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

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 Sφ(f)
Examples ☹
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Amplifier white noise
0
Noise figure F power law Sϕ = bi f i
Input power P0 i=−4
V0 cos ω0 t
∑ g F kT0
white
phase noise
b0 =
P0
nrf (t)
Sφ(f)
low P0
high P0
P0
f
Cascaded amplifiers (Friis formula)
(F2 − 1)kT0
N = F1 kT0 + 2 + ...
g1
As a consequence, (phase) noise is chiefly that of the 1st stage
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E. Rubiola – FCS 2004 2004
E. Rubiola – FCS
Amplifier flicker noise 4 4
parametric up-conversion of the near-dc noise
Flicker noise in RFRF and microwave amplifiers
Flicker noise in and microwave amplifiers
no carrier
no carrier noise
noise
S(f) S(f) S(f) S(f)
near-dc flicker
near-dc flicker up-conversion
up-conversion
no flicker
f f f f
ω0 = ? ω0
random modulation fromfrom near-dc noise
random modulation near-dc noise
carrier + near-dc noise
,
n , !t " n !t " n, ,
n , , !t " !t " noise-free
noise-free
the parametric nature of 1/f
v
v o !t " o !t "
vi (t) =vVii cos!$0 t "n 0 t "AM jn (t)
i !t ejω cos!$ (t) +
vi !t "#V "#V0it
+ AM
noise is hidden in n’ and n”
PM PM a1 a1
substitute
(careful, this hides
vo (t) = a1 vi (t) + a2 vi (t) + . . . i
2o % %
modulated signal: v !t the down-conversion) t "&m , " , !t " cos!$ t "'m , , n , , !t !$ t " t
modulated signal: v !t " o# " V cos!$ t "&m , n , !t ncos!$ t "'m , , n , !t "sin"sin !$ 0 a"
# V i cos!$0
0
carrier
carrier
Sφ(f) 0 0
AM noisenoise ≈ independent ofnoise
AM b PM
PM noise P0
0 1 ( (a 1
–1
non-linear amplifier
0
the simplest v # v the x&a
the simplest and select # aω0 terms x 2withwithx # V # V i cos!$0 t n &"n!t " *ni!t**+1+1
expand a1 x&a 2 x 2 2 x cos!$ t " & " !t Sϕ = i=−4 b f "n!t "*
i
o 1
nonlinearity o
nonlinearity i 0
carrier near-dc
carrier near-dc
f
yields: 2a2 n (t) + jn (t) ejω0 t
yields:
vo (t) = Vi a1 +
% % ( (
v o# " a# a1 V i cos!$0 t "&a2 cos2cos2 !$0 t "&2 V i"n !t " cos!$0 t "&n"2 !t "
2
v o !t " !t V cos!$0 t "&a 2 V i2 V i !$ t "&2 V n !t cos!$ t "&n 2 !t
1 i 0 i 0
get AM and PM noise m cascaded amplifiers
2 a 2 n!t " n!t "
2 2a 2 am2 a 2
AM noise: #
AM noise: )!t " )!t " # modulation index: m
modulation index: ) m # =# a
2
a2 a= 2aa2 n (t) (b−1 cascade a (b−1 )i
α(t) = 2 n (t) ϕ(t) 1 1 1 1
a1 noise originates in1the same way, but for a 90° phase i=1 in the product
a
PM originates in the same way, but for a 90° phase shift in the product
shift
PM noise
independent of Vi (!) In practice, each stage contributes ≈ equally
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Resonator in the phase space
1 – theory of linear systems
t δ(t) h(t) 1 H(s)
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[ω0 t + δ(t)] cos[ω0 t + b(t)]
resonator
☹☠☈≹∞⁈ cos[ ω 0 t] t=0
cos[ ω 0 t+ κ U(t)]
cos[ ω 0 t+ κ ]
cos ω0 t + U (t) cos ω0 t + b(t)dt
resonator κ −> 0
this is easy –> linearize
3 – the resonator phase response is a low-pass function
U (t) b(t) dt Laplace 1/s (1/s) B(s)
b(t) B(s)
transform
1 −t 2Q 1
b(t) = e τ τ= B(s) =
τ ω0 1 + sτ
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The Leeson effect
(input)
oscill main
basic out out
+ A
feedback Σ e jΨ A e jΨb 1 H(s) =
+ random noise random noise 1 − Aβ(s)
theory phase free phase free
β(s)
resonator
phase response – use the linear-feedback theory
Φ(s) 1 1 + sτ
Ψ(s) Φ(s) Ψb (s) Φo (s) H(s) = = =
Ψ(s) 1 − B(s) sτ
oscill main
+ out + out
+ 1 + ω2 τ 2
2 2Q
Σ 1 Σ 1 |H(jω)| = τ=
+ ω2 τ 2 ω0
B(s) 1 ν0
2
Sϕ o (f ) = 1 + 2 Sψ (f ) + Sψ b (f )
resonator f 4Q 2
Sφ(f)
1/f2
Sψ(f)
fL f
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Interpretation of Sφ(f) [1]
real phase-noise spectrum
after parametric estimation
Leeson effect?
b−3f−3 Sϕ(f) 1 ν0 2
= 1+
check S ψ (f) f 2 2Q
2 = 2ln(2)b−3
Sϕ(f)
σy
ν20
x f −2
check
FkT0
P0 =
b0
b−1f−1
b0 f 0
’
fL fc f
Sanity check:
– power P0 at amplifier input
– Allan deviation σy (floor)
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Interpretation of Sφ(f) [2]
take away the
buffer 1/f noise
b−3f−3 ’’
estimate fL
Sϕ(f)
evaluate
Leeson effect? ν0
sus Qs=
tain ’’
2fL
ing
am
pli buffer + sust.ampli
b−1a f−1
b0 f 0
’ ’’
fL fL fc f
~ 6dB
~
2–3 buffer stages => the sustaining amplifier contributes 25% of the total 1/f noise
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Interpretation of Sφ(f) [3]
technology => Q t
b−3f−3
ν0
resonator 1/f fL =
Sϕ(f)
freq. noise 2Qt
x f −2
the Leeson effect
is hidden
sus
tain
ing
am
pli
fL ’ ’’
fL fL fc f
Technology suggests a merit factor Qt. In all xtal oscillators we find Qt Qs
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Example – CMAC Pharao
−90
Courtesy of CMAC. Interpretation and mistakes are of the authors.
−100
(b −3) tot =−132dB
dBrad 2/Hz
−110
f’ =1.5Hz
L
−120
f"=3Hz
Sϕ(f)
L
−130 f’ =50Hz
c
−140
−150 b0=−152.5dB
technology
Q=2x10 6? (b −1) tot =−135.5dB
−160 => fL =1.25Hz
fc =13Hz
(b −1) osc=−141.5dB
−170
10−1 1 10 102 103 104 105
Fourier frequency, Hz
F=1dB b0 => P0=–20.5 dBm (b–3)osc => σy=5.9x10–14, Q=8.4x105 (too low)
Q≟2x106 => σy=2.5x10–14 Leeson (too low)
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Example – Oscilloquartz 8607
Courtesy of Oscilloquartz. Interpretation and mistakes are of the authors.
−67
Oscilloquartz 8600
dBrad 2/Hz
(specifications)
−87
(b −3) tot =−128.5dB
Sϕ(f)
−107
f’ =1.6Hz
L
fL =1.25Hz