9CHARACTERISTICS OFTUNED LCCIRCUITSCHAP. 1]Tocalculate Cmin, recall that:7<X-0f _ CroaxVmin)Crainwhere7o(raax)V./o(min) /8.8=TR =—1Solving for Cmin gives us:Crain=%8.8363.4pF8.8= 41.3pFBandwidth RemainsConstant When Cis VariedWhen Cisvariedin anLC circuit, the resonantfrequency changes butthe bandwidthremains constant.Thisisimportant because it isimperative that thebandwidth of anLCcircuit remain the samethroughoutthe tuningrange. The reason that thebandwidth does notchange when Cisvariedis that the Q ofthecircuitvariesin directproportion tothe resonantfrequency. That is,if the resonantfrequency isdoubled byreducing the valueof C,thecircuit Q alsodoubles. Conversely,if thecircuit’s resonantfrequency ishalvedby increasing C,thecircuit Q is alsohalved. Since BW=fo/Q, it is obvious that there canbe nochangein the circuit’sbandwidth asCis variedthroughout its tuning range. If L rather than Cisvaried, however,the bandwidth would vary throughoutthe tuning range ofL. Specifically,if thecircuit’s resonantfrequencyis doubled by reducing the valueof L, the Q ofthe circuit is halvedand the bandwidth is quadrupled.Conversely, if thecircuit’s resonantfrequency is halved byincreasing the value of L, Q doubles and thebandwidth is reduced by afactorof 4. Thecondition of avarying bandwidth is highlyundesirable.Thisis why anLCcircuit isusually tunedby varying C rather than L.1.2CHARACTERISTICS OFA PARALLEL RESONANTLC CIRCUITFigure 1-5shows aparallel LCcircuit which is sometimes called atank circuit. As with a series LCcircuit, the resonantfrequency /0 is the frequency at which XL =Xc. Unlike aseries LCcircuit however,the resonanteffect for aparallel LCcircuit is asharp increase in thecircuit’s total impedance at/0. Thereasonforthis is that the inductive andcapacitive branch currentsareequal atthe resonantfrequency asaresult ofXL andXc beingequal. Since theinductive currentIL and thecapacitive currentIc are180° outof phase, the netortotal line currentequals zero atthe resonantfrequency. With atotalline currentIT ofzero,the tankimpedance Ztank approaches infinity atthe resonantfrequency.Practical LC TankCircuitIn apractical LCtank circuit,the inductive branchimpedance isslightly greaterthan theimpedanceofthe capacitive branch at/0. This isowing tothe factthat acoil always contains acertain amountofinternalresistance. Whenthe coil has aQ of 10 or more,thebranch currentsarepracticallyequal since thecoilresistance r,contributes very little tothe overallimpedance ofthe inductive branch. Nevertheless, IL isalways slightlyless than lcat/0. This meansthat, at/0, the netline currentIT is neverexactly zero,andasaresult thetank impedance is neveractually infinity.