On the 1/f noise in ultra-stable quartz oscillators

Transcription

1 On the 1/ noise in ultra-stable quartz oscillators Enrico Rubiola and Vincent Giordano FEMTO-ST Institute, Besançon, France (CNRS and Université de Franche Comté) Outline Ampliier noise Leeson eect Interpretation o Sφ() Examples

2 Ampliier white noise 2 Noise igure F Input power P0 power law S ϕ = 0 i= 4 b i i V 0 cos ω 0 t n r (t) g white phase noise b 0 = F kt 0 P 0 Sφ() low P0 high P0 P0 Cascaded ampliiers (Friis ormula) N = F 1 kt 0 + (F 2 1)kT 0 g As a consequence, (phase) noise is chiely that o the 1st stage

3 Ampliier licker noise 3 parametric up-conversion o the near-dc noise S() no carrier near-dc licker S() noise up-conversion no licker ω0 =? ω0 carrier + near-dc noise v i (t) = V i e jω 0t + n (t) + jn (t) v o (t) = a 1 v i (t) + a 2 v 2 i (t) +... non-linear ampliier substitute (careul, this hides the down-conversion) expand and select the ω0 terms v o (t) = V i { a 1 + 2a 2 [ n (t) + jn (t) ]} e jω 0t get AM and PM noise α(t) = 2 a 2 a 1 n (t) ϕ(t) = 2 a 2 a 1 n (t) independent o Vi (!) the parametric nature o 1/ noise is hidden in n and n Sφ() b 1 independent o P0 m cascaded ampliiers (b 1 ) cascade = m (b 1 ) i i=1 S ϕ = In practice, each stage contributes equally 0 i= 4 b i i

4 Resonator in the phase space 4 1 theory o linear systems t Dirac δ(t) h(t) h(t) 1 H(s) H(s) Laplace Heaviside U(t) h(t) h(t) dt transorm 1/s (1/s) H(s) H(s) 2 resonator phase response cos[ω 0 t + δ(t)]?! cos [ ω 0 t + U(t) ] resonator resonator cos[ω 0 t + b(t)] cos [ ω 0 t + b(t)dt ] this is easy > cos[ ω 0 t] cos[ ω 0 t+ κ ] t=0 cos[ ω 0 t+ κ U(t)] κ > 0 linearize 3 the resonator phase response is a low-pass unction U(t) b(t) b(t) dt Laplace 1/s (1/s) B(s) transorm B(s) b(t) = 1 τ e t τ τ = 2Q ω 0 B(s) = sτ

5 The Leeson eect 5 basic eedback theory (input) + Σ + e jψ random phase A noise ree oscill out e jψ b random phase 1 noise ree main out H(s) = A 1 Aβ(s) β(s) resonator phase response use the linear-eedback theory Ψ(s) Φ(s) Ψ b (s) Φ o (s) + Σ + 1 oscill out + Σ + 1 main out H(s) = Φ(s) Ψ(s) = 1 1 B(s) = 1 + sτ sτ H(jω) 2 = 1 + ω2 τ 2 ω 2 τ 2 τ = 2Q ω 0 B(s) resonator S ϕ o () = [ ν 2 ] 0 2 4Q 2 S ψ () + S ψ b () Sφ() Sψ() 1/ 2 L

6 Interpretation o Sφ() [1] 6 real phase-noise spectrum ater parametric estimation ϕ() S x b Leeson eect? S ϕ() 1 ν = 1+ 0 check S 2 ψ () 2Q b 3 σ 2 y= 2ln(2) ν 2 0 b 1 1 b check FkT P 0 0 = b 0 L c Sanity check: power P0 at ampliier input Allan deviation σy (loor)

7 Interpretation o Sφ() [2] 7 take away the buer 1/ noise () ϕ S b 3 3 sustaining ampli 1 b 1a Leeson eect? estimate evaluate Q s = buer + sust.ampli b 0 0 ν 0 2 L L L L c ~ 6dB 2 3 buer stages => the sustaining ampliier contributes 25% o the total 1/ noise

8 Interpretation o Sφ() [3] 8 technology => Q t () ϕ S b 3 3 resonator 1/ req. noise L ν 0 = 2Qt x 2 the Leeson eect is hidden sustaining ampli L L L c Technology suggests a merit actor Qt. In all xtal oscillators we ind Qt Qs

9 Example CMAC Pharao 9 90 Courtesy o CMAC. Interpretation and mistakes are o the authors. ϕ () dbrad 2 /Hz S (b 3 ) tot = 132dB L =1.5Hz "=3Hz L c =50Hz b 0 = 152.5dB 160 technology Q=2x10 6? (b 1 ) tot = 135.5dB => L=1.25Hz c =13Hz (b 1 ) osc = 141.5dB Fourier requency, Hz F=1dB b0 => P0= 20.5 dbm (b 3)osc => σy=5.9x10 14, Q=8.4x10 5 (too low) Q 2x10 6 => σy=2.5x10 14 Leeson (too low)

10 Example Oscilloquartz ϕ () dbrad 2 /Hz S (b 3 ) tot = 128.5dB L =1.6Hz L=1.25Hz <= "=3.2Hz=> L Courtesy o Oscilloquartz. Interpretation and mistakes are o the authors. Q=2x10 6? Q=7.9x10 5 Oscilloquartz 8600 (speciications) 147 (b 1 ) tot = 132.5dB b 0 = 153dB 167 (b 1 ) osc = 138.1dB Fourier requency, Hz F=1dB b0 => P0= 20 dbm (b 3)osc => σy=8.8x10 14, Q=7.8x10 5 (too low) Q 2x10 6 => σy=3.5x10 14 Leeson (too low)

11 Example Wenzel Data are rom the manuacturer web site. Interpretation and mistakes are o the authors. 30dB/dec b 3 = 67 dbrad 2 /Hz Wenzel speciications Estimating (b 1)ampli is diicult because there is no visible 1/ region 2 /Hz Leeson eect (hidden) is about here Phase noise, dbrad ampli noise (?) b 0 = 173dBrad 2 /Hz L =3.5kHz guess Q=8x104 => L =625Hz Fourier requency, Hz F=1dB b0 => P0=0 dbm (b 3)osc => σy=5.3x10 12 Q=1.4x10 4 Q 8x10 4 => σy=9.3x10 13 (Leeson)

12 Other oscillators 12 Oscillator ν 0 (b 3 ) tot (b 1 ) tot (b 1 ) amp L L Q s Q t L (b 3 ) L R Note Oscilloquartz (1) Oscilloquartz (1) CMAC Pharao (2) FEMTO-ST LD prot (3) Agilent (4) Agilent prototype (5) Wenzel ? 138? (6) unit MHz db rad 2 /Hz db rad 2 /Hz db rad 2 /Hz Hz Hz (none) (none) Hz db rad 2 /Hz Notes (1) Data are rom speciications, ull options about low noise and high stability. (2) Measured by CMAC on a sample. CMAC conirmed that < Q < in actual conditions. (3) LD cut, built and measured in our laboratory, yet by a dierent team. Q t is known. (4) Measured by Hewlett Packard (now Agilent) on a sample. (5) Implements a bridge scheme or the degeneration o the ampliier noise. Same resonator o the Agilent (6) Data are rom speciications. db R = (σ y) oscill (σ y ) Leeson = loor (b 3 ) tot = Q t = L (b 3 ) L Q s L

13 13 Warning: an eect not accounted or still remains A luctuating impedance that aects the input without participating to the gain luctuating impedance This does not it general experience on ampliiers, yet it is to be reported

14 14 Conclusions The analysis o Sφ() provides insight in the oscillator The oscillator 1/ 3 phase noise (Allan variance loor) originates rom: ampliier 1/ noise, via the Leeson eect resonator instability In actual oscillators, the resonator instability turns out to be the dominant eect qed Full text available on Talk slides and ull text (20 pages, pd) available on We owe gratitude to J.-P. Aubry (Oscilloquartz), V. Candelier (CMAC), G.J. Dick (JPL), J. Grolambert (FEMTO-ST), L. Maleki (JPL), R. Brendel (FEMTO-ST)

15 Details 15 cos[ ω 0 t] cos[ ω 0 t+ κ ] t=0 cos[ ω 0 t+ κ U(t)] κ > 0 linearize v I (t) v I (t) initial phase κ t t v O (t) v(t) v O (t) v(t) envelope envelope τ t τ t cos(ω 0 t) e t/τ τ = 2Q cos(ω 0 t + κ) ω 0 [1 e t/τ ]

16 Details [2] cos[ ω 0 t] cos[ ω 0 t+ κ ] t=0 cos[ ω 0 t+ κ U(t)] κ > 0 linearize

17 Details [3]

18 Summary o the ampliier phase noise 18 Sϕ () b 1 ~ constant b = 0 FkT 0 P 0 b 1 1 b 0 0 low P 0 P 0 high P 0 c c depends on P 0 White PM noise is inversely proportional to P0 Flicker PM noise is about independent P0 The corner requency c ollows

19 The Leeson eect 19 A High Q, low (xtal) B Low Q, high ν 0 ν 0 (microw.) S ϕ () b 3 3 Leeson eect S ϕ () b 3 3 Leeson eect x 2 b 1 1 b 0 0 b 2 2 x b 0 0 L c c L two typical patterns

20 Example Oscilloquartz Courtesy o Oscilloquartz. Interpretation and mistakes are o the authors resonator instability b 3 = 124dBrad 2/Hz Oscilloquartz OCXO 8600 Leeson eect (hidden) sust. ampli + buer b = 131dBrad 2/Hz 1 b 0 = 155dBrad 2 /Hz sustaining ampliier b 1 = 137dBrad 2/Hz =1.4Hz L (guessed) =4.5Hz L =63Hz c F=1dB b0 => P0= 18 dbm (b 3)osc => σy=1.5x10 13, Q=5.6x10 5 (too low) Q 1.8x10 6 => σy=4.6x10 14 Leeson (too low)

21 Example FEMTO-ST prototype 21 S ϕ () dbrad 2 Hz resonator instability b 3 = 116.6dBrad 2 /Hz Leeson eect (hidden) b 3 = 123.2dBrad 2/Hz sustaining ampliier b 1 = 136dBrad 2 /Hz 3dB sust. ampli + buer b 1 = 130dBrad 2 /Hz 3dB FEMTO ST prototype LD cut, 10 MHz, Q=1.15E6 apparent b 0 = 147dBrad 2 /Hz L =4.3Hz calculated L =9.3Hz apparent Fourier requency, Hz F=1dB b0 => P0= 26 dbm (b 3)osc => σy=1.7x10 13, Q=5.4x10 5 (too low) (there is a problem) Q=1.15x10 6 => σy=8.1x10 14 Leeson (too low)

22 Example Agilent S ϕ() dbrad 2 /Hz b 3 = 103 db sust. ampli + bu b 1 = 131 db sust. ampliier b 1 = 137 db 147 b 0 = 162 db L ~ 7Hz (guess!) L =50Hz =320Hz c requency, Hz F=1dB b0 => P0= 11 dbm (b 3)osc => σy=8.3x10 13, Q=1x10 5 (too low) Q 7x10 5 => σy=1.2x10 13 Leeson (too low)

23 Example Agilent prototype 23 S ϕ() dbrad 2 /Hz b 3 = 102 db sust. ampli + bu b 1 = 126 db sust. ampliier b 1 = 132 db b0 = 158dB L =32Hz c =400Hz requency, Hz L ~ 7Hz (guess!) F=1dB b0 => P0= 12 dbm (b 3)osc => σy=9.3x10 13 Q=1.6x10 5 Q 7x10 5 => σy=2.1x10 13 (Leeson)