A Few Basics--Modulation, AM & FMYour favorite musicYour station’scarrier frequencyThe Amplitude Modulated(AM) version of your musicThe Frequency Modulated(FM) version of your musicDemodulation is about recovering theoriginal signal--Crystal Radio ExampleTuningCircuitDemodulationCircuitFilter(Mechanical)FMAMAntenna = Long WireA simple Diode!Basically a “tapped”Inductor (L) and variableCapacitor (C)(envelop ofAM Signal)Signal Flow in Crystal Radio--Circuit Level Issues+V-Vtimetimemusicground=0V+V (only)Wire=AntennaBWfoFilter:•fo set by LC•BW set by RLCfrequency“KX” “KY” “KZ”“tuning”Series Resonant “Tank” CircuitCLRZ jω( )=R1+jQsωωo−ωoω            where:Qs≡ωoLR=XsRs      ωo=1LCtRCLParallel Resonant “Tank” CircuitY jω( )=Gt1+jQpωωo−ωoω            where :Qp=ωoCGt=Rt1ωoC=RpXp      About notation and components:•We’ll use the “p” and “s” subscripts (per text) andtheir definitions in terms of respective R and X values(where X is either ωL or 1/ωC)•Components are NOT ideal! (see Chapter 6 reading)•Basically, when talking about resonant circuits wewant to have equivalent tank circuit (either series orparallel per above to cases)Example of non-ideal Inductor(and transforming back to equivalent tank circuit)tRCLRConversion…series-to-parallelLp=LQs2+1Qs2      Rt=R Qs2+1( )where :Qs=ωoLRtRCLpFinal equivalent resonant“tank” circuit:•transformed L now setsresonant frequency•let’s see how thesetransformations work...A bit more about “Q”We can think about (and/or measure) the quality factor (Q)in two ways--parallel (sub_p) or series (sub_s).As discussed in text (section 6.7.2) there is ultimately onlyone “Q” for the circuit, we’ll call it QcThe definitions of the parallel and series (component) Q’sare as follows:Qp=Rp/Xp (parallel or “shunt”)Qs=Xs/Rs (series)Based on simple math, comparing the series and parallelcases for either capacitor or inductor:Rp=(Qc^2+1) Rsand Qc=Qp=QsWe can derive all of the following relationships based onusing the above relationshipsConversion Relationships--CapacitorspCpRsRsCCs−eq=CpQp2+1Qp2      Rs−eq=Rp1+Qp2Qp=RpXpXp=1/ωCpCp−eq=CsQs2Qs2+1      Rp−eq=RsQs2+1( )Qs=XsRsXs=1/ωCsparallel-to-seriesseries-to-parallelpLpRsLsRLp−eq=LsQs2+1Qs2      Rp−eq=Rs1+Qs2( )Qs=XsRsXs= ωLsLs−eq=LpQp2Qp2+1      Rs−eq=Rp1+Qp2Qp=RpXpXp= ωLpparallel-to-seriesseries-to-parallelConversion Relationships--InductorsAnother Example…(a.k.a. Lab 1)original circuit...2L1LtR1RC2L1seLseRtRCconvert to series...Following steps:•Combine L1+L2(series equivalent)=L(new)•Convert L(new) & Rse back to parallel•Result gives final equivalent tank circuitDesign-Oriented Example(including use of transformer concepts*…text (Sect. 6.8.2) )2L1L1RtRN≡L1+L2L1RtR1=N2(per_book_notation:)N=nn1LT=L=L2+L1Example:R1=50ΩIn order to have Rt=2KΩ,N^2=40 or N=6.3=LT/L1*caution: this simple relationship works for Qp>10,otherwise one needs to do the transformations per above exampleUseful (more complete) Design Equations(per supplemental book by Krauss…last year’s text)2L1L1RtRCFor_Qc=Qt≈ωo/BW≥10C≈12π ⋅BW⋅RtL≈1ωo2CN=RtR1      12QtN=Qp...If _QtN≥10For_Qp≥10:Qp=QtNL1=LNL2=(N−1)L1=L−L1 For _Qp<10:Qp=Qt2+1N2−1     12L1=R1ωoQpL1_ se=L1Qp2Qp2+1L2=L−L1_