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How Phase Noise Appears in Oscillators

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					This chapter appears in the book "Analog Circuit Design-RF Analog-to-
Digital Con-verters; Sensor and Actuator Interfaces; Low-Noise Oscillators,
PLLs and Synthesizers", published by Kluwer Academic Publishers, Boston,
November 1997, 428 pp. Hard-bound, ISBN 0-7923-9968-4




                                                                               frequency (IF), then the phase noise around fLO will
    How Phase Noise Appears in Oscillators                                     downconvert nearby channels on to the same IF. As will
                                                                               shortly emerge, fundamental reasons dictate that the phase
                              Asad A. Abidi                                    noise spectral density is always highest at fLO and falls off at
                Integrated Circuits & Systems Laboratory                       frequencies away from it. Thus the largest expected strengths
                   Electrical Engineering Department                           of the immediately adjacent channels set a limit on the
                  University of California, Los Angeles                        tolerable phase noise close-in to the receiver LO frequency. In
             abidi@icsl.ucla.edu   http://www.icsl.ucla.edu/aagroup            modern wireless communication systems with power control
                                                                               such as GSM and DECT, the allowed strength of nearby
                                   ABSTRACT                                    channels must conform to a template roughly inverse to the
     Wireless transceivers closely specify the phase noise                     oscillator phase noise specification.
     in the local oscillator. Yet it is not very well                             Phase noise in the transmitter LO — very often this is the
     understood how phase noise is predicted, especially                       same oscillator as is used in the
     in oscillators which do not use a passive resonator. It                   receiver — may overwhelm nearby
     is also difficult to model flicker noise in the close-in                  weak channels. This is because the




                                                                                                                                            TX
                                                                                                                                        ph
     phase noise spectrum. This is a qualitative                               phase noise spectral density grows                          a se noi
                                                                                                                                                   se

     discussion of the various physical processes                              directly with the transmitted signal                   Nearby       Frequency
                                                                                                                                      Channels
     responsible for phase noise production, particularly                      power, and at a given point in space,
     in CMOS oscillators, and it offers a common                               the noise sidebands of a strong Figure 2: Phasenearbyin transmitter LO
                                                                                                                          can overwhelm
                                                                                                                                         noise
                                                                                                                                                  weak channels.
     treatment of resonator-based oscillators, ring                            transmitter may be greater than
     oscillators, and relaxation oscillators.                                  another faded or attenuated signal occupying the same
                                                                               frequency (Figure 2).
Why is Oscillator Phase Noise Important?                                          Although the topic of noise in oscillators has engaged
The phase noise in the radio receiver’s local oscillator limits                classical investigations of a qualitative nature [1-3], Leeson in
immunity against nearby interference                                           1966 was the first to propose a simple intuitive
signals. In the transmitter, phase noise                                       phenomenological model [4, 5] relating the level of phase
can swamp out nearby channels. In the                         Unwanted         noise in a widely used class of resonator-based oscillators to
                                                              Channel
receiver (Figure 1), the local oscillator        LO                            voltage and current noise sources in the circuit elements. This
(LO) frequency, fLO, is tuned to a                                             model has been widely embraced, and serves well to predict
                                                                  Frequency
certain frequency offset from the                      IF                      oscillator phase noise induced by sources of white noise.
desired channel (zero offset in the case Figure phase noise indownconversion
                                                1: Undesired                   However, while Leeson admits that device flicker (1/f ) noise
                                          by LO                a receiver.
of direct conversion) to downconvert it to the intermediate                    may determine the phase noise very close to the oscillation
frequency, his model cannot explain why. More recently,                                           Then, the waveform of the noisy oscillation is sampled at the
phase noise has been discussed in the relaxation oscillator, and                                  grid points, t k , and deviations in the samples from the
even in the ring oscillator which is of some interest for CMOS                                    baseline are translated into phase deviations, f( t k ) , by the
implementations. However, there is no one satisfactory                                            inflection          slope,     S,    of   the
                                                                                                                                                         2
                                                                                                                                                 waveform .   Thus,
method to predict phase noise in the latter types of oscillators,                                 f( t k ) = 2pfc ◊ x( tk )/ S , where fc is the nominal oscillation
nor one treatment which unifies and differentiates the                                            frequency. This works well for small fluctuations in phase.
mechanisms as to how voltage and current noise in the circuit
components of an oscillator transform into phase noise. The                                          Phase noise is defined by the variance of f( t k ) , E f( tk )2 .
purpose of this paper is to present a progress report on the                                      This is characterized in the frequency domain by the spectral
state of qualitative models explaining the production of phase                                    density, Sf ( f ) . After some simplifying assumptions to
noise. Ongoing work on this topic seeks to develop these into                                     surmount the problem that the phase noise defined above is a
quantitative models validated by measurements of phase noise                                      nonstationary random walk process, it follows from purely
in the various types of oscillators.                                                              statistical reasoning [6] that

Definition of Phase Noise                                                                                                Sf ( f ) µ
                                                                                                                                       FG fc / S IJ 2                   (1)
When asked to visualize phase noise, we first invoke the
                                                                                                                                        H f K
                                                                                                  and the result is the same for every voltage and current in the
picture of a noisy sinewave whose
                                                                                                  oscillator circuit.
phase, measured relative to a time
                                                                                                     The use of an oscillator with phase noise requires the
grid set by a noiseless sinewave, is
                                                                                                  statistics of phase to be converted into the equivalent
randomly perturbed at the zero
                                                                                                  sidebands the phase modulations induce around the
crossings (Figure 3). This may be
                                                                                                  oscillation frequency. We define the phase noise spectral
generalized to the non-sinusoidal
                                                                                                  density, ( f ) , as the ratio of the power density in one phase
periodic waveforms x(t) in practical
                                                                                                  modulation sideband relative to the oscillation power. From
                                                     Figure 3: A noiseless sinewave compared
                                                     with a sinewave with phase noise (jittered   phase modulation theory for small angles, it follows that:
                                                     zero crossings).
                                       oscillator circuits [6] by first defining                                                      i2 f = Sf ( f )/ 2
          S




                                                                                                                           d
    Slope=




                     x(tk)                                                                                          ( f ) = f pk / 2
                             tk        a zero crossing baseline and periodic
                                       time grid at every inflection point on,
                                                                                                  whose waveform inflects at some point. The period and phase noise is
                                       say, the rising edge of the noiseless                      exactly the same for all waveforms within an oscillator, so the most
    Figure 4: Defining phase jitter.                            1
                                       oscillation waveform (Figure 4).                           convenient waveform may be chosen for measurement.
                                                                                                  2
                                                                                                    By accounting only for deviations in the zero crossing times, this
1
  Even in oscillation waveforms such as a perfect triangle wave with no                           definition captures the phase noise by after passing the noisy oscillation
inflection, there is bound to be another voltage or current in the circuit                        through a limiter.
Voltage and Current Noise into Phase Noise                           zero dependence on voltage, current, and temperature. Then
It is convenient to think of any oscillator as an unstable closed    we can definitively say that noise cannot modulate the
loop circuit, comprising one or more active devices to provide       resonant frequency. At most, noise sources within the
gain or negative resistance and one or more passive reactances       oscillator loop will experience some gain and add to the
to set timing. In any practical electronic system, the oscillator    oscillation, thereby inducing phase noise. Leeson has given a
will connect to other circuits, such as an amplifier or a mixer.     simple and satisfactory model [4] for these conditions.
Voltage and current noise sources exist everywhere in the               Let us model a harmonic oscillator as a frequency-
system, and after adding to the oscillation they will all create     dependent block with transfer function            vn
phase noise in the sense of the above definition. However, the       F( jw ) in negative feedback with an
                                                                                                                            1




                                                                                                                                           Oscillator
noise sources within the oscillator loop exert a much more           amplifier whose gain is absorbed into this




                                                                                                                                           output
profound influence on phase noise than do the noise sources          transfer function (Figure 5). In keeping
                                                                     with the Barkhausen criterion for                     F(jw )
outside the loop. This is for the following distinct reasons:
                                                                     oscillation, the loop has infinite gain at Figure 5: Block Diagram of feed-
• Noise in the oscillator loop may be substantially enhanced                                                      back harmonic oscillator.
                                                                     the oscillation frequency, fc , and finite
  by the sharp frequency selectivity of the loop, and thus
  become the dominant source of phase noise                          gain at all other frequencies, i.e. F( j 2pfc ) = 1 . Consider a
                                                                     white noise source, vn , within the loop. Then the noise
• Noise in the oscillator loop may directly modulate the
                                                                     appearing at the output in series with the oscillation is:
  oscillation frequency
                                                                                                 1                        1
• Noise in the oscillator loop may modulate a reactance, and                       vout =               vn = -                        vn     (2)
                                                                                            1 - F( jw )        F ¢( jw ) ◊( w - w c )
   thereby the oscillation frequency                                                                       This is more conveniently expressed in
   These three processes are now illustrated at work in three                                              terms of the frequency offset from




                                                                      Noise T.F.
different types of oscillators: a resonator-based oscillator, such
                                                                                                           oscillation, fm ∫ f - fc .
as with an LC tuned circuit; a relaxation oscillator, such as a
                                                                                                           The frequency selection of the
multivibrator; and a ring oscillator, comprising a cascade of
                                                                                                           oscillating loop leads to a noise
active delay stages operating in the large signal mode.
                                                                                      fc         Frequency transfer function which is singular at

Phase Noise in Resonator-Based Oscillators                           Figure 6: Noise transfer function      fc , and for small offsets declines
                                                                     around the oscillation frequency.
                                                                                                           inversely with fm (Figure 6). Thus,
Additive Noise
                                                                     noise frequencies around fc are preferentially amplified, and
Harmonic oscillators use passive reactances, such as an LC           add to the oscillation waveform to create phase noise (Figure
tuned circuit or an RC network, to define the oscillation
frequency. Let us suppose these reactances are constant, with
7).   The      phase     noise      in   an     LC     resonator,      where         Although this phenomenon had been observed all along at
                             2Q                                                   very small offset frequencies in oscillators, it was attributed to
F ¢( jw c ) ª d –F( jw c ) =    , is given by
              dw             wc                                                   parametric flicker fluctuations in the circuit component
                               LM F I 2 1 OP vn2                                  values, such as fluctuations in resistance. However, when this
                              1     f
                   ( fm ) =     1+ c
                              2 MN GH JK fm2 PQ A2
                                   2Q
                                                                          (3)     was found to be the typical character of phase noise in
                                                                                  microwave GaAs FET oscillators at offset frequencies beyond
This expression gives the two principal factors regulating                        1 MHz [7], a search began in earnest to explain how the
                                               phase noise. First is the net      baseband flicker noise known to exist in MESFETs could
                Additive noise with
                strong fc spectrum             noise-to-signal ratio within the   appear at the oscillation frequency. It was recognized that
                                               oscillator found by dividing the   Leeson’s linear model does not account for the effects of
                                               output due to all the noise        nonlinearity on noise in an oscillator which self-limits the
                                               sources by the oscillation         oscillation amplitude. The oscillator can no longer be treated
  Figure 7: Noise accentuated around fc causes amplitude, A. Second is the        as operating at a fixed bias point, with small noise signal
  phase jitter.
                                               half-power bandwidth of the        superimposed. In reality, the bias point varies considerably
                                               loaded resonator, fc / 2Q . This   over the oscillation amplitude,
suggests that aside from lowering the noise levels in the                         and owing to the voltage and               v            n
                                                                                                                                           2
                                                                                                                                                      Composite noise
                                                                                                                                                   adding to oscillator
components, the oscillation amplitude must be designed                            current dependence of the
                                                                                                                                pco
large, and a high Q resonator must be used.




                                                                                                                                          U




                                                                                                                                                           )
                                                                                                                                                               lo
                                                                                  transconductance,            output




                                                                                                                                                          ss
                                                                                                                                   nve
                                                                                                                                       rsi o n (
                                                                                                                                                 wit h co
                                                                                                                                                          n v.     Frequency


Upconverted Noise                                                                 conductance,         FET       gate
                                                                                  capacitance, and of noise itself,
Additive noise explains the observed phase noise spectrum                                                                      9: Upconversion of baseband (1/f)
                                                                                  the baseband flicker noise is Figureto oscillation frequency.
with a 20 dB/decade slope close to the                                                                                 noise
                                                                                  upconverted to fc (Figure 9) [8, 9]. The upconverted noise
                                                            (fm)




oscillation frequency. However, in practice at     ~1/f3
                                                                                  then enters into the oscillator loop according to Leeson’s
small fm a 30 dB/decade slope is observed                ~1/f2                    model. As in any mixing process where neither signal drives
(Figure 8). This is associated with flicker
                                                                                  the circuit into clipping, the upconverted noise depends on
(1/f ) noise in the oscillator’s active devices.            fm(log)
                                                                                  the oscillator signal, and grows with oscillation amplitude, A.
However, the additive noise model cannot Figure 8: Typical close-in
                                                                                  The resultant phase noise spectral density is therefore
explain how flicker noise at low frequencies phase noise spectrum.
                                                                                  independent of A.
appears around the oscillation frequency, fc .
    In contrast to MESFETs, the insulating gate of the                                 independent of the oscillation amplitude A and the final phase
MOSFET supports much larger oscillation amplitudes. Even                               noise spectral density depends inversely on A2 .
at frequencies as high as 1 GHz, CMOS resonator-based
oscillators will support rail-to-rail voltage swings. This makes                       Phase Noise in Relaxation Oscillators
it somewhat easier to analyze the noise upconversion process.
                                                                                       It is instructive to contrast phase noise production in
Let’s use as an example a CMOS oscillator comprising a
                                                                                       resonator-based oscillators with a
cross-coupled       differential     pair
                                                                                       sharply contrasting oscillator                I
negative resistor across an LC
                                                                                       type, the relaxation oscillator.
resonant circuit (Figure 10). Assume                                                                                                   C             DV          F/F
                                                                       Out             Generically this comprises a
that the self-limited oscillation M1                      M2                           single reactance, almost always a
amplitude is so large as to completely                                                                                            2I
                                                                                       capacitor, a regenerative memory
switch the differential pair, M1-M2,                 M3
                                                                   In                  element such as a flip-flop or
while the current source M3 always
                                               Oscillator             Mixer
                                                                                       Schmitt trigger, and a means of Figure 11: Typical relaxation oscillator.
remains in saturation. Then M1-M2
                                                                                       charging and discharging the capacitor (Figure 11).
acts as a commutating mixer, which Figure 10: CMOS LC Oscillator. A simple
                                           mixer is shown to illustrate the inherent      There are two fundamental differences between this and
upconverts baseband flicker and mixing action in the oscillator.
                                                                                       the harmonic oscillator discussed in the previous section. The
white noise from M3 to fc , and downconverts white noise at                            single reactance is not frequency selective like the resonator,
2 fc in M3 to fc . If the noise bandwidth exceeds 2 fc and the                         and the regenerative element makes this into a discrete-time
up- and down-conversion gains are equal, the net noise                                 feedback loop. The two different means by which circuit noise
delivered by M3 into the resonant circuit resembles its own                            converts into phase noise are now discussed.
baseband noise spectrum, except with a flicker noise corner                               The frequency of oscillation is set by the charge/discharge
frequency lowered by 2×.                                                               rate of the capacitor, and two separate trip points at the input
    The differential pair delivers noise to the resonant circuit                       of the regenerative element separated by some voltage DVt .
over that fraction of the oscillation period when both M1 and                          Thus, the oscillation frequency is
M2 are ON. This again consists of upconverted baseband
                                                                                                                            I
noise and downconverted noise from 2 fc . Otherwise, while                                                        fc =                                         (5)
                                                                                                                         2C ◊ DVt
only one of the pair FETs is ON it acts like a cascode on M3                           Low-frequency noise in in the charging current, I, or vn on
and contributes no net noise of its own.                                               the reference voltages which set the regeneration trip points
    With complete commutation, the frequency translated                                will directly frequency modulate the oscillation to create close-in
noise (at least the main component from M3) is once again                              phase noise [10]. Invoking textbook FM theory, if a
                                                                                       modulating sinewave of frequency fm causes a peak deviation
Df pk in the carrier frequency fc , then at a small modulation                     flicker noise remains unchanged as long as the flicker noise
index the main sideband in the FM spectrum lies at fm with                         corner frequency lies below          1 f .
                                                                                                                        2 c
strength
                      Vssb
                           = J1
                               FDf pkIª
                                        Df pk                                      Phase Noise in Ring Oscillators
                       A
                               GHfm
                                     JK 2 fm
                                                                             (6)
                                                                                   The ring oscillator at RF has raised interest among CMOS IC
Thus, the SSB phase noise produced by a noise spectral                             designers because it is simple, fast, and readily yields output
density S( in ) and S( vn ) is                                                     phases in quadrature. However, as is often true in
                                                                                   engineering, the simplest circuits bundle so many nonlinear
      ( fm ) =
                       FG IJ 2 and
                 S( in ) fc
                                      ( fm ) =
                                                      S( vn )   FG fc IJ 2   (7)   effects that they are quite difficult to analyze. This seems to
                        H K
                  2I 2 fm              2( DVt )             2    H fm K            be true in the matter of predicting phase noise in ring
Flicker noise in either S( in ) or S( vn ) will produce, through                   oscillators. Time-domain [12] and frequency-domain [13]
this straightforward process of FM, a phase noise spectrum of                      models for noise production in ring oscillators have been
1 / f 3 . High frequency components of S( in ) are filtered out by                 proposed. We present a new model here, which refines and
the integrating capacitor, but the regenerative element reacts                     extends a previously published model [13] to more accurately
to high frequencies in S( vn ) somewhat unusually. The                             capture the process of phase noise production.
transition times in a relaxation oscillator are known to be set                       Without loss of generality, consider a ring oscillator
by the first crossing of a voltage ramp across a noisy                             comprising             four
threshold, which irreversibly triggers the regenerative element                    differential delay stages            Delay       Delay        Delay  Delay
                                                                                                                        Stage       Stage        Stage  Stage
to change state. The ramp cannot distinguish between an                            (Figure 13). A short ring
intersection with high frequency                                                   will always be the case
or low frequency noise (Figure                                                     for oscillation at RF. The                                      td              td
                                           Noisy DV




12). For instance, the effect will                                                 steady-state oscillation is
be the same if the noise on the                                                    very close to a large
                                                                                   sinewave,       with     an                                 Model
threshold is of frequency fm , Figure 12: Regenerative element samples noise
                                                                                   amplitude set by full
 fc - fm , or fc + fm . In other on thresholds.
                                                                                   switching       of                                  Oscillator.
                                                                                                          each Figure 13: CMOS Ring function. A model to calculate the
                                                                                                               additive noise transfer
words, the relaxation oscillator samples wideband noise on the                     differential pair current
threshold [11]. This causes the noise spectrum of S( vn ) at                       into the loads. The oscillator derives its frequency from the
frequencies above       1 f   to alias back to the Nyquist band from               cumulative delay in the stages making up the ring. It follows
                        2 c
                                                                                   by symmetry that if all the stages are identical, then as the
0 to 1 fc and raise the white noise floor appearing in the
     2                                                                             sinewave traverses each stage of the ring its amplitude remains
second equation in (7). However, the phase noise caused by
                                                                                   unchanged, and it experiences a phase lag of 45°.
    For ease of visualization, assume that only one of the delay         So far the analysis applies to high frequency noise entering
stages in the ring is noisy, and the others are noiseless. Then       the oscillating loop. We now note that the ring oscillator is
for frequencies around fc , the ring oscillator may be modelled       also akin to the relaxation oscillator, in that low frequency noise
as a single noisy differential pair with negative feedback from       in the charge/discharge currents will modulate the delay of
the output to the input via an ideal delay line, td (Figure 13).      each stage, and therefore the td . Again, there is a linear
The unity gain delay line models the other three noiseless            dependence       between     the     noise      currents    which
stages because its gain is one (specifically at the oscillation       charge/discharge the output capacitor in each delay stage and
frequency), and we lump into it the delay of the entire ring,         the frequency of oscillation. Suppose the deviation coefficient
i.e. td = 1 / ( 2 fc ) .                                              is K v1 Hz/A for the tail current and K v2 Hz/A for the load
    We now draw from the preceding analyses of phase noise            current; the differential pair does not contribute any
in harmonic and relaxation oscillators to understand the ring         substantial low frequency noise. Then, using FM theory as
oscillator. In that the ring oscillator comprises a continuous-       before, the spectral density of phase noise is:
time feedback loop with a delay line resonator, it is similar to                                    2
                                                                                                  K v1S( in1 )         2
                                                                                                                     K v2S( in2 )
the harmonic oscillator. Noise propagates around this loop,                            ( fm ) =                  +                  (10)
                                                                                                      fm2                fm2
and Leeson’s model applies to how additive and upconverted               The ring oscillator, in conclusion, is a hybrid of the
noise converts into phase noise. The noise transfer function is:      harmonic oscillator and the relaxation oscillator.
                   Sv / A2                        e jwt d / 2
           Sf =                    = Sv / A2                    (8)   The Spectral Linewidth of an Oscillation
                         - jwt d               2cos(wtd / 2 )
                  1+ e
As in the harmonic oscillator, noise at the oscillation               The foregoing analysis concerns itself with noise spectra at
frequency experiences infinite gain, and the phase noise              frequency offsets from the carrier frequency. In all cases, the
spectrum for small offset frequencies fm from fc is:                  spectral density becomes infinite at the oscillation frequency.
                                                                      However, in practice the spectral density of an oscillation is
            ( fm ) = 1
                                   Sv
                                 =1 v
                                     S    fc          FG IJ 2   (9)   never infinite, even if one were to measure it on a spectrum
                     2 2
                      A (w mtd )2 2 A2p 2 fm           H K            analyzer with almost zero resolution bandwidth. Over any
By comparing this with the expression for an LC tuned                 finite observation period, the oscillation frequency will waver
circuit, we may ascribe an effective Q of p / 2 to the delay-line     around a mean value due to drifts in temperature, vibration,
resonator. The additive noise entering the oscillator is almost       etc. One may think of this wavering as a residual FM caused
all frequency-converted noise. For instance, the tail current         by environmental factors. Associated with each source of
flicker noise is upconverted by the switching differential pair.      modulation, such as temperature or vibration, is a deviation
The large sinusoidal bias current in the load FETs upconverts         constant. The mean-square value of the oscillator output
their flicker noise. Similarly the oscillating bias in the            voltage (or current) will then spread over this usually quite
differential pairs upconverts their flicker noise.
small frequency range of wavering, which, borrowing a term                       into AM and PM phasors (Figure 14). The two types of
from optics, is called the spectral linewidth.                                   modulation will produce superimposed phase noise sidebands
   Calculations of the spectrum of a carrier frequency                           at fm which are indistinguishable on a spectrum analyzer.
modulated by Gaussian noise identify this spectral linewidth.                        In practice the random AM may be removed by a limiter
If a wideband white Gaussian noise voltage with spectral                         or when the oscillator output is applied to a commutating
              2
density Sv V /Hz modulates the frequency of a sinewave with                      mixer. The PM will remain and produce the net phase noise.
deviation constant K Hz/V, then the resulting frequency                          Therefore, it is of interest to measure the random PM
           3
spectrum is :                                                                    separately from the AM. It is possible to do this by mixing the
                           A2       p 2 K 2Sv                                    noisy oscillation with an appropriately delayed version of
                ( fm ) =                                                (11)     itself. If the delay is exactly a quarter period, then the
                                2 2
                           2 ( p K Sv )2 + ( 2pfm )2
                                                                                 oscillation will self-downconvert to zero, and any amplitude
The spectral density is almost constant over the offset                          modulations will also downconvert to zero. However, phase
                                                                                 modulations will downconvert to their respective offset
frequency range         e0,pK 2Sv / 2j   which defines the spectral              frequency.
linewidth.                                                                           [ A + a(t )]sin(wt + f(t )) ¥ [ A + a(t - T )]cos(wt + f(t - T ))
                                                                                                                                         4            4
                                                                                                                                                               (12)
AM Noise, FM Noise, and True Measurement of Phase Noise                                A2[f(t ) - f(t - T )]
                                                                                                         4
Noise in an oscillator produces fluctuations not only in phase,                  using the small angle approximation, and assuming no
but also in amplitude. Consider, for                                             correlation between f(t ) and f(t - T ) .
                                                                  fm fm                                                   4
instance, noise in the reference trip points                               fm       An automated phase noise measurement system drives a
of a relaxation oscillator. This modulates                                 fm    variable delay element with
the amplitude of the capacitor waveform,         FM                              a feedback loop to hold the
and also the frequency. As another                                               average DC at the mixer
example, consider a random phasor at an                                  AM




                                                                                                                            Oscillator
                                                                                 output to zero [15]. A                                                       Spectrum
                                             Figure 14: Additive noise creates                                                                                Analyzer
offset frequency fm adding to the FM and AM components.                          spectral analysis of the                   Pwr
                                                                                                                            Split
oscillation phasor. This may be resolved                                         output      fluctuations    at
                                                                                 frequencies offset from DC Figure 15: Delay discriminator method to accurately meas-
                                                                                 will directly read out ( f m ) . ure phase noise.
                                                                                 As the measurement system automatically tracks out the
3                                                                                oscillation frequency, it is insensitive to slow drifts in the
  This expression was derived from first principles by Masoud Djafari of
UCLA, and corrects for an error in an original calculation by Stewart
[14].
frequency4. This makes it convenient to accurately measure
phase noise in an oscillator with a large spectral linewidth,           Conclusions
such as in a relaxation or ring oscillator. The large drifts in         Measured          phase        noise        -70
these oscillators make it difficult to measure noise sidebands




                                                                                                                               Phase Noise, dBc/Hz
                                                                        characteristics of RF-CMOS                               Fli
                                                                                                                                     ck
                                                                                                                                        er
                                                                                                                    -80                    No
on a spectrum analyzer, but a phase noise measurement                   LC (Figure 16) and ring                                               ise
                                                                                                                                                  Do
                                                                                                                                                     mi
                                                                                                                    -90                                 na
instrument will produce repeatable results.                             oscillators (Figure 17) show                                                       nt

                                                                                                                  -100
                                                                        that flicker noise dominates at
                                                                                                                  -110
                                                                        the frequency          offsets    of
                                                                        interest.     The      design     of      -120
                                                                                                                      10                              10 4   2       3   4 5 6 78   5       2   3   4

                                                                                                                            Offset Frequency, Hz
                                                                        oscillators to certain phase
                                                                        noise specifications will require Figure 16: Measured Phase Noise in 900 MHz
                                                                                                             RF CMOS LC Oscillator (Q=3 to 4).
                                                                        a priori modelling of FET
                                                                        flicker noise, an art which is
                                                                        still at an early empirical stage [16, 17]. Although the various
                                                                        frequency conversion effects may be qualitatively anticipated,
                                                                        it only seems feasible to model the various conversion gains
                                                                        and the noise modulations caused by large oscillations in the
                                                                        bias currents and voltages through recently developed
                                                                        numerical techniques [18].
                                                                        Circuit      designers      have         20




                                                                                                                    L(fm), dBc/Hz
                                                                        successfully      used      hand          0
                                                                                                                -20
                                                                        calculations       to     predict       -40      Flic
                                                                        amplifier noise, and even the           -60          ker
                                                                                                                                 No
                                                                                                                                      ise
                                                                        phase noise caused by                   -80
                                                                                                               -100
                                                                        additive noise in nearly               -120

                                                                        linear oscillators, but it                10 2 3 4 10 2 3 4 10 2 3 4 10 2 3 4 10
                                                                                                                                                     2           3            4         5               6


                                                                        seems that the prediction of                     Offset Frequency, Hz
                                                                        CMOS         oscillator    phase Figure 17: Measured Phase Noise in 800 MHz 4-stage
                                                                        noise, when it is finally CMOS Ring Oscillator.
                                                                        possible, will heavily rely on
                                                                        advanced CAD tools.
4
 As long as the rate of drift is within the tracking bandwidth of the
delay-adjustment feedback.
References                                                                 [14] J. L. Stewart, “The Power Spectrum of a Carrier Frequency
                                                                                Modulated by Gaussian Noise,” Proc. of the IRE, vol. 42, no. 10,
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