On the 1/f noise in ultra-stable quartz oscillators Enrico Rubiola - PowerPoint PPT Presentation

On the 1/f noise in ultra-stable quartz oscillators Enrico Rubiola and Vincent Giordano FEMTO-ST Institute, Besançon, France (CNRS and Université de Franche Comté) Outline ☺ ☺ Amplifier noise Leeson effect Interpretation of S φ (f) ☹ Examples

2 Amplifier white noise 0 � Noise figure F b i f i power law S ϕ = Input power P 0 i = − 4 V 0 cos ω 0 t ∑ b 0 = FkT 0 g white phase noise P 0 n rf ( t ) S φ (f) low P 0 high P 0 P 0 f Cascaded amplifiers (Friis formula) N = F 1 kT 0 + ( F 2 − 1) kT 0 + . . . g 2 1 As a consequence, (phase) noise is chiefly that of the 1st stage

3 Amplifier flicker noise parametric up-conversion of the near-dc noise no carrier noise S(f) S(f) up-conversion near-dc flicker no flicker f f ω 0 = ? ω 0 carrier + near-dc noise the parametric nature of 1/f v i ( t ) = V i e j ω 0 t + n ′ ( t ) + jn ′′ ( t ) t t noise is hidden in n’ and n” a substitute (careful, this hides the down-conversion) v o ( t ) = a 1 v i ( t ) + a 2 v 2 S φ (f) i ( t ) + . . . b –1 ≈ independent of P 0 non-linear amplifier 0 � b i f i S ϕ = expand and select the ω 0 terms i = − 4 f � �� e j ω 0 t � n ′ ( t ) + jn ′′ ( t ) v o ( t ) = V i a 1 + 2 a 2 m cascaded amplifiers get AM and PM noise m � ( b − 1 ) cascade = ( b − 1 ) i α ( t ) = 2 a 2 ϕ ( t ) = 2 a 2 n ′ ( t ) n ′′ ( t ) a 1 a 1 i =1 independent of V i (!) In practice, each stage contributes ≈ equally

4 Resonator in the phase space 1 – theory of linear systems δ ( t ) h ( t ) H ( s ) 1 t h ( t ) H ( s ) Dirac Laplace transform � U ( t ) h ( t ) dt 1 /s (1 /s ) H ( s ) h ( t ) H ( s ) Heaviside 2 – resonator phase response cos[ ω 0 t + b ( t )] cos[ ω 0 t + δ ( t )] resonator ☹ ☠☈ ≹ ∞⁈ t=0 cos[ ω 0 t] ω 0 t+ κ U(t)] cos[ ω 0 t+ κ ] cos[ � � � � � cos ω 0 t + U ( t ) cos ω 0 t + b ( t ) dt κ − > 0 resonator this is easy –> linearize 3 – the resonator phase response is a low-pass function � U ( t ) b ( t ) dt Laplace (1 /s ) B ( s ) 1 /s b ( t ) B ( s ) transform b ( t ) = 1 τ = 2 Q 1 τ e − t B ( s ) = τ 1 + s τ ω 0

5 The Leeson effect (input) oscill main out out basic + A j A e j Ψ Ψ 1 H ( s ) = e feedback Σ b 1 − A β ( s ) + random noise random noise theory phase free phase free β (s) resonator phase response – use the linear-feedback theory H ( s ) = Φ ( s ) 1 − B ( s ) = 1 + s τ 1 Ψ ( s ) = Ψ (s) Φ (s) Ψ b (s) Φ o (s) s τ oscill main out out + + |H ( j ω ) | 2 = 1 + ω 2 τ 2 τ = 2 Q + 1 1 Σ Σ ω 2 τ 2 ω 0 + ν 2 � 1 + 1 � B(s) 0 S ϕ o ( f ) = S ψ ( f ) + S ψ b ( f ) resonator f 2 4 Q 2 S φ (f) 1/f 2 S ψ (f) f L f

6 Interpretation of S φ (f) [1] real phase-noise spectrum after parametric estimation Leeson effect? −3 −3 2 b f S (f) ν 0 1 ϕ = 1 + f 2 S ψ (f) 2Q check b −3 ϕ (f) σ 2 = 2ln(2) y ν 2 0 S −2 f x check FkT 0 P 0 = b 0 −1 −1 b f b 0 f 0 f L ’ f c f Sanity check: – power P 0 at amplifier input – Allan deviation σ y (floor)

7 Interpretation of S φ (f) [2] take away the buffer 1/f noise ’’ estimate f L −3 b f −3 (f) ϕ evaluate S ν 0 Leeson effect? Q s = sustaining ampli 2f’’ L buffer + sust.ampli f −1 b −1a b 0 f 0 ’ ’’ f L f L f c f ~ 6dB ~ 2–3 buffer stages => the sustaining amplifier contributes ≲ 25% of the total 1/f noise

8 Interpretation of S φ (f) [3] technology => Q t −3 b f −3 ν 0 f L = 2Q t resonator 1/f (f) freq. noise ϕ S −2 f x the Leeson effect is hidden s u s t a i n i n g a m p l i ’ ’’ f L f L f L f c f Technology suggests a merit factor Q t . In all xtal oscillators we find Q t ≫ Q s

9 Example – CMAC Pharao −90 Courtesy of CMAC. Interpretation and mistakes are of the authors. −100 (b −3 ) tot =−132dB (f) dBrad 2 /Hz −110 f’ L =1.5Hz −120 f"=3Hz L ϕ S f’ c =50Hz −130 −140 b 0 =−152.5dB −150 technology 6 ? (b −1 ) tot =−135.5dB Q=2x10 −160 f =1.25Hz => f =13Hz L c (b −1 ) osc =−141.5dB −170 10 −1 10 2 10 3 10 4 5 1 10 10 Fourier frequency, Hz (b –3 ) osc => σ y =5.9x10 –14 , Q=8.4x10 5 (too low) F=1dB b 0 => P 0 =–20.5 dBm Q ≟ 2x10 6 => σ y =2.5x10 –14 Leeson (too low)

10 Example – Oscilloquartz 8607 Courtesy of Oscilloquartz. Interpretation and mistakes are of the authors. −67 Oscilloquartz 8600 (f) dBrad 2 /Hz (specifications) −87 (b −3 ) tot =−128.5dB −107 f’ L =1.6Hz ϕ S 6 ? <= Q=2x10 f =1.25Hz L −127 Q=7.9x10 5 => f"=3.2Hz L (b −1 ) tot =−132.5dB −147 b 0 =−153dB (b −1 ) osc =−138.1dB −167 2 10 3 10 4 10 5 10 6 1 10 10 Fourier frequency, Hz (b –3 ) osc => σ y =8.8x10 –14 , Q=7.8x10 5 (too low) F=1dB b 0 => P 0 =–20 dBm Q ≟ 2x10 6 => σ y =3.5x10 –14 Leeson (too low)

11 Example – Wenzel 501-04623 Data are from the manufacturer web site. Interpretation and mistakes are of the authors. −100 Wenzel 501−04623 −30dB/dec b −3 =−67 dBrad 2 /Hz specifications −110 Estimating (b –1 ) ampli −120 is difficult because there is no visible 1/f region −130 Leeson effect (hidden) /Hz is about here −140 2 Phase noise, dBrad ampli noise (?) −150 −160 dBrad 2 /Hz −170 b 0 =−173 −180 10 1 10 3 10 4 5 10 10 2 ’’ =3.5kHz f L Q=8x10 4 => guess f L =625Hz Fourier frequency, Hz (b –3 ) osc => σ y =5.3x10 –12 Q=1.4x10 4 F=1dB b 0 => P 0 =0 dBm Q ≟ 8x10 4 => σ y =9.3x10 –13 (Leeson)

12 Other oscillators f ′ f ′′ Oscillator ( b − 3 ) tot ( b − 1 ) tot ( b − 1 ) amp Q s Q t f L ( b − 3 ) L R Note ν 0 L L Oscilloquartz 5 . 6 × 10 5 1 . 8 × 10 6 5 − 124 . 0 − 131 . 0 − 137 . 0 2 . 24 4 . 5 1 . 4 − 134 . 1 10 . 1 (1) 8600 Oscilloquartz 7 . 9 × 10 5 2 × 10 6 5 − 128 . 5 − 132 . 5 − 138 . 5 1 . 6 3 . 2 1 . 25 − 136 . 5 8 . 1 (1) 8607 CMAC 8 . 4 × 10 5 2 × 10 6 5 − 132 . 0 − 135 . 5 − 141 . 1 1 . 5 3 1 . 25 − 139 . 6 7 . 6 (2) Pharao FEMTO-ST 5 . 4 × 10 5 1 . 15 × 10 6 4 . 3 10 − 116 . 6 − 130 . 0 − 136 . 0 4 . 7 9 . 3 − 123 . 2 6 . 6 (3) LD prot. Agilent 1 × 10 5 7 × 10 5 10 − 103 . 0 − 131 . 0 − 137 . 0 25 50 7 . 1 − 119 . 9 16 . 9 (4) 10811 Agilent 1 . 6 × 10 5 7 × 10 5 10 − 102 . 0 − 126 . 0 − 132 . 0 16 32 7 . 1 − 114 . 9 12 . 9 (5) prototype Wenzel 1 . 4 × 10 4 8 × 10 4 100 − 67 . 0 − 132 ? − 138 ? 1800 3500 625 − 79 . 1 15 . 1 (6) 501-04623 dB dB dB dB unit MHz Hz Hz (none) (none) Hz dB rad 2 / Hz rad 2 / Hz rad 2 / Hz rad 2 / Hz Notes (1) Data are from specifications, full options about low noise and high stability. (2) Measured by CMAC on a sample. CMAC confirmed that 2 × 10 6 < Q < 2 . 2 × 10 6 in actual conditions. (3) LD cut, built and measured in our laboratory, yet by a di ff erent team. Q t is known. (4) Measured by Hewlett Packard (now Agilent) on a sample. (5) Implements a bridge scheme for the degeneration of the amplifier noise. Same resonator of the Agilent 10811. (6) Data are from specifications. � � ( σ y ) oscill ( b − 3 ) tot = f ′′ = Q t � L R = = � ( σ y ) Leeson ( b − 3 ) L Q s f L � floor

13 Warning: an effect not accounted for still remains A fluctuating impedance that affects the input without participating to the gain fluctuating impedance This does not fit general experience on amplifiers, yet it is to be reported

14 Conclusions The analysis of S φ (f) provides insight in the oscillator The oscillator 1/f 3 phase noise (Allan variance floor) originates from: – amplifier 1/f noise, via the Leeson effect – resonator instability In actual oscillators, the resonator instability turns out to be the dominant effect ☺ ☺ qed ☹ Full text available on http://arxiv.org/abs/physics/0602110 Talk slides and full text (20 pages, pdf) available on http://rubiola.org 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 t=0 cos[ ω 0 t] ω 0 t+ κ U(t)] cos[ ω 0 t+ κ ] cos[ κ − > 0 linearize v’ I (t) v’’ I (t) initial phase κ t t v’ O (t) v(t) v’’ (t) v(t) O envelope envelope τ τ t t τ = 2 Q � 1 − e − t/ τ � cos( ω 0 t ) e − t/ τ cos( ω 0 t + κ ) ω 0

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

Details [3]

18 Summary of the amplifier phase noise FkT 0 b −1 ~ ~ constant b = 0 S ϕ (f) P 0 −1 −1 b f P 0 low b 0 f 0 P 0 high P 0 f c f f c depends on P 0 • White PM noise is inversely proportional to P 0 • Flicker PM noise is about independent P 0 • The corner frequency f c follows

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