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Summary Basics how the oscillator works heuristic approach to the - - PowerPoint PPT Presentation

Jan 30, 2024 282 likes •987 views

Enrico Rubiola The Leeson Effect 1 The Leeson Effect Enrico Rubiola Dept. LPMO, FEMTO ST Institute Besanon, France e mail rubiola@femto st.fr or enrico@rubiola.org D. B. Leeson, A simple model for feed back

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Enrico Rubiola – The Leeson Effect –

  • D. B. Leeson, A simple model for feed back oscillator noise, Proc. IEEE 54(2):329 (Feb 1966)

1

The Leeson Effect

Enrico Rubiola

  • Dept. LPMO, FEMTO­ST Institute – Besançon, France

e­mail rubiola@femto­st.fr or enrico@rubiola.org

noise of electronic circuits

  • scillator noise

S!" f # f

Leeson effect

resonator

  • ut

fL = !0/2Q

S!" f # $ %1&" '0 2Q#

2

1 f 2( S)" f #

  • scillator

noise ampli noise

www.rubiola.org

you can download this presentation, an e-book on the Leeson effect, and some other documents

  • n noise (amplitude and phase) and on precision electronics from my web page
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Enrico Rubiola – The Leeson Effect –

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

2

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Enrico Rubiola – The Leeson Effect –

Oscillator

3

Basics

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Enrico Rubiola – The Leeson Effect – basics – oscillator 4

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Enrico Rubiola – The Leeson Effect – 5 basics – oscillator

ω = 2πν

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Enrico Rubiola – The Leeson Effect – 6 basics – oscillator

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Enrico Rubiola – The Leeson Effect –

Heuristic approach to the Leeson effect

7

Basics

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Enrico Rubiola – The Leeson Effect – 8 basics – heuristic approach

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Enrico Rubiola – The Leeson Effect – 9 basics – heuristic approach

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Enrico Rubiola – The Leeson Effect – 10 basics – heuristic approach

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Enrico Rubiola – The Leeson Effect – 11 basics – heuristic approach

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Enrico Rubiola – The Leeson Effect – 12 basics – heuristic approach

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Enrico Rubiola – The Leeson Effect – 13 basics – heuristic approach

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Enrico Rubiola – The Leeson Effect –

Resonator instability

14 basics – heuristic approach

f > f intersection

L

f > f

total noise

intersection

L

total noise ϕ(f)

  • f the resonator

frequency rw

fL < fc Type 2 fL > fc Type 1

frequency flicker

  • f the resonator

frequency flicker

  • f the resonator
  • f the resonator

frequency rw

f

amplifier amplifier

f

Leeson effect Leeson effect

f lower−noise resonat.

L

fc fL fc Sϕ(f) S

The resonator instability may be larger than the noise of the electronics (high-performance xtal oscillators)

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Enrico Rubiola – The Leeson Effect – 15 basics – heuristic approach

Sϕ(f) h0 h1f h2f2 Sy(f) b0

2

ν0 f2/

x

f−4 b−4 b−3 b−2 b−1 f−2 f−1 h−2 h−1 f−2 f−1 f−3

flicker freq. white freq. white phase flicker phase.

  • r. w. freq.

f

  • r. w. freq.

white freq. flicker phase white phase

f

  • r. w. freq.

white freq. flicker freq. white phase flicker phase

freq. drift τ

flicker freq. y 2

σ (τ)

frequency noise phase noise Allan variance

white σ2(τ) = h0/2τ flicker σ2(τ) = 2ln(2) h-1 r.walk σ2(τ) = ((2π)2/6) h0τ

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Enrico Rubiola – The Leeson Effect –

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.

16

Basics

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Enrico Rubiola – The Leeson Effect –

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

17 basics – commercial oscillators

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Enrico Rubiola – The Leeson Effect –

Poseidon Scientific Instruments − Shoebox 10 GHz sapphire whispering-gallery resonator oscillator fL = v0/2Q = 2.6 kHz → Q = 1.8×106

This incompatible with the resonator technology. Typical Q of a sapphire whispering gallery resonator: 2×105 @ 295K (room temp), 3×107 @ 77K (liquid N), 4×109 @ 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.

the plot is reconstructed from data available on the manufacturer's web site http://www.psi.com.au 18 basics – commercial oscillators

phase noise, dBc/Hz Fourier frequency, Hz

−180 −170 −160 −150 −140 −130 −120 −110 −100 −90 100 1000 10000 100000 Poseidon Shoebox 10 GHz sapphire WG resonator noise correction fL=2.6kHz b−1 f−1 instrument background

  • scillator

b−3 f−3 d ~ 6dB

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Enrico Rubiola – The Leeson Effect –

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/f2 are present

white noise −169 dBrad2/Hz, guess F = 5 dB (interferometer) → P0 = 0 dBm buffer flicker −120 dBrad2/Hz @ 1 Hz → good microwave amplifier fL = v0/2Q = 25 kHz → Q = 2×105 (quite reasonable) fc = 850 Hz → flicker of the interferometric amplifier −139 dBrad2/Hz @ 1 Hz

19 basics – commercial oscillators

Poseidon Scientific Instruments − Shoebox 10 GHz sapphire whispering-gallery resonator oscillator

−180 −170 −160 −150 −140 −130 −120 −110 −100 −90 1000 10000 100000 100

phase noise, dBc/Hz Fourier frequency, Hz

fL=25kHz f0 to f−2 conversion (b−1 )ampli=−140dBrad2/Hz dBrad2/Hz (b0)ampli=−169 (b−1 )buffer=−120dBrad2/Hz fc =850Hz

  • instr. background
  • scillator

f−1 to f−3 conversion noise correction 10 GHz sapphire WG resonator Poseidon Shoebox

the plot is reconstructed from data available on the manufacturer's web site http://www.psi.com.au

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Enrico Rubiola – The Leeson Effect –

Slopes are not in agreement with the theory. In this case I am unable to say what is going on inside the oscillator.

20 basics – commercial oscillators the plot is reconstructed from data available on the manufacturer's web site http://www.psi.com.au

Poseidon Scientific Instruments 10 GHz dielectric resonator oscillator (DRO)

106 107 105 102 103 104 −50 −180 −170 −160 −150 −140 −130 −120 −110 −100 −90 −80 −70 −60

SSB phase noise, dBc/Hz Phase noise of two PSI DRO−10.4−FR

D R O − 1 . 4 − F R D R O − 1 . 4 − X P L

Fourier frequency, Hz

3dB difference −30dB/dec −25 dB/dec slope fL =3.2MHz −20dB/dec slope close to −25dB/dec b0=−165dBrad2/Hz fc =9.3kHz 7dB b−1=−165dBrad2/Hz dBrad2/Hz b−3 =+4

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Enrico Rubiola – The Leeson Effect – 21 basics – commercial oscillators

ANALYSIS 1 – floor S!0 = –155 dBrad2/Hz, guess F = 1 dB P !

0 = –18 dBm

2 – ampli flicker S! = –132 dBrad2/Hz @ 1 Hz good RF amplifier ! 3 – merit factor Q = "0/2fL = 5·106/5 = 106 (seems too low) 4 – take away some flicker for the output buffer: * flicker in the oscillator core is lower than –132 dBrad2/Hz @ 1 Hz * fL is higher than 2.5 Hz * the resonator Q is lower than 106 This is inconsistent with the resonator technology (expect Q > 106). The true Leeson frequency is lower than the frequency labeled as fL The 1/f3 noise is attributed to the fluctuation of the quartz resonant frequency

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

Oscilloquartz OCXO 8600

  • utstanding 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

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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 P !

0 = 0 dBm

2 – merit factor Q = "0/2fL = 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 fL The 1/f3 noise is attributed to the fluctuation of the quartz resonant frequency

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Enrico Rubiola – The Leeson Effect – 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 23 basics – commercial oscillators

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Enrico Rubiola – The Leeson Effect –

Laplace transforms

24

Theory

Z = R Z = sL Z = 1 sC R L C

most (electrical) engineers are familiar with the generalized impedance

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Enrico Rubiola – The Leeson Effect – 25 theory – Laplace transforms

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Enrico Rubiola – The Leeson Effect – 26 theory – Laplace transforms

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Enrico Rubiola – The Leeson Effect – 27 theory – Laplace transforms

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Enrico Rubiola – The Leeson Effect – 28 theory – Laplace transforms

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Enrico Rubiola – The Leeson Effect – 29 theory – Laplace transforms

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Enrico Rubiola – The Leeson Effect – 30 theory – Laplace transforms

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Enrico Rubiola – The Leeson Effect –

Proof of the Leeson formula

31

Theory

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Enrico Rubiola – The Leeson Effect – 32 theory – proof of the Leeson formula

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Enrico Rubiola – The Leeson Effect – 33 theory – proof of the Leeson formula

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Enrico Rubiola – The Leeson Effect –

Let’s introduce the phase space

34 theory – proof of the Leeson formula

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
  • f 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
  • f 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

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Enrico Rubiola – The Leeson Effect – 35 theory – proof of the Leeson formula

Resonator response in the phase space

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Enrico Rubiola – The Leeson Effect – 36 theory – proof of the Leeson formula

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Enrico Rubiola – The Leeson Effect – 37 theory – proof of the Leeson formula

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Enrico Rubiola – The Leeson Effect – 38 theory – proof of the Leeson formula

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Enrico Rubiola – The Leeson Effect – 39 theory – proof of the Leeson formula

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Enrico Rubiola – The Leeson Effect – 40 theory – proof of the Leeson formula

Feedback loop in the phase space

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Enrico Rubiola – The Leeson Effect – 41 theory – proof of the Leeson formula

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Enrico Rubiola – The Leeson Effect –

Detuned resonator

42

Advanced topics

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Enrico Rubiola – The Leeson Effect – 43 advanced – detuned resonator

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Enrico Rubiola – The Leeson Effect – 44 advanced – detuned resonator

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Enrico Rubiola – The Leeson Effect – 45 advanced – detuned resonator

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Enrico Rubiola – The Leeson Effect – 46 advanced – detuned resonator

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Enrico Rubiola – The Leeson Effect – 47 advanced – detuned resonator

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Enrico Rubiola – The Leeson Effect – 48 advanced – detuned resonator

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Enrico Rubiola – The Leeson Effect – 49 advanced – detuned resonator

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Enrico Rubiola – The Leeson Effect – 50 advanced – detuned resonator

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Enrico Rubiola – The Leeson Effect –

Delay-line oscillator

51

Advanced topics

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Enrico Rubiola – The Leeson Effect – 52 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 53 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 54 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 55 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 56 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 57 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 58 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 59 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 60 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 61 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 62 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 63 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 64 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect – 65 advanced – delay-line oscillator

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Enrico Rubiola – The Leeson Effect –

Phase noise in lasers

66

Advanced topics

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Enrico Rubiola – The Leeson Effect –

A laser is a delay-line oscillator

67 advanced – delay-line oscillator

gain

reference plane

2

R2 T1

i

V’

  • V’

Σ

+ + T2 T1 Vi

i

V’

A’

Vo

  • V’

roundtrip time τ R1 R2 R1

τ

delay in

  • ut

(out) gain A’ mirror 1 mirror 2 reference plane T

τ Σ A

mirror mirror

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Enrico Rubiola – The Leeson Effect –

Oscillation modes and noise depend

  • n the gain mechanism

68

ω (Αβ) =0 largest gain large signal small signal arg(Αβ) =0 ω1 ω2 ω3 ω4 ω5 ω1 ω2 ω3 ω4 ω5 arg(Αβ) =0 |Αβ|=1 |Αβ| |Αβ| arg A − single gain B − gain cluster 1 ω 1 |Αβ|=1 large signal small signal

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Enrico Rubiola – The Leeson Effect –

References

69

  • 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 express my gratitude to Vincent Giordano and Jacques Groslambert (FEMTO-ST), and to John Dick and Lute Maleki (JPL/ NASA/Caltech), for numerous discussions 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. 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

  • ne's work by one's colleagues.

David B. Leeson