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Voltage-Controlled Oscillator (VCO)

f osc. f max. slope = K vco. f min. V C. ^. ^. ^. Voltage-Controlled Oscillator (VCO). Desirable characteristics: Monotonic f osc vs. V C characteristic with adequate frequency range Well-defined K vco. +.

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Voltage-Controlled Oscillator (VCO)

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  1. fosc fmax slope = Kvco fmin VC ^ ^ ^ Voltage-Controlled Oscillator (VCO) • Desirable characteristics: • Monotonic fosc vs. VC characteristic with adequate frequency range • Well-defined Kvco + Noise coupling from VC into PLL output is directly proportional to Kvco. Prof. M. Green / U.C. Irvine

  2. Oscillator Design loop gain Barkhausen’s Criterion: If a negative-feedback loop satisfies: then the circuit will oscillate at frequency 0. Prof. M. Green / U.C. Irvine

  3. 1 inverter V2 feedback V1 V2 feedback 2 inverters V1 Inverters with Feedback (1) 1 inverter: V1 V2 1 stable equilibrium point 2 inverters: 3 equilibrium points: 2 stable, 1 unstable (latch) V1 V2 Prof. M. Green / U.C. Irvine

  4. Inverters with Feedback (2) 3 inverters forming an oscillator: V2 V1 V2 1 unstable equilibrium point due to phase shift from 3 capacitors V1 Let each inverter have transfer function Loop gain: Applying Barkhausen’s criterion: Prof. M. Green / U.C. Irvine

  5. tp tp tp VA VB VC Ring Oscillator Operation VA tp VB tp VC tp VA Prof. M. Green / U.C. Irvine

  6. Variable Delay Inverters (1) Current-starved inverter: Inverter with variable load capacitance: Vin Vout VC Vin Vout VC Prof. M. Green / U.C. Irvine

  7. ISS R R Vout- Vout+ + VC Vin+ Vin- Vin+ Vin- _ RG RG Ifast Islow Variable Delay Inverters (2) Interpolating inverter: • tp is varied by selecting weighted sum of fast and slow inverter. • Differential inverter operation and differential control voltage • Voltage swing maintained at ISSR independent of VC. Prof. M. Green / U.C. Irvine

  8. Differential Ring Oscillator − + + + + VA VD VC VB VA − − − − + additional inversion (zero-delay) VA tp tp VB VC tp tp VD Use of 4 inverters makes quadrature signals available. VA Prof. M. Green / U.C. Irvine

  9. 1  r r  Resonance in Oscillation Loop At dc: At resonance: Since Hr(0) < 1, latch-up does not occur. Prof. M. Green / U.C. Irvine

  10. Vin Vout Vout Vin LC VCO L C 2L C C realizes negative resistance Prof. M. Green / U.C. Irvine

  11. Variable Capacitance varactor = variable reactance Cj A. Reverse-biased p-n junction VR + – VR Cg B. MOSFET accumulation capacitance p-channel – VBG + n diffusion in n-well VBG accumulation region inversion region Prof. M. Green / U.C. Irvine

  12. LC VCO Variations IS IS 2L C C 2L 2L C C C C ISS 2L C C Prof. M. Green / U.C. Irvine

  13. 1 nH 3.8  108 fF 400 fF 400 fF 108 fF Cg = 108fF 1 nH 3.8  400 fF 400 fF Effect of CML Loading 1. 1. ideal capacitor load 2. 2. CML buffer load Prof. M. Green / U.C. Irvine

  14. CML Buffer Input Admittance (1) where: (note p < z) Substantial parallel loss at high frequencies  weakens VCO’s tendency to oscillate Prof. M. Green / U.C. Irvine

  15. CML Buffer Input Admittance (2) Yin magnitude/phase: Yin real part/imaginary part: magnitude imaginary phase real Contributes 2k additional parallel resistance Prof. M. Green / U.C. Irvine

  16. Cg = 108 fF 1 nH 3.8  3.8 nH 400 fF 400 fF CML Buffer Input Admittance (3) 3. CML tuned buffer load imaginary real Contributes negative parallel resistance Prof. M. Green / U.C. Irvine

  17. Cg = 108 fF 1 nH 3.8  3.8 nH 400 fF 400 fF ideal capacitor load CML Buffer Input Admittance (4) CML buffer load Loading VCO with tuned CML buffer allows negative real part at high frequencies  more robust oscillation! CML tuned buffer load Prof. M. Green / U.C. Irvine

  18. Differential Control of LC VCO Differential VCO control is preferred to reduce VC noise coupling into PLL output. Prof. M. Green / U.C. Irvine

  19. Oscillator Type Comparison Ring Oscillator LC Oscillator – slower – low Q  more jitter generation + Control voltage can be applied differentially + Easier to design; behavior more predictable + Less chip area + faster + high Q  less jitter generation – Control voltage applied single-ended – Inductors & varactors make design more difficult and behavior less predictable – More chip area (inductor) Prof. M. Green / U.C. Irvine

  20. PX(x) 1 0.5 x Random Processes (1) Random variable: A quantity X whose value is not exactly known. Probability distribution function PX(x): The probability that a random variable X is less than or equal to a value x. Example 1: Random variable Prof. M. Green / U.C. Irvine

  21. Random Processes (2) x1 x2 PX(x) 1 0.5 x Probability of X within a range is straightforward: If we let x2-x1 become very small … Prof. M. Green / U.C. Irvine

  22. Random Processes (3) Probability density function pX(x): Probability that random variable X lies within the range of x and x+dx. PX(x) pX(x) 1 0.5 x x dx Prof. M. Green / U.C. Irvine

  23. Random Processes (4) Expectation value E[X]: Expected (mean) value of random variable X over a large number of samples. Mean square value E[X2]: Mean value of the square of a random variable X2 over a large number of samples. Variance: Standard deviation: Prof. M. Green / U.C. Irvine

  24. Gaussian Function Provides a good model for the probability density functions of many random phenomena. Can be easily characterized mathematically . Combinations of Gaussian random variables are themselves Gaussian. 2 x Prof. M. Green / U.C. Irvine

  25. Joint Probability (1) Consider 2 random variables: If X and Y are statistically independent (i.e., uncorrelated): Prof. M. Green / U.C. Irvine

  26. Joint Probability (2) Consider sum of 2 random variables: y dy = dz determined by convolution of pX and pY. x dx Prof. M. Green / U.C. Irvine

  27. Joint Probability (3) * Example: Consider the sum of 2 non-Gaussian random processes: Prof. M. Green / U.C. Irvine

  28. Joint Probability (4) * 3 sources combined: Prof. M. Green / U.C. Irvine

  29. Joint Probability (5) * 4 sources combined: Prof. M. Green / U.C. Irvine

  30. Joint Probability (6) Noise sources Central Limit Theorem: Superposition of random variables tends toward normality. Prof. M. Green / U.C. Irvine

  31. Fourier transform of Gaussians: F Recall: F F -1 Variances of sum of random normal processes add. Prof. M. Green / U.C. Irvine

  32. Autocorrelation function RX(t1,t2):Expected value of the product of 2 samples of a random variable at times t1 & t2. For a stationary random process, RX depends only on the time difference for any t Note Power spectral density SX(): SX() given in units of [dBm/Hz] Prof. M. Green / U.C. Irvine

  33. Relationship between spectral density & autocorrelation function: infinite variance (non-physical) Example 1: white noise   Prof. M. Green / U.C. Irvine

  34. Example 2: band-limited white noise   For parallel RC circuit capacitor voltage noise: x Prof. M. Green / U.C. Irvine

  35. Random Jitter (Time Domain) Experiment: CLK data source RCK DATA CDR (DUT) analyzer Prof. M. Green / U.C. Irvine

  36. trigger Jitter Accumulation (1) Experiment: Observe N cycles of a free-running VCO on an oscilloscope over a long measurement interval using infinite persistence. NT Free-running oscillator output t3 t2 t4 t1 Histogram plots Prof. M. Green / U.C. Irvine

  37. Jitter Accumulation (2) proportional to  proportional to 2 Observation: As  increases, rms jitter increases. Prof. M. Green / U.C. Irvine

  38. Noise Spectral Density (Frequency Domain) Single-sideband spectral density: Power spectral density of oscillation waveform: Sv(f) 1/f3 region (-30dBc/Hz/decade) 1/f2 region (-20dBc/Hz/decade) fosc f (log scale) fosc+f Ltotal(f) given in units of [dBc/Hz] Ltotalincludes both amplitude and phase noise Prof. M. Green / U.C. Irvine

  39. Noise Analysis of LC VCO (1) noise from resistor + vc C L C L R -R inR _ active circuitry Consider frequencies near resonance: Prof. M. Green / U.C. Irvine

  40. Noise Analysis of LC VCO (2) + Spot noise current from resistor: vc C L inR _ spot noise relative to carrier power Leeson’s formula (taken from measurements): dBc/Hz Where F and1/f3 are empirical parameters. Prof. M. Green / U.C. Irvine

  41. _ Vosc + Oscillator Phase Disturbance ip(t) ip(t) Current impulse q/t ip(t) t t Vosc(t) Vosc(t) Vosc jumps by q/C • Effect of electrical noise on oscillator phase noise is time-variant. • Current impulse results in step phase change (i.e., an integration). •  current-to-phase transfer function is proportional to 1/s Prof. M. Green / U.C. Irvine

  42. change in phase charge in impulse (normalized to signal amplitude) Example 1: sine wave Example 2: square wave t t   Impulse Sensitivity Function (1) The phase response for a particular noise source can be determined at each point  over the oscillation waveform. Impulse sensitivity function (ISF): Note  has same period as Vosc. Prof. M. Green / U.C. Irvine

  43. Recall from network theory: LaPlace transform: Impulse response: time-variant impulse response Recall: ISF convolution integral:  can be expressed in terms of Fourier coefficients: from q Impulse Sensitivity Function (2) Prof. M. Green / U.C. Irvine

  44. Impulse Sensitivity Function (3) , m = 0, 1, 2, … Case 1: Disturbance is sinusoidal: (Any frequency can be expressed in terms of m and .) significant only for m = k negligible Prof. M. Green / U.C. Irvine

  45. Impulse Sensitivity Function (4) For Current-to-phase frequency response: I  osc 2osc  osc osc 2osc 2osc    Prof. M. Green / U.C. Irvine

  46. Impulse Sensitivity Function (5) Case 2: Disturbance is stochastic: MOSFET current noise: A2/Hz thermal noise 1/f noise 1/f noise thermal noise osc 2osc osc 2osc    Prof. M. Green / U.C. Irvine

  47. Impulse Sensitivity Function (6) due to thermal noise due to 1/f noise Total phase noise: n osc 2osc   Prof. M. Green / U.C. Irvine

  48. Impulse Sensitivity Function (7) noise corner frequency n (dBc/Hz) 1/(3 region: −30 dBc/Hz/decade 1/(2 region: −20 dBc/Hz/decade (log scale) Prof. M. Green / U.C. Irvine

  49. Impulse Sensitivity Function (8) t t  is higher  will generate more 1/(2 phase noise   will generate more 1/(3 phase noise Example 1: sine wave Example 2: square wave Example 3: asymmetric square wave t  Prof. M. Green / U.C. Irvine

  50. Impulse Sensitivity Function (9) Effect of current source in LC VCO: Due to symmetry, ISF of this noise source contains only even-order coefficients − c0 and c2 are dominant. _  Noise from current source will contribute to phase noise of differential waveform. Vosc + Prof. M. Green / U.C. Irvine

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