Impact of AAC design on phase noise performance of VCOs

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 48, NO. 6, JUNE 2001 537

Impact of AAC Design on Phase Noise Performanceof VCOs

Alfio Zanchi, Member, IEEE, Carlo Samori, Member, IEEE, Andrea L. Lacaita, Senior Member, IEEE, andSalvatore Levantino, Student Member, IEEE

Abstract—The automatic amplitude control (AAC) system is aservo-circuit of voltage-controlled oscillators (VCOs) that provesimportant for the practical implementation of fully integrated RFfront-ends. Nevertheless, the noise contributed by the AAC can de-grade the phase noise performance of the oscillator. In this paper,the time-varying noise transfer of the peak detector is analyzed,since it is recognized as the critical noise source in the simple AACloop taken as reference. A fast rule to predict the noise spectral den-sity at the peak detector output is given that substantially facilitatesthe estimate of the phase noise degradation on the VCO output.The analysis has been carried out both theoretically and by simu-lation; experimental validation has also been provided by means ofa discrete-components test board. Guidelines for low-noise designof the peak detector of AAC are finally proposed.

Index Terms—Bandlimited stochastic processes, bipolar tran-sistor oscillators, communication circuits, integrated circuit noise,mobile communications, peak detectors, phase noise, voltage-con-trolled oscillators.

I. INTRODUCTION

I N THE FIELD of modern mobile communications, cellularstandards are progressively emerging that are characterized

by more and more stringent requirements concerning the wholesystem, from the input low-noise amplifier (dynamic range) tothe final power amplifier (battery consumption). The compli-ance of RF transceiver circuits with these protocols relies sub-stantially onto the oscillator performance. In fact, a lot of pub-lications have been devoted to the problem of phase noise involtage-controlled oscillators (VCOs), and more in general todesign techniques aimed at reducing the noise sources whichaffect the oscillation stability [1], [2]. The latter is usually quan-tified via theSingle-Sideband-to-Carrier Ratio(SSCR), a figurethat weights the phase noise in 1-Hz bandwidth at a given offsetagainst the carrier power.

Most of the literature on this topic concerns howeverthe phase noise performance of a standalone core oscillator,whereas in the realization of a complete RF front-end additionalissues must be considered. Among these is the need for an

Manuscript received September 1999; revised May 2001. This work was sup-ported in part by the Italian Ministry of University and Research and by CNR(the Italian National Research Council) under the “MADESS II” Project. Thispaper was recommended by Associate Editor H. Helfenstein.

A. Zanchi was with the Dipartimento di Elettronica ed Informazione, Politec-nico di Milano, Milan, Italy. He is now with Texas Instruments, Incorporated,Dallas, TX 75243 USA.

C. Samori, A. L. Lacaita, and S. Levantino are with the Dipartimento di Elet-tronica ed Informazione, Politecnico di Milano and CEQSE-CNR, 20133 Milan,Italy.

Publisher Item Identifier S 1057-7130(01)07496-1.

AAC (Automatic Amplitude Control) system, especially feltfor fully integratedLC-resonator VCOs. This servo-circuit isgaining popularity in the RFIC field and has been addressed inrecent papers that discuss the design of oscillators embeddedin complete transceivers [3], [4]. The main reasons why anamplitude regulator is called for can be summarized as follows.

— It ensures a fast and reliable startup. Such a feature isessential for example in some FHSS (frequency-hop-ping spread spectrum) applications, where the activeVCO is periodically selected and turned on from anarray of oscillators tuned to different carrier frequen-cies. In any case of sudden change in supply voltage,quite common in mixed-signal chips with stand-bycapability, the AAC keeps the oscillation amplitudestable thus tightening the switching delays to theprogrammed values (whose tolerances are set forth bytransmission standards, e.g., see [5]).

— It allows to set the oscillator to its optimal bias pointin terms of noise, while still biasing it with the op-timal current at startup. These two conditions do con-flict without an AAC, as reported in [3].

— The well-defined level of output power set via the AACis important for the design of the stages that follow theoscillator in a frequency synthesizer. For instance, thenoise and power dissipation of the frequency dividerin a PLL are influenced by the amplitude of the VCOoutput signal [6]. Even if in current-limited oscillators[7] the output amplitude is defined by the bias currentand the resonator loss, the process tolerances—duemainly to the inductor manufacturing—are far fromguaranteeing the required accuracy.

— Last, but not least, the adoption of the amplitude loopallows for smaller values of the VCOs excess loop gainneeded for startup [3], [8]. This eventually leads to re-duce distortion in the oscillation, removing harmfulharmonics from the local oscillator input of the down-conversion mixer.

Despite the practical relevance of this circuit, the impact ofits noise sources on the oscillator phase noise performance hasbeen so far neglected. The aim of this paper is to overcome theproblem of figuring out the nonlinear time-varying noise trans-fers that operate within the control loop, by adopting the mostgeneral procedure: that is, first the autocorrelation of the noisyprocess is computed, and then its power spectrum is inferred.The results presented allow for the prediction of the spectrum ofthe noise that couples from the AAC to the VCO bias current;after this, the SSCR estimate comes straightforward. Finally, the

1057–7130/01$10.00 © 2001 IEEE

538 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 48, NO. 6, JUNE 2001

Fig. 1. LC-tank VCO equipped with the AAC servo-circuit. Note theR C

pole of the peak detector and theR C dominant pole of the loop.

analysis carried out suggests the guidelines for the design of alow-noise amplitude control loop.

The paper is organized as follows. Section II outlines thestructure of the system taken as reference for the discussion.Section III briefly recalls how the noise injected in the VCOfrom the AAC can degrade the phase noise performance. In Sec-tion IV will be analyzed the noise transfer mechanisms into theregulator loop; the main sources of disturbance are pointed outthere, the worst being the peak detector embedded in the loop.Since it works in a nonstationary time-varying way, its noisewill be treated with time-dependent stochastic analysis in Sec-tion V. Section VI includes the results of the noise simulationsperformed on a mathematical model; they fully support the be-havior predicted by the theory. A simple approximated formulacapable of predicting the peak detector noise level at low fre-quencies is provided in Section VII. Eventually, the correctnessof the theoretical and simulation approaches has been tested viaa discrete-components circuit board: experimental data are re-ported in Section VIII. Section IX summarizes the conclusionsand proposes some design tips.

II. STRUCTURE OF THEVCO–AAC CIRCUIT

Fig. 1 presents one of the simplest schematic structures foran oscillation amplitude regulator [9]–[11]; this all-bipolar con-figuration will be addressed for reference in the following. Theoscillation magnitude is sensed by the peak detector circuit (rec-tifier and filter), whose output drives the error amplifierstage. The signal is then low-pass-filtered by the looppole, both for stability concerns and ripple cutoff, before con-trolling the current generator that biases the transconductor ofthe VCO. This control system is particularly critical from thestability standpoint [12], since high loop gain is required for ef-fectively rejecting the amplitude oscillation variations. Domi-nant pole compensation is often adopted ( pole in Fig. 1),hence the loop gain will feature a low-pass profile with the firstpole standing well below the other singularities.

From Fig. 1 it is evident that the various noise sources in theAAC stages directly affect the oscillator bias current. For ex-ample, the base-referred voltage noise of the differential stagegets amplified by the differential couple and gives rise to tailcurrent fluctuations. The mechanisms through which this tailcurrent noise affects the phase jitter of the oscillator were dis-

Fig. 2. Block diagram of the control system.A indicates the oscillationamplitude. The output noise of the peak detector is not compensated for by theloop, as it comes embedded in the feedback path.

cussed in [13], [14]; therefore, the impact of the AAC onto theSSCR of the oscillator can be easily assessed, once the transferbetween the noise sources and the tail current has been evalu-ated. The latter is indeed the main difficulty, given the presenceof a time-varying element (namely, the peak detector) whosenoise can be the dominant term since it is directly amplified bythe differential stage.

III. EFFECT OF THEAAC ON THE VCO PHASE NOISE

Fig. 2 shows a simplified diagram of the whole system. Theloop controls the oscillation amplitude ; therefore it compen-sates for the variations of due to noise and disturbances in-jected in its direct path, such as the ones in the VCO. However,the loop is totally ineffective against the output noise of the peakdetector, because the block is embedded in the feedback path;i.e., the fluctuations of the peak detector output and ofhave the same transfer function toward the oscillator output.

Since the controller tunes the VCO tail current, the noisecoming from the AAC blocks (peak detector and differentialgain stage) gives rise to random fluctuations of the bias current.The dominant pole filters the AAC noise sources, thusthe noise injected on the oscillator bias current is low-passshaped. This noise is upconverted by the “mixer” action ofthe switching transconductor, and in principle its contributionshould be only amplitude noise at the output of the synthesizer[13]. However the bias current noise contributed by the AACdoes not produce only amplitude modulation [15], which maybe otherwise rejected with a limiter on the output. It also affectsthe oscillation frequency, and thus the phase, through twodistinct mechanisms.

The former is the well-known AM-to-PM conversion due tovaractors and other voltage-dependent parasitic capacitors of thetransconductor [16]. Since the– characteristic of these de-vices is nonlinear, the effective value of their capacitance de-pends on the oscillation amplitude; any AM noise then modu-lates the tank capacitance too, causing variations in the oscil-lator output frequency. This effect can be accounted for througha suitable sensitivity coefficient [17] and may worsen the SSCRmuch more than the tank loss does, especially when the oscilla-tion amplitude increases.

The second mechanism is related to the so-calledindirect sta-bility of the oscillator. According to the Barkhausen criterion theVCO frequency is not merely set by the tank resonance, but italso depends on the phase delay of the oscillator’s active ele-ment [18]. Since this delay is a function of the bias current, any

ZANCHI et al.: IMPACT OF AAC DESIGN ON PHASE NOISE PERFORMANCE 539

Fig. 3. Peak detector operation: (a) when one of the transistors is on, the two main noise sources are shunted byg ; (b) when both transistors are off, theIshot noise vanishes and the noise reaching the differential stage is4kT=R .

low-frequency noise from the tail generator will produce fre-quency modulation. This latter effect contributes a phase noiseamount that can easily become the dominant one for fully in-tegrated oscillators embedded in RF transceivers, as shown in[14].

IV. THE ROLE OF THEPEAK DETECTOR

As previously outlined, the output noise of the peak detectoris the only source in the AAC that can practically affect theSSCR performance of VCOs. The noise of the voltage reference

is equally important, but, as it is out of the signal loop, itis much easier to reduce.

The voltage noise of the peak detector at the input of thedifferential stage in Fig. 1 is then to be evaluated. Let us startby taking into account the thermal noise of the resistor.The input resistance of the differential stage is considered muchhigher than and neglected; it can be anyway added in par-allel to . The input current generator forces animpedance that is periodically varying, due to the switching be-havior of the rectifier:

— at the oscillation peak one of the two transistors is on,Fig. 3(a), and the noise generator forces the parallel of

and ;— when the driving signal is lower both transistors turn

off, Fig. 3(b), and the noise current is injected inand .

In the first case, the dc value of the impedance is witha pole ; in the second, the dc impedanceraises to and the pole lowers to . Note that theimpedance gain times bandwidth product GBWP always keepsconstant to the value .

Another source that affects the voltage driving the differentialamplifier is the shot noise of the current in the rectifier.When one of the transistors is on, Fig. 3(a), this noise is higherthan ; whereas it vanishes when both transistors areoff. The effect of this term is expected to be negligible for lowvalues of the rectifier duty cycle1 (that is usually 10%), butit could play a role if this figure increases. Finally, the equivalent

1Here and throughout the paper, the duty cycleD designates the ratio be-tween the “on” time window of the rectifier,T , and the periodT of the rec-tified waveform. Note that for a differential rectifier (Fig. 1)T is half the VCOoscillation period, whereas for a single-ended rectifier (Fig. 13)T is equal tothe period of the oscillator.

Fig. 4. Relative values of the time constants involved in AAC operation.

input noise of the differential stage is negligible with respect tothe two sources considered. In any case, its effect on the VCObias current can be easily quantified, since it does not involve aswitching operation.

The time-varying nature of the transfer function for the noisesources considered makes it quite difficult to evaluate their ef-fects: the transfer depends on duty cycleand transconduc-tance of the rectifier, frequency of the input signal, etc. How-ever, for the AAC to work properly, only one suitable sequenceof the time constants involved in the transfer exists:

— the time constant of the rectifier in “on” state ( )must be smaller than the periodof the rectified wave,in order to track its envelope;

— the time constant in “off” state ( ) must be insteadlonger than the period, to hold the input peak;

— an effective ripple-filtering low-pass stage must featurea time constant much longer than the period, andpossibly than the (for stability concerns also).

In the following, we will pursue a theoretical analysis leading tofinal prediction of the spectrum of the switching noise process.The time constants of the AAC will be arranged as indicatedabove, and depicted in Fig. 4.

V. SIMPLIFIED ANALYSIS OF THETIME-VARYING NOISE

The noise of the two sources identified in Section IV can beconveniently considered as a white noise, periodically modu-lated in amplitude, followed by a low-pass filter that changes itspole synchronously to the input waveform. In fact, the steadynoise from enters a transfer function having both varyinggain and pole. shot noise is even more complicated tohandle, since it switches between zero and the case representedin Fig. 3(a), before filtering occurs; its effect will be brieflyoutlined in Section VII.

A. Theoretical Background

The classical approach to noise evaluation relies on autocor-relation concept. Given a stochastic process and taken two


time instants two definitions can be given for its autocor-relation:

— ensemble autocorrelation,calculated over a represen-tative set of different realizations through thejoint probability density

(1)

— time autocorrelation,calculated over one single time-bounded realization of the process

(2)

It is well known that the ensemble average of (2), ,is equal to the time average of (1), , when theprocess is ergodic2 (Birkhoff’s theorem, [19]).

To obtain the output from the input noise we mustconsider that the system is switching. The impulse response

used in time-invariant systems is to be replaced byits time-varying counterpart, the weighting function

(3)

where runs over the entire time span into which is signif-icant. Let us take now a white input noise, with spectral density

. The input autocorrelation is a Dirac-function

(4)

From (1) to (4), it is finally obtained [20]

(5)

Equation (5) highlights that the sought autocorrelation of theoutput process between the instants depends on both

and , and not simply on their distance. Instead of manip-ulating two-dimensional (2–D) autocorrelations and spectra,the analysis can be simplified by exploiting the properties oflow-pass filtering. In fact, by definition a time-varying processcannot be ergodic; but a tightly low-pass-filtered time-varyingprocess recovers the ergodicity [20]. Provided the time constantof the filter encompasses some periods of the switching noisewe have again , i.e., the couple( ) reduces to the time difference – . The one-di-mensional (1-D) autocorrelation is now suitable for Fouriertransformation, and the computed noise power spectrum can beaccounted for in the phase noise evaluation of the VCO.

2A stochastic process isergodicif, with probability equal to 1, all of its statis-tics can be determined from a single realizationx(t) of the process [20]. In otherterms, the observation of one single experiment over time allows to describe thebehavior of all the other possible realizations. Although heuristic, this definitionis the most suitable for our purposes.

In the actual AAC, the narrow loop filter after the peakdetector performs this time averaging of the switching noise[21]. The filter action is equivalent to the cascade of twodistinct effects: besides performing time-domain average, it ob-viously introduces a pole in the spectrum profile. However, inorder to expunge unessential effects from the analysis, the soletime average will be considered in the next sections. The spectracorrected by introducing the additional pole cutoff will be pre-sented in Section VI.

B. Application of the Theory to the Peak Detector

To better clarify the steps that lead to figure out the spectrumof the noise, let us try to predict the 2-D shape of the ensembleautocorrelation function for the thermal noise of

. Despite the case considered in the following is an oversim-plified one, other simulated situations will be easily understoodby extending the results drawn.

First of all, the process is recognized to be cyclosta-tionary.3 The simplest case occurs when the “on” time constantof the peak detector approaches zero: this implies an almostcomplete loss of the filter memory, thus be-comes a short exponential decay with negligible area. In sucha case, one can concentrate onto what happens within a singleperiod, i.e., for . Given a duty cycleof the peak detector of 50%, the weighting functions can be rep-resented as in Fig. 5(a)–(f). The corresponding 2-D autocorre-lation is computed as follows.

1) When , the product ofunder the (5) integral is zero [Fig. 5(a), dashed area inFig. 5(g)].

2) Moving along the main diagonal of the graph means, i.e., the autocorrelation in (5) is proportional to the

integral of :

a) in “off” state, from 0 to rises asthe area of a truncated exponential

(6)

then the rise shows negative concavity [Fig. 5(b),and the diagonal locus of maxima in Fig. 5(g)];

b) in “on” state, from to is con-stant and equal to the variance of the incomingwhite current noise, attenuated by the effect of[Fig. 5(c), and the low diagonal relief in Fig. 5(g)].

3) Fixing one of the time variables, say :

a) rises until , since the twoweighting functions tend to overlap completely[Fig. 5(d)];

b) it falls exponentially for , because the func-tions progressively disoverlap [Fig. 5(e)].

In both intervals the concavities are directed upward [asrepresented by the dotted line embedded in Fig. 5(g)].

3A stochastic process is defined ascyclostationary(in wide sense) when itsstatistical characteristics (mean, variance, ensemble autocorrelation) vary withtime according to a periodical pattern.


Fig. 5. Derivation of the 2-D autocorrelation. (a)–(f) depict the relative posi-tions of the weighting functions; (g) is the ideal final profile ofR (t ; t ).

4) Outside the square theautocorrelation is zero, since one of the weighting func-tions is surely the short exponential decay [Fig. 5(f), andgray plane surface in Fig. 5(g)].

5) Obviously the diagram is symmetrical with respect tothe diagonal of the horizontal plane, because

.

The resulting three-dimensional (3-D) graph of the ensembleautocorrelation is to be periodically reproduced over the maindiagonal of plane, as cyclostationarity requires.

Since the “on” interval shows a fast but not instantaneouspole response, in reality Fig. 5(g) should be slightly modified. Inparticular, all the vertical surfaces become exponential decays,and the surface is not tapered to zero on the axis boundaries, butit tends instead to approach the height of the diagonal relief [thiseffect is striking in Fig. 6(c)].

VI. V ALIDATION OF THE THEORY THROUGHSIMULATIONS

A crosscheck of the characteristics discovered in the formeranalysis for the thermal noise from can be performedthrough behavioral simulations. It is to be remarked that simplyswitching the noisebetween two different filtering paths (fastand attenuating during “on” intervals, slow and not attenuatedfor the “off” state) is not a fair reproduction of the real working

Fig. 6. Results of the Matlab-Simulink simulation of the peak detector noisedue toR . The noise values are already scaled to account for the successiveamplifications. (a) Single-realization autocorrelation forD = 50%; (b) auto-correlation averaged over 1000 statistically independent simulation trials (after90 counterclockwise rotation).

of peak detector. In fact, in this way the filter in the slowerbranch would keep its memory between one cycle and the next,even if the time constant of the faster path is zero. Instead, awhite noise sequence of length (or ) is generatedwith Matlab [22] and fed to aswitched filter,according to theinterval being “on” or “off.” It is of course necessary to recordthe filter’s state at the end of every session, and restore it atthe start of the next run, so as to ensure a continuous outputtransition. Finally, the filtered noise is saved and appended toprevious outputs.

The simulation parameters were chosen in order to minimizethe computation run times, while reproducing noise results com-parable to the ones of a test board. Namely:

— period of the rectified wave: s ( MHz)— “off” time constant: s (or slow

pole kHz; k , pF)— “on” time constant: ns (or fast pole

MHz; ).

Impedance gains k and are neededto keep the gain-bandwidth product GBWP equal to 100 kHz inboth cases.


Fig. 7. Time average in 2-D representation. Averaging overt and t isequivalent to taking the mean on the+45 diagonal.

The column-by-row product of one realization of outputnoise, , gives the surface in Fig. 6(a); again,

% was chosen for sake of clarity. Fig. 6(b) shows theensemble average taken over 1000 statisticallyindependent realizations: the simulated 3-D autocorrelationprofile resembles very closely the theoretical prediction.

Once is known, it must be averaged over timeto get the noise spectrum. The resulting function is dependentonly from the time difference – , then it will be located ontothe diagonal of plane. The direction in which theaveraging must be carried out isperpendicularto this one, henceoriented as the main diagonal of the same plane, as shownin Fig. 7.

In the following are considered the output spectra obtainedwith different values, respectively, of the input duty cycleand of . Of course the behavior of the peak detector dependson many circuit parameters, but for clarity only one of them isvaried at a time. For example, as differentvalues imply vari-ations in the charging should vary in turn;has been instead supposed to be constant. The designer caneasily accomplish this condition by adopting a current gener-ator in place of , or suitably changing the voltage drop on

through voltage translation techniques.

A. Modulation of the Duty Cycle

In these simulations the duty cycle was progressively risenfrom 1% to 99% to verify the correctness of theoretical anal-ysis; actually, classic implementations of peak detectors for si-nusoidal inputs seldom show duty cycles beyond 10%. The rmsvalue of the white current noise density injected in thefilter was 200 nA/ Hz, in order to get 54 dBV /Hz of noisepower density when the detector is “off”; when in “on” state, theattenuation drops the level down to78 dBV /Hz. These levelswere chosen in the same range of magnitude of the experimentalones reported in Section VIII for ease of comparison only, butdo not cause the discussion to lose in generality.

Fig. 8 shows the autocorrelations obtained. Allthe shapes are similar to bilateral exponential decay, the typical

Fig. 8. Simulated shrinking of the autocorrelation for increasing duty cycles.Despite relative to switched noise, they closely follow the classic exponentialdecay.

Fig. 9. Simulated noise power density spectra for increasing duty cycles. The0%D level has been set to�54 dBV /Hz.

autocorrelation function of the single-pole low-pass filter. De-spite this seems obvious for the curves corresponding to verylow or very high duty cycles, it is surprising for the interme-diate cases. The peak detector analysis led to recognize thatGBWP holds constant in “on” and “off” states; but the time-varying characteristics of noise prevent this property from triv-ially holding in the output noise spectra.

Fig. 9 depicts the power spectra computed from the simu-lations for different duty cycles. It shall be remarked that thespectra are obtained after the sole time averaging of ;hence, they could not be found at the output of a real low-passfilter, that would introduce its own pole. All of the spectra showa constant gain-bandwidth product. Given the impulse responseof a low-pass filter with gain and pole 1/ , and its Fouriertransform magnitude

(7)

It is apparent that, by keepingk constant, the area of alsokeeps, and a fixed GBWP is maintained; so happens of course


Fig. 10. Simulated noise power density spectra for increasingg . Forg =

625 (g = 1:6mA/V) a level of�62 dBV /Hz is expected, sinceD = 10%(see Fig. 9).

for . Fig. 9 indicates that GBWP keeps constant whenswitching between two processes with equal GBWPs. Althoughthe autocorrelation profiles in Fig. 8 derive from complicatefiltering, it can be stated that their areas and decay time con-stants inherit the building blocks’ original property, maintaininga fixed value for GBWP.

B. Modulation of of the Peak Detector

Other simulations investigated the behavior of the peak de-tector when the transconductance of the bipolar rectifier ischanged, e.g., by using a current generator fordischarge,or tailoring the voltage drop across with a suitable trans-lator. The numeric setup adopts a realistic duty cycle of 10%;the effects due to the increase in are modeled by means ofgain shrinking and band widening. Fig. 10 shows the resultingpower spectra.

Since separately affects the attenuation factor () andthe position of the pole ( ) but still leaves GBWP un-affected, the resulting spectra are of the same type illustratedin Fig. 9. As expected, noise power spectra feature a constantGBWP. For the upper curve s; an almostideal low-pass filtering is thus performed, and the spectrum fea-tures the usual 20 dB/dec downfall beyond 100 kHz. The twolowest curves refer to very fast “on” filtering, with of 16and 8 ns. By taking into account that is 100 ns, or 6.25and 12.5 times , respectively, it is easy to understand howthe memory of the filter is completely lost in the first case, andeven overkilled in the second one. The two noise spectra areconsequently identical.

C. Correction for Additional Low-Pass Pole

The previous results are easily adjusted to take into accountthe effect of the dominant pole of the AAC loop: the spectradepicted so far must include the new singularity. This has beenverified by simulation: a time-invariant low-pass filter has beeninserted in cascade to the switching blocks that generate thetime-varying noise. The pole must be slower than

kHz: for this test, 10 kHz were set. The plot in Fig. 11 illus-

Fig. 11. The noise spectra corrected with the addition of the low-pass filterpole at 10 kHz. Finally a constant G(BW)P locus is always reached.

trates the noise power spectra with different duty cycles. It con-firms that the low-pass filter action can be decomposed into thecascade of two separate effects: the time average ofand the spectrum cutoff at 10 kHz.

It is worth observing that the crossings between the two linesat 20 and 40 dB/dec follow a constant G(BW)P slope; themathematical proof for this rule is effortless. For, once the polesof the peak detector and of the successive filter are known, thecurve locus is uniquely determined: the duty cycle of the recti-fier will finally set the noise level at low frequency.

VII. PREDICTION OFNOISEDENSITY VERSUSDUTY CYCLE

The noise of the AAC eventually becomes bias noise for theVCO, and is upconverted around the carrier. The low-frequencyregion of the spectra previously analyzed must be lowered, sinceit contributes to limit the phase noise performance of the oscil-lator; in fact, phase noise is usually rated at small frequency off-sets from the carrier.

Finding a relation to link the peak detector parameters to theflat noise levels of Figs. 9 and 11 is not straightforward. Notein fact that the flat zone of the spectrum with 50% duty cycle isabout 69 dBV /Hz. By simply compounding “on” and “off”noise powers, weighted with their half-period spans, we get in-stead dBV /Hz! A first-order noise estimationcan be more rigorously found by computing the average area ofthe impulse response. By squaring it, the area of [i.e.,

] is obtained.Regarding noise, the impulse response looks like a

sequence of exponential decays, as reported in the Appendix(Fig. 16). The spectrum magnitude computation eventuallyleads to

(8)

where and .


Fig. 12. (a) Comparison between the simulation results (triangles—R

noise, circles—I shot noise) and the predictions given by the approximatedformulas (solid line—R , dashed line—I ). As expected, at high dutycyclesI plays a major role. (b) Comparison of the total peak detector noisesimulated (triangles) and estimated with the rule of thumb (solid line); a worstcase error of 1.5 dB is achieved.

The noise of can be approximated by weighting the areaof impulse responses generated only during “on” intervals, withthe span of

(9)

In Fig. 12(a) the noise levels at low frequency found withprevious simulations (see Fig. 9) are compared with the estima-tion given by (8). The highest error is less than 2.5 dB and occursfor high values of the duty cycle, while the prediction is almosterrorless for duty cycles 20%.

The analysis of the shot noise is more complex, be-cause the input noise is also switched, together with thefilter. The shot noise contribution has been simulated startingfrom the known case of 100% duty cycle. Indicating by thevoltage drop across , it is in this simplecondition; the usual expression for the “on” noise at the output,

, can be rewritten as , tobe weighted against the thermal noise term. Theratio between the two contributions is therefore .By choosing as done in Section VI, is setto 40 A, or V; hence, when the peak detector is

Fig. 13. Circuit schematic of the discrete-components test board. The powersupply was set to 12 V. For every measurement, the differential stage wasbalanced by adjusting the trimmer shown to the left.

“on” the shot noise is 9 dB higher than its thermal counterpart[ 69 dBV /Hz with % in Fig. 12(a)]. was thenkept constant over the entire sweep. The noise predic-tions of (9) compare less satisfactorily with the relative simu-lations. To keep our estimation simple, we ought to neglect theeffect of noise that is still decaying inside the filter during ,when the input noise goes to zero; especially at low duty cycles,such an effect is instead remarkable. The simulated behavior isanyway approximated with acceptable accuracy.

The overall output noise and the sum of the estimations arecompared in Fig. 12(b): the worst case error is 1.5 dB. Thusthe (8) and (9) provide simple while accurate formulas for pre-dicting the low-frequency noise power density after the peakdetector.

VIII. T ESTBOARD SETUP AND NOISEMEASUREMENTS

Experimental confirmation of the noise transfer has beenachieved by designing a test board, inclusive of a peak detectorand a noise amplifier stage. The rectifier is implemented witha simple – parallel connection, and one bipolar transistor;the schematic is represented in Fig. 13. The low testing fre-quency chosen (1 MHz, as in simulation) allows neglectingany impedance matching problem; the driving signal is feddirectly to the base of the bipolar rectifier . The output hasto be connected to a spectrum analyzer with standard 50-input impedance: a wide-band amplifier with 14 dB of gain and1 M input impedance was adopted as buffering stage.

With k , the bias current of is about 1 mAwith V; this voltage reference is insulated fromthe common low rail, in order to vary the transistor . Thecapacitance value is nF to give a ripple 50 mV;then the rectifier pole is 370 Hz. The differential stage has beenbiased with 1 mA, and by choosing k its gainreaches 27 dB, while its input-referred noise is negligible. Thethermal noise from , amplified by the cascade of differentialstage and wideband amplifier, results in117 dBV /Hz, or1.39 V/ Hz. This level is only slightly over the sensibilityof the spectrum analyzer, therefore it proved convenient toinject a known amount of white noise into the peak detectornode. For this purpose we synthesized a white Gaussian noise


Fig. 14. Experimental noise power spectra at the output of the board. The noisefloor is due to the poor noise performance of the square waveform synthesizerthat drives the rectifier.

with 200 V/ Hz power density with a signal generator.The external noise was coupled to the circuit via a resistor

k , so that the chosenpole was not displaced. The 1 : 10 voltage partition attenuatesthe noise to 20 V/ Hz on the filter ( 53 dBV /Hz, or2.2 mV/ Hz, after amplification).

The measurements reproduce the simulations described inSection VI: duty cycle variation and tuning of the rectifier’s

. The noise fed to the analyzer has not been low-pass filteredon the board, because the narrow sliding filter internal to theheterodyne analyzer performs exactly the time average functionsimulated in Section VI, without featuring the frequency cutoffinherent to fixed-frequency filters. We expect since then to ob-serve the same behavior shown in Figs. 9 and 10.

A. Duty Cycle Sweep

The first measurements are relative to duty cycle. waskept to ground level to ensure low values for. A duty cyclesweep ranging from 0.2 up to 99.8% was applied to the circuitby adopting a square waveform input. The curves in Fig. 14 aretenfold averages of the instantaneous noise spectra, and refer toa 1-MHz wave of amplitude 0.7-V peak.

The circuit shows the predicted behavior, keeping GBWPconstant to 370 Hz 2.02 mV/ Hz ( 54 dBV /Hz; the ac-tual amplification is about 70). A very rapid decrease in the flatpart of the spectra is observed for duty cycles10%; thereafterthe curves tend to superimpose one over another. The measuredspectra end up in a noise floor of76 dBV /Hz, clearly visibleat high frequencies. This is due to the noise directly injected inthe circuit by the square wave generator during intervals; apolynomial synthesizer with rather poor noise performance wasused in fact. Next measurements require a sinusoidal input thatwas derived from a much cleaner generator, then the noise floorwill disappear (Fig. 15).

B. Rectifier Sweep

This test concerns variation of , here imposed by changingthe amplitude of the input. The duty cycle is always around 10%.

Fig. 15. Experimental results when theg of the rectifier is varied. Since theinput generator is much cleaner the noise floor has disappeared.

GBWP remains constant over all of the measurements, as is ap-parent from Fig. 15. was kept rather low in this test; this en-tails that the autocorrelation function of the filter is quite narrow,leaving only minor room for further lowering settlements. Thespectral downsettling becomes even more pronounced when the

supply is lowered: after it is decreased under2 V, allthe spectra get almost stuck over the 940-mV curve (sinceincreases and the “on” pole of the filter gets faster: a memory“overkill” for the peak detector).

IX. DESIGN HINTS AND CONCLUSION

In order to improve the low-frequency noise coming from thepeak detector, the analyses performed above suggest that thedesigner works:

— toward a minimization of the impedance gain-band-width product GBWP;

— and both toward and (i.e., duty cycle) increase.

This paper highlighted that low-duty cycle switched implemen-tations of the peak detector (the so-called C-class stages), whileeffective for low-power purposes, are not suitable for a low-noise approach to the oscillator design. It is instead preferableto operate the rectifier device keeping it active as much as pos-sible: for instance, average peak detectors instead of envelopepeak detectors [18] can be conveniently exploited in the AAC.

Moreover, the rectifying device transconductance should beincreased through appropriate biasing. When this is accom-plished via a classic shunt, using the lowest stillcompatible with the ripple specification can enhance the noiseperformance. Eventual biasing constraints can be adjustedby means of simple diodes, since they introduce the minimalamount of noise into the network and do not alter the currentshunt mechanisms that lead to these results. For these rulesto be valid, the AAC structure must also carefully reject thenoise coming from the stages driven by the peak detector,e.g., by assuring signal amplification immediately after it. Afully differential amplifier is best suited for this task, as thenoise coming from its tail’s bias reference is often remarkable,and can otherwise add significantly to the one of the peak


detector. The single-ended driver for the VCO tail can thenbe implemented with a further stage, whose noise is less of aconcern since it comes after the amplification has taken place.A properly filtered replica bias like the one proposed in [3] canbe also employed to generate , appreciably improvingthe symmetry of the circuit.

In this paper, two simple formulas were given that are capableof predicting the output noise lowering versus the duty cycle ofthe rectifier. From the equations and simulations presented, ithas emerged that, with the little labor required to take the dutycycle up from a few percent to 10%, the SSCR of the AAC-regulated oscillator can improve by as much as 5 dB. This wasshown to be due to the nonlinearity of the noise process; forexample, in the benchmark case of 50% duty cycle it does notreduce its power by 3 dB only but as much as 12 dB instead.

The measurements performed on a board fabricated for thepurpose demonstrated that the enhancement can be even moredramatic in the actual implementation. Were it to be obtainedonly by modifying the VCO core cell, such a SSCR improve-ment would require a substantially greater design effort.

Following these guidelines, the designer can tailor the recti-fier and the low-pass filter to achieve a suitable tradeoff betweenthe power consumption and the phase noise issues.

APPENDIX

Let us briefly outline the procedure followed to determinethe area of an impulse response having the shape illustrated inFig. 16. For our convenience, we suppose the plot to start duringthe “off” interval. Provided the response begins at , it canbe written

(10)

Fig. 16. Peak detector response to an impulse occurring at the beginning ofT interval.

then

(11)

(12)

A converging geometric series withratio is recognized, then it

is stated in (13), shown at the bottom of the page. If theresponse starts at the beginning of an “on” interval, we mustsimply swap the position of the exponential scaling term, nowaccounting for the initial “on” decay as stated in (14), shown atthe bottom of the page. Of course, when displacing the impulseoccurrence within the period, the exponential scaling factorvaries, and the two cases transform one into the other. Thesimplest technique to predict the mean autocorrelation area inall the cases, is to take the arithmetic average ofand . Performing little approximations, a morecompact rule of thumb for estimating the noise power densityresults in (15), shown at the bottom of the page. After termredefinition, (15) eventually becomes (8).

(13)

(14)

(15)


(16)

The shot noise acts only during intervals (evenif it decays inside the filter also during ). To a firstorder, it can be written as stated in (16), shown at the top of thepage, that becomes (9).

ACKNOWLEDGMENT

The authors wish to thank Prof. A. S. Spinelli for partakingin useful discussions on the stochastic processes theory.

REFERENCES

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Alfio Zanchi (S’94–M’01) was born in AlzanoLombardo (Bergamo), Italy in 1971. He received thedegree in electronics engineering from the Politec-nico di Milano, Milan, Italy, in 1996, with a thesison the design of the digital decimation filter forSigma-Delta ADCs. He received the Ph.D. degreein electronics and communications engineering,also at the Politecnico di Milano, in 2000, wherehe developed analog integrated circuits for wirelesscommunications front-end.

From 1996 to 1999 he was a Consultant with theRF Group of STMicroelectronics, Catania, Italy. He is now with Texas Instru-ments Incorporated, Dallas, TX, working on the design of high-speed data con-verters.

Carlo Samori (M’98) was born in 1966, in Perugia,Italy. He received the Laurea degree in electronicsengineering in 1992, and the Ph.D. degree in elec-tronics and communications at the Politecnico di Mi-lano, Milan, Italy, in 1995.

In 1996 he was appointed assistant professor ofelectronics at the Politecnico di Milano. He workedon solid-state photodetectors and the associatedfront-end electronics. His current research interestsinclude design and analysis of integrated circuits forcommunications in bipolar and CMOS technologies,

noise analysis in oscillators, frequency synthesizer architectures. Since 1997 hehas been a consultant with the wireless Circuit Research Department at LucentTechnologies, Bell Labs.

Andrea L. Lacaita (M’89–SM’94) was born in 1962.He received the Laurea degree in nuclear engineeringfrom the Politecnico di Milano, Milan, Italy, in 1985.

In 1987 he joined the Italian National ResearchCouncil as Researcher. In 1990 he was a visiting sci-entist at the AT&T Bell Laboratories, Murray Hill,NJ, where he worked on photorefractive effects in su-perlattices. He was appointed associate professor in1992, and full professor of electronics in 1999, at thePolitecnico di Milano. His current research interestsare physics of carrier transport in semiconductor de-

vices, development of new experimental methods for characterization of semi-conductor material an devices, and low-noise design of integrated circuit forwireless applications. He is a coauthor of more than 100 papers published injournals or presented in international conferences.

Salvatore Levantino (S’98) was born in Palermo,Italy, in 1973. He received the degree of Ingegnerefrom the Politecnico di Milano, Milan, Italy, in 1998.He is currently working toward the Ph.D. degree inelectronics and communications at the Politecnico diMilano.

His activity is focused on the design of oscillatorsand frequency synthesizers for 3G applications. In1999, he was an exchange visitor at STMicroelec-tronics (RF group, Catania), working inS-parame-ters measurement of in-fixture devices. Since 2000,

he has been a consultant at Lucent Technologies, Bell Labs, Murray Hill, NJ.

Documents

Impact of AAC design on phase noise performance of VCOs