EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 1 Voltage-Controlled Oscillator (VCO) VCVC f osc f min f max slope = K vco Desirable characteristics: Monotonic f osc vs. V C characteristic with adequate frequency range Well-defined K vco ^ ^ Noise coupling from V C into PLL output is directly proportional to K vco. ^ 

EECS 270C / Spring 2014Prof. 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.

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 3 Inverters with Feedback (1) V1V1 V2V2 V1V1 V2V2 1 inverter feedback V1V1 V2V2 2 inverters feedback 1 stable equilibrium point 3 equilibrium points: 2 stable, 1 unstable (latch) 1 inverter: V1V1 V2V2 2 inverters:

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 4 Inverters with Feedback (2) 3 inverters forming an oscillator: 1 unstable equilibrium point due to phase shift from 3 capacitors V1V1 V2V2 V1V1 V2V2 Let each inverter have transfer function Loop gain: Applying Barkhausen’s criterion:

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 5 Ring Oscillator Operation VAVA VBVB VCVC tptp tptp tptp VAVA VBVB VCVC VAVA tptp tptp tptp

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 6 Variable Delay Inverters (1) VCVC V in V out Current-starved inverter: Inverter with variable load capacitance: V in V out VCVC

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 7 Variable Delay Inverters (2) RR V in+ V in- V in+ V in- V out- V out+ I fast I slow RGRG RGRG I SS VCVC + _ Interpolating inverter: t p is varied by selecting weighted sum of fast and slow inverter. Differential inverter operation and differential control voltage Voltage swing maintained at I SS R independent of V C.

VAVA VBVB VCVC VDVD tptp tptp tptp tptp VAVA EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 8 Differential Ring Oscillator additional inversion (zero-delay) VAVA + − Use of 4 inverters makes quadrature signals available. VBVB + − VCVC + − VDVD + − VAVA − +

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 9 Resonance in Oscillation Loop  rr rr 1 At dc: Since H r (0) < 1, latch-up does not occur. At resonance:

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 10 LC VCO V in V out V in V out C L realizes negative resistance 2L2L CC

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 11 A. Reverse-biased p-n junction +– VRVR VRVR CjCj B. MOSFET accumulation capacitance + – V BG varactor = variable reactance Variable Capacitance V BG CgCg accumulation region inversion region p-channel n diffusion in n-well

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 12 LC VCO Variations 2L2L CC 2L2L CC 2L2L CC I SS 2L2L CC ISIS ISIS

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine ideal capacitor load 2. CML buffer load Effect of CML Loading  1 nH 400 fF C g = 108fF 1 nH 3.8  400 fF 108 fF 2.

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 14 Substantial parallel loss at high frequencies  weakens VCO’s tendency to oscillate (note p < z) where: CML Buffer Input Admittance (1)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 15 Y in magnitude/phase: Y in real part/imaginary part: magnitude phase imaginary real Contributes 2k  additional parallel resistance CML Buffer Input Admittance (2)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 16 imaginary real Contributes negative parallel resistance C g = 108 fF 3.8 nH 3.8  1 nH 400 fF CML Buffer Input Admittance (3) 3. CML tuned buffer load

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 17 Loading VCO with tuned CML buffer allows negative real part at high frequencies  more robust oscillation! ideal capacitor load CML buffer load CML tuned buffer load CML Buffer Input Admittance (4) C g = 108 fF 3.8 nH 3.8  1 nH 400 fF

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 18 Differential Control of LC VCO Differential VCO control is preferred to reduce V C noise coupling into PLL output.

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 19 Ring OscillatorLC 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) Oscillator Type Comparison

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 20 Random Processes (1) Random variable: A quantity X whose value is not exactly known. Probability distribution function P X (x): The probability that a random variable X is less than or equal to a value x x PX(x)PX(x) Example 1: Random variable

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine x PX(x)PX(x) x1x1 x2x2 Probability of X within a range is straightforward: If we let x 2 -x 1 become very small … Random Processes (2)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 22 Probability density function p X (x): Probability that random variable X lies within the range of x and x+dx x PX(x)PX(x) x pX(x)pX(x) dx Random Processes (3)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 23 Expectation value E[X]: Expected (mean) value of random variable X over a large number of samples. Mean square value E[X 2 ]: Mean value of the square of a random variable X 2 over a large number of samples. Variance: Standard deviation: Random Processes (4)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 24 Gaussian Function x 22 1.Provides a good model for the probability density functions of many random phenomena. 2.Can be easily characterized mathematically. 3.Combinations of Gaussian random variables are themselves Gaussian.

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 25 Joint Probability (1) If X and Y are statistically independent (i.e., uncorrelated): Consider 2 random variables:

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 26 Consider sum of 2 random variables: x y dx dy = dz determined by convolution of p X and p Y. Joint Probability (2)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 27 * Example: Consider the sum of 2 non-Gaussian random processes: Joint Probability (3)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 28 3 sources combined: * Joint Probability (4)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 29 4 sources combined: * Joint Probability (5)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 30 Central Limit Theorem: Superposition of random variables tends toward normality. Noise sources Joint Probability (6)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 31 Fourier transform of Gaussians: F Recall: F F -1 Variances of sum of random normal processes add.

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 32 Autocorrelation function R X (t 1,t 2 ): Expected value of the product of 2 samples of a random variable at times t 1 & t 2. For a stationary random process, R X depends only on the time difference for any t Note Power spectral density S X (  ): S X (  ) given in units of [dBm/Hz]

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 33 Relationship between spectral density & autocorrelation function: Example 1: white noise   infinite variance (non-physical)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 34 Example 2: band-limited white noise  x For parallel RC circuit capacitor voltage noise:

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 35 Random Jitter (Time Domain) Experiment: data source CDR (DUT ) analyzer CLK DATA RCK

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 36 Jitter Accumulation (1) Free-running oscillator output Histogram plots Experiment: Observe N cycles of a free-running VCO on an oscilloscope over a long measurement interval using infinite persistence. NT 11 22 33 44 trigger

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 37 Observation: As  increases, rms jitter increases. proportional to  2 proportional to  Jitter Accumulation (2)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 38 Noise Spectral Density (Frequency Domain) f osc f osc +  f Sv(f)Sv(f)  f (log scale) 1/  f 2 region (-20dBc/Hz/decade) Power spectral density of oscillation waveform : Single-sideband spectral density : L total includes both amplitude and phase noise L total (  f) given in units of [dBc/Hz] 1/  f 3 region (-30dBc/Hz/decade)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 39 Noise Analysis of LC VCO (1) active circuitry C L R -R C L + _ vcvc i nR Consider frequencies near resonance: noise from resistor

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 40 Spot noise current from resistor: C L + _ vcvc i nR Noise Analysis of LC VCO (2) Leeson’s formula (taken from measurements): Where F and  1/f 3 are empirical parameters. dBc/Hz spot noise relative to carrier power

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 41 Oscillator Phase Disturbance Current impulse  q/  t _ + V osc t t ip(t)ip(t) V osc (t) V osc 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 ip(t)ip(t) ip(t)ip(t)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 42 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): (normalized to signal amplitude) change in phase charge in impulse t  Example 1: sine wave t  Example 2: square wave Note  has same period as V osc.

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 43 Impulse Sensitivity Function (2) Recall from network theory: LaPlace transform: Impulse response: time-variant impulse response Recall: ISF convolution integral: from  q  can be expressed in terms of Fourier coefficients:

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 44 Case 1: Disturbance is sinusoidal:, m = 0, 1, 2, … negligible significant only for m = k (Any frequency can be expressed in terms of m and .) Impulse Sensitivity Function (3)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 45  I 2  osc   Impulse Sensitivity Function (4) Current-to-phase frequency response:  osc  osc      osc   2  osc   2  osc   For

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 46   osc 2  osc Case 2: Disturbance is stochastic: Impulse Sensitivity Function (5) MOSFET current noise: thermal noise 1/f noise A 2 /Hz    osc 2  osc thermal noise 1/f noise

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 47 Impulse Sensitivity Function (6)    osc 2  osc due to 1/f noise due to thermal noise Total phase noise: nn

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 48 Impulse Sensitivity Function (7) noise corner frequency  n  (log scale) (dBc/Hz) 1/(  3 region: −30 dBc/Hz/decade 1/(  2 region: −20 dBc/Hz/decade

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 49 t  t  Example 1: sine waveExample 2: square wave Impulse Sensitivity Function (8) Example 3: asymmetric square wave t   will generate more 1/(  3 phase noise is higher  will generate more 1/(  2 phase noise

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 50 Impulse Sensitivity Function (9) Effect of current source in LC VCO: V osc + _ Due to symmetry, ISF of this noise source contains only even-order coefficients − c 0 and c 2 are dominant.  Noise from current source will contribute to phase noise of differential waveform.

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 51 Impulse Sensitivity Function (10) I D varies over oscillation waveform Same period as oscillation Let Thenwhere We can use

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 52 ISF Example: 3-Stage Ring Oscillator M1AM1A M1BM1B M2AM2A M2BM2B M3AM3A M3BM3B MS1MS1 MS2MS2 MS3MS3 R1AR1A R1BR1B R2AR2A R2BR2B R3AR3A R3BR3B + V out − f osc = 1.08 GHz PD = 11 mW

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 53 ISF of Diff. Pairs for each diff. pair transistor

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 54 ISF of Resistors for each resistor

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 55 ISF of Current Sources ISF shows double frequency due to source-coupled node connection. for each current source transistor

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 56 Phase Noise Calculation Using:C out = 1.13 pF V out = 601 mV p-p q max = 679 fC = −112  f = 10 MHz

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 57 Phase Noise vs. Amplitude Noise (1)  osc t  v vv Spectrum of V osc would include effects of both amplitude noise v(t) and phase noise  (t). How are the single-sideband noise spectrum L total (  ) and phase spectral density S  (  ) related?

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 58 Phase Noise vs. Amplitude Noise (2) t t i(t)i(t) i(t)i(t) Vc(t)Vc(t)Vc(t)Vc(t) Recall that an input current impulse causes an enduring phase perturbation and a momentary change in amplitude: Amplitude impulse response exhibits an exponential decay due to the natural amplitude limiting of an oscillator...

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 59  + Phase noise dominates at low offset frequencies. Phase Noise vs. Amplitude Noise (3) 

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 60  osc Phase & amplitude noise can’t be distinguished in a signal. Sv()Sv() Amplitude limiting will decrease amplitude noise but will not affect phase noise. Phase Noise vs. Amplitude Noise (4) noiseless oscillation waveform phase noise component amplitude noise component phase noise amplitude noise 

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 61 Sideband Noise/Phase Spectral Density noiseless oscillation waveform phase noise component

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 62 Jitter/Phase Noise Relationship (1) autocorrelation functions Recall R  and S  (  ) are a Fourier transform pair: NT

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 63 Jitter/Phase Noise Relationship (2)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 64 Let Consistent with jitter accumulation measurements! Jitter/Phase Noise Relationship (3) Jitter from 1/(  noise: 2 3 ^ ^ ^ ^ ^

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 65 Jitter/Phase Noise Relationship (4) ff (dBc/Hz) dBc/Hz per decade Let f osc = 10 GHz Assume phase noise dominated by 1/(   ) 2 Setting  f = 2 X 10 6 and S  = : Let  = 100 ps (cycle-to-cycle jitter):    = 0.02ps rms (0.2 mUI rms) Accumulated jitter: 2 MHz

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 66 More generally: ff (dBc/Hz) fmfm NmNm -20 dBc/Hz per decade  rms jitter increases by a factor of 3.2 Jitter/Phase Noise Relationship (5) Let phase noise increase by 10 dBc/Hz:

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 67 Jitter Accumulation (1) K pd phase detector loop filter K vco VCO  in  out  vco  fb Open-loop characteristic: Closed-loop characteristic:

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 68 Jitter Accumulation (2) Recall from Type-2 PLL:  |G| zp  |1 + G| -40 dB/decade  (dBc/Hz) 1/(  3 region: −30 dBc/Hz/decade 1/(  2 region: −20 dBc/Hz/decade  1 80 dB/decade  As a result, the phase noise at low offset frequencies is determined by input noise...

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 69 f osc = 10 GHz Assume 1-pole closed-loop PLL characteristic Jitter Accumulation (3) ff (dBc/Hz)  f 0 = 2 MHz dBc/Hz per decade

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 70 For large  :   = 0.02 ps rms cycle-to-cycle jitter Jitter Accumulation (4)  f 0 = 2 MHz f osc = 10 GHz For small  : (log scale)    = 1.4 ps rms Total accumulated jitter

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 71 The primary function of a PLL is to place a bound on cumulative jitter:  (log scale) proportional to  (due to thermal noise) proportional to   (due to 1/f noise)  Jitter Accumulation (5)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 72 L(  ) for OC-192 SONET transmitter Closed-Loop PLL Phase Noise Measurement

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 73 Other Sources of Jitter in PLL Clock divider Phase detector Ripple on phase detector output can cause high-frequency jitter. This affects primarily the jitter tolerance of CDR.

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 74 Jitter/Bit Error Rate (1) Histogram showing Gaussian distribution near sampling point 1UI Bit error rate (BER) determined by  and UI … LR Eye diagram from sampling oscilloscope

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 75 R 0T Probability of sample at t > t 0 from left- hand transition: Probability of sample at t < t 0 from right- hand transition: Jitter/Bit Error Rate (2)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 76 Total Bit Error Rate (BER) given by: Jitter/Bit Error Rate (3)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 77 t 0 (ps) log BER Example: T = 100ps (64 ps eye opening) (38 ps eye opening) log(0.5) Jitter/Bit Error Rate (4)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 78 Bathtub Curves (1) The bit error-rate vs. sampling time can be measured directly using a bit error-rate tester (BERT) at various sampling points. Note: The inherent jitter of the analyzer trigger should be considered.

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 79 Bathtub Curves (2) Bathtub curve can easily be numerically extrapolated to very low BERs (corresponding to random jitter), allowing much lower measurement times. Example: BER with T = 100ps is equivalent to an average of 1 error per 100s. To verify this over a sample of 100 errors would require almost 3 hours! t 0 (ps)

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 80 Equivalent Peak-to-Peak Total Jitter BER , T determine BER BER determines effective Total jitter given by: Areas sum to BER