LECTURE 130 – VOLTAGE-CONTROLLED OSCILLATORS

1 LECTURE 130 VCOs (6/10/03)Page 130-1 ECE 6440 - Frequency Synthesizers Allen - 2003 LECTURE 130 VOLTAGE-CONTROLLED OSCILLATORS (READING: [4,6,9])ObjectiveThe objective of this presentation is examine and characterize the types of VOLTAGE-CONTROLLED OSCILLATORS compatible with both discrete and integrated Characterization of VCO s OSCILLATORS - RC- LC- Relaxation OSCILLATORS - Ring OSCILLATORS - Direct digital synthesis (DDS) Varactors SummaryLecture 130 VCOs (6/10/03)Page 130-2 ECE 6440 - Frequency Synthesizers Allen - 2003 CHARACTERIZATION OF VOLTAGE-CONTROLLED OSCILLATORSI ntroduction to VOLTAGE-CONTROLLED OscillatorsWhat is an oscillator?An oscillator is a circuit capable of maintaining electric oscillator is a periodic function, f(x) = f(x+nk) for all x and for all integers, n,and k is a OSCILLATORS use positive feedback of one form or of OSCILLATORS :OscillatorsTuned OscillatorsUntuned OscillatorsRCOscillatorsSCOscillatorsLCO scillatorsCrystalOscillatorsRelaxationOs cillatorsRingOscillatorsFig.

2 Osc-01 What are tuned OSCILLATORS ?A tuned oscillator uses a frequency-selective or tuned-circuit in the feedbackpath and is generally untuned or oscillator uses nonlinear feedback and is generally non-sinusoidalLecture 130 VCOs (6/10/03)Page 130-3 ECE 6440 - Frequency Synthesizers Allen - 2003 Types of OscillatorsRing Oscillator:Cascade of invertersFrequency of oscillation = 1 of stage delays controlled by current or power supplyHigher powerLC Oscillator:Frequency of oscillation = 1LC controlled by voltage dependent capacitance (varactor)Medium powerRelaxation Oscillator:Frequency determined by circuit time constantsControlled by currentMedium powerRC OSCILLATORS :Don t require inductorsOperate at lower frequencies (1-100 MHz) LECTURE 130 VCOs (6/10/03)Page 130-4 ECE 6440 - Frequency Synthesizers Allen - 2003 Characteristics of OSCILLATORS Frequency of oscillation Frequency tuning range as a function of the controlling variable (either voltage orcurrent) Frequency stability phase noise and jitter Amplitude stability (adjustable?)

3 Purity (harmonics) LECTURE 130 VCOs (6/10/03)Page 130-5 ECE 6440 - Frequency Synthesizers Allen - 2003 Linear Feedback Oscillator SystemSimplified block diagram:-VinVoutA(jw)F(jw)VfFig. Osc-02 The loop gain of this diagram is,LG(j ) = A(j )F(j )When the loop gain is equal to 1, oscillation [LG(j )] + Im[LG(j )] = 1 + j0 The frequency of oscillation is found from,Im[LG(j )] = 0and the gain necessary for oscillation is found from,Re[LG(j )] = 1 LECTURE 130 VCOs (6/10/03)Page 130-6 ECE 6440 - Frequency Synthesizers Allen - 2003 Linear Oscillator Amplitude StabilizationWhat determines the amplitude of the oscillator? Good (j ) and/or F(j ) must have an output-input characteristic that looks like an small amplitudes, the magnitude of the loop gain is greater than one and theoscillation the amplitude grows, the effective gain decreases and stabilizes at just theright amplitude to give an effective loop gain of :vinvoutEffective LoopGain = 1 Peak-to-peakamplitudej Pole Locations as a function of amplitudeFig.

4 Osc-03 LECTURE 130 VCOs (6/10/03)Page 130-7 ECE 6440 - Frequency Synthesizers Allen - 2003 Van der Pol Equations for OscillatorsBasic RLC oscillator and negative resistance circuit:-RiLiCiRi+-vviVanderPolLCRpiL + iC + iR + i = 0andi = f(v) = -a1v + a3v3vL = LdiLdt diLdt = vL didv = -a1 + 3a3v2iC = Cdvdt diCdt = C d2vdt2 didv |v=0 = -a1 = - 1Rn iR = vRp diRpdt = 1Rp dvdt didt = -a1dvdt + 3a3v2dvdt diLdt + diCdt + diRpdt + didt = 0 vLL + C d2vdt2 + 1Rp dvdt + -a1dvdt + 3a3v2dvdt = 0C d2vdt2 + 1Rp - a1 + 3a3v2 dvdt + vL = 0 d2vdt2 + 1 CRp - a1C + 3a3v2C dvdt + vLC = 0 LECTURE 130 VCOs (6/10/03)Page 130-8 ECE 6440 - Frequency Synthesizers Allen - 2003 Van der Pol EquationsAt start up, v is very small so that 3a3v2 0 s2V(s) + 1Rp - a1C sV(s) + V(s)

5 LC = 0 s2 + ms + 1LC = 0 Poles = - - 4LC = - m2 For j axis poles, m = steady-state, the following relationship must = 1 CRp - a1C + 3a3v2C = 0We see that the amplitude of oscillation ( osc = 1LC ) will be,V = a1 - 1Rp3a3 = 1Rn - 1Rp3a3 For V = 1V, 1Rn = 1Rp - 3a3 LECTURE 130 VCOs (6/10/03)Page 130-9 ECE 6440 - Frequency Synthesizers Allen - 2003 Open-Loop Concept of an OscillatorBasic closed-loop oscillator:H(j )X(j )Y(j )Frequency shaping network & amplifierFig. ++Oscillator oscillates when H(j ) = 1+j0 Open-loop Q:The open-loop Q is a measure of how much the closed loop system opposesvariations in the oscillation frequency. The higher the Q, the lower the phase of Q:1.) Q = o where o is the frequency of ) Q = 2 Energy StoredEnergy Dissipated per Cycle3.

6 Q = o2 dd Arg[H(j )] o Fig. 130 VCOs (6/10/03)Page 130-10 ECE 6440 - Frequency Synthesizers Allen - 2003 voltage controlled OSCILLATORS - TuningA voltage controlled oscillator (VCO) is an oscillator whose frequency can be varied by avoltage (or current).In local oscillator applications, the VCO frequency must be able to be varied over the Rxor Tx range (quickly).FrequencyTx or RxRangeVoltage tuning rangeLinear tuningNonlineartuningFig. variables: Capacitance (varactor) Current Power supplySpeed of tuning will be determined by the bandwidth of the phase lock 130 VCOs (6/10/03)Page 130-11 ECE 6440 - Frequency Synthesizers Allen - 2003 OSCILLATORSRC OSCILLATORS - Wien-Bridge OscillatorCircuit:Open-Loop Gain: For simplicity, let R1 = R2 = R and C1 = C2 = C.

7 LG(s) = K sRCs2 + 3RC s + 1(RC)2 LG(j ) = K j RC 1(RC)2 - 2 + 3RC j Equating the loop gain to 1+j0 givesK j RC 1(RC)2 - 2 + 3RC j = 1 + j0 The only way this equation can be satisfied is if o2 = 1RC and K = Osc-04 LECTURE 130 VCOs (6/10/03)Page 130-12 ECE 6440 - Frequency Synthesizers Allen - 2003 Wien-Bridge Oscillator ContinuedHow do you realize the amplifier of K = 3?VoutVin = 1+ R2R1 How does the amplitude stabilize? Thermistor (a resistor whose resistance decreases with increasing temperature) Nonlinear transfer functionExample:vinvout+-V1D1V2D2R1R1R2R 2R3R4+-RKRvoutvinSlope = R4R3R42R2+Slope = R4R3R42R1+Slope = R42R1R42R1+R4R3+2V12V2 Fig. Osc-06 VinVout+-R1R2 Fig. Osc-05 LECTURE 130 VCOs (6/10/03)Page 130-13 ECE 6440 - Frequency Synthesizers Allen - 2003 Other RC OscillatorsRC Phase-Shift Oscillator:If R1 = R2 = R3 = R and C1 = C2 = C3 = C, thenLG(j ) = (R4R3)(j RC)( RC)2[1-6( RC)2] + j RC [5 -( RC)2] osc = 16 RC and K = R4R3 = 29 Quadrature Oscillator:LG(j ) = -1(j )2R1C2R3C4 osc = 1R1C2R3C4 Other RC OSCILLATORS : Twin-tee RC oscillator, Sallen-Key bandpass filter with Q = ,Infinite gain, bandpass filter with Q = How do you tune the RC oscillator?

8 Must vary either R or C or +-R1R2R3R4C1C2C3 Fig. Osc-07 Vout+-R3C4+-RR+-R1C2 Fig. Osc-08 LECTURE 130 VCOs (6/10/03)Page 130-14 ECE 6440 - Frequency Synthesizers Allen - 2003 ExampleThe circuit shown is a RC oscillator. Find thefrequency of oscillation in Hertz and the voltagegain, K, of the voltage amplifiers necessary foroscillation. The voltage amplifiers have infiniteinput resistance and zero output loop gain can be found from the schematicshown:T(s) = VrVx = K2 1sRC+1 sRCsRC+1 = K2sRCs2R2C2 + 2sRC + 1 T(j ) = K2j RC1- 2R2C2 + j 2RC = 1 + j0We see from this equation that for oscillation to occur, the following conditions must besatisfied:1- 2R2C2 = 0andK2 = 2or osc = 1RC = 1104 10 9 = 105 radians/sec. fosc = and K = 2 = = 1nFC =1nFR = 10k R =10k S03 FEP1 KKVrC = 1nFC =1nFR = 10k R =10k S03 FES1 VxLecture 130 VCOs (6/10/03)Page 130-15 ECE 6440 - Frequency Synthesizers Allen - 2003Gm-C OscillatorsSame the quadrature oscillator only implemented in a more IC friendly +-gm2+-C2C1Vo1Vo2 Fig.

9 130-02Vo1tVo2tOpen Loop Gain = L(s) = gm1sC1 -gm2sC2 = -gm1gm2s2C1C2 Letting s = j and setting L(j ) = 1 gives, osc = gm1gm2C1C2 = gm1 C1 = gm2C2 if gm1 = gm2 and C1 = C2 This circuit is much easier to tune. If the transconductors are MOS transistors, thengm = 2K'IDWL Varying the bias current will vary gm and tune the 130 VCOs (6/10/03)Page 130-16 ECE 6440 - Frequency Synthesizers Allen - 2003 Switched Capacitor OscillatorsConceptTheoretically, the R s of any RC oscillator can be replaced by switches andcapacitors to create an SC SC Oscillator+-V1(z)C1 1 2 1C1+-C2 1 2 2C2 2 1 Vout(z)eeFig. Osc-09 1 2 osc = 1 2T2 = 1 2 fclock(really a frequency translator)The output is a sinusoid at frequency of 1 2 130 VCOs (6/10/03)Page 130-17 ECE 6440 - Frequency Synthesizers Allen - 2003LC OscillatorAll LC OSCILLATORS require feedback for oscillation to ) Hartley or Colpitts OSCILLATORS +-GmLC1C2R+-voutColpitts+-GmC L1L2R+-voutHartleyFig.

10 O = 1L C1C2C1+C2 and C2C1 = gmR o = 1(L1+L2)C and L2L1 = gmR 2.) Negative resistance LC tank o = 1LC LC-RRp+-GmC1L+-voutFig. ControlRLLecture 130 VCOs (6/10/03)Page 130-18 ECE 6440 - Frequency Synthesizers Allen - 2003 Example Clapp LC OscillatorA Clapp oscillator which is a version of the Colpitt soscillator is shown. Find an expression for the frequency ofoscillation and the value of gmRL necessary for that the output resistance of the FET, rds, and RLargecan be neglected (approach infinity).SolutionThe small-signal model for this problem is shown loop gain will be defined as Vgs/Vgs . Therefore,Vgs = -gmVgs' RL||(1/sC3) RL||(1/sC3) + 1sC1+ 1sC2 + sL 1sC2 = -gmVgs' RL(1/sC3) RL +(1/sC3) 1sC2 RL(1/sC3) RL +(1/sC3) + 1sC1+ 1sC2 + sL T(s) = VgsVgs = -gm RLsRLC3+1 1sC2 RL sRLC3+1 + 1sC1+ 1sC2 + sL = -gm RLsC2 RL + (sRLC3+1) 1sC1 + 1sC2 + sL VDDRL argeVBiasLC1C2C3 IBiasF02 FEP5 RLgmVgs'VgsRLC3C2C1L+-F02 FES5 LECTURE 130 VCOs (6/10/03)Page 130-19 ECE 6440 - Frequency Synthesizers Allen - 2003 Example ContinuedT(s) = -gm RL sC2RL + (sRLC3+1)(s2LC2+ C2C1 + 1) T(s) = -gm RL sC2RL +s3 RLC3LC2+ sRLC2C3C1 + sC3RL + s2LC2 +C2C1 +1 T(j ) = -gm RL[1+ C2C1 - 2LC2] + j [RL (C2+C3)]