Jitter and Pha Noi in Ring Oscillators Ali Hajimiri,Sotirios Limotyrakis,and Thomas H.Lee,Member,IEEE
Abstract—A companion analysis of clock jitter and pha noi of single-ended and differential ring oscillators is prented.The impul nsitivity functions are ud to derive expressions for the jitter and pha noi of ring oscillators.The effect of the number of stages,power dissipation,frequency of oscillation,and short-channel effects on the jitter and pha noi of ring oscillators is analyzed.Jitter and pha noi due to substrate and supply noi is discusd,and the effect of symmetry on the upconversion of 1/f noi is demonstrated.Several new design insights are given for low jitter/pha-noi design.Good agreement between theory and measurements is obrved.
Index Terms—Design methodology,jitter,noi measurement, oscillator noi,oscillator stability,pha jitter,pha-locked loops,pha noi,ring oscillators,voltage-controlled oscillators.
I.I NTRODUCTION
D U
E to their integrated nature,ring oscillators have be-
come an esntial building block in many digital and communication systems.They are ud as voltage-controlled oscillators(VCO’s)in applications such as clock recovery circuits for rial data communications[1]–[4],disk-drive read channels[5],[6],on-chip clock distribution[7]–[10],and integrated frequency synthesizers[10],[11].Although they have not found many applications in radio frequency(RF), they can be ud for some low-tier RF systems. Recently,there has been some work on modeling jitter and pha noi in ring oscillators.References[12]and[13] develop models for the clock jitter bad on time-domain treatments for MOS and bipolar differential ring oscillators, respectively.Reference[14]propos a frequency-domain approach tofind the pha noi bad on an linear time-invariant model for differential ring oscillators with a small number of stages.
In this paper,we develop a parallel treatment of frequency-domain pha noi[15]and time-domain clock jitter for ring oscillators.We apply the pha-noi model prented in[16] to obtain general expressions for jitter and pha noi of the ring oscillators.
The next ction briefly reviews the pha-noi model prented in[16].In Section III,we apply the model to timing jitter and develop an expression for the timing jitter of oscilla-tors,while Section IV provides the derivation of a clod-form expression to calculate the rms value of the impul nsitivity function(ISF).Section V introduces expressions for jitter and pha noi in single-ended a
nd differential ring oscillators Manuscript received April8,1998;revid November2,1998.
A.Hajimiri is with the California Institute of Technology,Pasadena,CA 91125USA.
S.Limotyrakis and T.H.Lee are with the Center for Integrated Systems, Stanford University,Stanford,CA94305USA.
Publisher Item Identifier S0018-9200(99)04200-6.in long-and short-channel regimes of operation.Section VI describes the effect of substrate and supply noi as well as the noi due to the tail-current source in differential struc-tures.Section VII explains the design insights obtained from this treatment for low jitter/pha-noi design.Section VIII summarizes the measurement results.
II.P HASE N OISE
The output of a practical oscillator can be written
as
is periodic in
2
modelfluctuations in amplitude and pha due to internal and external noi sources.The amplitudefluctuations are significantly attenuated by the amplitude limiting mechanism, which is prent in any practical stable oscillator and is particularly strong in ring oscillators.Therefore,we will focus on pha variations,which are not quenched by such a restoring mechanism.
As an example,consider the single-ended ring oscillator with a single current source on one of the nodes shown in Fig.1.Suppo that the current source consists of an impul of current with
area
is proportional to the injected
charge
(3)
where is the voltage swing across the capacitor
and
thus reprents the nsitivity of every point of the waveform to a
perturbation,
Fig.1.Five-stage inverter-chain ring oscillator.
Being interested in its pha
(4)
where is a unit step.
Knowing the respon to an impul,we can calculate
where reprents the noi current injected into the node of interest.Note that the integration aris from the clod-loop nature of the oscillator.The single-sideband pha-noi spectrum due to a white-noi current source is given by[16]
1
(6)
where is the single-
sideband power spectral density of the noi current source,
and is the frequency offt from the carrier.In the ca
of multiple noi sources injecting into the same
node,
noi sources
is
times for a differential ring
oscillator).
From(5),it follows that the upconversion of low-frequency
noi,such as
1
(7)
where is the dc value of the ISF.Since the height of the
positive and negative lobes of the ISF is determined by the
slope of the rising and falling edges of the output waveform,
respectively,symmetry of the rising and falling edges can
reduce and hence the upconversion of
1
where
is the variance of the uncertainty introduced by one stage
during one transition.Noting
that
is a proportionality constant determined by circuit
parameters.
Another instructive special ca that is not usually consid-
ered is when the noi sources are totally correlated with one
another.Substrate and supply noi are examples of such noi
sources.Low-frequency noi sources,such as
1
is another proportionality constant.Noi sources
such as thermal noi of devices are usually modeled as
uncorrelated,while substrate and supply-noi sources,as
well as low-frequency noi,are approximated as partially
or fully correlated sources.In practice,both correlated and
uncorrelated sources exist in a circuit,and hence a log–log
plot of the timing
jitter
In most digital applications,it is desirable for
or where
calculated to be
becomes
larger,since each transition occupies a smaller fraction of the
period.Bad on the obrvations,we approximate the ISF
as triangular in shape and with symmetric rising and falling
edges,as shown in Fig.6.The ca of nonsymmetric rising
and falling edges is considered in Appendix B.
The ISF has a maximum of1where is the
maximum slope of the normalized waveform
,and hence the
slopes of the sides of the triangles are
Fig.8.RMS values of the ISF’s for various single-ended ring oscillators versus number of stages.
On the other hand,stage delay is proportional to the ri
time
(14)
where
is a proportion-ality constant,which is typically clo to one,as can be en in Fig.7.
The period is
2
(16)
Note that the
1
dependence
of
becau the effect of variations in other parameters,such
as
,and thus the ISF is a unitless,frequency-and
amplitude-independent function.
Equation (16)is valid for differential ring oscillators as well,since in its derivation no assumption specific to single-ended oscillators was made.Fig.9shows
the
changes.Members of the
cond t of oscillators have a fixed total power dissipation and fixed load resistors,which result in variable swings and for whom data are shown with circles.The third ca is that of a fixed tail current for each stage and constant load resistors,who data are illustrated using cross.Again,in spite of the diver variations of the frequency and other
circuit parameters,the
1
dependency
of This is shown with the solid line in Fig.9.A similar result
can be obtained for bipolar differential ring oscillators.
Although
