Stanford EE 133 - Oscillations ad Voltage-Controlled Oscillators

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Oscillations ad Voltage-Controlled Oscillators• Feedback perspective A=a/(1-af), if af=1 we get infinite gain…or oscillations• From EE 122 the phase-shift oscillator specifically uses series-parallel RC network to:– Make |f|=1/|a| and– Guarantee exact 0-degree phase shift• Timing-based oscillations--this can be “ring oscillator” type or “charge-discharge” (of C) type• Transistor level oscillations (which we’ll do now)s (=jω)σ−σ +σReminder abouts-plane and polesmoving into eitherLHP or RHPR1R2(-)(+)InvertingGain amp.Av~ -R2/R1Phase-shiftNetworkφ=0 and foand attenuatingby 1/Av+VccControl Logic:S1 on (S2 off)ThenS2 on (S1 off)S1S2CTimer Circuits:•Schmitt Trigger•555 IC•Many others...I = CdVdtC∆VxI=xT“x” is the portion of thetotal period for which therespective “Ix” is in controlColpitts Oscillator (Analysis and Design):Read Chapter 5 (especially 5-5) in Krauss!The following is a combination:•First-order, small-signal analysis (Dutton)•Improved “large signal” version (T. Lee)•Discussion of Krauss’ version (ala 5-5)real groundacgroundacgroundCommon Base Amp.:•Biasing like CE but!•BIG Cap at base=ac ground•Cap divider (ala Ch. 3) from collector back to emitter•TANK circuit at collectorBottom-line:CB=>non-inverting GAIN stage + Cap divider closes loop with φ=0 (I.e. oscillations)C1C2real groundacgroundacgroundC1C2rπgmvπfeedback to starting point...vπ-+LC1C2going to small-signal modelrπgmvπfeedback to starting point...vπ-+LC1C2LC1C2RinGmV1V1VtankVo=nVtankwhere “n”is Cap dividerAnd creating anequivalent two-portfor transistor (which iswritten generally…eithersmall-sig. or equivalent as a large-signal behavior...)Assume a V1:Vtank = +Gm V1 Ztankat resonance, Ztank=Req=RiT || R(where R is all other resistances* and RiT comes from the impedance Transform of Rin based on C1 & C2)Vo = n Vtank= [C1/(C1+C2)] Vtankand, if Vo = V1 we will have condition for oscillations*Footnote: This notation follows T. Lee (copy from text attached); Krauss uses a different notation: Rt=RL||Rp (where Rp is from inductance L and RL is an actual LOAD) and Ri includes both the intrinsic transistor (1/gm) and external added resistor ReAt the highest level, we can use simple feedback theory to emphasize a couple of points:a= GmReqf=C1/(C1+C2)af=1--> denominator is zero *Footnote: It turns out that, as shown in Fig. 16.6 (T. Lee book) the current flow in the device is highly non-linear (spiked in time as VBE turns on) and we really can’t use normal small-signal parameters for Gm. How to cope with that problem is discussed in Sect. 16.3.2 of T. Lee text (Ch.16)Note*: this doesn’tSpecify where itComes from...12iTRiR()=1n∴iTR=iR2n=12nmGeqR=R||iTR=R||12nmGat..resonance:tan kυ=mG1υR||12nmGæ è ç ö ø ÷ tan kυ= 2biasI()R1 +2nmGRthe..next..not − so − obvious..step..usesmG=2biasI1υ=2biasIntan kυtan kυ=2biasIR1+2n⋅ R⋅2biasIn⋅tan kυÞThis is a highlightsummary of the T. Leediscussion, Ch. 16,Section 16.3.4...LoadingVoltage gainClosing theLoop...tan kυ+n⋅ R⋅ 2biasI= 2biasIR∴tan kυ= 2biasIR⋅ 1− n()Other..notation :eqC=1C2C1C+2C;..ω=1LeqC;..n =1C1C+2CThis is the bottom-line result, giving the final tank voltage in terms of the bias current, R and the voltage divider ration n.Incidentally, if one were using the notation from Krauss, then R would actually be given by


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Stanford EE 133 - Oscillations ad Voltage-Controlled Oscillators

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