Linearity, Time Variation, and Oscillator Phase Noise

3Linearity, Time Variation, andOscillator Phase NoiseThomas H. Lee and Ali Hajimiri

3.1 Introduction

The theoretical and practical importance of oscillators has motivated thedevelopment of numerous treatments of phase noise. The sheer number ofpublications on this topic underscores the importance attached to it. At thesame time, many of these disagree on rather fundamental points, and it maybe argued that the abundance of such conflicting research quietly testifiesto the inadequacies of many of those treatments. Complicating the searchfor a suitable theory is that noise in a circuit may undergo frequency transla-tions before ultimately becoming oscillator phase noise. These translationsare often attributed to the presence of obvious nonlinearities in practicaloscillators. The simplest theories nevertheless simply ignore the nonlinearitiesaltogether and frequently ignore the possibility of time variation as well.Such linear, time-invariant (LTI) theories surprisingly manage to provideimportant qualitative design insights. However, these theories are understand-ably limited in their predictive power. Chief among the deficiencies of anLTI theory is that frequency translations are necessarily disallowed, beggingthe question of how the (nearly) symmetrical sidebands observed in practicaloscillators can arise.

Despite this complication, and despite the obvious presence of non-linearities necessary for amplitude stabilization, the noise-to-phase transfer

59

60 RF and Microwave Oscillator Design

function of oscillators nonetheless may be treated as linear. However, aquantitative understanding of the frequency translation process requires aban-donment of the principle of time invariance implicitly assumed in mosttheories of phase noise. In addition to providing a quantitative reconciliationbetween theory and measurement, the time-varying phase noise model pre-sented here identifies an important symmetry principle, which may beexploited to suppress the upconversion of 1/f noise into close-in phase noise.At the same time, it provides an explicit accommodation of cyclostationaryeffects, which are significant in many practical oscillators, and of amplitude-to-phase (AM-PM) conversion as well. These insights allow a reinterpretationof why certain topologies, such as the Colpitts oscillator, exhibit good perfor-mance. Perhaps more important, the theory informs design, suggesting noveloptimizations of well-known oscillators, as well as the invention of newcircuit topologies. Tuned LC and ring oscillator circuit examples are presentedto reinforce the theoretical considerations developed. Simulation issues andthe topic of amplitude noise are considered as well.

We first revisit how one evaluates whether a system is linear or time-invariant. Indeed, we find that we must even take care to define explicitlywhat is meant by the word ‘‘system.’’ We then identify some very generaltradeoffs among key parameters, such as power dissipation, oscillation fre-quency, resonator Q , and circuit noise power. These tradeoffs are first studiedqualitatively in a hypothetical ideal oscillator in which linearity of the noise-to-phase transfer function is assumed, allowing characterization by an impulseresponse. Although the assumption of linearity is defensible, we shall seethat time invariance fails to hold even in this simple case. That is, oscillatorsare linear, time-varying (LTV) systems, where ‘‘system’’ is defined by thenoise-to-phase transfer characteristic. Fortunately, complete characterizationby an impulse response depends only on linearity, not time invariance. Bystudying the impulse response, we discover that periodic time variation leadsto frequency translation of device noise to produce the phase noise spectraexhibited by real oscillators. In particular, the upconversion of 1/f noise intoclose-in phase noise is seen to depend on symmetry properties that arepotentially controllable by the designer. Additionally, the same treatmenteasily subsumes the cyclostationarity of noise generators, and helps explainwhy class-C operation of active elements within an oscillator can be beneficial.Illustrative circuit examples reinforce key insights of the LTV model.

In general, circuit and device noise can perturb both the amplitudeand phase of an oscillator’s output. Because amplitude fluctuations are usuallygreatly attenuated as a result of the amplitude stabilization mechanismspresent in every practical oscillator, phase noise generally dominates, at least

61Linearity, Time Variation, and Oscillator Phase Noise

at frequencies not far removed from the carrier. Thus, even though it ispossible to design oscillators in which amplitude noise is significant, wefocus primarily on phase noise here. We show later that a simple modificationof the theory allows the accommodation of amplitude noise as well, permittingthe accurate computation of output spectrum at frequencies well removedfrom the carrier.

3.2 General Considerations

Perhaps the simplest abstraction of an oscillator that still retains some connec-tion to the real world is a combination of a lossy resonator and an energyrestoration element. The latter precisely compensates for the tank loss toenable a constant-amplitude oscillation. To simplify matters, assume thatthe energy restorer is noiseless (see Figure 3.1). The tank resistance is thereforethe only noisy element in this model.

To gain some useful design insight, first compute the signal energystored in the tank:

E 2 =12

CV 2pk (3.1)

so that the mean-square signal (carrier) voltage is:

V 2sig =

E storedC

(3.2)

where we have assumed a sinusoidal waveform.The total mean-square noise voltage is found by integrating the resistor’s

thermal noise density over the noise bandwidth of the RLC resonator:

Figure 3.1 ‘‘Perfectly efficient’’ RLC oscillator.


V 2n = 4kTRE

∞

0

| Z ( f )R |2df = 4kTR

14RC

=kTC

(3.3)

Combining (3.2) and (3.3), we obtain a noise-to-carrier ratio (thereason for this ‘‘upside-down’’ ratio is simply one of convention):

NC

=V 2

n

V 2sig

=kT

E stored(3.4)

Sensibly enough, one therefore needs to maximize the signal levels tominimize the noise-to-carrier ratio.

We may bring power consumption and resonator Q explicitly intoconsideration by noting that Q can be defined generally as proportional tothe energy stored, divided by the energy dissipated:

Q =v0E stored

Pdiss(3.5)

Therefore,

NC

=v0kTQPdiss

(3.6)

The power consumed by this model oscillator is simply equal to Pdiss,the amount dissipated by the tank loss. The noise-to-carrier ratio is hereinversely proportional to the product of resonator Q and the power consumed,and directly proportional to the oscillation frequency. This set of relationshipsstill holds approximately for real oscillators and explains the near obsessionof engineers with maximizing resonator Q , for example.

Other important design criteria become evident by coupling the forego-ing with additional knowledge of practical oscillators. One is that oscillatorsgenerally operate in one of two regimes that may be distinguished by theirdiffering dependence of output amplitude on bias current (see Figure 3.2),so that one may write

Vsig = IBIASR (3.7)


Figure 3.2 Oscillator operating regimes.

where R is a constant of proportionality with the dimensions of resistance.This constant, in turn, is proportional to the equivalent parallel tank resistance[1], so that

Vsig ~ IBIASR tank (3.8)

implying that the carrier power may be expressed as

Psig ~ (IBIASR tank)2 (3.9)

The noise power has already been computed in terms of the tankcapacitance as

V 2n =

kTC

(3.10)

but it may also be expressed in terms of the tank inductance:

V 2n =

kTC

=kT

S 1

v20LD

= kTv20L (3.11)

An alternative expression for the noise-to-carrier ratio in the current-limited regime is therefore


NC

~kTv0L

(IBIASR tank )2 (3.12)

Assuming operation at a fixed supply voltage, a constraint on powerconsumption implies an upper bound on the bias current. Of the remainingfree parameters, then, only the tank inductance and resistance may be practi-cally varied to minimize the N /C ratio. That is, optimization of such anoscillator corresponds to minimizing L /(R tank)2. In many treatments, max-imizing tank inductance is offered as a prescription for optimization. How-ever, we see that a more valid objective is to minimize L /(R tank)2 [2]. Since,in general, the resistance is itself a function of inductance, identifying (andthen achieving) this minimum is not always trivial. An additional consider-ation is that, below a certain minimum inductance, oscillation may cease.Hence, the optimization prescription here presumes oscillation, and in aregime where the output amplitude is proportional to the bias current.

3.3 Detailed Considerations: Phase Noise

To augment the qualitative insights of the foregoing analysis, let us nowdetermine the actual output spectrum of the ideal oscillator.

3.3.1 Phase Noise of an Ideal Oscillator

Assume that the output in Figure 3.1 is the voltage across the tank, as shown.By postulate, the only source of noise is the white thermal noise of the tankconductance, which we represent as a current source across the tank with amean-square spectral density of

i 2n

D f= 4kTG (3.13)

This current noise becomes voltage noise when multiplied by theeffective impedance facing the current source. In computing this impedance,however, it is important to recognize that the energy restoration elementmust contribute an average effective negative resistance that precisely cancelsthe positive resistance of the tank. Hence, the net result is that the effectiveimpedance seen by the noise current source is simply that of a perfectlylossless LC network.


For a relatively small offset frequency Dv from the center frequencyv0, the impedance of an LC tank may be approximated by

Z (v0 + Dv ) ≈ jv0L

2Dvv0

(3.14)

We may write the impedance in a more useful form by incorporatingan expression for the unloaded tank Q :

Q =R

v0L=

1v0GL

(3.15)

Solving (3.15) for L and substituting into (3.14) yield:

|Z (v0 + Dv ) | ≈1G

v0

2Q |Dv |(3.16)

Thus, we have traded an explicit dependence on inductance for adependence on Q and G.

Next, multiply the spectral density of the mean-square noise currentby the squared magnitude of the tank impedance to obtain the spectraldensity of the mean-square noise voltage:

v 2n

D f=

i 2n

D f|Z |2 = 4kTRS v0

2QDv D2

(3.17)

The power spectral density of the output noise is frequency-dependentbecause of the filtering action of the tank, falling as the inverse-square ofthe offset frequency. This 1/f 2 behavior simply reflects the fact that thevoltage frequency response of an RLC tank rolls off as 1/f to either side ofthe center frequency, and that power is proportional to the square of voltage.Note also that an increase in tank Q reduces the noise density, when allother parameters are held constant, underscoring once again the value ofincreasing resonator Q .

In our idealized LC model, thermal noise affects both amplitude andphase, and (3.17) includes their combined effect. The equipartition theoremof thermodynamics tells us that, in equilibrium, amplitude and phase noise


power are equal. Therefore, the amplitude-limiting mechanism present inany practical oscillator suppresses half the noise given by (3.17).

It is traditional to normalize the mean-square noise voltage density tothe mean-square carrier voltage, and report the ratio in decibels, therebyexplaining the ‘‘upside down’’ ratios presented previously. Performing thisnormalization yields the following equation for phase noise:

L {Dv } = 10 logF2kTPsig

S v02QDv D

2G (3.18)

The units of phase noise are thus proportional to the log of a density.Specifically, they are commonly expressed as ‘‘decibels below the carrier perhertz,’’ or dBc/Hz, specified at a particular offset frequency Dv from thecarrier frequency v0. For example, one might speak of a 2-GHz oscillator’sphase noise as ‘‘−110 dBc/Hz at a 100-kHz offset.’’ It is important to notethat the ‘‘per Hz’’ actually applies to the argument of the log, not to thelog itself; doubling the measurement bandwidth does not double the decibelquantity. As lacking in rigor ‘‘dBc/Hz’’ is, it is common usage [1].

Equation (3.18) tells us that phase noise (at a given offset) improvesas both the carrier power and Q increase, as predicted earlier. These dependen-cies make sense. Increasing the signal power improves the ratio simply becausethe thermal noise is fixed, while increasing Q improves the ratio quadraticallybecause the tank’s impedance falls off as 1/QDv .

Because many simplifying assumptions have led us to this point, itshould not be surprising that there are some significant differences betweenthe spectrum predicted by (3.18) and what one typically measures in practice.For example, although real spectra do possess a region where the observeddensity is proportional to 1/(Dv )2, the magnitudes are typically quite a bitlarger than predicted by (3.18), because there are additional important noisesources besides tank loss. For example, any physical implementation of anenergy restorer will be noisy. Furthermore, measured spectra eventually flattenout for large frequency offsets, rather than continuing to drop quadratically.Such a floor may be due to the noise associated with any active elements(such as buffers) placed between the tank and the outside world, or it caneven reflect limitations in the measurement instrumentation itself. Even ifthe output were taken directly from the tank, any resistance in series witheither the inductor or capacitor would impose a bound on the amount offiltering provided by the tank at large frequency offsets and thus ultimately


produce a noise floor. Finally, there is almost always a 1/(Dv )3 region atsmall offsets.

A modification to (3.18) provides a means to account for these discrep-ancies:

L {Dv } = 10 logF2FkTPsig

HS1 +v0

2QDv D2JS1 +

Dv1/f 3

|Dv | DG (3.19)

These modifications, due to Leeson, consist of a factor F to accountfor the increased noise in the 1/(Dv )2 region, an additive factor of unity(inside the braces) to account for the noise floor, and a multiplicative factor(the term in the second set of parentheses) to provide a 1/ |Dv |3 behaviorat sufficiently small offset frequencies [3]. With these modifications, thephase noise spectrum appears as in Figure 3.3.

It is important to note that the factor F is an empirical fitting parameterand therefore must be determined from measurements, diminishing thepredictive power of the phase noise equation. Furthermore, the model assertsthat Dv1/f 3, the boundary between the 1/(Dv )2 and 1/ |Dv |3 regions, isprecisely equal to the 1/f corner of device noise. However, measurementsfrequently show no such equality, and thus one must generally treatDv1/f 3 as an empirical fitting parameter as well. Also it is not clear whatthe corner frequency will be in the presence of more than one noise source,each with an individual 1/f noise contribution (and generally differing 1/fcorner frequencies). Finally, the frequency at which the noise flattens out isnot always equal to half the resonator bandwidth, v0 /2Q .

Figure 3.3 Phase noise: Leeson versus (3.18).


Both the ideal oscillator model and the Leeson model suggest thatincreasing resonator Q and signal power are ways to reduce phase noise.The Leeson model additionally introduces the factor F, but without knowingprecisely what it depends on, it is difficult to identify specific ways to reduceit. The same problem exists with Dv1/f 3 as well. Finally, blind applicationof these models has periodically led to earnest but misguided attempts bysome designers to use active circuits to boost Q . Sadly, increases in Q throughsuch means are necessarily accompanied by increases in F as well, generallypreventing the anticipated improvements in phase noise. Again, the lack ofanalytical expressions for F can obscure this conclusion, and one continuesto encounter various doomed oscillator designs based on the notion of activeQ boosting.

That neither (3.18) nor (3.19) can make quantitative predictions aboutphase noise is an indication that at least some of the assumptions used inthe derivations are invalid, despite their apparent reasonableness. To developa theory that does not possess the enumerated deficiencies, we need to revisit,and perhaps revise, these assumptions.

3.4 The Roles of Linearity and Time Variation in PhaseNoise

The foregoing derivations have all assumed linearity and time invariance.Let us reconsider each of these assumptions in turn.

Nonlinearity is clearly a fundamental property of all real oscillators, asits presence is necessary for amplitude limiting. Several phase noise theorieshave consequently attempted to explain certain observations entirely as aconsequence of nonlinear behavior. One of these observations is that a single-frequency sinusoidal disturbance injected into an oscillator gives rise to twoequal-amplitude sidebands, symmetrically disposed about the carrier [4].Since LTI systems cannot perform frequency translation and nonlinear sys-tems can, nonlinear mixing has often been proposed to explain phase noise.Unfortunately, the amplitude of the sidebands generally must then dependnonlinearly on the amplitude of the injected signal, and this dependency isnot generally observed. One must conclude that memoriless nonlinearitycannot explain the discrepancies, despite initial attractiveness as the culprit.As we shall see momentarily, amplitude-control nonlinearities certainly doaffect phase noise, but only indirectly, by controlling the detailed shape ofthe output waveform.


An important insight is that disturbances are just that: perturbationssuperimposed on the main oscillation. They will always be much smaller inmagnitude than the carrier in any oscillator worth designing or analyzing.Thus, if a certain amount of injected noise produces a certain amount ofphase disturbance, we ought to expect that doubling the injected noise willproduce double the disturbance. Linearity would therefore appear to be areasonable assumption as far as the noise-to-phase transfer function is concerned.It is therefore particularly important to keep in mind that, when assessinglinearity, it is essential to identify explicitly the input-output variables. Linearrelationships may exist between certain variable pairs at the same time nonlin-ear ones exist between others. This assumption of linearity is not equivalentto a neglect of the nonlinear behavior of the active devices. It is rather alinearization around the steady-state solution and therefore takes the effectof device nonlinearity into account. There is thus no contradiction here withthe prior acknowledgment of nonlinear amplitude control. We see that theword system is actually ill-defined. Most take it to refer to an assemblage ofcomponents and their interconnections, but a more useful definition is basedon the particular input-output variables chosen. With this definition, thesame circuit may possess nonlinear relationships among certain variables andlinear ones among others. Time invariance is also not an inherent propertyof the entire circuit; it is similarly dependent on the variables chosen.

We are left only with the assumption of time invariance to reexamine.In the previous derivations, we have extended time invariance to the noisesources themselves, meaning that the measures that characterize noise (e.g.,spectral density) are time-invariant (stationary). In contrast with linearity,the assumption of time invariance is less obviously defensible. In fact, it issurprisingly simple to demonstrate that oscillators are fundamentally time-varying systems. Recognizing this truth is the main key to developing a moreaccurate theory of phase noise [5].

To show that time invariance fails to hold, consider explicitly how animpulse of current affects the waveform of the simplest resonant system, alossless LC tank (Figure 3.4). Assume that the system is oscillating withsome constant amplitude until the impulse occurs; then consider how thesystem responds to an impulse injected at two different times, as seen inFigure 3.5.

If the impulse happens to coincide with a voltage maximum (as in theleft plot of Figure 3.5), the amplitude increases abruptly by an amountDV = DQ /C , but because the response to the impulse superposes exactly inphase with the preexisting oscillation, the timing of the zero crossings doesnot change. On the other hand, an impulse injected at some other time


Figure 3.4 LC oscillator excited by current pulse.

Figure 3.5 Impulse responses of LC tank.

generally affects both the amplitude of oscillation and the timing of the zerocrossings, as in the right-hand plot. Interpreting the zero-crossing timingsas a measure of phase, we see that the amount of phase disturbance for agiven injected impulse depends on when the injection occurs; time invariancethus fails to hold. An oscillator is therefore a linear but (periodically) time-varying system. It is especially important to note that it is theoreticallypossible to leave unchanged the energy of the system (as reflected in aconstant tank amplitude), if the impulse injects at a moment near the zerocrossing, such that the net work performed by the impulse is zero. Forexample, a small positive impulse injected when the tank voltage is negativeextracts energy from the oscillator, and the same impulse injected when thetank voltage is positive delivers energy to the oscillator. Just before the zerocrossing, an instant may be found where such an impulse performs no workat all. Consequently, the amplitude of oscillation cannot change, but thezero crossings will be displaced.

Because linearity of noise-to-phase conversion remains a good assump-tion, the impulse response still completely characterizes that system, evenwith time variation. Noting that an impulsive input produces a step changein phase, the impulse response may be written as:

hf (t , t ) =G(v0 t )

qmaxu (t − t ) (3.20)


where u (t ) is the unit step function. Dividing by qmax, the maximum chargedisplacement across the capacitor, makes the function G(x ) independent ofsignal amplitude. G(x ) is called the impulse sensitivity function (ISF), and isa dimensionless, frequency- and amplitude-independent function periodicin 2p . As its name suggests, it encodes information about the sensitivity ofthe oscillator to an impulse injected at phase v0 t . In the LC oscillatorexample, G(x ) has its maximum value near the zero crossings of the oscillation,and a zero value at maxima of the oscillation waveform. In general, it ismost practical (and most accurate) to determine G(x ) through simulation,but there are also analytical methods (some approximate) that apply in specialcases [6, 7]. In any event, to develop a feel for typical shapes of ISFs, considertwo representative examples, first for an LC and a ring oscillator in Figure3.6.

Once the ISF has been determined (by whatever means), we maycompute the excess phase through use of the superposition integral. Thiscomputation is valid here since superposition is linked to linearity, not timeinvariance:

f (t ) = E∞

−∞

hf (t , t ) i (t )dt =1

qmaxEt

−∞

G(v0t ) i (t )dt (3.21)

This computation can be visualized with the help of the equivalentblock diagram shown in Figure 3.7.

Figure 3.6 Example ISF for (a) an LC oscillator and (b) a ring oscillator.


Figure 3.7 The equivalent block diagram of the process.

To cast this equation in a more practically useful form, note that theISF is periodic and therefore expressible as a Fourier series:

G(v0t ) =c02

+ ∑∞

n=1c n cos (nv0t + un ) (3.22)

where the coefficients cn are real, and un is the phase of the n th harmonicof the ISF. We will ignore un in all that follows because we will be assumingthat noise components are uncorrelated, so that their relative phase is irrele-vant.

The value of this decomposition is that, like many functions associatedwith physical phenomena, the series typically converges rapidly, so that itis often well approximated by just the first few terms of the series.

Substituting the Fourier expansion into (3.21), and exchanging summa-tion and integration, one obtains:

f (t ) =1

qmax 3 c02 E

t

−∞

i (t )dt + ∑∞

n=1cn E

t

−∞

i (t ) cos (nv0t )dt4 (3.23)

The corresponding sequence of mathematical operations is showngraphically in the left half of Figure 3.8. Note that the block diagram containselements that are analogous to those of a superheterodyne receiver. Thenormalized noise current is a broadband ‘‘RF’’ signal, whose Fourier compo-nents undergo simultaneous downconversions (multiplications) by ‘‘localoscillator’’ signals at all harmonics of the oscillation frequency. It is importantto keep in mind that multiplication is a linear operation if one argument isheld constant, as it is here. The relative contributions of these multiplicationsare determined by the Fourier coefficients of the ISF. Equation (3.23) thusallows us to compute the excess phase caused by an arbitrary noise current


Figure 3.8 The equivalent system for ISF decomposition.

injected into the system, once the Fourier coefficients of the ISF have beendetermined (typically through simulation).

Earlier, we noted that signals (noise) injected into a nonlinear systemat some frequency may produce spectral components at a different frequency.We now show that a linear, but time-varying system can exhibit qualitativelysimilar behavior, as implied by the superheterodyne imagery invoked inthe preceding paragraph. To demonstrate this property explicitly, considerinjecting a sinusoidal current whose frequency is near an integer multiple,m , of the oscillation frequency, so that

i (t ) = Im cos [(mv0 + Dv )t ] (3.24)

where Dv << v0. Substituting (3.24) into (3.23) and noting that there isa negligible net contribution to the integral by terms other than whenn = m , one obtains the following approximation:

f (t ) ≈Im cm sin (Dv t )

2qmaxDv(3.25)

The spectrum of f (t ) therefore consists of two equal sidebands at±Dv , even though the injection occurs near some integer multiple of v0.This observation is fundamental to understanding the evolution of noise inan oscillator.

Unfortunately, we are not quite done: (3.25) allows us to figure outthe spectrum of f (t ), but we ultimately want to find the spectrum of theoutput voltage of the oscillator, which is not quite the same thing. The twoquantities are linked through the actual output waveform, however. Toillustrate what we mean by this linkage, consider a specific case where the


output may be approximated as a sinusoid, so that vout(t ) = cos [v0 t + f (t )].This equation may be considered a phase-to-voltage converter; it takes phaseas an input, and produces from it the output voltage. This conversionis fundamentally nonlinear because it involves the phase modulation of asinusoid.

Performing this phase-to-voltage conversion, and assuming ‘‘small’’amplitude disturbances, we find that the single-tone injection leading to(3.25) results in two equal-power sidebands symmetrically disposed aboutthe carrier:

PSBC (Dv ) = 10 logS Im cm4qmaxDv D

2

(3.26)

To distinguish this result from nonlinear mixing phenomena, note thatthe amplitude dependence is linear (the squaring operation simply reflectsthe fact that we are dealing with a power quantity here). This relationshipcan be, and has been, verified experimentally for an exceptionally wide rangeof practical oscillators.

The foregoing result may be extended to the general case of a whitenoise source [8]:

PSBC (Dv ) ≈ 10 log1 i 2n

D f ∑∞

m=0c2

m

(4qmaxDv )22 (3.27)

Equations (3.26) and (3.27) imply both upward and downward fre-quency translations of noise into the noise near the carrier, as illustrated inFigure 3.9. Figure 3.9 summarizes what the foregoing equations tell us:Components of noise near integer multiples of the carrier frequency all foldinto noise near the carrier itself.

Noise near dc gets upconverted, with relative weight given by coefficientc0, so 1/f device noise ultimately becomes 1/f 3 noise near the carrier; noisenear the carrier stays there, weighted by c1; and white noise near higherinteger multiples of the carrier undergoes downconversion, turning into noisein the 1/f 2 region. Note that the 1/f 2 shape results from the integrationimplied by the step change in phase caused by an impulsive noise input.Since an integration (even a time-varying one) gives a white voltage or currentspectrum a 1/f character, the power spectral density will have a 1/f 2 shape.


Figure 3.9 Evolution of circuit noise into phase noise.

It is clear from Figure 3.9 that minimizing the various coefficients cn(by minimizing the ISF) will minimize the phase noise. To underscore thispoint quantitatively, we may use Parseval’s theorem to write:

∑∞

m=0c2

m =1p E

2p

0

|G(x ) |2dx = 2G2rms (3.28)

so that the spectrum in the 1/f 2 region may be expressed as:

L {Dv } = 10 log1 i 2n

D fG2

rms

(2qmaxDv )22 (3.29)

where Grms is the rms value of the ISF. All other factors held equal, reducingGrms will reduce the phase noise at all frequencies. Equation (3.29) is therigorous equation for the 1/f 2 region and is one key result of this phase-noise model. Note that no empirical curve-fitting parameters are present in(3.29).


Among other attributes, (3.29) allows us to study quantitatively theupconversion of 1/f noise into close-in phase noise. Noise near the carrieris particularly important in communication systems with narrow channelspacings. In fact, the allowable channel spacings are frequently constrainedby the achievable phase noise. Unfortunately, it is not possible to predictclose-in phase noise correctly with LTI models.

This problem disappears if the new model is used. Specifically, assumethat the current noise behaves as follows in the 1/f region:

i 2n ,1/f = i 2

nv1/f

Dv(3.30)

where v1/f is the 1/f corner frequency. Using (3.27) we obtain the follow-ing for the noise in the 1/f 3 region

L {Dv } = 10 log1 i 2n

D fc2

0

(8q2maxDv2)

v1/f

Dv2 (3.31)

The 1/f 3 corner frequency is then

Dv1/f 3 = v1/fc2

0

4G2rms

= v1/fG2

dc

G2rms

(3.32)

from which we see that the 1/f 3 phase noise corner is not necessarily thesame as the 1/f device/circuit noise corner; it will generally be lower. In fact,since Gdc is the dc value of the ISF, there is a possibility of reducing bylarge factors the 1/f 3 phase noise corner. The ISF is a function of thewaveform, and hence potentially under the control of the designer, usuallythrough adjustment of the rise and fall time symmetry. This result is notanticipated by LTI approaches and is one of the most powerful insightsconferred by this LTV model. This result has particular significance fortechnologies with notoriously poor 1/f noise performance, such as CMOSand GaAs MESFETs. A specific circuit example of how one may exploitthis observation follows shortly.

One more extremely powerful insight concerns the influence of cyclo-stationary noise sources. As alluded to earlier, the noise sources in manyoscillators cannot be well modeled as stationary. A typical example is the


nominally white drain or collector noise current in a MOSFET. Noisecurrents are a function of bias currents, and the latter vary periodicallyand significantly with the oscillating waveform. The LTV model is able toaccommodate a cyclostationary white noise source with ease, since such asource may be treated as the product of a stationary white noise source anda periodic function [9]:

in (t ) = in0(t )a (v0 t ) (3.33)

Here, in0 is a stationary white noise source whose peak value is equalto that of the cyclostationary source, and a (x ) is a periodic unitless functionwith a peak value of unity. Substituting this into (3.21) allows us to treatcyclostationary noise as a stationary noise source provided we define aneffective ISF as follows:

Geff (x ) = G(x )a (x ) (3.34)

Cyclostationarity is thus easily accommodated within the frameworkalready established. None of the foregoing conclusions changes as long asGeff is used in all of the equations.

Having identified the factors that influence oscillator noise, we arenow in a position to articulate the requirements that must be satisfied tomake a good oscillator. First, in common with the revelations of LTI models,both the signal power and resonator Q should be maximized, all other factorsheld constant. In addition, note that an active device is always necessary tocompensate for tank loss, and that active devices always contribute noise.Note also that the ISFs tell us that there are sensitive and insensitive momentsin an oscillation cycle. Of the infinitely many ways that an active elementcould return energy to the tank, this energy should be delivered all at once,where the ISF has its minimum value. In an ideal LC oscillator, therefore,the transistor would remain off almost all of the time, waking up periodicallyto deliver an impulse of current at the signal peak(s) of each cycle. Theextent to which real oscillators approximate this behavior determines in largepart the quality of their phase noise properties. Since an LTI theory treatsall instants as equally important, such theories are unable to anticipate thisimportant result.

The prescription for impulsive energy restoration has actually beenpracticed for centuries, but in a different domain. In mechanical clocks, astructure known as an escapement regulates the transfer of energy from a springto a pendulum. The escapement forces this transfer to occur impulsively, and


only at precisely timed moments that are chosen to minimize the disturbanceof the oscillation period. Although this historically important analog ishundreds of years old, having been designed by intuition and trial and error,it was not analyzed mathematically until 1826, by Astronomer Royal GeorgeAiry [10].

Finally, the best oscillators will possess the symmetry properties thatlead to small Gdc for minimum upconversion of 1/f noise. After examiningsome additional features of close-in phase noise, we consider in Section 3.4.1several circuit examples of how to accomplish these ends in practice.

3.4.1 Close-In Phase Noise

From the development so far, one expects the spectrum Sf (v ) to have aclose-in behavior that is proportional to the inverse cube of frequency. Thatis, the spectral density grows without bound as the carrier frequency isapproached. However, most measurements fail to show this behavior, andthis failure is often misinterpreted, either as being the result of some newphenomenon, or as evidence of a flaw in the LTV theory. It is thereforeworthwhile to spend some time considering this issue in detail.

The LTV theory asserts only that Sf (v ) grows without bound. Most‘‘phase’’ noise measurements actually measure the total output spectrum ofthe oscillator voltage. That is, what is often measured is actually SV (v ). Insuch a case, the output spectrum will not show a boundless growth as theoffset frequency approaches zero, reflecting the simple fact that a cosinefunction is bounded, even for unbounded arguments. This bound causesthe measured spectrum to flatten as the carrier is approached, with a resultingshape that is Lorentzian [11, 12] (see Figure 3.10).

Figure 3.10 Lorentzian spectrum.


Depending on the details of how the measurement is performed, the−3-dB corner may or may not be observed. If a spectrum analyzer is used,the corner will typically be observed. If an ideal phase detector and a phase-locked loop were available to downconvert the spectrum of f (t ) and measureit directly, no flattening would be observed at all. If a real phase detector,possessing limited range, is used in such a measurement, a −3-dB cornerwill generally be observed, but the precise value of the corner will now bea function of the instrumentation; the measurement will no longer reflectthe inherent spectral properties of the oscillator. The lack of consistency inmeasurement techniques has been a source of great confusion in the past.

3.5 Circuit Examples

3.5.1 LC Oscillators

Having derived expressions for phase noise at low and moderate offsetfrequencies, it is instructive to apply to practical oscillators the insightsgained. We examine first the popular Colpitts oscillator and its relevantwaveforms (see Figures 3.11 and 3.12). An important feature is that thedrain current flows only during a short interval coincident with the mostbenign moments (the peaks of the tank voltage). Its corresponding excellentphase noise properties account for the popularity of this configuration. Ithas long been known that the best phase noise occurs for a certain narrowrange of tapping ratios (e.g., a 4:1 capacitance ratio), but before the LTVtheory, no theoretical basis existed to explain a particular optimum.

Figure 3.11 Colpitts oscillator (simplified).


Figure 3.12 Approximate incremental tank voltage and drain current for the Colpitts oscil-lator.

Both LTI and LTV models point out the value of maximizing signalamplitude. To evade supply voltage or breakdown constraints, one mayemploy a tapped resonator to decouple resonator swings from device voltagelimitations. A common configuration that does so is Clapp’s modificationto the Colpitts oscillator (Figure 3.13). Differential implementations ofoscillators with tapped resonators have recently made an appearance in theliterature [13–15]. These types of oscillators become increasingly attractiveas supply voltages scale downward, where conventional resonator connectionslead to VDD -constrained signal swings. The use of tapping allows signalenergy to remain high even with low supply voltages.

Figure 3.13 Clapp oscillator.


Phase-noise predictions using the LTV model are frequently moreaccurate for bipolar oscillators due to the availability of better device noisemodels. In [15] impulse response modeling (see Appendix 3A) is used todetermine the ISFs for the various noise sources within the oscillator, andthis knowledge is used to optimize the noise performance of a differentialbipolar VCO. A simplified schematic of this oscillator is shown in Figure3.14(a). A tapped resonator is used to increase the tank signal power, Psig.The optimum capacitive tapping ratio is calculated to be around 4.5 (corre-sponding to a capacitance ratio of 3.5) based on simulations that take intoaccount the cyclostationarity of the noise sources. Specifically, the simulationaccounts for noise contributions by the base spreading resistance and collectorshot noise of each transistor, as well as the resistive losses of the tank elements.The ISFs (taken from [15], in which these are computed through directevaluation in the time domain, as described in Appendix 3A) for the shotnoise of the core oscillator transistors, and for the bias source, are shown inFigure 3.14(b) and 3.14(c), respectively. As can be seen, the tail currentnoise has an ISF with double the periodicity of the oscillation frequency,owing to the differential topology of the circuit. Noteworthy is the observationthat tail noise only at even multiples of the oscillation frequency thus contrib-ute to phase noise. The individual ISFs are used to compute the contributionof each corresponding noise sources, and the contributions summed.

The reduction of 1/f noise upconversion in this topology is clearly seenin Figure 3.15, which shows a predicted and measured 1/f 3 corner of3 kHz, in comparison with an individual device 1/f noise corner of200 kHz. Note that the then-current version of one commercial simulationtool, Spectre, fails in this case to identify a 1/f 3 corner within the offsetfrequency range shown, resulting in a 15-dB underestimate at a 100-Hzoffset. The measured phase noise in the 1/f 2 region is also in excellentagreement with the LTV model’s predictions. For example, the predictedvalue of −106.2 dBc/Hz at 100-kHz offset is negligibly different from themeasured value of −106 dBc/Hz. As a final comment, this particular VCOdesign is also noteworthy for its use of a separate automatic amplitude controlloop, which allows for independent optimization of the steady-state andstart-up conditions in terms of phase noise.

As mentioned, a key insight of the LTV theory concerns the importanceof symmetry, the effects of which are partially evident in the precedingexample. A configuration that exploits this knowledge more fully is thesymmetrical negative resistance oscillator shown in Figure 3.16 [8]. Thisconfiguration is not new by any means, but an appreciation of its symmetryproperties is. Here, it is the half-circuit symmetry that is important, because


Figure 3.14 (a) Simplified schematic of the VCO in [15], (b) the ISF for the shot noise ofeach core transistor, and (c) the ISF for the shot noise of the tail current.


Figure 3.15 Measured and predicted phase noise of VCO in [15].

Figure 3.16 Simple symmetrical negative resistance oscillator.

noise in the two half circuits is only partially correlated at best. By selectingthe relative widths of the PMOS and NMOS devices appropriately to mini-mize the dc value of the ISF (Gdc) for each half -circuit, one may minimizethe upconversion of 1/f noise. Through exploitation of symmetry in thismanner, the 1/f 3 corner can be dropped to exceptionally low values, eventhough device 1/f noise corners may be high (as is typically the case for


CMOS). Furthermore, the bridge-like arrangement of the transistor quadallows for greater signal swings, compounding the improvements in phasenoise. As a result of all of these factors, a phase noise of −121 dBc/Hz atan offset of 600 kHz at 1.8 GHz has been obtained with low-Q (estimatedto be 3 to 4) on-chip spiral inductors, on 6 mW of power consumption ina 0.25-mm CMOS technology [8]. This result rivals what one may achievewith bipolar technologies, as seen by comparison with the previous example.With a modest increase in power, the same oscillator’s phase noise becomescompliant with specifications for GSM1800.

3.5.2 Ring Oscillators

As an example of a circuit that does not well approximate ideal behavior,consider a ring oscillator. First, the ‘‘resonator’’ Q is poor since the energystored in the node capacitances is reset (discharged) every cycle. Hence, ifthe resonator of a Colpitts oscillator may be likened to a fine crystal wineglass, the resonator of a ring oscillator is mud. Next, energy is restored tothe resonator during the edges (the worst possible times), rather than at thevoltage maxima. These factors account for the well-known terrible phasenoise performance of ring oscillators. As a consequence, ring oscillators arefound only in the most noncritical applications, or inside wideband PLLsthat clean up the spectrum.

However, there are certain aspects of ring oscillators that can beexploited to achieve better phase noise performance in a mixed-mode inte-grated circuit. Noise sources on different nodes of an oscillator may bestrongly correlated due to various reasons. Two examples of sources withstrong correlation are substrate and supply noise, arising from current switch-ing in other parts of the chip. The fluctuations on the supply and substratewill induce a similar perturbation on different stages of the ring oscillator.To understand the effect of this correlation, consider the special case ofhaving identical noise sources on all the nodes of the ring oscillator as shownin Figure 3.17.

If all the inverters in the oscillator are the same, the ISF for differentnodes will differ only in phase by multiples of 2p /N, as shown in Figure3.18.

Therefore, the total phase due to all the sources is given by (3.21)through superposition [16]:

f (t ) =1

qmaxEt

−∞

i (t )3 ∑N−1

n=0GSv0t +

2pnN D4dt (3.35)


Figure 3.17 Five-stage ring oscillator with identical noise sources on all nodes.

Figure 3.18 Phasors for noise contributions from each source.

Expanding the term in brackets in a Fourier series, it can be observedthat it is zero except at dc and multiples of Nv0—that is,

f (t ) =N

qmax∑∞

n=0c nN E

t

−∞

i (t ) cos (nNv0t )dt (3.36)

which means that for fully correlated sources, only noise in the vicinity ofinteger multiples of Nv0 affects the phase. Therefore, every effort shouldbe made to maximize the correlations of noise arising from substrate andsupply perturbations. This result can be achieved by making the inverterstages and the noise sources on each node as similar to each other as possibleby proper layout and circuit design. For example, the layout should be keptsymmetrical, and the inverter stages should be laid out close to each other


so that substrate noise appears as a common-mode source. This latter consid-eration is particularly important in the case of a lightly doped substrate,since such a substrate may not act as a single node [17]. It is also importantthat the orientation of all the stages be kept identical. The interconnectingwires between the stages must be identical in length and shape, and a commonsupply line should feed all the inverter stages. Furthermore, the loading onall stages should be kept identical, perhaps by using dummy buffer stagesas necessary, for example. Use of the largest number of stages consistentwith oscillation at the desired frequency will also be helpful because, as apractical matter, fewer cn coefficients will then affect the phase noise. Finally,as the low frequency portion of the substrate and supply noise then dominates,one should exploit symmetry to minimize Gdc.

Another common conundrum concerns the preferred topology forMOS ring oscillators (i.e., whether a single-ended or differential topologyresults in better jitter and phase-noise performance for a given center fre-quency, f0, and total power dissipation, P ). Facilitating an analysis of thesechoices is an approximate expression for the ISF (see Figure 3.19). The ISFis approximated by two triangles, as shown in Figure 3.19. The rms valueof the ISF has a value given by

G2rms =

13p S 1

f ri′seD3

(1 + A3) (3.37)

Figure 3.19 Derivation of approximate analytical expression for ring oscillator ISF.


where f ri′se and f fa′ll are the maximum slope during the rising and fallingedges, and A is the ratio of f ri′se to f fa′ll [6].

Coupling the ISF equation with the noise equations for transistors,one may derive expressions for the phase noise of MOS differential andsingle-ended oscillators [16]. Based on these expressions the phase noise ofa single-ended (inverter chain) ring oscillator is found to be independent ofthe number of stages for a given power dissipation and frequency of operation.However, for a differential ring oscillator, the phase noise (jitter) grows withthe number of stages. Therefore, even a properly designed differential CMOSring oscillator underperforms its single-ended counterpart, with a disparitythat increases with the number of stages. The difference in the behavior ofthese two types of oscillators with respect to the number of stages can betraced to the way they dissipate power. The dc current drawn from thesupply is independent of the number and slope of the transitions in differentialring oscillators. In contrast, inverter-chain ring oscillators dissipate powermainly on a per transition basis and therefore have better phase noise for agiven power dissipation. However, a differential topology may still be pre-ferred in ICs with a large amount of digital circuitry because of the lowersensitivity to substrate and supply noise, as well as lower noise injection intoother circuits on the same chip. The decision of which architecture to useshould be based on both of these considerations.

Yet another commonly debated question concerns the optimum num-ber of inverter stages in a ring oscillator to achieve the best jitter and phasenoise for a given f0 and P. For single-ended CMOS ring oscillators, thephase noise and jitter in the 1/f 2 region are not strong functions of thenumber of stages [16]. However, if the symmetry criteria are not well satisfied,and/or the process has large 1/f noise, a larger N will reduce the jitter. Thisreduction results from the faster edge speeds that must accompany the useof a larger number of stages to achieve the same oscillation frequency. Thefaster edge speeds reduce the effect of asymmetries in rise and fall times,and thus reduce the upconversion of 1/f noise. In general, the choice of thenumber of stages must be made on the basis of several design criteria, suchas 1/f noise effect and the desired maximum frequency of oscillation, as wellas the influence of external noise sources, such as supply and substrate noise,that may not scale with N. A symmetry-based reduction in 1/f noise cansignificantly augment the reduction that may be provided by the trap-resetting that can attend operation of MOSFETs in the switching regime[18, 19].

The jitter and phase-noise behavior is different for differential ringoscillators. Jitter and phase noise increase with an increasing number of


stages. Hence, if the 1/f noise corner is not large, and/or proper symmetrymeasures have been taken, the minimum number of stages (3 or 4) shouldbe used to give the best performance. This recommendation holds even ifthe power dissipation is not a primary issue. It is not fair to argue thatburning more power in a larger number of stages allows the achievementof better phase noise, since dissipating the same total power in a smallernumber of stages with larger devices results in better jitter and phase noise,as long as it is possible to maximize the total charge swing.

3.6 Amplitude Response

While the close-in sidebands are dominated by phase noise, the far-outsidebands are greatly affected by amplitude noise. Unlike the induced excessphase, the excess amplitude, A (t ), due to a current impulse decays withtime. This decay is the direct result of the amplitude restoring mechanismsalways present in practical oscillators. The excess amplitude may decay veryslowly (e.g., in a harmonic oscillator with a high-quality resonant circuit)or very quickly (e.g., a ring oscillator). Some circuits may even demonstratean underdamped second-order amplitude response. The detailed dynamicsof the amplitude control mechanism have a direct effect on the shape of thenoise spectrum.

In the context of the ideal LC oscillator of Figure 3.4, a current impulsewith an area, Dq , will induce an instantaneous change in the capacitorvoltage which, in turn, will result in a change in the oscillator amplitudethat depends on the instant of injection, as shown in Figure 3.5. Theamplitude change is proportional to the instantaneous normalized voltagechange, DV /Vmax, for small injected charge Dq << qmax—that is,

DA = L(v0 t )DV

Vmax= L(v0 t )

Dqqmax

(3.38)

where L(v0 t ) is a periodic function that determines the sensitivity of eachpoint on the waveform to an impulse and is called the amplitude impulsesensitivity function. It is the amplitude counterpart of the phase impulsesensitivity function, G(v0 t ). From a development similar to that of Section3.3, the amplitude impulse response can be written as

hA (t , t ) =L(v0 t )

qmaxd (t − t ) (3.39)


where d (t − t ) is a function that defines how the excess amplitude decays.Figure 3.20 shows two hypothetical examples of d (t ) for a low-Q oscillatorwith overdamped response, and a high-Q oscillator with underdamped ampli-tude response.

For most oscillators, the amplitude-limiting system can be approxi-mated as first- or second-order [20], again for small disturbances. The func-tion d (t − t ) typically will thus be either a dying exponential or a dampedsinusoid.

For a first-order system,

d (t − t ) = e −v0(t− t )/Q u (t − t ) (3.40)

Therefore, the excess amplitude response to an arbitrary input current,i (t ), is given by the superposition integral,

A (t ) = Et

−∞

i (t )qmax

L(v0t ) e −v0(t− t )/Q dt (3.41)

If i (t ) is a white noise source with power spectral density i 2n , the output

power spectrum of the amplitude noise, A (t ), can be shown to be

L amplitude {Dv } =L2

rms

q2rms

i 2n /Df

2Sv20

Q2 + Dv2D(3.42)

where Lrms is the rms value of L(v0 t ). If L total is measured, the sum ofboth L amplitude and Lphase will be observed and hence there will be a pedestalin the phase noise spectrum at v0 /Q as shown in Figure 3.21. Also note

Figure 3.20 Overdamped and underdamped amplitude responses.


Figure 3.21 Phase, amplitude, and total sideband powers for the overdamped amplituderesponse.

that the significance of the amplitude response depends greatly on Lrmswhich, in turn, depends on the topology.

As a final comment on the effect of amplitude-control dynamics, anunderdamped response would result in a spectrum very similar to Figure3.21, except for the presence of peaking in the vicinity of v0 /Q .

3.7 Summary

The insights gained from LTI phase noise models are simple and intuitivelysatisfying: One should maximize signal amplitude and resonator Q . Anadditional, implicit insight is that the phase shifts around the loop generallymust be arranged so that oscillation occurs at or very near the center frequencyof the resonator. This way, there is a maximum attenuation by the resonatorof off-center spectral components.

Deeper insights provided by the LTV model are that the resonatorenergy should be restored impulsively at the ISF minimum, instead of evenlythroughout a cycle, and that the dc value of the effective ISF should bemade as close to zero as possible to suppress the upconversion of 1/f noiseinto close-in phase noise. The theory also shows that the inferior broadbandnoise performance of ring oscillators may be offset by their potentially superiorability to reject common-mode substrate and supply noise.

References

[1] Lee, T. H., The Design of CMOS Radio-Frequency Integrated Circuits, New York:Cambridge University Press, 1998.


[2] Ham, D., and A. Hajimiri, ‘‘Concepts and Methods in Optimization of IntegratedLC VCOs,’’ IEEE J. Solid-State Circuits, June 2001.

[3] Leeson, D. B., ‘‘A Simple Model of Feedback Oscillator Noise Spectrum,’’ Proc. IEEE,Vol. 54, February 1966, pp. 329–330.

[4] Razavi, B., ‘‘A Study of Phase Noise in CMOS Oscillators,’’ IEEE Journal of Solid-State Circuits, Vol. 31, No. 3, March 1996.

[5] Hajimiri, A., and T. Lee, ‘‘A General Theory of Phase Noise in Electrical Oscillators,’’IEEE Journal of Solid-State Circuits, Vol. 33, No. 2, February 1998, pp. 179–194.

[6] Hajimiri, A., and T. Lee, The Design of Low-Noise Oscillators, Boston, MA: Kluwer,1999.

[7] Kaertner, F. X., ‘‘Determination of the Correlation Spectrum of Oscillators with LowNoise,’’ IEEE Trans. on Microwave Theory and Techniques, Vol. 37, No. 1, January1989.

[8] Hajimiri, A., and T. Lee, ‘‘Design Issues in CMOS Differential LC Oscillators,’’ IEEEJournal of Solid-State Circuits, May 1999.

[9] Gardner, W. A., Introduction to Random Processes, New York: McGraw-Hill, 1990.

[10] Airy, G. B., ‘‘On the Disturbances of Pendulums and Balances, and on the Theoryof Escapements,’’ Trans. of the Cambridge Philosophical Society, Vol. III, Part I, 1830,pp. 105–128.

[11] Edson, W. A., ‘‘Noise in Oscillators,’’ Proc. IRE, August 1960, pp. 1454–1466.

[12] Mullen, J. A., ‘‘Background Noise in Nonlinear Oscillators,’’ Proc. IRE, August 1960,pp. 1467–1473.

[13] Craninckx, J., and M. Steyaert, ‘‘A 1.8GHz CMOS Low-Phase-Noise Voltage-Con-trolled Oscillator with Prescaler,’’ IEEE Journal of Solid-State Circuits, Vol. 30,No. 12, December 1995, pp. 1474–1482.

[14] Ahrens, T. I., and T. H. Lee, ‘‘A 1.4-GHz, 3-mW CMOS LC Low Phase NoiseVCO Using Tapped Bond Wire Inductance,’’ Proc. ISLPED, August 1998, p. 16-9.

[15] Margarit, M. A., et al., ‘‘A Low-Noise, Low-Power VCO with Automatic AmplitudeControl for Wireless Applications,’’ IEEE Journal of Solid-State Circuits, Vol. 34,No. 6, June 1999, pp. 761–771.

[16] Hajimiri, A., S. Limotyrakis, and T. H. Lee, ‘‘Jitter and Phase Noise in Ring Oscilla-tors,’’ IEEE Journal of Solid-State Circuits, Vol. 34, No. 6, June 1999, pp. 790–804.

[17] Blalack, T., et al., ‘‘Experimental Results and Modeling of Noise Coupling in a Lightlydoped Substrate,’’ Tech. Dig. IEDM, December 1996.

[18] Bloom, I., and Y. Nemirovsky, ‘‘1/f Noise Reduction of Metal-Oxide-SemiconductorTransistors by Cycling from Inversion to Accumulation,’’ Appl. Phys. Lett., Vol. 58,April 1991, pp. 1664–1666.

[19] Gierkink, S. L. J., et al., ‘‘Reduction of the 1/f Noise Induced Phase Noise in aCMOS Ring Oscillator by Increasing the Amplitude of Oscillation,’’ Proc. 1998 Int.Symp. Circuits and Systems, Vol. 1, May 31–June 3, 1998, pp. 185–188.

[20] Clarke, K. K., and D. T. Hess, Communication Circuits: Analysis and Design, Reading,MA: Addison-Wesley, 1971.


Appendix 3A: Notes on Simulation

Exact analytical derivations of the ISF are usually not obtainable for any butthe simplest oscillators. Various approximate methods are outlined in [5, 6],but the most generally accurate method is direct evaluation of the time-varying impulse response. In this direct method, an impulsive excitationperturbs the oscillator, and the steady-state phase perturbation is measured.The timing of the impulse with respect to the unperturbed oscillator’s zerocrossing is then incremented and the simulation repeated until the impulsehas been ‘‘walked’’ through an entire cycle.

The impulse must have a small enough value to ensure that the assump-tion of linearity holds. Just as an amplifier’s step response cannot be evaluatedproperly with steps of arbitrary size, one must judiciously select the area ofthe impulse, rather than blindly employing some fixed value (e.g., 1 coulomb).If one is unsure if the impulse chosen is sized properly, linearity may alwaysbe tested explicitly by scaling the size of the impulse by some amount andverifying that the response scales by the same factor.

Finally, some confusion persists about whether the LTV theory properlyaccommodates the phenomenon of amplitude-to-phase conversion that someoscillators exhibit. As long as linearity holds, the LTV theory provides thecorrect answer, provided that an exact ISF has been obtained. This is dueto the fact that changes in the phase of oscillator arising from an amplitudechange appear in the impulse response of the oscillator. As noted in thepreceding paragraphs, the direct impulse response method is the most reliableone, as it makes no assumptions other than linearity. This reliability is incontrast with the approximate analytical approaches offered in the appendixto [1].

Acknowledgments

The authors of Chapter 3 are grateful to Professor David Leeson of Stanfordfor his gracious assistance and encouragement when the LTV theory was inits formative stages.

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Linearity, Time Variation, and Oscillator Phase Noise