Enrico Rubiola The Leeson Effect 1 The Leeson Effect Enrico Rubiola Dept. LPMO, FEMTO ST Institute Besançon, France e mail rubiola@femto st.fr or enrico@rubiola.org D. B. Leeson, A simple model for feed back oscillator noise, Proc. IEEE 54(2):329 (Feb 1966) resonator out S! " f # $ % 1& " ' 0 2Q oscillator noise #2 1 f 2 ( S ) " f # ampli noise oscillator noise S! " f # Leeson effect noise of electronic circuits f L =! 0 /2Q f www.rubiola.org you can download this presentation, an e-book on the Leeson effect, and some other documents on noise (amplitude and phase) and on precision electronics from my web page
Enrico Rubiola The Leeson Effect 2 Summary Basics how the oscillator works heuristic approach to the Leeson effect analysis of commercial oscillators Theory Laplace transforms proof of the Leeson formula Advanced topics detuned resonator delay-line oscillator phase noise in lasers
Enrico Rubiola The Leeson Effect 3 Basics Oscillator
Enrico Rubiola The Leeson Effect 4 basics oscillator
Enrico Rubiola The Leeson Effect 5 basics oscillator ω = 2πν
Enrico Rubiola The Leeson Effect 6 basics oscillator
Enrico Rubiola The Leeson Effect 7 Basics Heuristic approach to the Leeson effect
Enrico Rubiola The Leeson Effect 8 basics heuristic approach
Enrico Rubiola The Leeson Effect 9 basics heuristic approach
Enrico Rubiola The Leeson Effect 10 basics heuristic approach
Enrico Rubiola The Leeson Effect 11 basics heuristic approach
Enrico Rubiola The Leeson Effect 12 basics heuristic approach
Enrico Rubiola The Leeson Effect 13 basics heuristic approach
Enrico Rubiola The Leeson Effect 14 basics heuristic approach Resonator instability The resonator instability may be larger than the noise of the electronics (high-performance xtal oscillators) frequency rw of the resonator Type 1 f L > f c total noise frequency rw of the resonator Type 2 f L < f c total noise S ϕ (f) lower noise resonat. Leeson effect frequency flicker of the resonator ϕ(f) S frequency flicker of the resonator intersection f > f L Leeson effect amplifier intersection f > f L amplifier f f f c L f L f c f
Enrico Rubiola The Leeson Effect 15 basics heuristic approach S ϕ (f) r. w. freq. b 4 f 4 phase noise b 3 f 3 flicker freq. white freq. b 2 f 2 flicker phase. b 1 f 1 white phase b 0 f x f 2 / ν2 0 frequency noise S y (f) h 2 f 2 r. w. freq. h 1 f 1 h 0 flicker freq. white freq. h 1 f h 2 f 2 flicker phase white phase f Allan variance white σ 2 (τ) = h0/2τ flicker σ 2 (τ) = 2ln(2) h-1 r.walk σ 2 (τ) = ((2π) 2 /6) h0τ σ y 2 (τ) flicker phase white phase white freq. flicker freq. r. w. freq. freq. drift τ
Enrico Rubiola The Leeson Effect 16 Basics Analysis of some commercial oscillators The purpose of this section is to help to understand the oscillator inside from the phase noise spectra, plus some technical information. I have chosen some commercial oscillators as an example. The conclusions about each oscillator represent only my understanding based on experience and on the data sheets published on the manufacturer web site. You should be aware that this process of interpretation is not free from errors. My conclusions were not submitted to manufacturers before writing, for their comments could not be included. I have modified some the web pages reproduced here, with the only purpose of making logos visible after zooming in. Please look for the original on the manufacturer web page.
Enrico Rubiola The Leeson Effect 17 basics commercial oscillators Courtesy of Miteq (handwritten notes are mine). The DRO test data are available at the URL http://www.miteq.com/micro/fresourc/d210b/dro/drotyp.html
the plot is reconstructed from data available on the manufacturer's web site http://www.psi.com.au Enrico Rubiola The Leeson Effect 18 basics commercial oscillators Poseidon Scientific Instruments Shoebox 10 GHz sapphire whispering-gallery resonator oscillator 90 100 Poseidon Shoebox 10 GHz sapphire WG resonator noise correction 110 phase noise, dbc/hz 120 130 140 150 160 oscillator b 3 f 3 instrument background d ~ 6dB b 1 f 1 170 180 100 1000 f L =2.6kHz 10000 100000 Fourier frequency, Hz fl = v0/2q = 2.6 khz Q = 1.8 10 6 This incompatible with the resonator technology. Typical Q of a sapphire whispering gallery resonator: 2 10 5 @ 295K (room temp), 3 10 7 @ 77K (liquid N), 4 10 9 @ 4K (liquid He). In addition, d ~ 6 db does not fit the power-law. The interpretation shown is wrong, and the Leeson frequency is somewhere else.
Enrico Rubiola The Leeson Effect 19 basics commercial oscillators Poseidon Scientific Instruments Shoebox 10 GHz sapphire whispering-gallery resonator oscillator 90 100 Poseidon Shoebox 10 GHz sapphire WG resonator noise correction phase noise, dbc/hz 110 120 130 140 150 160 oscillator f 1 to f 3 conversion f c =850Hz (b 1 ) ampli = 140 dbrad 2 /Hz instr. background f 0 to f 2 conversion (b 1 ) buffer = 120dBrad 2 /Hz (b 0 ) ampli = 169 dbrad 2 /Hz 170 180 100 1000 10000 f L =25kHz 100000 Fourier frequency, Hz The 1/f noise of the output buffer is higher than that of the sustaining amplifier (a compex amplifier with interferometric noise reduction) In this case both 1/f and 1/f 2 are present white noise 169 dbrad 2 /Hz, guess F = 5 db (interferometer) P0 = 0 dbm buffer flicker 120 dbrad 2 /Hz @ 1 Hz good microwave amplifier fl = v0/2q = 25 khz Q = 2 10 5 (quite reasonable) fc = 850 Hz flicker of the interferometric amplifier 139 dbrad 2 /Hz @ 1 Hz the plot is reconstructed from data available on the manufacturer's web site http://www.psi.com.au
Enrico Rubiola The Leeson Effect 20 basics commercial oscillators Poseidon Scientific Instruments 10 GHz dielectric resonator oscillator (DRO) SSB phase noise, dbc/hz 50 60 70 80 90 100 110 120 130 140 150 160 170 180 30dB/dec b dbrad 2 3 =+4 /Hz DRO 10.4 XPL slope 25 db/dec DRO 10.4 FR b 1 = 165dBrad 2 /Hz 10 2 10 3 10 4 Phase noise of two PSI DRO 10.4 FR f c =9.3kHz 7dB b 0 = 165dBrad 2 /Hz slope close to 25dB/dec 10 5 Fourier frequency, Hz 20dB/dec 3dB difference 10 6 10 7 f L =3.2MHz Slopes are not in agreement with the theory. In this case I am unable to say what is going on inside the oscillator. the plot is reconstructed from data available on the manufacturer's web site http://www.psi.com.au
Enrico Rubiola The Leeson Effect 21 basics commercial oscillators Oscilloquartz OCXO 8600 outstanding stability oscillator based on a 5 MHz AT-cut BVA (electrodless) resonator stability # y ($) = 3 10-13 for $ = 0.2 30 s aging 3 10-12 /day Courtesy of Oscilloquartz (handwritten notes are mine). The specifications, which include this spectrum, are available at the URL http://www.oscilloquartz.com/file/pdf/8600.pdf ANALYSIS 1 floor S!0 = 155 dbrad2/hz, guess F = 1 db! P0 = 18 dbm 2 ampli flicker S! = 132 dbrad2/hz @ 1 Hz! good RF amplifier 3 merit factor Q = " 0 /2f L = 5 10 6 /5 = 10 6 (seems too low) 4 take away some flicker for the output buffer: * flicker in the oscillator core is lower than 132 dbrad 2 /Hz @ 1 Hz * f L is higher than 2.5 Hz * the resonator Q is lower than 10 6 This is inconsistent with the resonator technology (expect Q > 106). The true Leeson frequency is lower than the frequency labeled as f L The 1/f 3 noise is attributed to the fluctuation of the quartz resonant frequency
Enrico Rubiola The Leeson Effect 22 basics commercial oscillators Wenzel 501-04623 G - Lowest phase noise 100 MHz SC-cut oscillator manufacturer specs, phase noise -130 dbc/hz @ 100 Hz -158 dbc/hz @ 1 khz -176 dbc/hz @ 10 khz -176 dbc/hz @ 20 khz 1 floor S!0 = 173 dbrad2/hz, guess F = 1 db! P0 = 0 dbm 2 merit factor Q = " 0 /2f L = 108/7 103 = 1.4 104 (seems too low) From the literature, one expects Q ~ 105. The true Leeson frequency is lower than the frequency labeled as f L The 1/f 3 noise is attributed to the fluctuation of the quartz resonant frequency
Enrico Rubiola The Leeson Effect 23 basics commercial oscillators Courtesy of OEwaves (handwritten notes are mine). Cut from the oscillator specifications available at the URL http://www.oewaves.com/products/pdf/tdalwave_datasheet_012104.pdf
Enrico Rubiola The Leeson Effect 24 Theory Laplace transforms most (electrical) engineers are familiar with the generalized impedance Z = R Z = sl Z = 1 sc R L C
Enrico Rubiola The Leeson Effect 25 theory Laplace transforms
Enrico Rubiola The Leeson Effect 26 theory Laplace transforms
Enrico Rubiola The Leeson Effect 27 theory Laplace transforms
Enrico Rubiola The Leeson Effect 28 theory Laplace transforms
Enrico Rubiola The Leeson Effect 29 theory Laplace transforms
Enrico Rubiola The Leeson Effect 30 theory Laplace transforms
Enrico Rubiola The Leeson Effect 31 Theory Proof of the Leeson formula
Enrico Rubiola The Leeson Effect 32 theory proof of the Leeson formula
Enrico Rubiola The Leeson Effect 33 theory proof of the Leeson formula
Enrico Rubiola The Leeson Effect 34 theory proof of the Leeson formula Let s introduce the phase space the Laplace-transform of the phase is commonly used in PLLs (previous) voltage space the amplifier gain (impulse response) is A phase noise is a modulation process β(s) is the transfer function of the resonator (now) phase space the amplifier gain (impulse response) is 1 the phase Ψ(s) is additive noise B(s) is the phase transfer function of the resonator The phase space simplifies the representation of phase noise, which turns into additive noise Still need a phase-space representation of the resonator
Enrico Rubiola The Leeson Effect 35 theory proof of the Leeson formula Resonator response in the phase space
Enrico Rubiola The Leeson Effect 36 theory proof of the Leeson formula
Enrico Rubiola The Leeson Effect 37 theory proof of the Leeson formula
Enrico Rubiola The Leeson Effect 38 theory proof of the Leeson formula
Enrico Rubiola The Leeson Effect 39 theory proof of the Leeson formula
Enrico Rubiola The Leeson Effect 40 theory proof of the Leeson formula Feedback loop in the phase space
Enrico Rubiola The Leeson Effect 41 theory proof of the Leeson formula
Enrico Rubiola The Leeson Effect 42 Advanced topics Detuned resonator
Enrico Rubiola The Leeson Effect 43 advanced detuned resonator
Enrico Rubiola The Leeson Effect 44 advanced detuned resonator
Enrico Rubiola The Leeson Effect 45 advanced detuned resonator
Enrico Rubiola The Leeson Effect 46 advanced detuned resonator
Enrico Rubiola The Leeson Effect 47 advanced detuned resonator
Enrico Rubiola The Leeson Effect 48 advanced detuned resonator
Enrico Rubiola The Leeson Effect 49 advanced detuned resonator
Enrico Rubiola The Leeson Effect 50 advanced detuned resonator
Enrico Rubiola The Leeson Effect 51 Advanced topics Delay-line oscillator
Enrico Rubiola The Leeson Effect 52 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 53 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 54 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 55 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 56 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 57 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 58 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 59 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 60 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 61 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 62 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 63 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 64 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 65 advanced delay-line oscillator
Enrico Rubiola The Leeson Effect 66 Advanced topics Phase noise in lasers
Enrico Rubiola The Leeson Effect 67 advanced delay-line oscillator A laser is a delay-line oscillator mirror mirror gain reference plane reference plane V i T 1 V o T 2 V o out R 1 gain A R 2 V i in T 1 V i + Σ + A T 2 V o (out) R 2 R 1 delay τ roundtrip time τ mirror 1 mirror 2 Σ A τ
Enrico Rubiola The Leeson Effect 68 Oscillation modes and noise depend on the gain mechanism A single gain B gain cluster Αβ arg(αβ) =0 largest gain small signal large signal Αβ small signal large signal Αβ =1 arg(αβ) =0 1 1 ω 1 ω 2 ω 3 ω 4 ω 5 ω ω 1 ω 2 ω 3 ω 4 ω 5 ω Αβ =1 arg(αβ) =0
Enrico Rubiola The Leeson Effect 69 References E. Rubiola, The Leeson effect (e-book, 117 pages, 50 figures) free downloaded from http://arxiv.org arxiv:physics/0502143 or http://rubiola.org work is in progress, check for updates Acknowledgements I wish to thank Vincent Candelier (CMAC), Mark Henderson (OEwaves), Art Faverio, Mike Greco and Charif Nasrallah (Miteq), and Jean-Pierre Aubry (Oscilloquartz), for kindness and prompt support. I wish to express my gratitude to Vincent Giordano and Jacques Groslambert (FEMTO-ST), and to John Dick and Lute Maleki (JPL/ NASA/Caltech), for numerous discussions As the intellectual pursuits of academia are largely in the realm of the gift economy, by far the greatest reward is found in the appreciation of one's work by one's colleagues. David B. Leeson