostan where indicator : symbolizedpart X1in series Xof and this by with R effectiveness theare the letterin inductorohms. Q. TheThereby,and or resistanceiscapacitor. termed Q = 1.8 RESONANCE From18, X, Equation= 1/c0C = 1-20, 398 ohms. X, = 40,212 - 398 = 39,814 ohms. where, RLf isis inin hertz,henrys,ohms. Q = 27 fL R (1-22) Atandtion, a capacitive sufficientlyinductive reactance low frequencyWhen X,XL isinductance is dominant wheredominant XL and at at capacitancelow are operated in combina- >> X, thehigh phase frequencies. frequencies For the capacitor : Q = 2,771CR 1 (1-23) zerothesomewhereangleand L (since andat (0)willa sufficientlyC of at values,reach between thethat circuit apoint the maximumhigh these totalwill X, frequency two =reach XL valueextremes, -a whereX,maximum of = -90°. which 0),X, and At value the ofphase reactance, X,, of the circuit is >> XL, the phase depends upon a frequency +90°; where, TheRf is Cis inisinseries hertz, inohms. farads, resistance of a capacitor is virtually impossible 26tiontheangle resonant: is zero. frequencyThe frequency (fr) atof whichthat this situation particular LC combina- occurs is fromto ductormeasure Q measurements at directly,dc and low and (Rfrequencies usually = X,./Q). is determined is The quite resistance entirely by calculation ofthe an dc in- re - 27 suredresistancesistance dc of resistance, component the wire in at hasthe thecoil. While very high frequencies the resistance same value as the easily mea- at low frequencies, the willpure consider becapacitance, saved thefor effect separatethe ideal of purediscussion LC inductancecircuit. in The Section in effect combination 1.12. of resistance with 1.10neticarisingis a NATUREfield, combination from and OFinfluenceskin PRACTICAL of effect, de of resistance influence INDUCTOR of dielectrics in shielding. and all in -phase opposition the mag- isInductance equalvaried, to the XLand phase - CapacitanceX, and angleIn is thezero increases inseries atSeries resonance. LC to circuit+90° As at (see thethe Fig.frequency 1-15A), the total reactance Thesehasthe internal idealstray componentspure capacitance inductance,It was has mentioned inherent earlier that (termed distributed capacitance). a practical inductor, unlike resistance. It also at InductanceXLatwhich which = X,. X, XL >>and >> XL. Capacitance Xe, The and angle increases in isParallel zero to at -90° resonance, at the frequency where wire.usingcies,wireductance, The withalso thick shunting bywhichin wire Fig.skin the or,capacitance1-14.effect coilespecially Theandis wound resistanceother results at radiofactors. is occasioned It is are shown in relationship to the in- and, at very high frequen- frequencies, braided minimized by by the actance is equal to L [C In(XL the - X,)]parallel LC circuit (see Fig. 1-15B), the total re- , and is infinite at resonance. windingisbetween minimized inadjacent which by spacing turnsthe turns and the arebetween turns crisscrossed to and by special styleslayers of of turns, andfrom it capacitor effect Xt Xt = XLo -= Xe ± 90°= 0 =AT 0° RESONANCE AT RESONANCE resistancetionsimpedance,adjacency. it must has rather be already dealt than withbeen a Internalsimple pointed reactor, R andout in and in Cd combine with L to make an inductor as such. The detrimental effect of the discussion of Q many applica-destroy their an (A) L and C in series. resonatepacitancein Section with can 1.9. alimit selected Depending the range external over capacitor. which upon frequency, the distributed ca- a given inductor can Xt Xt = = ± 90° = 0° AT RESONANCE L - Xc) = 00 AT RESONANCE L Fig. 1-14. Equivalent circuit of practical inductor. (B) L and C in parallel. 1.11 PURE L AND C IN COMBINATION formanceXLangleAs the>> is X,frequency zero approachesand increasesat resonance, is varied thatAs to theof-90° over the andlosses when ideala it sufficient increasesin X, inductor inductors >>Fig. XL. 1-15.range, to and Basicand+90° capacitor.the capacitors combinations whenphase are of pure reduced, inductance per- and capacitance. 28ways present, in howeverFor small the moment, amount, wein LCwill circuits neglect and the resistance that is al- sometimesavailableIndeed, when for is asoextremely given small application, as high to be -Q ignored inductors their remainingso thatand capacitorscircuit resistance design are 29 section.cannotused.may proceed In be most ignored, as instances, if pure and inductance its however, influence and the capacitanceis resistive treated in were the being component next Z = i/R2 + (XL - k)2 = R AT RESONANCE 1.12 PRACTICAL L ANDSince C the IN COMBINATIONinherent resistance in an LC circuit usually can- = ARC0° AT TAN RESONANCE k - Xc thusindesignnot isa be practicalmost neglected,of LCoften circuits.LC an circuit itLCR must At circuit resistanceresonance, be dealt with doesreactance in most disappears, but not. An LC circuit analyses and (A) Series circuit. Z = k R2 + )(1,2 LCRpedancewe must arrangements. Zoluns talk =about A/ Rohms2 impedance, not just simple reactance. Im- X0Inns2. Fig. 1-16 shows the basic ; and when resistance enters, )6 =ARC TAN (XL R2 + (XL - k)2 (XL - k) + R2 inductorentirelySeries to andCircuit inductor capacitor. L or In Seeit may Fig. be 1-16A. the combined In this resistancecircuit, resistance of R may be due (B) Parallel circuit (inductor resistive). 1-16Aseries resistance. that the impedance Note from of thethis circuit is the any event, it is shown as the single impedance equation in Fig. simple vector z= ze + zc Zi %/ReiRcz xc2Xc2 x 01.2 + X12 iRL2 xL2 theequationsum arc oftangent forresistance the of phase the (R) total angle and reactance alsototal is reactance (XL relativelyto simple, the resistance. being Note - X). The = R1 + Rc AT RESONANCE xt.2)1 Parallelcircuitalso that resistance Circuit at No. (the 1 reactanceSee Fig. 1-16B. In this circuit,- resonance, the impedance is R. the value of the having disappeared).only the inductor has sig- 0 = ARC TAN !Riffleik (Re + xe)i - ixdR,2 + xc2)I +IRA! + xL2)i complicatedfinalwillenteringhasnificant suchneedlessly result. resistance. thea high thanNote tiny complicate Q,that capacitorthat Thiscompared of the Fig.is impedanceoften them 1-16A.resistance with the and that contribute of into of this the the calculations case, where the capacitor circuit is moreinductor, thatlittle to the 1.13 INDUCTIVE COUPLING (C) Parallel circuit (inductor and capacitor both resistive). Fig. 1-16. Basic LCR circuits. instancesoftenParallel the casein Circuit which in practice,No. the 2 inductorSee it isFig. notand 1-16C. capacitor While each the have condition shown in always so. There are many Fig. 1-16B is sivnifi- sultingonecutsone fromthe to from turnsthe the othercurrent other.of the or, Thus,flowother conversely,Two in inso inductorsthe Fig.that primary so1-17,energy that are theL1C1 energycoupledis magnetictransmitted circuit is when absorbed inducesflux thefrom re- magnetic by field of one 30all.equationcant resistance and the (Rr,phase and -angle R(.). equation Note here are thethat most the compleximpedance of a voltageclosein the together secondary, across asinductor possible L2C2, L2, circuit.and and correctly this If the causes inductorsoriented, a current so are as placed toto flowutilize as 31 MAGNETIC FLUX Capacitor -Resistor Time Constant Cl constantaresistor. standard The(T) percentage oftime the interval RCA circuit.of capacitor therequired final takesvalueto charge timeis termed or to discharge charge the time or to discharge through a as much of the flux as possible, the transfer of energy between Fig. 1-17. Inductive coupling. 36.79%voltagetoagedefined 63.2% equal equalofas of toinitialthe final 1to time - 1/E 1/Evoltage). voltage) requiredof of Numerically,the the Forinitial, orfinal, to practicalto charge discharge fullyfully the charged the purposes,charged time capacitor the constant voltagecapacitor voltage these to (i.e.,figuresa of volt-(i.e., to the toa RC combination is coupled.caseslooseofinstead,the the two coupling90°flux circuitsthe orientation), escapes, inductors results. is maximum the are Atthe transfer spaced extreme two and circuits of farther tight separation couplingare apart, completely so results. that much de - energy is minimum and (or in some If, lowingarewhere, usually manner rounded : offThe to 63% time and constant 37%, respectively.of an RC circuit is calculated in the fol- T = RC (1-25) torscoupling, is : k. The coefficientThe degree of coupling of coupling between is expressedtwo induc- by the coefficient k = V L1L2 (1-24) of ktFThus, and the 100,000 time constant ohms isCRT Tof is = an the100,000 RC timeresistancecapacitance circuit constant (2 x 10-")incontaining inohms, in farads. =seconds, 0.0002 0.002 s = where, L2L1Mk is is the the coefficient inductancemutual inductance of ofcoupling,of henrys,thethe second first between inductor, inductor, the in two in henrys, henrys. inductors, in andeitheras0.2 1-18A)for ms. C charge.R= orT/R. and C forFig.Thus,of dischargea 1-18desired theTheFor illustrates resistance thevalue(Fig.time same -constant 1-18B).of the that timeRC progress mustcircuit, constant equation be of T used charge has: Rcan thewith= Tbe(Fig. same /C, anrewritten value forto determine discharge lineresponds1.14 ofTIME force to CONSTANT 100% in the coupling.linkingThe flux This maximum is figureutilized. can value result which only ifk can reach is 1, which cor- every monconstantthestantavailable capacitance RC= T/C ofcombinations 0.005-µF 48 = seconds(1 required x 10-6)/(5capacitor Appendix =ranging withT/R x =afor 10-9)50,000 48/50,000from Da 1 gives -microsecond=0.0001 -ohm 200 =the 960ohms.resistor microfarad time µ,F. constantsSimilarly, timefor a withcon-time of a number of com- capacitortheasponse capacitor charge when to toisequal they chargedbe completed,the are source operatedthroughCapacitors voltage. i.e., ain resistor, seriesfor Similarly,and the withinductors timevoltage resistors. whenis required both a voltage When are for subject to delayed re- across the withcurrentInductor1 ohm a resistor. to to reach-Resistor 1000 Numerically, microfaradsa maximum TimeAfter Constant with application inthe an 10inductor inductor megohms. of -resistor voltage,operated timea intime series con- interval is required for i.e.,quiredis32 applied to risefor the toto anitscurrent final,inductor flowing steady in series value.through with the a inductor resistor, to time stabilize, is re- final,throughstant issteady defined the inductorvalue as (i.e.,the to time to reach 63% required a of value final for equalvalue). the currentto 1 - 1/E flowing of its 33 100% 100% 100% 80% 80% 80% O 60% ti 60%L., 60% 63% 20% 20% 40% 0 0 NUMBER OF TIME CONSTANTS1RC 2RC 3RC 4RC 0 0 1RC 2RC 3RC 4RC 20% (A) Charge. Fig. 1-18. RC time constant. NUMBER OF TIME CONSTANTS(B) Discharge. 0 0 (LIR) NUMBER OF TIME CONSTANTS 21L/R) 3(LIR) 4(L R) where,lowing manner : The time constant of an LR circuit is calculated in the fol- T = R (1-26) 1.15 OSCILLATIONS INIf LCa capacitor CIRCUIT is charged (say, with its upper plate positive, Fig. 1-19. L/R time constant Thus, the time constantL of RTis an isthe theLR inductance timecircuitresistance constant containing in in henrys,the ohmsin seconds,inductor). (including internal resistance of a 20 -henry toarrow.platedischargeductor,as theshown passingother, ThisL, throughbyconnected action,recharges the through solid the transferring acrossinductor, the+the in capacitor inductor,Fig. it, the the the1-20A) capacitor electrons electronsto as the shownand opposite thenon willfrom bythe proceedhas theonenegativepolarity, an solidplate to in- eitherLRohmsinductor circuit. is R T or = (internal L20/ for (900 a desired resistance,+ 5000)TheFig. value =time1-19 20/5900 900 of-constantillustrates ohms)time = 0.0034 constant inequation the series progress with can be5000 of rewritten current growthto determine in s = 3.4 ms. : L = RT, an andcausinginductorlapsingtheinductor. the current current fielda terminals, Butcurrent ceases then causeswhen toinduces and and flowthe a themagnetic thiscapacitor ina magnetic voltagethevoltage opposite field becomes of dischargesfield oppositeto direction expandcollapses. fully polaritythe about recharged,(see capacitor, The dotted atthe col- the ondsrequiredisavailableplusand R R inductor=is =L/TL L/with8 -henry=T. 8/0.01-coil aThus, 27 inductor resistance)-ohm =the 800 totalresistor for ohms. resistancea 10 for Similarly,-millisecond a time (external constant the time inductance resistance ofconstant 2 RT = 27 (2) = 54 H. that must be used with an sec- natinginductorpacitorstored.arrow) current chargesTheand field actionthe thus alternatelyin original first flowsthen one repeatsOnce inpolarity directionexpands the started, circuit. itself of andand the periodically thisthen collapses. capacitor action the other, would asAnto and the bealter- continueca-re-the forever, but the 34withductor 1 ohm-resistor to 1000 combinations henrysAppendix with ranging 10 E megohms. gives from the 100 time constants of a number of in- microhenrys circuitThus,which has whenabsorbs internal a singleenergy resistance impulse and eventually (see starts dotted the stifles chainR in the Fig. of oscillations. events, 1-20A) the 35 (A) Circuit. (B) Damped wave. TIME -0- CHAPTER 2 Wo (C) Sustained oscillation. TIME Fig. 1-20. Oscillations in LC circuit. The first business of the LC circuit was the tuning of radio Tuned Circuits circuitdampedoneonlyamplitude (see a singlewill waveFig. "ring" of push,1-20B) iseach the in will cycle,result. untilIn be a complete like lowertheThe the practical response to a single exciting pulse.oscillator circuit, a tube oscillationshigher finallyswing its die Q, thanout.of the a Apendulumlongerthat of thethe given or transistor preceding modernandtionthisequipment; audiosimpleof tuning applications frequencies circuitand, television. after is farstill ; more however, removed doingIt isthan doing that even three-quartersfrom job-with other in simple its jobs,most the tuning, ofboth sophisticatedadded a century, at the func-radio LC flywheel,but1-20B.atthesupplies itsconstant losses action Thus,by just oflittle amplitudemust theenoughthe pushes LC be circuit ofto circuit,energy and this in thesustains propersustained, the phase to like overcome that of a pendulumcorrect the dampingenergyactually from shown isa tubethe in oscillating or transistor. medium, oscillations Fig. or Butmanysentspresentcircuit even more, a stillnumber thischapter ofis, brief course,at of describescore, survey itsAdvancing applications thanproviding should thecan LC be highlightselectivity.the includedincircuit explanationthat function.qua the in tuner,versatilitythis introduced chapter.There and pre-of in Chapter 1, the are applications.1.16 RANGE A OFfew APPLICATION of theBoth fixed and variable LC common devices in which these cir- circuits enjoy a wideOF LC CIRCUITS range of materialChapterthis simple 1in ifcombinationthe he presentneeds toThe chapter. of strengthen passivereader shouldcomponents. his understanding return often to of Sections the 1.11 and 1.12 in wavetraps,supply),networks,cuits provide voltage and tuned test theregulators, transformers, instruments basis of filters operation arediscriminators, of and rf tuning ratio detectors,(bridges, comparators, wave (both signal and power - in2.1 this SERIES arrangement, -RESONANT generatorFig. CIRCUIT 2-lA GEN,shows inductancea series -resonant circuit, so called because L, capaci- 36meters, null devices, etc.). LThetance and resonant CC, values and internalfrequency in the following resistance (fr) of relationshipthe R circuitare connected depends in series. : upon the 37 27r\/LC 1 (2-1) where, 7rCLfr is 3.1416.the inductanceresonantcapacitance frequency in in henrys, farads, in hertz, f - (A) Circuit. FromIllustrativeHere,capacitanceof 43.75a Equation series L =Example:x 5 -resonant10-15 ofx 2-1, 10-575 pF.fr= Calculate H,=circuit1/6.283 1/6.283and Ccontaining (6.124the= V5 7.5 resonant X x 10-5 10-11 an inductance (7.5frequency F. ofin 50 x 10-8) = 1/3.845 x 10-7 = X 10-11) = 1/6.283 megahertz/LH and a rangeChas or a both singleof resonant are resonant variable, frequencies.259,895 frequency.When and Hz the = the 2.599 circuit inductanceOften, MHz. and capacitance can be adjusted overhowever, aare fixed, either the L circuitor theeachthe current.(This(E), other,inductive forcing is leaving Maximumanother reactance a current only way currentIn the and of(I)Fig. internal sayingthrough the therefore2-1A, capacitive resistancethat thethe flowstheac circuit. generator reactanceimpedance atto At delivers cancel of the a constant voltage resonance.resonance,determine lectivitysponsearesharpestmanceseries obtained and -resonant(sharpisresponse shownlowest when tuning) inheighttunerandresistance Fig. correspondsgreatest iswhen2-1B. lowest is resistance Notelowest,height at to from high (highestand is theseQ, highest. the and broadestcurvesselectivity) vice High that se- the resonance.) This perfor- versa. re- looksestingholdingLThe and likeresponseto theC note constantan frequency inductorthat in Fig.to and the constant 2-1B(in varyinggenerator series and the with the varying seriesthe internal L -resonant or C. resistanceIt is circuit inter- may be obtained by either holding generator frequency, or by Fig. 2-1. Series -resonant circuit. (B) Performance graph. FREQUENCY (I) --- isnance. resonance,ofresonant the(in latter) series voltage and at with likefrequencies step-up. thea resistor Aninternal Thisimportant above (the means resistance) internal effect that resistance)at observedat frequencies in at the reso- belowseries -resonant circuit resonance, like a capacitor resonance, the andfrequencyment,supplythelaw, suminternal for the voltage. E1, of ofinput the andresistancethe voltageFig. LCE, arecombination. 2-2 R180° dropsillustrates (E,)= 10 out is ohms. around1 of Thus,V phasethis at Since 1000XLtheeffect. with =circuitthe Hz,X. Ineachtotal = thethis 62.83 equals other,reactance resonant arrange- ohms, andthe 38However,thevoltage capacitor (EL) this acrosseachphenomenon will the be inductor higherdoes not andthan violate the the voltage Kirchhoff's (E,) second across generator voltage. thisentirelyat resonance current upon flowsis the zero, resistance; through the current the i.e., flowinginductor I = E/R in =and the 1/10 capacitor.circuit = 0.1 depends A, Theand 39 age,ducedThus,ELvoltage E.= across IXL IfEr, dropthe =and the0.1 Q produced E,of(62.83)capacitor eachthe inductor =byis 6.28 ismore this E, V current isthan high, 6 times the extent of this ; = IX, = 0.1and (62.83) the voltage = 6.28 dropV. across the inductor is the input volt- pro- puttowill beinstance,transformerlesscause voltage. be 1 A,62.8 puncture and ifV inductorinfor Fig. 1voltageor -volt flashover2-2, voltage amplificationresistance in R is 1 input. Sometimes, E, is high enoughEL and capacitor voltage E, each a capacitor rated at the in- can be surprising. For ohm, current I will nancerequiring and high that impedance the circuitThe off have series resonance. its -resonant lowest tuner circuit finds L = 20 H R = 10!! impedanceuse in applications at reso- (A) Circuit. E, = I AC INPUT V. 1000 Hz EcXc ==62.83 62.83 V SI 2.2 PARALLEL -RESONANT CIRCUIT Fig. 2-2. Resonant voltage step-up circuit. tancecapacitancecause of in this this C circuit arrangement,are connected is mainlyFig. generator2-3A in in parallel. the shows GEN, Thea parallel inductance -resonant inductor leg, as showncircuit, so called be- internal resis- L, and Fig. 2-3. Parallel -resonant circuit. (B) Performance graph. FREQUENCY --e- hasappliesCby a(fr)values, R.single alsoAsof thein toresonantin the parallelthe series parallelsame frequency. When-resonant -resonant circuit. the inductancecircuitcircuit, the and relationship. Therefore, Equation 2-1 capacitance are fixed, the circuit depends upon the L and resonant frequency Withinshown)(IL(E), thein forcing whichthe circuit, inductive area thiscurrent out current Inlegof Fig.phase and divides2-3A, I,, with in the intoeach capacitiveac two other.generator components The leg-neither delivers net re- a constant voltage (I1) through the parallel circuit. 40a rangeC or both of resonant are variable, frequencies. and the circuit canOften, be adjusted however, either L or over throughcircuitsult of the theseas inductora resulttwo currents of (See charge Section is current and 1.15 discharge 12 in flowing Chapter of withinthe 1, capacitor"Oscil- the LC 41 nance.toThelations be opposition high Atin resonance,LC across Circuit")of IL the and therefore,parallel andL. to termed each circuit, the other input the being causescirculating current, highest the I1, impedance current. isat lowerreso- Inductancebestances, rewritten either unknown for L that or C component will be unknown, : and the equation may currentthoughinfinity).cellationallelthan offcircuit (I1) theresonance of The thecirculatingishas veryresult reactance; internal (it low is would thecurrentat resistance, therefore,resonance. interesting fall (12)to the R,is This phenomenon remainingverycurrent mechanism high, cannot afterthe that inputreachis of-al- zero, except that the par- can- where, CL is the unknownavailable capacitanceinductance inin henrys,farads, L = 4ir2f2C 1 (2-2) nance.dipparallelten used,of this -circuit as current in tuningradio indicates transmitters,:If the thatimpedance the as LC a sensitivetankof the is parallel tuned indicator to -resonant reso- of circuit is plotted current Il is monitored, and a sharp aWhatIllustrative frequency471-2f valueis the is Example: of39.5. desired 1inductance MHz? An resonant accurate (in mH) frequency0.0047 will resonate -AF incapacitor thishertz, capacitance is available. to sponseareresonantagainst obtainedobtained. and frequency, circuit,least when Note height resistancefromthe the greatestwhen responsethese isresistance curves lowest,height curves that, and andis shown highest. as sharpestthe in broadest thein HighFig. series 2-3B se-re- - response Capacitance unknown10-6HFromHere,(1 CEquation= x =0.00539 1012) 4.7 x 10-9F,4.7 mH. x 10-9) and f = = 1/30.5 1 X 106 (4.7 Hz. x 103) 2-2, L = 1/(39.5 (1 x 106)2 4.7= x1/185,650 10-9) == 1/(39.55.39 X nance,ternalresonantItlectivity is interesting resistanceand (sharpcircuit like tuning) a looksto capacitorof note the Aslike corresponds thatlatter)was an at toinductorexplainedfrequencies the to generator high (in in series aboveQ,Section and the withresonance. vice 1.15parallel the versa. in in- Chapter 1, it is at frequencies below reso- neces- - where, LC is the availableunknown inductance capacitance in inhenrys, farads, 4r2f 2L 1 (2-3) circulatingsuppliesto tosarykeep an only oscillatingthe the circuitto currentneeded supply LCoscillating. pulses (I2) enoughcircuit remains to to additionalaThus, overcomeparallel constant. when -resonantenergy circuit a transistor In inlosses, this circuit,proper way,or in tube phaseorderthe a ofIllustrative 120capacitance4r2f isHz? theis Example:39.5. desired (in AF) An resonant will 8.5 resonate-henry frequency inductor this inductance is in available. hertz, to a What frequency value cuit,acwhatputlarge energy current or ascirculating ajust flywheel. in (I1)a this tank. currentmanner, BecauseThe (I2) itparallel the is frequentlyparallel -resonant -resonant termed tuner circuit acircuit tank stores cir-finds use in applications , the parallel -resonant circuit behaving some- is maintained for a small in- allel.)made=From 1/4,834,800Appendix upEquation by connecting =2-3, F2.068 gives C =one x 1/(39.5(1202) 10-7F the0.2- resonantand = 0.2068one 8.5)0.0068-µF frequency µF. = 1/(39.5(14,400)8.5) (This capacitor value of a caninnumber par- be of 2.3resonancerequiring RESONANT thatand -CIRCUIT thelow tuned impedance CONSTANTS circuit at have frequencies its highest off resonance.impedance at 2.4henrysnantcombinations SELECTIVITY andand parallel1000 frommicrofarads, -resonant 1 microhenry circuits. and applies and 10 both picofarads to series -reso-to 1000 42from Equation 2-1 when TheL and resonant C both frequency are known. of In a tuned circuit can be determined some in - It Selectivityexpresses isthe the ability sharpness of a pass of response -type circuit of a to tuned select circuit. a signal 43 circuitobtainedquencyof adesired reject andand by tuning pass -typeapplyingfrequency all circuitthe others. agenerator andconstantFig. to 2-4 aboveshows anda selectivity curve. A suppresssuppress a signal all others, of undesired or the fre-ability input voltage to the tuned plot of this type is below resonance. point,ofAfrom sharplythe maximum) saybandwidth 60tuned dB). circuit isat equal one haspoint, to goodthe frequency skirt selectivity difference fb - fa. say 3 dB, to that at another (the ratio alongcircuit.mum2-4)The dip theresonantof Voltage, (wherevoltagevertical point Fig.current, oraxis, current is2-4 indicatedfrequency wouldor in a be along inverted) the in arbitrary units may bepass plotted -typeby peak circuit, upswing or by maxi- (as in Fig. horizontal. The a reject -type and2.52-1B CIRCUIT vice and versa. 2-3B-that This permits highIt is selectivity clearthe value from of isthe circuitthe resonance result curves shown earlier-Figs. of high Q, pointed.sidesofselectivityheight100% the (called circuit,(current, point curvethe someinskirts) voltage, thisconsists illustration etc.) of its of tip themerely (called the curves are blunt nosed and .others Depending are upon the characteristics curve under study. The designates maximum nose) and its andoutputinductivelyproximated (fr)below -voltage to thebandwidth closelycoupling resonant signal by the generator,finding(BW).frequency, tuned Thisthe circuit tuning ratio is doneloosely of the resonant in tothe noting the current or volt- generator above laboratory by Q ato constant be ap- - frequency at theselectivebuttuneda selected bandwidthoperates circuit the point. circuit,overis of not Thisthe a completely thecurve-in pointIt is is apparent from the width of the narrow band of frequencies. The more narrower this band. At any point, Hz, kHz, or MHz-is thesingle width -frequency responsive, curve that a practical andtoquencyage (f.)Fig. (2)response,note at 2-4, Detune towhich the bandwidth tune value and thethe the calculating ofcircuitgenerator generator circuitat the voltage -3 below Qto-dB asthe point:the resonant ratio (1) ofWith resonant reference fre- voltage or current at this point. or currentresonance falls to to 70.7% the frequency of frequency (fr) 2-4, the bandwidth at 70.7% of maximum rise 100% usually stated. Thus, in Fig. (i.e., 3 dB down againfrequencyrent.its resonant(3) to 70.7%.Next, (fb) value. detuneat (4) which Then,This the theis calculate 3 circuitdB below voltageQ: resonant generator above resonance to the or current fallsvoltage or cur- 70.7% L - 3 dB DOWN FROM TIP OF CURVE where, all frequencies MHz). (f's) are in the same unit (Hz, kHz, or BW Iff f,. fb - f fa (2-4) thegeneratorputonantIllustrative inductor voltage frequency and thenvoltage Example:is high adjustedis (fr)detuned -inputfalls is A foundforto certain -impedancebelow 71exactly mVto resonancebetuned 1002.5 electronic circuitMHz,mV to across the andis ac testedfrequency voltmeter.the the inductor. with atThe which The (approximately 70.7% of reso- generator out- a signal res- FREQUENCY -.- fa fb ductoratorMHz.nant is detunedvoltagevoltage), Calculate again above and the fallsthis Qresonance of frequencyto this 71 circuit.mV, to the (ft)and frequencyis this 2.3 MHz. at Next,which the the in- frequency (fb) gener-is 2.7 44 Fig. 2-4. Selectivity curve. From Equation 2-4, Q = 2.7 - 2.3 - 0.4 2.5 2.5 6.25. 45 cuit) tends to reduce theDrawing selectivity energy and fromQ, because a tuned the circuit drain ("loading" the cir- C2 C4 byreflectedconstitutes lightly backloading a loss into whichthe the circuit tuned resembles whenever circuit. the This possible.equivalent action isresistance reduced INPUT 0 LI c T1 L2 - C3' L3 C5 OUTPUT 2.6 COUPLED RESONANTWhen CIRCUITS identical resonant circuits are cascaded, the selectiv- (A) Cascaded, capacitance coupled. T I forenedcircuits.ity ofganged (bandwidth the Thus, combination tuning. the reduced)tuning Often, is of higher an byan amplifyingcascadingelectronic than that equipmentseveral device-tube, of any LC one is circuits sharp-of tran- the CI C2 Cl broadennantcoupled,manysistor, circuits instancesor astuningIC-operates shown are theunder usedin resonant Fig.between controlled also 2-5A. circuitsfor successive Paradoxically, the conditions. areopposite simply LC circuits, Forcoupled capacitance example, but reso- in purpose : to L2 OUTPUT INPUT LI bandtheone other, (broader of two and circuits -nosedthe two iscurve) circuits resonated than together isat afforded a higher offer byfrequencya wider one circuit than pass - (6) Series in, series out. C2 (C) Series in, parallel out. arealone. spaced far enough apartIn Fig. to eliminate2-5A, the mutualresonant inductance circuits (L1,C1, L2,C3, L3C5) OUTPUT INPUT ofducedcuitbetweenother C2 narrows andacross is them, entirelyC4. successivethe and passband, via the capacitors couplingtanks, the depending resonant C2of and C4.voltage Although this cir- energy from one to the upon the reactance may be re- (D) Parallel in, series out. (E) Parallel in, parallel out. outputinductors.plishedmany forms. the by secondary.meansThe In inputevery of the iscase,Fig.Inductive mutualcalled 2-5 however, (B theinductancecoupling to primary E) the shows (alsocoupling between circuit, several called is theand accom-transformerexamples. two the In coupling) Fig. 2-5B, takes the loadsgreatly the influences primary, theand selectivity Inthe inductively primary of theloads coupled circuit. the secondary The circuits, secondary ; theand degree of coupling Fig. 2-5. Coupled resonant circuits. resonant.resonantin5C, orinputFig. inputoutput, 2-5D, and ; Series isand andseriesoutput input in resonance parallelFig. resonant iscircuit 2-5E,parallel resonance is each inputand used resonant outputis and for seriesfor outputlow high isand resonant;parallel-impedance -impedance outputeach isresonant; isinparallel series inputFig. input 2- tion,secondarytunedencestendingthis mutual Curve whichcircuit to broadenare action A-the resultand tuned the sharpest-correspondswhenincreasesthe tocoupling response.the a constantsame as is the changed.frequency.Fig. voltagecoupling 2-6 to showsThe is looseIn growsapplied primarythis the coupling illustra- tighter,differ- to and the 46andone,or output. in being tuned Thethe audio type parallel-in/parallel-out transformers. found in rf, if, and detector circuit transformersis a familiar circuit.condition(coils Here, inwell which the separated). coils maximum are more Curve energy closely B shows is drawnspaced, critical from so the coupling,the peak tuned is the 47 CLOSE COUPLING 1 TIGHT COUPLING IOVERCOUPLING) OUTPUT CRITICAL COUPLING LOOSE COUPLING (A) Interwound. LI = OUTER COIL INPUT L2 = INNER COIL OUTPUT FigFREQUENCY 2-6. Effect --o- of coupling. LI Fig. 2-7. Unity coupling. L4 (B) Coaxial. betweencriticalhigher, the couplingbut primary broader. and and Forare secondary saidCurve to C,be now theclose coilscauses coupled. are two closer Thepeaks-one reactionthan for (A) Link. COAXIAL LINE L3 C2 OUTPUT INPUT (B) Autotransformer. cientbroader.on each of couplingsideIn Curve of the approachesD, resonance-to we have 1).tight Here, appear. coupling the Thecurve (i.e., curve is the broadest alsocoeffi- is Cl C3 OUTPUT CI C2 L2aretremelyand (dotted shownthe twotight inline) peaks inductiveFig. are 2-7.are interwound most Ascoupling shown, prominent. is with the unity turns those A coupling. special of of secondary primary case Two of types coil coilex- INPUT INPUT anhollowL1 insulated(solid tubing, line). wire and In threaded Fig.secondary 2-7B, through primarycoil L2 this (dotted coil tubing L1 line) is coil. wound consists In unity with of (C) Common capacitor. Cl C2 (D) Common inductor. tunedin2-8Ator,coupling, radio Cl,circuits shows tunes transmitters mutual mustthe both familiar inductance be Ll spacedand andFig. link other L2.2-8 M farcoupling, showsis equipment apart. so high several Thewhich that in"links" additional awhichis single widely consist thecapaci- methodsused two of coupling. Fig. 48inductivetheof coils "cold" couplingL2 end and ofL3, which the each tank is usuallyimpossible coil 1 to to 3 obtainturns wound directly around be - (LI., L4). They provide the Fig. 2-8. Miscellaneous coupling methods. (E) Common resistor. 49 nected2-8B,aretween together stepdown, energy L1 throughand is the coupledL4. links aSince coaxial are into the low linethe turns impedance tuned ratios circuit and at low or twisted pair. In Fig. (L1:L2 and L4 :L3) may be con- imped- interposedbeingForhumps a widerconnected to some amplifiers. band, extent as severalshown and The givein tunedseparate Fig. a flatter, 2-5A,circuits resonant broadereither may withfrequencies be-nosed, employed, or without curve. re- odsshownwhich,tanksecondaryance of coil coupling.inthroughacting Li.Figs. The creates in 2-8C, a conjunction Inincludedtap each takenD,an and autotransformer.of turns athese, E fewwith thus turns therethe establish entire from is The coil,the "cold" Ll, end of are common -impedance meth- a common imped- a primaryarrangements coil as the 2.8Fig.sult 2-9B.inRANGE somewhat COVERAGE of a ripple in the passband, as shown in torFig.cuit.flowingandance, Rl. 2-8D,In the Fig.i.e., voltageinit one2-8C,isthe inductor firstwhich developedthe tuned shared isL2, shared circuitand across in by excitesFig. each each 2-8E, ofthe of the itthesesecond is tuned resistor tuned circuits, cir- impedance is capacitor C2, in by current R1. employed.atof 10 thea tunedshows upper The circuita and few capacitor lower of and these InThereoffrequency Fig.hasmethods.presetting area 2-10A, minimum many limits the a ways fixedof circuitcapacitance that ofinductor range. determining(aligning (Crni)Fig. and it) 2-variable , the operating capacitor range are pass2.7 filtering,BROADBAND a wider TUNINGIn band some of applications, frequencies such must as be passed high-fidelity radio and band- quencyasmine well as(f,,)the maximum tuningmust be range. calculated, capacitance From using Equation (C..), Cmin and 2-1, ;these then, the two upper fre- the lower deter- ingdevices,suchFig.than the broadbanding. 2-6, isseparate, such afforded overcoupling as filterscoupled, by A a more andsingle has resonant hi-fi satisfactoryalready tuned if amplifiers,circuits beencircuit. method shown each consists to of tun- From Curve D in as a means ofin fixed -tune a different (A) Fixed L, variableLI C. (B) Veriable L, fixed C. frequency.tofrequency, the lower Fig. fr., frequency 2-9Aand fr.. illustrates Oneto be of passed, the the two resulting and frequencies the other"double to thecorresponds upper -hump" LI response. Careful adjustment of the coupling will smooth the 1 fr2 fr2 1 1 fr4 Irl 1 fr3 1 fri 1 I 1 (C) Variable L, variable C. (D) Parallel trimmer capacitor. Li LI 50 (A) Two circuits. Fig. 2-9. Broadband response. (B) Several circuits. (E) Series padder capacitor. Fig. 2-10. Variable tuning arrangements. (F) Trimmer plus padder. 51 range,frequency Af, then (f,,n) is fmust be calculated, using C,. The tuning -max theder. opposite Occasionally, is true. however, as in amateur bandspread tuning, tancepoweris +fixed. Af. is rf adjusted Inductanceapplications by means may andIn bein ofFig. somevaried a slug,2-10B, of in asapplications, several indicatedinductor inL1 the the is induc- variable,sche- and faun, and the frequency coverage is fnun ways. In lowcapacitor - Cl tanceofis employed.main(C3) (Cd) tuning are of employed Boththe capacitor inductor a seriesIn for Fig.eachCl. has closepadder 2-10F, ofbeen pruning the (C2) neglected, aforegoing combination and of the parallel since capacitanceexamples, of ittrimmer theis usu- twothe range distributedpreceding methods capaci- maximum10B,twoation,havematic coils a the; or seriesin inductance the coilintransmitters series, coilof has taps can a (Lmax)one minimumor be anda rotatable aturns variometer industrialwhich -contacting inductance insidewill oscillators, (adetermine thedevice slider (L),other). consisting forthe In thiscoil of as well as a Fig.tuning 2- vari-may frequencyallyignored.pacitancesand small padder compared Forcircuits-especially are capacitances example, employed-distributed to the a capacitance popularin the where circuit. 10-mH veryof capacitance the But, smallinductor tuning, in sometuning trimmer, cannothas radio aca- dis- be - fmaxmustberange. calculated, be From calculated, Equation using using L,, In2-1, Fig. Lm. the 2-10C, upperThe tuning inductance frequency and(fmnx) capacitance must fmin,both and the frequency coverage is fd + Af. ; then, the lower frequency range, Af, then is are variable. (f,,) circuit,+circuitwhosetributed Cd =such minimumwhen 11.5 capacitance as +the Fig. 4 =variablecapacitance 2-10A,15.5 of pF. 4capacitor with pF. Residual is Ifa11.5 100-pFthis is pF, capacitance setcoil the tovariable is "zero"capacitance used suchcapacitor isin C a as= tunedin C,,,,this the ductor.quency,sincerangements,Either itandone is moremaythe however, other be amenable used Inused the Fig. to capacitor forset 2-10Dto tuning.thea dial circuit a usually smallthan In mostto istrimmer is the the practical variabletuning capacitor ar-unit, in- (C2) is connected a range -limit fre- 2.9ofis verya SELF tuning important -RESONANCE range. in determining the upper frequency limit in FromtrackingL1C1rangeinChapter parallel thecircuit.of of 1),thenature withseparate the latterThis tuning capacitanceof is andparallel tuned the capacitorthus method circuits capacitance the frequencycommonly Cl in to ganged limit coverage the-tuning setups. of the range in this circuit is AC = (see Equation 1-17 employed in the capacitance thesesomebyin parallelthis frequency,units combination. withhave the itsself selfinductance, -resonantEveryASince -resonant search inductorthe frequencies of a distributedfrequency. simple manufacturers' therefore LC ranging capacitance circuit is resonantratings fromis set (Cd)138on up at inductors of an inductor shows actsthat isally,preset true.the(Clmax however, tuning value capacitor,for as C2. in amateur InIn andthis Fig. C2arrangement, bandspread 2-10E, the a small tuning, Cl padderconventionally the opposite capacitor (C2) is connected C2) - (C11.1, + C2). This assumes, of preset trimmer. Occasion- course, a single fromductorwinding.kHz tothe whose690 intendedWhen MHz, self designing operating-resonantdepending a tunedrangefrequency upon circuit,of inductance the is circuit. asone far selects andas possible type an in- of 1-16cuits.Thiscapacitance.rangein seriesschemein ofFrom Chapter the with lattertheSometimes,is often 1),naturetuning and the found to capacitanceof capacitor reduceC2 series in is superheterodyne its capacitance Cl maximumin tothis limit circuit (see and theoscillator isminimum AC =cir- 1/ fixed, rather than variable. capacitance Equation mentended.2.10(or SYMMETRICAL is However,both)double tuned ended, a circuits. respectableCIRCUITS Thei.e., requiringtuned These amount circuits arrangements balanced of shown electronic input are up or known to outputequip- this alsopoint are all single 52conventionallycourse,{(1/ (CI.) a single - 1/C2]is preset the tuning- value[(1/Cimin) capacitor, for C2. - In1/C2]. and this C2 arrangement, This the preset pad Cl - assumes, of as eachbalanced, of these, pushpull, inductor or full Fig.Ll is-wave. 2-11 divided shows into two two formsidentical of thehalves, symmetrical circuit. In 53 the common point (center tap) usually being grounded. The LI LI CI H - halfthevoltage voltage of the between inductorbetween COMMON COMMONis tuned by and and an A identical B.is 180'In Fig. out capacitor-CI 2-11A, of phase each with VOLTAGEDC CONTROL INPUT L2 VOLTAGE INPUTDC CONTROL - L2 halfcircuitwillsplitmustfor the beof -statorbe thusofferedinductanceupper continuously isvariable half,identical for eachLlC2 capacitor, tunable,resonates for toat theallthe settings. L2C2lowerCl so with andthat half. half. C2 capacitanceThethe are That same WhenL1C1 the is, capacitance halves halfthethe Cl circuittoof of the upper a (A) Saturable reactor method, parallel resonant. (B) Saturable reac or method, series resonant. bepacitancesame in afrequency single C2. -ended For as thisthe circuit lowerreason, forhalf Ll the of is same thetwice inductance frequency. the size itwith Inwould Fig. ca- + VOLTAGE DC CONTROL INPUT COMMONtheplicationsentireductor2-11B, capacitor coil. (L1),a single requiring andThis and may fromcapacitance latterthe exhibit resonantaB variablearrangementto COMMON. unequal(C1) frequency capacitor, tunes iscapacitance notthe is since suitablecentercalculated the -tappedfrom inframe forall A thein- ofto ap- Fig. 2-12. Tuned, dc controlled circuits. (C) Varactor method. tappedisbe employed,employed coil. withthe link the symmetricalcoilThe iscoupling wound tanks. methodsaround When the shown linkcenter in ofFigs. the 2-5E and 2-8A coupling may I,andpacitorsecondaryondaryitymary are of for windingtheconnected areserieswinding core coilconnected resonance and(L1 L3 (L3). in therebyand series for ThecapacitorL2 in parallel changes inbucking,Fig.resonant series) 2-12B. Cl. resonancethe changescircuitsoThe inductance Primarythat secondary isno inthe comprised acFig.coils permeabil- of from the2-12A,and LI sec-and theca- of LI COMMON LI COMMON ternatingflowingsensitivitysecondary through current isof fed the theback incircuit primarythe to L3C1 theis such dcwinding tank.control that Special acan -voltagesmall control core direct source. amaterials large current The al- (A) Dual capacitor. Fig. 2-11. Symmetrical -tuned circuits. (B) Single capacitor. minedodeisFig.afford operated positive).2-12C. operationby the in This inductance theIn at thisisreverse higha speciallycircuit,A frequencies. varactorof-biased coil the processed Ll resonantmode(voltage and (anodethe silicon frequency-variable dc -controllednegative, diode capacitor), is whichdeter- cath- ca- D1, is employed in varioustroldc2.11 of voltage. DCapparatus, automation -TUNED This CIRCUITSautomaticis processesconvenientCertain frequency aid LCin telemetering.remote circuits control tuning, can be remotetuned by means of an adjustable (afc), andFig. 2-12 con- Isolatingthevoltage;whichpacitance varactorvaractor-rather prevents its ofresistance capacitance the(usually varactor.the coil R1than 1000 fromis Capacitor verycapacitorchosen x) short-circuiting for high lowest muchCl (usually Cl-tunesis a higher reactance,blocking the1 thethanto dc 10capacitor circuit. control sothat meg- that of 54Inshows this twotransformerlike methods.A device, saturable dc reactor flowing (T,) through is employed the pri- in Fig. 2-12A and B. inglythereohms) isno ; virtuallyand voltage since no dropthe current varactor across through it.is Theessentially thisvaractor resistor voltage and and dc operated, accord-control 55 tovoltage tune theare circuitselected over to providethe desired the capacitancefrequency range. range needed L1C1whichThislittle loss.trap tank.it captures offersA Removal parallel high as a -resonant ofcirculatingimpedance the signal trap current tocorresponds is the shown interferingflowing in toFig. insidereduction 2-13B.signal the resonantterfering2.12 WAVE signal.circuit, TRAPS It though mayA be waveusually either trap athe series is latter, a simple-resonant and LCeither device the for eliminating an in- or parallel - ca- desiredQtheciesresonantof mustthe receiver see line besignals.the circuit ascurrent, trapwith high (seeAnotheras little as awhich Sectionpracticable,lower loss. familiar is impedance,In characteristic2.2). either to positionSignals minimize type and of at forof passtrap, other attenuation theone throughthe parallel frequen-or circuit more ofto - ittenna isinstalledthepacitor shownpresent. frequencyand orthe atin inductor A anyFig.input familiar of point2-13. theof may a interferingin positionradio a besystem variable is signal. wherein the for Aline tuning between precisely to or tv receiver, the point where an interfering signalwave trap may be an an- 2.13trapwavetraps WAVEMETERSremoves isundesired the output harmonics circuit of of a the transmitter, radiated signal. where the asceivertuned,offers a higher lowtoand ground. impedance theimpedance, signal Signals accordinglytoA andseriesthe of interferingpassother -resonant by isfrequencies shuntedit to trap the aroundisreceiver, shown the inwith Fig. 2-13A. This signal to which it is see the trap trap re- strumentfromdependsis the theabsorption is uponcircuitalso calledits to wavemeter,absorption which Anotheran absorption it is familiar ofso inductively a called small frequency application becauseamount coupled. meter. itsof of rfoperation theThis energy simple in- LC tuned circuit (A) Basic. LI LI Cl (A) Series -resonant type. (B) Meter type. (C) Lamp type. LAMP orsistingBasically, amplifier(C1). of aThe fixed it,under coillike -inductance istest,the loosely Fig.wave by 2-14holding trap, coupledcoil shows (L1)is ita tosinglenear andthree the thevariable tank-tunedcommon latter, of circuitthecapacitor andversions oscillator thecon- of this instrument. Fig. 2-14. Wavemeters. 56 (B) Parallel -resonantFig. type. 2-13. Wave traps. iswavemeter read(fr) fromby adjusting is thetuned calibrated tothe resonance capacitor. dial ofat The the unknown unknowncapacitor. frequency frequency The fre- then 57 quency range (band) is changed by plugging in a different 2.14 VARACTOR FREQUENCY MULTIPLIER typeinhavecuitcoil. a tube in source),a Fig.current -type 2-14A and source,meter the is used, indeflectionaResonance collectorits outputthe unknown of milliammeter canstagethis be meter (a indicated-signal plate will in milliammeter rise in severalsharply ways. When the cir- a transistorsource must - propertywithimportantdc -variable LC arisestuned large capacitor -signalascircuits a result Theforapplication is resonantvaractorharmonicof the pronouncedof -circuitwas the generation. introducedvaractor tuning. distortion in The Theincompany Section latter most oc- 2.11 in its role as a 2-14B,andingas the resonance.C-the wavemeter a germanium wavemeter In is the tuned, diodehastwo a theother self-contained(D1) peak arrangements-Figs. rectifies point of indicator.the this picked rise indicat- In2-14B Table 2-1. Wavemeter Coil Data -up rf Fig. radioplers,highofcurring nonlinear -efficiency-frequency and when higher response. the equipment.passive -order varactor Thisfrequencymultipliers is property operated doublers, in istransmitters over utilized triplers,its entire in and modernquadru- range other (A) 1.1-3.8 MHz 72 turns No. 32 enameled1" -diameter wire closewound on Variable Capacitor C, = 140 pF plug-in form. Tap 18th turn CI LI L2 C2 (B) 3.7-12.5 MHz 21 turns No. 22 enameledbottom.from1" bottom. -diameterwire closewound plug-in form.on Tap 7th turn from FUNDAMENTAL INPUT if) HARMONIC OUTPUT Intl (D)(C) 37-150 12-39 MHz MHz Hairpin6 turns loop No. of 22 No. enameled 16 bareinginch.plug-in ofcopper wireTap form. 3rdon wire. 1"Space turn -diameter Spac- from to winding bottom. length of 3/8 1/2" between legs of hairpin. Total nometerflectionenergy and ofmay this deflects be meter used a indicatingin0-50-dc place ofmicroammeter this diode and (M1), meter peak combi- de- oflength bend. including bend: 2 inches Tap center resonance. (A thermogalva- Seriescuit.(f) Input -resonant (driving) circuit currentFig. L1C1 2-15 isat tunedtheshows fundamental to a this basic frequency. flowsvaractor frequency through Flowfrequency the leftmultiplier loop (C1L1D1) cir- of the circuit. Fig. 2-15. Basic varactor multiplier. terminalcanpilotminimizeisnation, improved belamp, plug-inbutused loading suchmay by only coil. tappingnotas ofthewhen beIn the 2-V,soFig. the tunedsensitive.)the 60-mA2-14C,indicator signal circuit. Typethe Insource-such circuit thisindicatorBut 48. circuit,this This down calls isarrangement asselectivitycoil fora radioL1 to a small a 3 - simpleHARMONICondharmonics.of this series currentarrangement -resonantThe OUTPUT through desired shown circuitEvery terminals.theharmonic invaractor (L2C2)varactor Fig. (nf) 2-15; andmultiplier is selected some,is delivered forcircuit by instance, the to is thesome adaptation of the generates a number of sec- frequencyinglamptotransmitter light a indicates140-pF the spectrum lamp.or tuning resonance.industrial In of thiscapacitor. 1.1Table arrangement, oscillator-suppliesto 1502-1 The givesMHz four coilin peak four inductors -winding brilliance bands.enough coverdataOther of for thethein- a wavemeter employ- power quency),powerPnotemploy is only output requiredbut parallelefficient it also for-resonant (PO/P, requires its operation approaches circuits.no local is suppliedpowerThe100% varactor forsupply. by doublers, the multiplier The input onlywhere sig- is power, and P, is input power, both radio -fre- 58forductance other frequenciesand capacitance (see combinationsEquations 2-1, 2-2, and may be worked out 2-3). nal itself. 59 L, XL I = EIXL CONSTANTVOLTAGE. AC CURRENT CHAPTER 3 FREQUENCYVARIABLE INPUT METER vicefrequency, versa, soas Ishown decreases in Fig. as the3-1B. frequency (If E = is1V increased, and L = 1and H, (A) Circuit. Fig. 3-1. Basic filtering action of inductor. (B) Performance graph. FREQUENCY ---- whichfilterscircuits arethe has nowLC been filter widely filters; isNext preferred, used,and toalthough theresimple often remain activetuned because circuits, (amplifierapplications it needsthe -type) most forno extensive use of LC Filters isX.meterofau,Acurrent increased,variabledecreases at M I10 isin kHz.)1.59 frequencyseries. andwith mA viceSimilarly, frequency, Here, at versa, 100is appliedthe Hz, inas resulting soFig. 0.159shown I toincreases 3-2A capacitormA in current Fig.theat 1000 asconstant 3-2B. CIthe = Hz,and E/X,..frequency (Here, andcurrentvoltage But159 by solid-statedemandpower supply continuously active and filter. usually variable The LCbecause tuningtype isthe norstill application the dominant small size indoes of notthe power - atcomparison, 100 Hz, 6.28 if mAE = at1 V1000 and Hz, C =and 1µF, 62.8 current mA at I10 is kHz.) 0.628 mA C, Xc I = E/Xc filtering.supply filtering and isFilters quite frequentlyare conveniently used in classified interference as wave filters (those CONSTANT -VOLTAGE. AC CURRENT METER todescribesremovewhich them. process ripple each fromsignals) type theand and de offers outputpower simple of-supply a rectifier).design filters data (thoseThis applicable chapter which FREQUENCYVARIABLE INPUT Since3.1 BASIC their FILTERINGreactance changesThe PROPERTIES inductor with frequency, and OF the L AND capacitor each C transmits are both basically filters. From Figs. 3-1B and 3-2B, respectively, it is seen that the (A) Circuit. Fig. 3-2. Basic filtering action of capacitor. (B) Performance graph. FREQUENCY-. capacitorresistance.ductorThisfrequencies action andin Fig. Thus,capacitor unequally,is shown3-2. in Fig. areforThese tending 3-1Apurethe examplesinductor reactance, ato constant separate inassume i.e.,Fig. voltage some that 3-1for from neitherillustrativeandof variable others.for hasthe purposes that the in- arepointfilteringquenciesfrequencies,inductor transmitted (cutoff tendsandaction whereastransmitfrequency, to withis transmit useful, little highthe fe) capacitor loss,lowitfrequencies. onis frequencieslimited,oneand tendssideon theNow, forof to other which and thereattenuate while attenuate side isfrequencies this no of low simple singlewhich high fre- 60series.frequency The isresulting applied current to inductor I = E/XL. L and But current XL increases meter M with in casefrequenciescapacitors and uniform are are attenuated. combined, (Figs. 3-1B Thehowever, and action 3-2B). into is When continuous filter inductors sections, in each and each 61 enhances the filtering action of the other. Filter sections may proaches the ideal. The constant -k type is so called because be3.2 used FILTER singly SECTIONS or cascaded. the product of the impedance (Z1) of its series arm and the CI Ll ofsuppressionbandpass, these sections. or or band bandstop As -elimination).to configuration,A (thefilter latter section Fig. type 3-3is named, isshows also as idealcalled to function action band : low-pass, high-pass, - INPUT I LI OUTPUT INPUT OUTPUT figurations.resemblescording to : the Greek Figs.or Roman 3-5 to letter3-15 givewhich circuits its schematic and data for L -type L, T, or pi. Fig. 3-4 illustrates these basic a section is named ac- con- (A) L -type. derivedhigh-pass,of these (both classes,bandpass, series two and and types shunt bandstop type). filter The sections.constant For each are shown : constant -k and m - -k type islow-pass, CI C2 LI L2 the simpler, but performance of the PASS BAND m type more closely ap- INPUT OUTPUT INPUT OUTPUT BANDSTOP STOP BAND PASS BAND (B) T -type. FREQUENCY -4. FREQUENCY -- Cl LI (A) Low pass. PASS BAND (B) High pass. STOP BAND OUTPUT INPUT Cl C2 OUTPUT BANDSTOP - - BANDSTOP BANDPASS BANDPASS Fig. 3-4. Basic filter configurations. (C) Pi -type. FREQUENCY -- fcl f 2 FREQUENCY -0- f fc2 Inimpedance the m type, (Z2) the of itsfactor shunt m arm governs equals the a constant: ratio of cutoffZ1Z2 =k2.fre- 62 (C) Bandpass. Fig. 3-3. Ideal filter action. (D) Bandstop. andquency,quency generally f.for to instance, ahas given the selectedathigh which -attenuation value transmission 0.6. frequency approaches (the zero) fre- 63 LI LI INPUT OUTPUT INPUT L2 OUTPUT O (A) Circuit. (B) Typical performance graph. FREQUENCY 1, (A) Circuit. (B) Typical performance graph. FREQUENCY , 3.3 WAVE FILTERS These are primarily signal filters and are regarded as dissi- Fig. 3-5. Low-pass filter (constant -k type). Low -Pass, Shunt -Derived m -Type Fig. 3-6. Low-pass filter (series -derived m -type). = 1 - m2 7rfei (3-6) takenpacitors).canpationless beto beapproached a networks resistance by (while Inemploying(R all in thisof the the idealequations), highsections state -Q inductorsisshown, andunattainable, the the andinput terminating ca- it impedance is See Fig. 3-7. Equations 3-7 to 3-10 describe this filter. LI selectedmuchimpedance of bythe of the selected the designer filter frequency assumes to suit the hisband. same demands Cutoff resistance forfrequencies the value filter. over are In INPUT C2 OUTPUT fci Lowinall farads,of -Pass, the equations,resistance Constant -k Rinductance Type in ohms, andL is frequency in henrys, f capacitance in hertz. C (A) Circuit. 1 1 (B) Typical performance. FREQUENCY -0- See Fig. 3-5. Equations 3-1 and 3-2 describe this filter. Ll = 7rf, R (3-1) Fig. 3-7. Low-pass filter (shunt -derived m -type). m = fe22fe12 (3-7) Low -Pass, Series -Derived m -Type Cl = 7rfell 1 (3-2) Cl = 4/rmfeiR- 1 - M2 rfeimR - (3-9)(3-8) See Fig. 3-6. Equations 3-3 to 3-6 describe this filter. m = V 1 - = mR (3-4)(3-3) High -Pass, Constant -k TypeSee Fig. 3-8. Equations 3-11 and 3-12 describe this filter. C2 = rfe2R m (3-10) 64 L2- (1 - m2)R 471-mfe, (3-5) Ll 47rfe R (3-11) 65 CI INPUT LI OUTPUT LI (A) Circuit. (13) Typical performance graph. FREQUENCY -0- lc Fig. 3-8. High-pass filter (constant -k type). Cl = 47rfeR 1 (A) Circuit.Fig. 3-10. High-pass filter (shunt -derived m -type). (B) Typical performance graph. FREQUENCY High -Pass, Series -DerivedSee m Fig. -Type 3-9. Equations 3-13 to 3-16 describe this filter. M = - iT1'22f 2 (3-17) CI Ll = L2 = (1 - m2)7rfc2 mR R (3-18) INPUT Cl = 47rmfc2R47rmfe2 1 (3-20)(3-19) lc 1 1c2 Bandpass, Constant -kSee Type Fig. 3-11. Equations 3-21 to 3-24 describe this filter. (A) Circuit.Fig. 3-9. High-pass filter (series -derived m -type). (B) Typical performance graph. FREQUENCY -- L1 = L2 = 47rfeif1c2 - felc2 R (3-22)(3-21) m= L1 = 47Tmfe2 R fe22f 2 Ll CI Cl = 477-mfe2R 1 INPUT 12 C2 OUTPUT High -Pass, Shunt -Derived m -Type C2 = (1 - m2)rfc2R (3-16) FREQUENCY --o- fci 66 See Fig. 3-10. Equations 3-17 to 3-20 describe this filter. (A) Circuit. Fig. 3-11. Bandpass filter (constant -k type). (B) Typical performance graph. 67 C2 - C 1 - Tr(fa - fei )R 117Tfo fe2Rfc2 - fel1 (3-24)(3-23) Bandpass, Shunt -Derived m -Type C3 - 47rfe2fe3zR f 3 fc2 (3-34) Bandpass, Series -DerivedSee m Fig. -Type 3-12. Equations 3-25 to 3-34 describe this filter. identify(For all x, fe's,y, z, see and Fig. m,See 3-13B.)see Fig. Equations 3-13. Equations 3-25, 3-27, 3-35 and to 3-28. 3-40 Todescribe this filter. (f 47T1,2faz f 2)R (3-35) x = 1(1 m- 1 (fc2fc3 x - fel;) (3-25)(3-26) L2Ll = - (f 3 471-fe,fay f.o)Rf 0 ) R(- (3-36) LI fe42 C1 = L3 = r (fa - fa ) R(f'3 477-fc2fom (3-38)(3-37) LI C2 = r (fa - fa) R L2 (3-39) (A) Circuit. (B) Typical performance graph. fci FREQUENCYfc2 fc3 fc4 Fig. 3-12. Bandpass filter (series -derived m -type). Y (1 - In2)fe2fe3 z _ L1 - (1 - M2) (14f,i2x 4x mR fc42/f (.12) fe42/fel2) (3-29)(3-28)(3-27) (A) Circuit. L3L2 - r (fa - fa) (fe3 yRzR fc2) (3-31)(3-30) (B) Typical performance graph. Cl - 4771,2fe3mR (fe3fa -- fa) fa (3-32) fc2 1c4 68 C2 - 477-fe2fe3YR fe3 - fc2 (3-33) Fig. 3-13. Bandpass filter (shunt -derived m -type). FREQUENCY 69 Bandstop, Constant -k Type C3 - m- 1,2)R (3-40) LI stopband.)Equation 3-41 and Fig. 3-14B,See Fig. fm is3-14. the Equationscenter 3-41 to 3-45 describe frequency of the this filter. (In INPUT L2 L3 OUTPUT L1 - fm = Vfclfe2 Trfeife2 (3-42)(3-41) fcl FREQUENCY -0- fc2 fc3 I, 4 L2 = 47 (fa (f- fe1) -f 1 R (3-43) (A) Circuit.Fig. 3-15. Bandstop filter (series -derived m -type). (fc4 (B) Typical performance graph. Cl = C2 = 47r (feife2)R(fe2 -f 2 f .1 )R (3-45)(3-44) L2 = Ll = rf f R fei(.4 )R (3-49) LI L3 = 47 (fe4 7._ fc, m (f3:4R- 1 fcl) (3-51)(3-50) INPUT L2 OUTPUT C1 - C2 = 47r (fe4 - rfeife4yRfe4 - fel fei)mR (3-52)(3-53) C2 T FREQUENCY fcl fm fc2 Bandstop, Shunt -Derived m -Type C3 = irfelfe4RXfe4 - fei (3-54) Bandstop, Series -DerivedSee Fig.m -Type 3-15. Equations 3-46 to 3-54, describe this filter. (A) Circuit. Fig. 3-14. Bandstop filter (constant -k type). (B) Typical performance graph. tify all(For fe's, m, see x, andFig. y, 3-16B.) seeSee Equations Fig. 3-16. 3-46, Equations 3-47, and 3-55 3-48. to To3-60 iden- describe this filter. L1 = (f' - fcl) R (3-55) m= fo2)(1fe32) fc4f - fe42/fe32) (3-46) L2 = (1 4- f rfclfc4xrtife4Y (3-56) x = 1(1 + felt') m 1 -ei fe32 2 (3-47) L3 = 417-4/7- (fe4 (fe4 - fei - fci) m )R (3-58)(3-57) 70 Y = 1(1 + f'3fcifc4 m (3-48) C2 = - 471-(fe4 - fei)R (3-59) 71 LI L2 Table 3-1. Ripple Frequency for Common Power Ripple Frequency Supplies SingleSingle -Phase -Phase.-Phase, Bridge CenterHalf -Wave -Tap Type of Supply (f = line -voltage frequency) 2ff (A) Circuit. safelyof the atsupply, the peak and output theFig. capacitor voltage 3-17 illustratesof (s) the 3must -Phase,-Phase rectifier. bethe DeltaStarFull able two-Wave, to basic Single types -Y of single -section 6f3f operate (B) Typical performance graph. typevaluecurrentprovidespower (Fig. of -supplythelevel 3-17B) the ac (high highervoltage filter. provides load dc Theinput outputresistance) lowercapacitor to voltage,the dc rectifiervoltage -input which type(Ede at (Fig.=low approxi- 3-17A) but has the poorer voltage regulation. The choke may approximate the peak (i.e., Ed, = 1.41 output - -input Fig. 3-16. Bandstop filter (shunt -derived m -type). Icl FREQUENCY -0 - fC2 Ica f Notemately(C2 that 0.9E,,), in inFig. each 3-17A,but instance has C1 superior in anFig. output output3-17B). capacitor -voltage Fig. 3-18 isregulation. shows typical LI required C3 = ni(f irfeife4R 4 f 1) (3-60) +0 0+ 3.4 POWER -SUPPLY FILTERSThe purpose of a power -supply filter is to remove the ripple UNFILTEREDDC INPUT CI C2 T DC OUTPUT1_ FILTERED Inacchokechokeisfrom thiscomponent) the blocks action,the coil;series dc thethe outputthe toarm flowcapacitor ground,choke and of thea cannotthe rectifier. hence,short-circuits ripple, capacitor block its In familiar suchthe isthe dethe name shunt of hence, its familiar name of a filter, the inductorripple (which is an component, and arm. The bypass. (A) Capacitor input. LI signedthethe pi capacitor or for L aconfiguration. cutoff cannot frequency short-circuitThe For power bestconsiderably -supplythe dc filter lower is a than the low-pass circuit, generally of results, it should be de- component. UNFILTEREDDC INPUT CI DC OUTPUT1_FILTERED 72mustandripple its be frequencyharmonics capable of (seeallhandling will Table be the severely3-1), maximum to ensureattenuated. direct that -current Thethe ripplechoke load Fig. 3-17. Typical single -section power -supply filters. (B) Choke input. 73 powertwo -section -supply filters filters, for capacitors increased function smoothing not only of asdc frequency output. In - LI theselective risecapacitor of elements, the chargesrectified but while also-voltage as current energy pulse is -storagefrom being zero delivereddevices to maximum, : duringto the DC + AC LI constant.outputto dczero, output load the loadcapacitor and ;thus discharges,maintaining delivering the filter its output energy voltage to the then, as the voltage subsequently decreases DC OUTPUT INPUT Cl DC OUTPUT tientto3-2.inglyfilters which (Load Emay all /I, the are wherebe R power low-pass, accomplishedin theseE is supply the A formulasconstant cursoryde delivers withload -k thewillvoltageinspection type.energy aid be of Theiranin orEquations volts,actualof will designFigs. andbe resistance 3-17 the 3-1 accord-I is andthe 3-18 shows that the quo- harmonic of the ac -supplyOccasionally, frequency-must a single be frequency-suchremoved from as a troublesome (A) Shunt type. Fig. 3-19. Resonant single -frequency filters. (B) Series type. cations.thatteamedinductanceproceduredc load the up.current filterSuch ischokesThe bypassedis an ineffectiveresulting arrangementamperes.) and highcompletely. cutoffin -capacitanceaFor large is frequencysome familiarly number applications, capacitors generally of termed possible simplythe isa brutesodesign appli- low are - Instead, on -hand high - ply,arecuitformingtheL1C1 seldom outputshort-circuitssince circuitthis theyofsatisfactory task a traps removerectifier. are the shownthe signal as signal;only Simplethe in to Fig.completeone inground. Fig. 3-19.LCfrequency combinations3-19B, filterIn These Fig. for a(at 3-19A, arrangementsseries a best power for L1C1 aa shuntvery per-sup- cir- force filter. 1m-6 LI L2 inareusednarrow Section immediately in bandconjunction 2.12, of frequencies), Chapterrecognized with 2. a regularas but the they simple power are wavetraps very-supply effective filter. described whenBoth UNFILTERED +0 C1 C2 0+ FILTERED DC INPUT -1 (A) Capacitor input. C3 1 DC OUTPUT +0 Odadal LI lira ICI L2 0+ UNFILTEREDDC INPUT Cl (B) Choke input. C2 1_ DC OUTPUT FILTERED 74 Fig. 3-18. Typical cascaded filters. I I 7' CHAPTER 4 Bridges and Other Fig. 4-1. Anderson bridge. andcillators filters, and and signal in special-purpose generators,Inductance where -capacitance meters, they wherefunction combinations they as servetanks are often seen in os- Measuring Devices calculated:The equivalent resistance (R.) of the inductor under test is (R1R2) GEN sorptioncombinationcomparedotheras selective testwavemeter equipmentwith filterswith a knownresistance),already and in as capacitance,which describedtuners. frequency an They unknownin suchSection aremeters asalso inductanceac2.13, (e.g., the bridges Chap-basis the ab-(in ofis 4.2 HAY BRIDGE See Fig. 4-2. In this bridge, unknown inductance L. is com- R = R4 (4-2) 4.1tertest 2),ANDERSON and and measuring component BRIDGE devices. testersThis chapterusing the describes resonance several method. of the better-known LC -based tanceinpared the -balanceconventionalwith standard rheostat manner capacitance R2 and by Q alternate -balance Cs. The adjustment rheostat circuit isR3. ofbalanced At induc- null: = R1R2R3 (4-3) athewith widerR2, more standard R3, range. common R4) capacitance in whichfour -impedance Seeunknown Cs. Fig. Although 4-1. inductancebridges, This harder is this a sixL. tocircuit adjustis-impedance compared offers than network (L., Cs, R1, where, All0)CsL. is isRs 27rf inin are henrys,farads, (f in is ohms. in hertz), 1 + (R32(02C82) where,pendent of each other andThe of inductance the generator balance frequency. (R3) andAt nullQ balance : (R1) are inde- L. = Cs [R3 ( 1 + R2/R4 ) + R2] (4-1) highersincedependent.inductance a) than also 10, appearsbalance However, the frequency inis thenot Sinceif equation,theindependent may R3Q of appears be the ignored, ofbalanceinductor thein theQ and isbalance. denominatorunder frequency Equation test Also, is of the fraction, the 76 AllCsL. is isRs inin are henrys,farads, in ohms. 4-3 may be simplified with an error of less than 1 percent : L. = CB(R1R2) (4-4) 77 Fig. 4-2. Hay bridge. Fig. 4-3. Maxwell bridge. der test is calculated :At null, the equivalent resistance (R8) of the inductor un- GEN calculatedThe equivalent : resistance (R8) of the inductor under test is GEN rheostatAgain note (R2) that and the Q settings-balance of rheostat both the (R3) inductance enter into -balance the cal- R. _ 1 + (co2C82R32)co2C82R1R2R3 (4-5) 4.4 OWEN BRIDGE R. = R1 (R2) R3 (4-8) thanquencyculation 10, dependent. Equation and that 4-5 However, the may R. determination,be ifsimplified the Q of :the like inductor that of is L., higher is fre- R8 = R1R2 R3 (4-6) employedpendentand Q balance asof thefrequency. Q are -balance independent TheSee component. Owen Fig. of 4-4. bridge each TheIn other providesthis inductance circuit,and eachan extremely balancea isvariable inde- capacitor (Cr) is ofdescribedwith4.3 the MAXWELL standard Q -balance in the BRIDGEcapacitor preceding rheostatSee C8. Fig. (R3)section It 4-3. differsand in This standard the from circuit parallel thecapacitor compares Hayconnection bridge (C8). unknown inductance L. quency.where,independent At null: of each otherIn this and bridge, each theis independent inductance balance of fre- and Q balance L. = Cs (R1R2) (4-7) are Fig. 4-4. Owen bridge. 78 BothC8L. isis inRsin henrys, farads,are in ohms. GEN 79 where,Atwide null inductance : range for a narrow range of Cs and R values. L, = (R1R2) (4-9) calculatedThe equivalent : resistanceBothCsL is is in (R,)Rsin henrys, farads, are of inthe ohms. inductor under test is Fig. 4-5. Resonant bridge (series type). where, Rx = R1 (--tC Cr) (4-10) GEN 4.5 RESONANCE BRIDGESCrR, and R1Cs are in farads.ohms, quency.ingtance the of circuit Atthe null, inductor a thefour unknown and-arm capacitor resistance frequency remain bridge in atthat that arm, one mak- fre- is : bridgementmeasurements,in ac bridgesof then frequency is used balancedthis for ifproperty allinductance,While offor the frequency.mayfrequency bridge becapacitance, utilized arms sensitivity In such are for and filledthean can resistanceinstance,measure- and be somewhat the of a nuisance where, f, Llis inis hertz,in henrys, f, - 271-\/L1C1 1 (4-11) andcision,valuesinpacitor)the R, balancesimple C, inso ormay thethat L.measurement -adjustment bridge beThethe made bridge resistance, arms direct becomescomponentofusually frequency. readingcapacitance, can a useful (rheostat bein Measurements knownfrequency device and/or or with forvariable inductance instead highaccurate of total ca- of pre- forvaluesCl variablethe toaudio change for spectrum tuning, ranges.A 7TCldesired whereand (Sinceis 3.1416.in by farads,thefrequency switchinghigh resonance capacitances spectrum L1 bridge to appropriate are may is needed effec- be covered by making theharmonicforreading thebridge fundamental distortion voltage),theinput amplitude -signal and,also frequency voltage.mayoffinally, the be outputof madecomparingThe a complex resonance atby null balancing this (whichtest bridgevoltage signal, the is isthebridge with thenone total Shunthightortive, decade.) -Qa -Type variable components. For C1 sharp would null for response, practical both purposes Cl and be L1 a mustcapaci- be Seriestomeasurementsof frequenciesseveral Type types of used20frequency Hz Seefor to 20theseFig. and kHz. 4-5. distortion This bridge are usually may be limited balanced only at the fre- purposes. Resonance -bridge Withmumthroughcircuitage someatand thisthe low inductors, resonantarm values and theespeciallyoffrequency See R1one areFig. containing unavoidable,when 4-6.and Because canhigh resistor reach input the the R1'high-signalresulting impedance is levels. maxi- volt- high arm in the series -type(Fig. 4-5) is a series -resonant circuit, the current 80quency1/o)Cin the determinedandupper the left total arm by reactance inductanceof the circuit. = 0, L1 and Atand only this the frequency,the capacitance internal coLresis- Cl = changedTocurrent eliminate can in Fig. introduce these 4-6 todifficulties, a enough shunt connection distortion the impedance of to the obscure inductance arm hasthe beennull. and 81 LI Fig. 4-6. Resonance bridge (shunt type). INPUT RI OUTPUT GEN (A) Circuit. totors.resonance,thecapacitance the current In series all approachingotherthrough type(parallel respects,described the -resonant impedancezero the in for theshunt circuit).high preceding arm -type-Q and inductorsIn bridgeR1this section. is arrangement, minimum isand identicalThe capaci- bal- at (B) Typical performance graph. 4.6ance BRIDGED equation -T likewiseNULL NETWORK is the same (see Equation 4-11). Fig. 4-7. Bridge null network. FREQUENCY fn tiveformednamehigh arm -Q from by inductorof capacitors thethe circuit,fact and that Cl capacitorsRx The andinductor is C2circuitthe and equivalent areLl shownresistor bridgesemployed. in seriesRl. Fig.the In TIt 4-7Aresistancethe networktakes induc- provides its a sharp null when The equivalent resistance (Rx) of inductor L1 is calculated: R, = R1 (c0C1) 2 1 (4-13) nately,inductancetively,mayof the(or beC1 inductor. theinductance) variedbeing capacitorsvaried.) equalsimultaneously Capacitances To toand may C2balance resistance at be all C1 heldto thepoints. andtune network,at R1 C2 atheAt fixedmust are nullnetwork. themade value,:be capacitances adjusted equal (Alterna- and andthe alter- where, TheClRx isand obridged inis R1 farads,27rf are -T(f in isnetwork ohms, in hertz). is used for frequency measurements where, fn is in hertz, fn - 7r-V2L1C1 1 (4-12) findsductanceforandis capacitancethe distortionused wide resonance measurementas use high measurementsas measurement aas bridgebandstop the (L1 lower (Section = filterin 1/272f2C1).(C1 vhfthe (notch orspectrum.same4.5). C2 Itfilter).manner= 1/87r2f2L),mayIn thisThis Anbe as function, deviceadvantageusedexplained or also in-also it 82 irClL1 is is 3.1416. in henrys,farads, erator,canof the be bridged network,used, is -T its andnetwork provision detector. over of anThis a commonequivalent removes ground bridge, the need for when forgen- ita 83 4.7coupling,shielded RESONANT transformer and permitsCIRCUIT atoperation ASeither MEASURING input at high or frequencies.output, DEVICE reduces cross where, L1frCl is is in in hertz, farads,henrys. Cl = 39.5fr2L1 1 (4-15) Directforfrequency, tests Measurement and as measurements. appropriate, ofFig.A Coils resonant is4-8 anda usefulshows Capacitors LC and acircuit simple dependable energized setup for adjunct atchecking either audio capacitance or or radio essarycircuitcount the capacitance.to adddistributed these Whencomponents capacitanceThis high direct accuracy tomethod of Cl, the and isinductor, of desired,usually measurement nor itthey is stray nec- are does not take into ac- capacitor.avariable vtvmForasinductance required) checking or -frequency, FETvm,The insupplies test anterms forfrequencyinductor low negligible theof -impedance resonanttest (L1), is signal. adjustedloading C1 frequency signal Meteris ofanfor generator the accurately M1peak ofL1C1 must the deflection (af circuit.setup.be knownor either rf, A herepensatesSubstitutiondifficult the forcapacitor to distributed find. Method The under for substitution andSee Capacitorstest wiring Fig. is connected4-9. capacitances.method This setupautomatically to terminals is similar X1com- to Fig. 4-8, except that noted.of meter The Ml, inductance and the correspondingthen is calculated resonant : frequency, fr, L1 = 39.5fr2C1 1 (4-14) justpacitorandcedure:The theX2 latter Cl testto (1)to place hasfrequency its With maximuma itdial terminalsin readingparallel for -capacitancepeak X1directly with deflection and variable X2in position: picofarads.open, of capacitormeter set Cl. tuning Ml.Test(2) Ad-Cl.(3) Pro- ca- where, Ll is in henrys, torUsing (C,) to the terminals shortest X1practicable and X2. leads,This additional connect the capacitance test capaci- MI AC VOLTMETERELECTRONIC C1fr is is in in hertz, farads. AC VOLTMETERELECTRONIC Fig. 4-9. Resonant circuit for substitution -type capacitance measurement. For checking a capacitor (Cl),Ll is an accurately known Fig. 4-8. Resonant circuit for component testing. pacitancenewtunedetunes capacitanceC1 theto : restore circuit, setting: peak causing deflection. Clb. the (6) meter Calculate (5) reading At this the topoint, unknown fall. read(4) De ca-the - C. = Cla - C1b (4-16) 84noted.ofinductor. meter The Ml,The capacitance and test the frequency corresponding then is is calculated: adjusted resonant for peak frequency, deflection fr, tanceited to (C1) capacitances of tuning that capacitorThe do substitution not C1.exceed methodthe maximum has the capaci- disadvantage that it is lim- 85 Angular Velocity APPENDIX A Reactance of Inductors APPENDIX B 1 Hz f 0, 6.28 5000 f 31,416 co at 1000 Hz 30252010 188.5157.1125.7 62.8 20151210 kHz 125,664 94,24875,39862,832 Inductance (L) 10 /LH 1 µ,FI Reactance (X, in ohms) 0.06280.00628 100 605040 628.3377314.1251.3 100 504025 628,318314,159251,327157,081 1000 p..H (1100 mH) µ,HmH 10 mH 628 62.8 6.280.628 200175150120 12561099 942.5754 500250200 1 MHz 6.283.141.571.257 >< 108x x108 108 1000 FlmH (1100 H) H 10 H 628,318 62,831 6283 6.28 x 108 500400300250 3142251318851571 4321.5 2.511.881.269.42 x 107108 (2)tance(1) InterpolateFor by a fx/1000.frequency for intermediate (f.) other than values 1000 of Hz,inductance. multiply the 1000 -Hz reac- 1600150012501000 10,053 942578546283 201510 5 9.426.283.14 x 107 x 107 4000300025002000 25,13318,84915,70812,566 100 5025 6.283.141.571.26 >

CD O CD z a: 211.11 rie O 0 0 0 (I) r CD

values. resistance and/or inductance intermediate for Interpolate seconds. in are above given constants Time NOTE: 0.0001 0.001 0.01 0.1 1.0 10 100 1000 1000 0.00001 0.0001 0.001 0.01 0.1 1.0 10 100 100

10-6 x 1 0.00001 0.0001 0.001 0.01 0.1 1.0 10 10

10-' X 1 -6 10 X 1 0.00001 0.0001 0.001 0.01 0.1 1.0 1.0

10-a x 1 10-7 x 1 10-6 x 1 0.00001 0.0001 0.001 0.01 0.1 0.1

10-9 X 1 10_a X 1 10-7 X 1 -6 10 X 1 0.00001 0.0001 0.001 0.01 0.01 X X X X X 10-70 1 10-9 1 10-8 1 10-7 1 10-6 1 0.00001 0.0001 0.001 0.001 X X X -" 10 1 10-10 1 10-9 1 10-8 x 1 10-7 x 1 10-6 x 1 0.00001 0.0001 0.0001

10M 1M 100K 10K 1K 100 10 1 (henrys) (ohms) Resistance Inductance Resonant Frequency Inductance ---1000pF 1 Capacitance 10 pF 100 pF (0.001 µF) 0.01 µF 0.1 µF 1 µF 10 µF 100 µF 1000 /IF 1 µH 50.3 MHz 15.9 MHz 5.03 MHz 1.59 MHz 503 kHz 159 kHz 50.3 kHz 15.9 kHz 5.03 kHz 10 µH 15.9 MHz 5.03 MHz 1.59 MHz 503 kHz 159 kHz 50.3 kHz 15.9 kHz 5.03 kHz 1591 Hz 100 µH 5.03 MHz 1.59 MHz 503 kHz 159 kHz 50.3 kHz 15.9 kHz 5.03 kHz 1591 Hz 503 Hz 1000 AH (1 mH) 1.59 MHz 503 kHz 159 kHz 50.3 kHz 15.9 kHz 5.03 kHz 1591 Hz 503 Hz 159 Hz 10 mH 503 kHz 159 kHz 50.3 kHz 15.9 kHz 5.03 kHz 1591 Hz 503 Hz 159 Hz 50.3 Hz 100 mH 159 kHz 50.3 kHz 15.9 kHz 5.03 kHz 1591 Hz 503 Hz 159 Hz 50.3 Hz 15.9 Hz 1000 mH (1H) 50.3 kHz 15.9 kHz 5.03 kHz 1591 Hz 503 Hz 159 Hz 50.3 Hz 15.9 Hz 5.03 Hz 10 H 15.9 kHz 5.03 kHz 1591 Hz 503 Hz 159 Hz 50.3 Hz 15.9 Hz 5.03 Hz 1.59 Hz 100 H 5.03 kHz 1591 Hz 503 Hz 159 Hz 50.3 Hz 15.9 Hz 5.03 Hz 1.59 Hz 0.503 Hz 1000 H 1591 Hz 503 Hz 159 Hz 50.3 Hz 15.9 Hz 5.03 Hz 1.59 Hz 0.503 Hz 0.159 Hz

-n-n -n0 0 0 0 K KKKKKK ClD CD CD CD CD .2 i.1 ci i.Oco 13) CD 7S iO .13ET ET ETc -m7 a;1. m mPmil' Fa' D (Si (S! 9 9 6' 6 6 6 6 6 0°2 2, ,. ,,,2 a a a_ O 0 o 0 0 0 o 0 0 0 co a> =- m-= N N4..'-, (4) (2), Cl) Cl) Cl) Cl) CD CD CD Cl) 7' =T 7' =T ..-^ Al7" 7' 7'm-3 3 w 0 0) a) co ...... (2) m E0 (/) ri Fr CA 0) M CD CD CD M 3 3 w CA O 7 77 7 IT) Cu Co 0, 0 ....74. O a a a N N n.Z ,.2 '2 ..-<.' Cl) a 0 Cl) Cl) co w w

3 K-0Z Ci)D3KG) K -CIZKincl.Z-nOMOKKIK Cl) Cl) Cl) co = CD 7' cDa) CD cr. CA' 3 Fi ,?.; a 2 21 0 - 6 7,3g.g.Bcoup.:ua 6(8 = 7, -a_,,ci. m g 2, 5 o ' 3 - .- , 03 o w- cl,,, N cn 07 N M = = 0 0 ,SD .- a, cD 7. E' FS cq." , . , .- a) = 92, cc) wa Om0 = 3 , wa CD a= a) = cDID , N 3 T, = a sl)2, O -, 7 -1 0.la) CA -. a NI -..< = -,..-,, a 7 ,.<,.2 ca N N w-G CA ..-<' cn FA- ca. a. 0) , 0 cn w N w

CO CD CD " CD -., -+ CD CD C) C) C) 5,::::. C) " Cp a ..., g x 8gxx8xxaax a acob x abxx8xxxx _, 0 " 0 CD CD CD _. _, CD -L a a -'....1 - , .D. i 8 8 8 8 8 0 0 8 8 8 8 CA 0. 0. 0. - . I CO I I . 0 0 cp CO To Convert FromMicroseconds Minutes To 1.667 x Multiply10-8 by MillihenrysMicroseconds NanohenrysMicrohenrysHenrysSeconds 0.0011 X1 10-8Xx 103106 MillisecondsMillihenrys SecondsMinutesMicrosecondsPicohenrys 0.0011.66710001 x 109 x 10-5 Minutes (angle) SecondsRadiansGradsDegrees 600.016670.0002910.01852 MinutesMinutes (time) (time) SecondsMillisecondsMicrosecondsHours 6060,00060.01667 x 107 NanohenrysNanofarads HenrysPicofaradsMicrofaradsFarads 110000.001 x 10-9 OhmsNanohenrys KilohmsPicohenrysMillihenrysMicrohenrys 0.001100010.001 x 10-8 PicohenrysPicofaradsOhms MicrofaradsMicrohenrysHenrysFaradsMegohms 1 x 10-1210-8 RadiansPicohenrys MinutesGradsDegreesNanohenrysMillihenrys 3437.7563.65457.29580.0011 x 10-9 SecondsRadians (angle) RadiansMinutesGradsDegreesSeconds 0.016670.00030860.000278206,265 Seconds (time)(angle) MinutesMillisecondsMicrosecondsHours 0.016667100010.00027774.848 x 106 x 10-6 96 inductiveThe LC circui and capacitive elements, , consisting 11111111111111111111111 of 11 firsttuningpopularity,electronicsas a found basic circuits widespreadselective however,state-of-the-art of radio unit was equipment.popularity of not electronics.advanced to be inThat restricted into new only areas, to radio so did devices the LC and circuit as pacitorsiestheexamples arepractical employed as are uncomplicatedLC offered circuits where Thisto presented.reinforce needed book as providespossible. the to Only keepprinciples the sufficient theWhere necessary subject involved. mathematics applicable ofmathematical inductors background and forr- ca- c- theory to support is used, of forandtypesousinductors easy tunedother of filterreference measuringandcircuits applications capacitors, thatto Thedevicesthe use usefulbookin LCresonant Chapter in combinationsbegins Chaptertables, 3,frequency with suchand 4. fundamentalSeven discusses inas Chapter angularof appendices LC manycombinations, theory2,velocity, presents special arein Chapterreactance provided variousbridges RC 1, covers the numer- andatalso.having the RL electronics advancedtime constants, abilitiesstudent, andAlthough inconversion technician, electronics the treatment factors. and may experimenter, find of theparts subject of the other and book thereaders useful level of difficulty is aimed engineerdegreePhD.Rufus (withdegrees P.in TurnerCalifornia honors) at isthe the atand University California author has both of of47State engineeringSouthern University California. and at collegeLos He Angeles, is-teaching a licensed and experience. his professional MA and and over 2500 articles. He earned his BA ICFrequencyGettingStylebook, IncludedTest Instruments Acquainted andabc'samong Its of Measurement, otherYouIntegrated With Can Sams the Build, Circuits, IC,books abc's andabc's by FETMetricsof Dr.of Integrated Circuits.Calculus, Turner for Millions, are Circuits, Solar Technical SolidState Cells abc's Writer's andof FET's, Components, Photocells, and Simple Editor's $5.95/21694 Howard4300 WEST 62ND W. ST. INDIANAPOLIS. Sams & INDIANA Co., 46268 Inc. USA ISBN: 0-672-21694-9