A Simple Model of Feedback Oscillator Noise Spectrum
I NTRODlTCTlON
This letter contaiw; brief thoughts on the following point;;.
a
1) The relationships among four commonh' used spect r l de­
sc riptio ns of oscilLltor short-term stabilit\· or Iloi:<e beh;n'ior.
2) :\ heuristic derivation, presented without formal proof, of the
expected spectrum of a feedback ,,:<cillator ill term, ()f known
oscillator parameter ,; .
3)
329
PROCEEDINGS LETTERS
1966
nonlinearities. The second component is due to parameter variations
at video frequencies w h ich affect the phase (such as variations in the
phase shift of a transistor due to car rier density fluctuations in the
base resistance). The additive noise component of S.\O(Wm) is iden­
tical to the spectral density of the noise voltage squared relative to
the mean square sig n a l voltage. For white additive noi,;e, this COI11'
ponent
is flat with frequency, For a feedback oscillator with an ef­
S.10(w)=2FKT/Ps; 1'8 is the
,igl l al level at os cil l at or active element input.
fective noise figure F, the two-sided
Some experimental result,; which illLhtrate the validity uf the
.J.) Comments on the dfel·t of nonlillcarity, spcci lic spectral re­
quirements for several applicatioJb, choice of resonator fre­
quency and active element, and expected spectrulll charac­
SPECTRAL
:\[ODELS
osci ll ator
1/"'m or l/i spectrum ) . The total power spectral density of oscil·
lator input phase errors is of the form S3.0(wm)=a/wm+f3, where
(a
a
is a constant determined by the level of
OF PHASE \'ARIATlO:\S
who:,c measurable o u tput can be ex­
It is common to treat
1>(t)
=
A cos
[wol
+
Wo
1>(1)].
w
as a zero-mean stationary random process
describing deviations of the phase from the ideal. The frequency do­
variations is ccllltained in
the "power" spectral density S,p(wm) of the phase 1>(t) or, alterna·
tively, in the "power" spectral density S¢(w",l of the frequcncy"i. By
analogy to modulation theory, we use w'" to mean the modulatioIl,
video, baseband, or offset frequency associated with the noi,;e-like
variations in
1>(t). The units of S¢(wm) are radians'/cps bandwidth
or dB relative to 1 radians'/cps BW; S<i>(w m ) is expressed in ( radialls/
sec)' per cis B\\, [1] [ 2 j. The two are related by S,p(wm) =w",'S¢(Wm I.
can also he expressed in terms of the equivalent rms fre­
quency deviation 6.j,,,,, in a given video bandwidth. Further, suhj e ct
to the limitatioJls that
1'«1
(small total modulatiol1 ind e� ) and tha t
AM «FM components, the normalized
T{F power spectrum G(w -w,,)
S¢(w " ) ; i.e., RF
sidebands relative to the carrier are down by S¢(wm) expressed ill
decibels relative to 1 radian'/B\\'.
,
is identical to the two-sided spectrum of the phase
RELATION TO OSCILLATOR
;\ basic requirement on a n oscillator noise model is that it show
S"(w,,.)
to the
known or expected noise and sig nal levels and resonator charac­
teristics of the oscillator.
Wo
> --2Q
[ (-2Qwm
"". ) 2J .
S.(w",) = S.10 1 +
Thi,; yield s an asymptotic model for
Fig.
-
S.p(w) shown on log-log scales i n
I.
The model can be summarized as follows.
decreases with w,"
9 dB/octave up to the point where l/i effects no longer
predom i nate.
S",(Wm)
at
at
6
dB/octave from that poi nt up to the feedback loop
half-bandwidth.
at
0 dR/octave above that frequency up to
a limit imposed
by subsequent filtering.
SJ,(w)
decrea'ics at 3 dB/octave up to the first breakpoint, is
/lat with frequency up to the feedback baseband bandwidth,
and increases at
I �TER:\AL � OISE
clearly the relationship of the spectrulll of the phase
m
,\ suitable cOl1lpo,;ite expression is
main infonriation about phase or frequency
S<i>(wm)
and
wm <-2Q
for
pressed as
v(i)
l/j variations
f3 is=2FKT/P.., for two-sided spectra.
To find S</>(wm) or S<f,(wm), we use the fact that
teristics of several 'bcillator types.
Consider a stable
The video spectrum of parameter variations is fOllnd typically to
have a power spectral density varying inversely with frequency
simpie Illodel.
6 dB octave above that point.
The case where l/f effects predominate only for frequencie,; sl1lall
compared with the feedback loop bandwidth is shown here as an
A simple picture can be comtrueted using
a model of a linear feedback oscillator. Minor corrections to the re­
sults are
ne cessary to account for nonlinear effects which must be
A ssume a single resonator feedback
network of fractional bandwidth 2B/wo= l/Q, where Q is the operat­
ing, or loaded, quality factor. For small phase deviations at video
rates which fall within the feedback half-bandwidth wc/2Q, a ph,hc
present in a physical oscillator.
error at the oscillator input due to noise or parameter
variatio ll s
S�,(�)2S68
S<j>'SM
In this region
in thiS region
reo
suIts in
a frequency error determined by the phase-frequency rela·
tionship of the feedback network, 6.0 = 2Q¢/wo Thus, for modulatioll
rates less than the half-bandwidth of the feedback loop, the spectrulll
of the frequency Sf,(w) is identical (with a scale factor) to the spec·
trum uf the uncertainty of the oscillator i n put phase due to noise and
parameter variations. This uncertainty will be denoted .10(1), and its
two· sided power spectral density
S.\e(wm).
large compared to the feedback bandwidth,
a series feedback loop is out of the circuit. :\t these modulation rate'.
the power spectral density of the outp u t phase S¢(w) is identical to
For modulation rates
the spectrum of the oscillator input phase uncertainty
For a physical oscillator the spectrum
S.\e(wm)
S.\o(wm).
of the i nput phil ' "
2FKT
P,
I
-1---.---
uncertainty 6.0(1) is expected to have two principal components Onc
component is due to phase ullcertainties resulting from additi\ 'e
whit(, noise at frequenci e , around the oscillator frequellcy, a,; well as
Iloi:<e at other f r equ encie s mixed int
()
the pi'"
hand
oj ill t P re :«
rvlanl!script received December 10, 1965; red�ed December 1'), 1965
h\"
h.�. I.
])('rl\-atiucl of ()scdlat[)r Spcctra. The lo.l'.kal s('(jupncp Ipading fWlll o..;cil­
idtor IJaran1f't('r� to S[lt'ctrdCl c:laracteri�tic.:; is fJ[e�)ented Il1:'re. The fJO\ver spe(tra
of OlttJltH pha5t' O[ fre-quency are d{'rh-ed frolll the spectmll1 of input phase
uncertainties and f rom the oscillator feedhack bandwidth. TIlf' calclilable con­
stants of the oscillator an" F KT, p.�, and :.tC11/20; the 1 /f constant a is not aCCLl­
rately pH'die-tahle hilt can bl? infe rred from data. The amplitllde 8pectrum of
frequency d("viation and the RF spectrum can be derived as shown, subject to
limitations discussed in the text.
PROCEEDtNGS OF THE IEEE
330
example. For a high-Q osci llator , 1 If effects in S..\9 call predominate
out to a modulation rate exceed ing wlI/2Q; in this case there is no
6
dB per octave regioll inS",(w) . .\ similar spectrum results where large
FEBRUARI
caused by the nonlineari ty. The excellent fit of the data implies that
this degradation of effective noise figure may well be an adequate
description of the effect of llolliinearity.
additive noise in fol lowing amplifier stages or measuring equipment
\'LDEO FNEQn,"ICY J{A:\GE OF
obscures the o;.;cillator internal noise, except at very low modulation
TKTEl1EST
:\ n um ber of app lication s which have been dealt with in this iScillc
frequencies.
:\ote that there is a port ion of the curve Sq,( w,,,) which is propor­
tional to 1 /W",2, leading to a l/w,., or l/f variation for rIlh pha,e de­
of the 1'1UJCEEDT:\(;S may be sllmmarized in terms of the v ide o fre­
quency range of interest. Space systems and Doppler radar applica­
viation. This is often confused with the true 1/f effects associated
tions are of particular intere,t to the author. For these two, illtere"t
with parameter variation" leading to the 1 If portion of the cllrve for
lie, in the rang e of
a
few cis up to 100 kcl,;. Space applications
Sj,8(w.,,) and S,;,iw",). These two are not the s;lIne thing; "1 (f" refers to
typically concentrate on the range \\'here, for a cry"tal os cil lat or ,
a power spectral density rather thall an amplitu de spectrum.
S;/w,,,) is proportio nal to S.l8(U:,,,)
I n practice, the me,hurable Sq,( W",) is always modilied by sub­
[3],
\"hile Doppler radar applica­
tions place additional empha,is on the region above the oscillator
[4 1 .
sequent bandlimiting filtering and by additive noise c ontr ibuted b\·
feedback loop bandwidth
following amplifiers. It is conceivable that. for a two-terminal (bcil­
microwave systems which employ multiplication from the ,,,cillator
lator, the filtering action of the resonator eliminates the additive
frequency.
phase noise component for w". >(wo)/2Q).
Both
a pplication s
typically require
CHOICE OF OSCILLATOR FREQUE:\CY FOR CRYSTAL
OSCILLATOR-l\IULTlPLlER
EXPERIME�TAL VERIFlCATIOX
:Yleasurements were taken on a stable microwave signal SOlIrcel
designed to have a spectral purity limited only by the oscillator,
which was a 100 \f/i crys ta l oscillator. This uni t employs two large­
It is of intere ,;t to in"pect the e ffec t of o'icillator frequency upon
the output spectrum of an oscillator-multiplier system having a
fixed output frequency. Two assllmptions which aid the calculation
jump step recovery diode llluitipliers with amplilication between
are a) constan t oscillator input signal-to-noise ratio, al�d b) res­
them. The data are presented ill Fig. 2 in comparison with a model
onator Q varying inversely with the oscillator frequency wu, ('nder
derived from the following constants:
the following results.
Feedback bandwidth = 16 kcls
1) For w'" «wo/2Q), of the lower frequency oscillator, the multi­
= -4 dBm
Ps
F
KT
plied output S6(W",) is identical for either choice.
2) For w",»(wn/2Q), the output S<t(wm) varies as the square of
the mu l tip lication ratio (i.e., inversely as the square of the
oscillator fre quency ) .
This can be verified by it simple graphical construction.
=9dB
= - 1 74 dBm in 1 cis B\\,
\1ultiplicat ion ratio = 100=40 dB
iV22FKT / P8
= +40 + 3 +9-174+4= -118 dB.
This leads to an a"'mptotic value for Sq,(wm) of -118 dB relative to
1 rad ian2/ B\Y in 1 cis bandwidth, i. e . , a carrier-to-sideband ratio of
118 dB. The "1(f" region (9 dB/oct ave ) constant" is estimated for
.0
E
.$
..,.
(f)
-70dB
CnOICE OF :\CTl\'I':
ELlc\U:XT 1:\ A TRANSISTOR OSCILLATOR
I t is apparent that l!f variations and nonlinearity can have
significant deleterious effecb on the attainable low levels of -"'<p( wm).
best data fit.
-60 dB
these assumptions a comparison of two oscillator frequencies yields
I
1�
"
I n the light of suggestions by O. \1 ueller that microthermal effects
[6] cont ribute to l/f noise in transistors, it is suggested that AGC
oscillators using large area transistors having high power capabilitie:;
may provide simultaneous improvements in 1Ij level and in non­
"
linear e ffec t s.
T
- CALCULATED
•
-BOdo
MEASURED
SPECTRUM CHARACTERISTICS OF MICROWAVE SOLID STATE SOURCES
9d6 per octo.. e
�
The spectrum model given here allows simple prediction of spec­
�
,
-90dB
trum shape and level for microwave sources of the types discussed by
2
lOOdB
-their measurements for crystal oscillator units extend to w".
IlOdB
such that, for the measurements cited, w'" «wo/2Q).
Johnson et al
'"
[5].
Comparison with their data shows good agreement
»(wo/2Q), while m i crowave oscillators are characterized by Q factors
�
120d6
250
1000
wm/171'
4000
16000
ACKNOWLEDGMENT
64000
cis
F ig. 2.
'\1.,(0.'11,) for �table 1..1icrowave �it!,llal :-:'nllrn'. ,T he d:1ta presented here is
the average of two independellt n1f'aSllrelllellt� which were in eXC'e"llent ag ree ­
ment. These measlirements were made at X Band on the mUltiplied Olltput of
a 100 �1('/s voltag.e controlled crystal oscillator having- a 16 kc/;; feerlhac k half­
ban(l\vidth. Since this bandwidth can be reduced by a considerable factor with­
out exceeding the present state of the art. the data is 110t intended to represent
ultimate attainable levels, but rather s er ves as an illustrative example. The tlf
con stant is dloEen for best data fit. SlopE'S al1(i other calculated p arame te rs are
rl erived from known o scillator characteristics.
The author is pleased to acknowledge helpful discussions with
members of IEEE Sllbcommittee 14.7, of wh ic h this letter may be
considered a brief summan·. Prepublication access to all of the papers
contained in this issue i, also freely acknowledged. The iniluence of
L. S. Cutler ,
J. :\.
\!tlllen, and \\'. L. O. Smith has been of sp eci a l
value in the preparation of this correspondence.
D. B. LEESOK
Applied Technology, Inc
Palo Alto, Calif.
:\ O:-lLlXEAR EFFECTS
The data was based on an estimated transistor noise figure of 9
dB. This was taken high to account for n onlin ear mixing of noise at
third harmonic and higher frequencies which is mixed into the pass
band of interest by second harmonic periodic parameter variations
19.5 Ge/s Solid
State Local Oscillator P::'\ 3t-OOi191, manufactured by Applied
Technology, Inc., Palo Alto, Calil. :Measurements are a\'erage of values measured
by the author and D. .1, l l ealey, III, \Yestinghouse Corp., Baltimore, :Md. . using
Spectra Electronics SE-200 and \Vestinghouse proprietary noise test sets.
REFEREXCES
[1] L, S. Cutler and C. L. Searle, "Some aspects of the theory and meClsurement of
frf"Quency fiu('tuations in frf'(Juency standards," this i8Slte, pa!.!:e 130.
[2] E. J. Baghdady, R. ;\1, Linc01n, and B. D. Xelin, "Short-term frequency sta­
bility: characterization, theory, and measurement" Proc. JJ£li.E. vol. 53,
I'll. 704 -722, July 1965.
[3] R. L. Sydnor,.T, .1. Calrlwell. and B. E. Ro se, "Frequency stability requirements
for space communkations and trdcking systems," this -issue, page 231.
[41 D. R. Lee son and (�. F. .Tolln�()n, (jShort-term stability for a Doppler radar:
requirements, measurements, and tec hnique s, " this issue, page 244.
[5] S. L. Johns-on, B. II. Smith, and D. .\. Calder. ":\oi�e �pect rll;ll characteristics
of low-noise microw<1\'e tube� and f;olili-state dedcf's,)) this issue, page 258,
[6] 0, l\i lleller, :'Thermal fel"dback and 1 /f-fHC'ker noise in semiconductor devices,"
1965 lnternai'l Solid Stal e Circll£ts Conference, Digest, p. 6g ff.