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Back to Phase Noise
Overview
Many descriptions of phase noise in
oscillators lack clear and explicit explanations as to the exact cause of phase
noise. Specifically, there is much confusion as to how low frequency amplitude
noise becomes up-converted to phase noise around the oscillator frequency.
Furthermore, although the general application of one well known analysis, the
Hajimiri-Lee approach, has been rigorously proven to be flawed, it is still
sometimes promoted as a useful point of view.
Introduction
The
usual starting point of an analysis of phase noise in oscillators is with
Leeson’s model. This model is empirical, and simply constructed as an attempt
to justify already known results. It has no calculating ability whatsoever,
other than modelling the general bode plot shape of the noise curve.
The
key part of the model assumes all oscillators have a dominate characteristic
described by an RLC tuned tank.
For
frequencies close to the oscillator frequency, the impedance of the LCR tank is
given by:
Z(Δω)=
R
1+j2
Q
l
Δω
ω
0
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwaiaacIcacqqHuoarcqaHjpWDcaGGPaGaeyypa0ZaaSaaaeaacaWGsbaabaGaaGymaiabgUcaRiaadQgacaaIYaGaamyuamaaBaaaleaacaWGSbaabeaakmaalaaabaGaeuiLdqKaeqyYdChabaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaaaaaaaaaa@484A@
For
oscillation the loop gain must equal unity at resonance, hence:
V
0
Gm=
V
0
R
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaaabeaakiaadEeacaWGTbGaeyypa0ZaaSaaaeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaamOuaaaaaaa@3CF7@
,
hence
GmR=1
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaad2gacaWGsbGaeyypa0JaaGymaaaa@3A0C@
To determine
the effect of injected noise, the closed loop gain with respect to the noise
signal is required.
V
0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaaabeaaaaa@3777@
due to I noise is therefore:
V
0n
=(
V
0n
Gm+
I
n
)Z
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaGaamOBaaqabaGccqGH9aqpcaGGOaGaamOvamaaBaaaleaacaaIWaGaamOBaaqabaGccaWGhbGaamyBaiabgUcaRiaadMeadaWgaaWcbaGaamOBaaqabaGccaGGPaGaamOwaaaa@4307@
V
0n
=
I
n
Z
GmZ−1
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiaadMeadaWgaaWcbaGaamOBaaqabaGccaWGAbaabaGaam4raiaad2gacaWGAbGaeyOeI0IaaGymaaaaaaa@40A5@
V
0n
=
I
n
R
1+j2
Q
l
Δω
ω
0
GmR
1+j2
Q
l
Δω
ω
0
−1
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiaadMeadaWgaaWcbaGaamOBaaqabaGcdaWcaaqaaiaadkfaaeaacaaIXaGaey4kaSIaamOAaiaaikdacaWGrbWaaSbaaSqaaiaadYgaaeqaaOWaaSaaaeaacqqHuoarcqaHjpWDaeaacqaHjpWDdaWgaaWcbaGaaGimaaqabaaaaaaaaOqaamaalaaabaGaam4raiaad2gacaWGsbaabaGaaGymaiabgUcaRiaadQgacaaIYaGaamyuamaaBaaaleaacaWGSbaabeaakmaalaaabaGaeuiLdqKaeqyYdChabaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaaaaaaGccqGHsislcaaIXaaaaaaa@573F@
V
0n
=
I
n
R
1−(1+j2
Q
l
Δω
ω
0
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiaadMeadaWgaaWcbaGaamOBaaqabaGccaWGsbaabaGaaGymaiabgkHiTiaacIcacaaIXaGaey4kaSIaamOAaiaaikdacaWGrbWaaSbaaSqaaiaadYgaaeqaaOWaaSaaaeaacqqHuoarcqaHjpWDaeaacqaHjpWDdaWgaaWcbaGaaGimaaqabaaaaOGaaiykaaaaaaa@4A9E@
V
0n
=−j
R
2
Q
l
ω
0
Δω
I
n
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaGaamOBaaqabaGccqGH9aqpcqGHsislcaWGQbWaaSaaaeaacaWGsbaabaGaaGOmaiaadgfadaWgaaWcbaGaamiBaaqabaaaaOWaaSaaaeaacqaHjpWDdaWgaaWcbaGaaGimaaqabaaakeaacqqHuoarcqaHjpWDaaGaamysamaaBaaaleaacaWGUbaabeaaaaa@46E3@
At zero carrier offset, the model
predicts infinite output voltage. This is essentially due to the divide by zero
of the transfer function of the feedback loop. Resolving this issue requires
that the limiting nature of real oscillators to be accounted for.
At high frequencies, real
oscillators have additional wideband noise due to various factors. Additionally,
the model simply assumes what the noise current is. Hence, these effects can be
included in the above model by simply forming a vector sum with this noise,
with a suitable empirical scale factor:
L(Δω)=F(1+
(
1
2
Q
l
ω
0
Δω
)
2
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacqqHuoarcqaHjpWDcaGGPaGaeyypa0JaamOraiaacIcacaaIXaGaey4kaSIaaiikamaalaaabaGaaGymaaqaaiaaikdacaWGrbWaaSbaaSqaaiaadYgaaeqaaaaakmaalaaabaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaaGcbaGaeuiLdqKaeqyYdChaaiaacMcadaahaaWcbeqaaiaaikdaaaGccaGGPaaaaa@4BAA@
And
to further account for the empirical fact that low frequency noise gets
up-converted, an additional factor is also included:
L(Δω)=F(1+
(
1
2
Q
l
ω
0
Δω
)
2
(1+
ω
f
(Δω)
))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacqqHuoarcqaHjpWDcaGGPaGaeyypa0JaamOraiaacIcacaaIXaGaey4kaSIaaiikamaalaaabaGaaGymaaqaaiaaikdacaWGrbWaaSbaaSqaaiaadYgaaeqaaaaakmaalaaabaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaaGcbaGaeuiLdqKaeqyYdChaaiaacMcadaahaaWcbeqaaiaaikdaaaGccaGGOaGaaGymaiabgUcaRmaalaaabaGaeqyYdC3aaSbaaSqaaiaadAgaaeqaaaGcbaGaaiikaiabfs5aejabeM8a3jaacMcaaaGaaiykaiaacMcaaaa@562A@
It
should be noted here that simple additive noise generates phase noise of
½
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=1laaaa@380D@
the value of the additive noise.
A
more mathematically rigorous approach shows that the main noise response is Lorentzian.
That is, of the form:
L(Δω)=
K
K+Δ
ω
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacIcacqqHuoarcqaHjpWDcaGGPaGaeyypa0ZaaSaaaeaacaWGlbaabaGaam4saiabgUcaRiabfs5aejabeM8a3naaCaaaleqabaGaaGOmaaaaaaaaaa@42C7@
A
similar result is achieved by feeding a noise current into a parallel RLC tank
circuit.
Hajimiri-Lee
Model
–
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@3784@
Up Conversion
The
Hajimiri-Lee Model is a model that,
attempts to explain phase noise, and very
specifically, up conversion of low frequency noise, by the proposal that it
there is an alleged time variant nature, to an otherwise, essentially, linear
system. This model is referred to as a LTV system rather than that of a linear
model regarded as a LTI. Hajimiri-Lee (HL) make several rather major claims of
the model’s correctness and explanatory power, despite these claims proving to
being incorrect. HL also make claims that alternative concepts produce
incorrect results, when some some of these actually do give the correct results.
The Hajimiri-Lee model is flawed for the following
reasons:
1
It fails
to correctly account for any frequency/phase modulation up-conversion generated
by non-linear capacitances and non-linear time constants. As all real
oscillators exhibit such effects, it is also a fatal flaw of the model.
2
It
predicts infinite noise at the oscillator frequency.
3
Various
researchers have shown that the model is invalid for flicker (up-converted)
noise and non-stationary noise, despite the HL paper claiming otherwise.
4
The
model can only be applied to specific oscillator types, despite claiming to be
a general theory. For example, its model collapses for quadrature oscillators.
5
A. Demi1,
with reference to the HL model, mathematically proves and states:
5.1
Is
the orthogonal decomposition valid in general?
5.2
Even if it is not strictly valid,
can it provide approximately correct results and intuition for practical
oscillator designs?
5.3
We show that the answer to both
questions is negative.
5.4
...it can predict results off by
as much as 50 dBc/Hz.
5.5
...the
argument [HL] in their “proof” is flawed, and the result they “proved” is
invalid.
5.6
… The
right-hand-side (RHS) of the differential equation (13) for the phase error is
non-linear. Thus, one can not use superposition to calculate the phase error
due to several perturbations, i.e., one can not calculate the phase errors due
to two perturbations separately and then sum them up to obtain the phase error
due to the two perturbations applied at the same
time.
6
The
root cause of up-conversion noise in real oscillators, as will be shown, is
non-linearity. The Hajimiri-Lee model specifically denies this, which makes it therefore,
essentially, useless in producing an optimum design that minimises
up-conversion of 1/f noise.
With the wealth of available information as to the non-validity of the HL
model, it is still somewhat interesting that so many references still refer to
the method in some positive manner.
The Hajimiri-Lee Model
The original paper should be referred to where necessary2.
The summary of the Hajimiri-Lee model as follows:
An LC oscillator
is excited by a noise current pulse.
An
argument is then presented that the phase of the oscillator system is,
typically maximally changed near the zero crossings of the oscillator, and
never changed at the oscillator peaks. It is also argued that effect of the
current pulse will persist indefinitely.
The essentials are that the Hajimiri-Lee model re-classify
a system from an assumed linear, time invariant system with one noise input signal
and one oscillator input signal, with a system block consisting of a linear a lossless
LC network, to that of an equivalent, alleged, time variant system, with one
noise input signal driving a block containing a “hidden” oscillator signal with
the linear LC circuit. A mathematical argument is then used that produces the
following result for the phase of the oscillator output signal:
ϕ(t)=
I
0
c
0
sin(Δωt)
2
q
max
Δω
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaaiikaiaadshacaGGPaGaeyypa0ZaaSaaaeaacaWGjbWaaSbaaSqaaiaaicdaaeqaaOGaam4yamaaBaaaleaacaaIWaaabeaakiGacohacaGGPbGaaiOBaiaacIcacqqHuoarcqaHjpWDcaWG0bGaaiykaaqaaiaaikdacaWGXbWaaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaGccqqHuoarcqaHjpWDaaaaaa@4EC8@
calculated
from what HL define as an Impulse Sensitivity Function (ISF):
Γ(
ω
0
τ)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCKaaiikaiabeM8a3naaBaaaleaacaaIWaaabeaakiabes8a0jaacMcaaaa@3CF9@
and
this result would then directly imply that
low frequency noise signals would be up-converted.
On a technical point, it should be noted that if
instead of a cos() noise input, a sin() noise input is assumed, The result for
phase noise becomes:
ϕ(t)=
I
0
c
0
cos(Δωt)
2
q
max
Δω
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaaiikaiaadshacaGGPaGaeyypa0ZaaSaaaeaacaWGjbWaaSbaaSqaaiaaicdaaeqaaOGaam4yamaaBaaaleaacaaIWaaabeaakiGacogacaGGVbGaai4CaiaacIcacqqHuoarcqaHjpWDcaWG0bGaaiykaaqaaiaaikdacaWGXbWaaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaGccqqHuoarcqaHjpWDaaaaaa@4EC3@
This
shows infinite noise at the oscillator frequency.
Overview
of the HL approach
Consider
a linear system in the Laplace transform domain.
v
¯
0
(s)=
v
i
¯
(s)h(s)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacaWG2baaamaaBaaaleaacaaIWaaabeaakiaacIcacaWGZbGaaiykaiabg2da9maanaaabaGaamODamaaBaaaleaacaWGPbaabeaaaaGccaGGOaGaam4CaiaacMcacaWGObGaaiikaiaadohacaGGPaaaaa@42C8@
Transforming
to the time domain is achieved by
V
0
(t)=inverse laplace[
v
i
(s)h(s)]
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaaabeaakiaacIcacaWG0bGaaiykaiabg2da9iaabMgacaqGUbGaaeODaiaabwgacaqGYbGaae4CaiaabwgacaqGGaGaaeiBaiaabggacaqGWbGaaeiBaiaabggacaqGJbGaaeyzaiaabUfacaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadohacaGGPaGaamiAaiaacIcacaWGZbGaaiykaiaac2faaaa@51E1@
However,
it can be shown, that if the Laplace transforms of the individual product terms
are known, then the output can be directly computed from the convolution
integral:
V
0
(t)=
∫
0
t
V
i
(τ)
H
v
(t−τ)dτ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaaabeaakiaacIcacaWG0bGaaiykaiabg2da9maapehabaGaamOvamaaBaaaleaacaWGPbaabeaakiaacIcacqaHepaDcaGGPaaaleaacaaIWaaabaGaamiDaaqdcqGHRiI8aOGaamisamaaBaaaleaacaWG2baabeaakiaacIcacaWG0bGaeyOeI0IaeqiXdqNaaiykaiaadsgacqaHepaDaaa@4DCC@
dimensionally,
H⇒
V
out
V
in
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabgkDiEpaalaaabaGaamOvamaaBaaaleaacaWGVbGaamyDaiaadshaaeqaaaGcbaGaamOvamaaBaaaleaacaWGPbGaamOBaaqabaaaaaaa@3FD0@
so that the integral calculates
V
0
⇒
V
out
V
in
V
i
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaaabeaakiabgkDiEpaalaaabaGaamOvamaaBaaaleaacaWGVbGaamyDaiaadshaaeqaaaGcbaGaamOvamaaBaaaleaacaWGPbGaamOBaaqabaaaaOGaamOvamaaBaaaleaacaWGPbaabeaaaaa@42CD@
Although,
the above completely specifies an output as a function of time, irrespective of
any “phase” parameter, the HL model makes an assumption of an additional
independent variable, the output “phase” of
V
0
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaaabeaakiaacIcacaWG0bGaaiykaaaa@39D3@
. The obvious
extension to the above convolution integral for this assumption would then be:
ϕ
0
(t)=
∫
0
t
ϕ
i
(τ)
H
ϕ
(t−τ)dτ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaaicdaaeqaaOGaaiikaiaadshacaGGPaGaeyypa0Zaa8qCaeaacqaHvpGzdaWgaaWcbaGaamyAaaqabaGccaGGOaGaeqiXdqNaaiykaaWcbaGaaGimaaqaaiaadshaa0Gaey4kIipakiaadIeadaWgaaWcbaGaeqy1dygabeaakiaacIcacaWG0bGaeyOeI0IaeqiXdqNaaiykaiaadsgacqaHepaDaaa@5073@
where
ϕ
i
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadshacaGGPaaaaa@3AF4@
is an input phase variable and
where
H
ϕ
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBaaaleaacqaHvpGzaeqaaOGaaiikaiaadshacaGGPaaaaa@3AD3@
a transfer function relating output phase to
an input phase variable:
H
ϕ
(t)⇒
ϕ
out
ϕ
in
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBaaaleaacqaHvpGzaeqaaOGaaiikaiaadshacaGGPaGaeyO0H49aaSaaaeaacqaHvpGzdaWgaaWcbaGaam4BaiaadwhacaWG0baabeaaaOqaaiabew9aMnaaBaaaleaacaWGPbGaamOBaaqabaaaaaaa@45FA@
The
equations HL write are
ϕ
0
(t)=
∫
−∞
t
I
i
(τ)
H
ϕ
(t,τ)dτ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaaicdaaeqaaOGaaiikaiaadshacaGGPaGaeyypa0Zaa8qCaeaacaWGjbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiabes8a0jaacMcaaSqaaiabgkHiTiabg6HiLcqaaiaadshaa0Gaey4kIipakiaadIeadaWgaaWcbaGaeqy1dygabeaakiaacIcacaWG0bGaaiilaiabes8a0jaacMcacaWGKbGaeqiXdqhaaa@50E0@
H
ϕ
(t,τ)=
Γ(
ω
0
τ)
q
max
u(t−τ)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBaaaleaacqaHvpGzaeqaaOGaaiikaiaadshacaGGSaGaeqiXdqNaaiykaiabg2da9maalaaabaGaeu4KdCKaaiikaiabeM8a3naaBaaaleaacaaIWaaabeaakiabes8a0jaacMcaaeaacaWGXbWaaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaOGaamyDaiaacIcacaWG0bGaeyOeI0IaeqiXdqNaaiykaaaa@4F9F@
where
dimensionally,
H
ϕ
⇒
ϕ
out
I
in
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBaaaleaacqaHvpGzaeqaaOGaeyO0H49aaSaaaeaacqaHvpGzdaWgaaWcbaGaam4BaiaadwhacaWG0baabeaaaOqaaiaadMeadaWgaaWcbaGaamyAaiaad6gaaeqaaaaaaaa@42AE@
so
that
ϕ
0
⇒
ϕ
out
Iin
I
i
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaaicdaaeqaaOGaeyO0H49aaSaaaeaacqaHvpGzdaWgaaWcbaGaam4BaiaadwhacaWG0baabeaaaOqaaiaadMeacaWGPbGaamOBaaaacaWGjbWaaSbaaSqaaiaadMgaaeqaaaaa@4457@
and
that therefore
H
ϕ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBaaaleaacqaHvpGzaeqaaaaa@3877@
is formulated as an output phase resulting
from an input current.
So,
if
Γ(
ω
0
τ)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCKaaiikaiabeM8a3naaBaaaleaacaaIWaaabeaakiabes8a0jaacMcaaaa@3CF9@
is non-zero, the output voltage automatically
has an inherent inbuilt phase variation dependant on another input signal.
For
technical reference, it should also be noted that there appears to be an
unstated argument that, as
Γ(
ω
0
τ)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCKaaiikaiabeM8a3naaBaaaleaacaaIWaaabeaakiabes8a0jaacMcaaaa@3CF9@
is supposed periodic, the integration is
independent of convolution folding.
There
are some mathematical inconstancies in the HL papers, in particular Reference
[2], p.183, equation 15 integrating cos() from negative infinity results in a
meaningless result, despite the result actually stated.
A
question immediately arises from this result of up-conversion, in that that nothing
has physically changed in the system from its equivalent linear, time invariant
system.
Mathematically
reformulating a system cannot change its basic physics. A LTI system, remodelled
as a LTV system cannot result in the existence of a something that can be proven
to not exist in the original formulation. Two signals driving a linear, time
invariant network cannot produce frequencies that are not present in the
driving signals, no matter how the system is reformulated mathematically.
Therefore
why does the HL model, seem to imply phase modulation for such a LTI system?
Cadence
Simulations
A
Cadence steady state simulation (PSS) for a linear RLC circuit was performed to
illustrate this point.
Linear
System ISF
A
main sine wave was generated with added square impulses at t=0 and t=π/2.
F
main=10Mhz, i=100ua
F
impulse=10Mhz, PW=20ns, TR=TF=1ns, i=100ua
Linear
ISF Schematics
Linear ISF
Waveforms
where
it can be seen (red) that the phase change is only of a transient nature.
So,
a driven circuit with loss, will only have a transient phase change, such that
the ISF will not be periodic, and decay, typically, exponentially to zero. This
means that the steady state ISF must go to zero, so that there is no steady
state phase modulation, as is required using standard LTI theory.
ISF
Oscillator Schematic
Consider
a simple tuned oscillator
ISF
Oscillator waveforms
the
above simulations results, show a persistent steady phase change arising from a
single impulse, as required for the HL model.
However,
despite the apparent nonzero steady state ISF, simulations have shown that without
time constant variation with signal amplitude, up conversion generated phase
noise is minimal. This implies that despite correction of oscillator losses, a
real oscillator can behaves as if its ISF is very low, or zero for low
frequencies.
The
above schematic shows a phase shift of around 0.3 radians, for a pulse current
of 500ua@10ns@20pf @1V, such that equation 15 of [2] becomes of the order of:
ϕ(t)=2×
10
−3
sin(Δωt)
Δω
per pa
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaaiikaiaadshacaGGPaGaeyypa0JaaGOmaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacqGHsislcaaIZaaaaOWaaSaaaeaaciGGZbGaaiyAaiaac6gacaGGOaGaeuiLdqKaeqyYdCNaamiDaiaacMcaaeaacqqHuoarcqaHjpWDaaGaaeiiaiaabchacaqGLbGaaeOCaiaabccacaqGWbGaamyyaaaa@528E@
or
-54 dBc for a 1pa @ 1hz noise signal.
Subsequent
simulations with a huge 15na low frequency noise injection, showed that any
up-conversion by this method would have to be < -50db, rather than the gross
+30dB calculated above.
What
went wrong? Why is there no up conversion?
From
the point of view of the HL theory, this is explained by the notion that the up
conversion gain constant is determined by the average value of the ISF
waveform. So, in this particular case the assumption is that C0 is
zero.
Linear/Non-Linear
Phase Noise Simulation
Many
different types of Cadence simulations were performed in order to confirm and
gain a correct understanding of up converted phase noise, only some of which
are included in this paper.
The
following open loop schematic shows a system that was adjusted to produce
linear and non-linear transfer functions. The system also allowed for the
injection of additional, low frequency noise to verify mixed up-conversion
effects.
Cadence
PSNoise analysis was then used to evaluate the conditions for up-conversion of
noise.
Schematic
This
schematic contains a square law component to generate square law non-linearities.
Low frequency noise was 15na/rthz into the tank.
The
above results show that without a non-linear procedure, up converted phase
noise is negligible
These
results were confirmed with a similar configuration, but as an actual oscillator.
Oscillator
Schematic
Oscillator
Waveforms
Simulations
showed that with minimal non-linearity (just enough to stabilise the
oscillator), up- conversion of low frequency was essentially zero. Maximum
up-conversion of noise occurred at maximum square law non-linearity, and heavy
diode clamping.
In
summary, the existence of baseband up converted noise frequencies in higher Q oscillators is
not because of an assumption that an oscillator is a linear, but a time variant
system with respect to a noise signal. Non-linearity is crucial in accounting
for up-conversion.
The
Correct Model
Consider
an oscillator signal;
V
0
(t)=
V
p
sin(
ω
0
(t)+ϕ(t))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaaabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaamiCaaqabaGcciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaaiikaiaadshacaGGPaGaey4kaSIaeqy1dyMaaiikaiaadshacaGGPaGaaiykaaaa@4B1B@
If
the phase or frequency of this expression is a function of that signal, that
is, the phase or frequency of the oscillator waveform is dependent on either
the circuit’s voltage or current at some point in the circuit, i.e. a non-linear
circuit. e.g.
V
0
(t)=
V
p
sin(
ω
0
(
s
mf
,t)+ϕ(
s
mp
,t))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaaIWaaabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaamiCaaqabaGcciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaaiikaiaadohadaWgaaWcbaGaamyBaiaadAgaaeqaaOGaaiilaiaadshacaGGPaGaey4kaSIaeqy1dyMaaiikaiaadohadaWgaaWcbaGaamyBaiaadchaaeqaaOGaaiilaiaadshacaGGPaGaaiykaaaa@529B@
and for any real
oscillator, this essentially, is always the case.
Then
mixed products will be generated.
Consider
an amplifier constructed using an ideal voltage controlled resistor:
Amplifier
with non-linear output resistance
Note
that this amplifier contains a fixed, linear capacitance. The block component
is a divider used to construct an ideal linear VCR with a gm source.
Additionally, there is a low frequency noise generator.
It
is often claimed that simple non-linear mixed products do not generate phase
noise, and sometimes the qualifier “memory less” is alluded to. It will now be
shown that it is precisely this non-linearity, with memory, that is the root
cause of up-conversion.
Signal
and phase noise waveforms of non-linear output resistance amplifier.
These
waveforms show a dramatic up-conversion of low frequency noise, dependant on
the value of load capacitor.
In
producing an output by varying a resister, the time constant of the output
network changes with signal voltage. This means that signal phase shift is
dependent on applied voltage, and hence phase shift is dependent on any applied
low frequency noise voltages. Therefore a phase modulator has been produced.
This results in up-conversion of low frequency noise.
Unfortunately,
the literature abounds with the notion that non-linear amplifiers are modelled
with Taylor series, when in reality, they need to be analysed with Volterra
series for any condition but zero frequency.
Non-linear
time constants can be produced in many ways. The most obvious is with
non-linear capacitors, of which all active devices exhibit to significant
extent, especially as most oscillators usually go from hard on to full cut-off.
One main offender are diode clampers. Time constants at a clamp point vary
significantly over the signal cycle.
References:
[1]
A. Demir - “On the Validity of
Orthogonally Decomposed Perturbations in Phase Noise Analysis” -
[2] Hanjmira and Lee
–
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@3783@
“A General Theory Of Phase Noise
in Electrical Oscillators”
–
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@3783@
IEEE Journal of Solid State
Circuits, vol 33, no 2, February 1998
[3] Hanjmira and Lee
–
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@3783@
“Oscillator Phase Noise
–
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A Tutorial”
[4] E. Hegazi, J. Rael, A. Abidi
–
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“The Designers Guide to High
Purity Oscillators”
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Aylward 2013
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