PHASE NOISE IN OSCILLATORS - CiteSeerX

PHASE NOISE IN OSCILLATORS
Donhee Ham, William Andress, and David Ricketts

Harvard University, Cambridge, MA 02138, USA

Abstract| In this tutorial paper, we review funda-
mentals of oscillator phase noise emphasizing its physi-
cal picture based on the concept of phase di®usion. This
review is intended to provide an essential understand-
ing of phase noise rather than a comprehensive phase
noise modeling. We then summarize several phase noise
reduction techniques for various types of oscillators.

Index Terms| oscillators, phase noise, integrated cir-
cuits, analog integrated circuits, radio-frequency (RF)
circuits.

1. INTRODUCTION Figure 1: Generic model for a self-sustained LC oscillator.

Self-sustained oscillators universally exhibit linewidth broad- Figure 2: Limit cycle in the V -I state-space.
ening of varying degree in their output power spectra. This
linewidth broadening, often referred to as phase noise, is R. The resonator loss and the active devices generate noise,
caused by noise inherent in the oscillator and is a measure which can be modeled as current sources as shown in Fig.
of spectral purity of the oscillator signal. Any system that 1.
requires an oscillator for reference frequency/timing gener-
ation can su®er from the phase noise. For instance, in wire- The dynamics of an LC oscillator can be visualized by
less transceivers which utilize local oscillators for frequency mapping the voltage, V , across the capacitor and the cur-
synthesis, the phase noise degrades several important perfor- rent, I, in the inductor onto a trajectory in the V -I state-
mance metrics such as receiver selectivity, compromising the space as shown in Fig. 2. The trajectory for steady-state
overall communication capacity. Due to this importance, os- oscillation is a closed curve due to the periodicity and is
cillator phase noise has been extensively studied in various called a limit cycle [18]. Regardless of its starting point,
¯elds of engineering and science (e.g., [1] - [17]). the state will be ultimately attracted to the limit cycle after
the initial transient fades away, as shown in Fig. 2. This
This paper is a tutorial which reviews the physics of peculiar property of the self-sustained oscillator directly af-
oscillator phase noise. The primary goal is to provide a fects its °uctuation behavior in the presence of noise. The
simple yet essential perspective of the oscillator phase noise °uctuations would remain small in the radial (amplitude)
rather than a comprehensive modeling of phase noise. This direction due to the tendency of the state to return to the
review is based on earlier pioneering works [1] - [4] which
elucidated the fundamental mechanism of phase noise and
recent work [5] which added several new physical insights.
The authors hopes that the physical perspective this paper
o®ers will complement the many important, yet rather phe-
nomenological modeling works (e.g., [6] - [17]). Section 2
is the review. In Section 3, we will summarize some of the
currently available phase noise reduction techniques.

2. PHASE NOISE FUNDAMENTALS

This section based on [1] - [5] reviews the physics of phase
noise from the viewpoint of phase di®usion, emphasizing
intuitive understanding.

2.1. Phase Di®usion

The LC oscillator of Fig. 1 will be used as a demonstra-
tive vehicle to discuss phase di®usion in this subsection. In
the ¯gure, the LC resonator provides a frequency selection
mechanism while the active devices form a positive feed-
back loop to compensate the resonator loss represented by

Figure 3: Ensemble of N identical oscillators.

Figure 4: (a) Phase di®usion in the state-space. (b) Phase di®usion

in the time-domain. (c) Time evolution of P (Á; t). In all cases, t1 < t2.

limit cycle. However, °uctuations in the direction along the delta function. As time elapses, the probability distribution
limit cycle do not experience any restoring force to return
the phase to its original value. Consequently, in the pres- of Á spreads out due to the phase di®usion. If the phase
ence of noise, the state point walks randomly along the limit
cycle, or, the phase undergoes a \di®usion" process [1] - [4]. di®usion is due to white noise, the variance of Á, which sig-

To see this phase di®usion more clearly, let us run a ni¯es the width of the probability distribution, is given by
thought experiment using an ensemble of N identical oscil-
lators shown in Fig. 3 where N is a very large number. All [1] - [4]
the oscillators are assumed to be at the same initial phase of
zero at t = 0. In the V -I state-space shown in Fig. 4(a), the hÁ2(t)i = 2Dt (2)
state points from the ensemble are all on top of one another
initially, rotating on the limit cycle together. However, the a key signature of any di®usion process subject to white
rotating oscillation points di®use along the limit cycle with noise [19]. The constant, D, called phase di®usion constant,
time, eventually spreading all over the limit cycle. indicates how fast the phase di®usion occurs. As will be
seen shortly, this phase di®usion constant is the sole factor
In the time-domain, the fundamental component of the that determines the oscillator phase noise.
voltage across the LC tank in Fig. 1 is expressed as
2.2. Oscillator Power Spectra and Phase Noise

V (t) = V0 cos[!0t + Á(t)] (1) When the oscillator phase undergoes di®usion satisfying (2)
subject to white noise, the power spectral density of V (t) in
p (1) is given by the familiar Lorentzian [1] - [4]:
where !0 = 1= LC is the oscillation frequency and V0 is

the amplitude of the fundamental component. We ignored

the amplitude noise due to its relative insigni¯cance as dis- V02 D
(¢!)2 +
cussed shortly before. The °uctuation along the direction SV (!) = D2 (3)

of the limit cycle in the state-space translates to the phase

°uctuation, Á(t), in (1), which assumes a di®usion process, where ¢! ´ ! ¡ !0 is the frequency o®set from the oscilla-
tion frequency, !0. Note that regardless of the value of D
as mentioned earlier. The time-domain picture of this phase the total oscillation energy remains the same value, V02=2, as
can be seen by integrating SV (!) over the whole frequency
di®usion is shown in Fig. 4(b). Initially, the oscillator sig- range. Figure 5 shows SV (!) versus ! for di®erent phase
di®usion constants for the ¯xed oscillator energy of V02=2.
nals from the ensemble are all on top of one another since the As can be seen, the phase di®usion directly translates into

oscillators are at the same initial phase. After a su±ciently the oscillator power spectrum broadening; for a larger D,
the Lorentzian shape is shorter and fatter, distributing the
long time, however, the signals become totally incoherent
same oscillation energy (the area under the power spectrum
due to the phase di®usion.
curves) more widely around the center frequency, hence in-
Based on the ensemble, we can de¯ne the time-dependent
creasing the errors in the oscillation frequency. This is the
probability distribution of the phase, P (Á; t), where P (Á; t)dÁ
frequency domain meaning of the phase di®usion.
represents the probability for the phase to be in (Á; Á + dÁ)

for a given time, t. Figure 4(c) shows the evolution of P (Á; t)

with time. Since all the oscillators in the ensemble are at

the same initial phase of zero, the initial distribution is a

Figure 5: Power spectral densities of the oscillator output for di®erent

di®usion constants for the same oscillation energy.

The degree of this energy spreading about the center Figure 6: Ensemble average of V (t) and virtual damping.

frequency for a given total energy is characterized by phase where we have used hcos Ái = e¡hÁ2i=2 and hsin Ái = 0 as-
suming that Á(t) has a Gaussian distribution [19]. Equation
noise. The phase noise at a given o®set frequency, ¢!, is (6) clearly shows the virtual damping with an exponential
de¯ned as the ratio of the power spectral density at the behavior. As guessed earlier, the virtual damping rate is
o®set frequency of ¢! to the total oscillation energy, V02=2: identical to the phase di®usion constant, D.

Lf¢!g ´ SV (!) = 2D (4) The virtual damping rate, D, and the phase noise of
V02=2 (¢!)2 + D2 the oscillator are related through (5). To convey an idea
of the size of the D, let us consider an example: a 1 GHz
This de¯nition of phase noise based on power spectrum oscillator whose phase noise is -121dBc/Hz at 600 kHz o®set
has D ¼ 5:6 Hz or D=f0 » 10¡9 according to (5). As
broadening is widely used due to its ease of measurement. this example shows, typical oscillators have very slow virtual
If ¢! À D, (4) simpli¯es to a familiar f ¡2 behavior [6]: damping rates as compared to oscillation frequencies.

Lf¢!g ¼ 2D (5) We can observe the virtual damping experimentally as
(¢!)2 well [5]. Figure 7 shows an example experimental setup us-
ing a ring oscillator.1 The ring oscillator's phase noise is
Equation (4) shows that phase noise solely depends on the signi¯cantly degraded by injection of a white noise current
phase di®usion constant, D; a slower phase di®usion corre- whose power can be controlled externally. A digital oscillo-
sponds to a smaller spectrum broadening. scope is used to sample the output waveform multiple times
and calculate the average over N samples, i.e., hV (t)iN .
2.3. Virtual Damping The N output waveforms are triggered at the same phase
at t = 0. To make this average as close to the mathematical
Now we introduce the concept of virtual damping [5] as an- ensemble average as possible, a considerably large N of 512
other manifestation of the phase di®usion. The time-domain was chosen.2 Figure 8 shows this average for 512 samples
picture of the phase di®usion in Fig. 4(b) is redrawn at the as a function of time. The exponential damping in the av-
top of Fig. 6. Initially, the oscillators in the ensemble have erage is apparent, con¯rming the virtual damping concept
the same phase and hence the ensemble average, hV (t)i, and (6).
is equal to V (t) of any single oscillator in the ensemble.
With time the oscillator signals become incoherent due to Using this experiment, the virtual damping rate, D, was
the phase di®usion and hV (t)i tends towards zero, as shown measured for di®erent injected noise power levels. The mea-
at the bottom of Fig. 6. We refer to this damping of the sured D was used to predict the phase noise at the o®set fre-
ensemble average as virtual damping. One can intuitively quency of 1 MHz resorting to (5). The phase noise was also
guess that the phase di®usion constant, D, is directly re- directly measured at the same o®set frequency using a spec-
lated to the virtual damping rate: a slower phase di®usion trum analyzer. Table 2, which summarizes the results, show
corresponds to a slower virtual damping.
1LC oscillators have relatively smaller D, making them less suitable
A quantitative veri¯cation of the virtual damping is as for experimental veri¯cation. However, the exponential virtual damp-
the following. In the presence of white noise, the oscillator ing is a general phenomenon in any types of oscillators in°uenced by
phase satis¯es (2) and the ensemble average of V (t) in (1) white noise, as the generic equation (6) suggests.
is given by
2Assuming ergodicity.
hV (t)i = V0e¡hÁ2(t)i=2 cos(!0t) = V0e¡Dt cos(!0t) (6)

2.4. D

As discussed earlier, the phase di®usion constant (virtual
damping rate) D is the sole factor that determines oscil-
lator phase noise in the presence of only white noise, and
hence in the design of oscillators, one should make every
e®ort to minimize D. To this end, it is important to have
an expression for D in terms of circuit parameters to obtain
insight into phase noise minimization. This subsection will
brie°y review the standard derivation of D taking into ac-
count time-varying e®ects, using the LC oscillator of Fig. 1
as an example. The essential features of this derivation are
commonly found in many works, e.g., [1], [5], [10], [13]-[16].

For the LC oscillator of Fig. 1, one can show that the
stochastic di®erential equation describing the time evolution
of Á(t) due to a k ¡ th noise source, in;k(t), is given by (e.g.
[1], [5], [10], [13]-[16])

Figure 7: Measurement setup for the virtual damping. dÁ = 1 (7)
dt CV0 in;k(t)pk(t)

Figure 8: A measurement example: measured hV (t)i512 versus t. where pk(t) is a deterministic periodic function associated
with the k¡th noise source and re°ects the time-variance in
oscillator noise dynamics. pk(t) is determined by detailed os-
cillator dynamics. While its analytical evaluation is di±cult
in general due to the nonlinearity of oscillators, computer-
aided simulations can numerically evaluate pk(t) [13]-[16].
The work in [13] broke down pk(t) into the impulse sen-
sitivity function and the noise modulating function, which
provided a great deal of insight into time-varying e®ects on
oscillator phase noise. This will be further discussed in Sub-
sec. 3.4.

Equation (7) shows that Á(t) is an integration of the
noise, in;k(t), and hence Á(t) undergoes di®usion if in;k(t)
is white noise. Assuming white noise and using a standard
stochastic theory, one can show that the phase di®usion con-
stant, Dk, due to the k¡th noise is given by [22]

Dk = 1 ¢ i2n;k ¢ pk2(t) (8)
4(C V0 )2 ¢f

consistent agreement between the two phase noise measure- where the overline signi¯es a time average. Note that the
ment methods, validating the virtual damping concept.

in2 =¢f measured PN from measured time-variance represented by pk(t) a®ects the di®usion rate
(A2 =H z ) measured D PN by modifying the average value pk2(t).
D
2:6 £ 10¡15 (sec¡1) (dBc/Hz) (dBc/Hz) The foregoing argument only dealt with a single, k¡th,
4:8 £ 10¡15
9:7 £ 10¡15 1:02 £ 104 -92.9 -93.0 white noise source, in;k(t). In the presence of multiple un-
2:1 £ 10¡14 1:56 £ 104 -91.0 -90.0
6:0 £ 10¡14 3:53 £ 104 -87.4 -86.5 correlated white noise sources as shown in Fig. 1, it can be
9:30 £ 104 -83.3 -81.7
1:90 £ 105 -80.2 -79.5 easily shown that the overall di®usion constant, D, is sum of

all the di®usion constants due to the multiple noise sources.

If there are total M uncorrelated Dno=isePsokMu=r1cDesk, the overall
di®usion constant, D, is given by :

D = 1 XM i2n;k ¢ pk2 (t) (9)
4(C V0 )2 ¢f
Table 2. Measured D, PN calculated from the measured D, and PN k=1
measured using a spectrum analyzer at di®erent injected noise power
levels. The o®set frequency used was 1 MHz. The following section is devoted to physical interpretation
of (9).

the white noise intensity of the k-th noise current as

in2 ;k = 4kBT (11)
¢f Rk

Figure 9: Brownian motion. where Rk is the e®ective equivalent resistance associated
with the k-th noise current. If the noise is of thermal origin,
2.5. Einstein Relation in Phase Noise Rk represents the loss associated with the thermal noise ac-
cording to the °uctuation-dissipation theorem [20]. For in-
The key to a meaningful interpretation of the phase di®usion stance, in the case of an ohmic resistor, R, such as the tank
constant, D, is to note that the rate of any di®usion process loss in Fig. 1, Rk simply becomes R and (11) is no more
is determined by two essential elements a®ecting the pro- than the Johnson-Nyquist formula for thermal noise. On
cess: the sensitivity of the physical quantity undergoing the
di®usion and the friction (energy loss) of the environment the other hand, for non-thermal noise (e.g., shot noise), Rk
in which the di®usion process occurs. To see this clearly let does not necessarily represent the loss since the °uctuation-
us digress for one paragraph and discuss Brownian motion, dissipation relation only holds true for the thermal noise
a typical di®usion process. sources. In this case, Rk can be simply thought of as an
alternative measure of the noise intensity in the form of re-
A Brownian particle of mass, m, immersed in a liquid at sistance.
temperature, T , with frictional coe±cient of ° shown in Fig.
9 experiences di®usion, due to incessant bombardment by By plugging (11) into (9) and ignoring the time-varying
thermally agitated liquid molecules [20]. When the random e®ects for simplicity, we obtain
force exerted by the liquid molecules has a white spectrum,
the displacement, x, of the Brownian particle satis¯es hx2i = D ¼ 1 ¢ kB T ¢ !0 (12)
2Dt, where D is the di®usion constant. This is analogous to V02 | {Cz } |Q{ezf f}
phase di®usion described by (2). The di®usion constant of
the Brownian particle is given by the Einstein relation [20]: sensitivity loss/noise

Here, Qeff , de¯ned by the authors as (13)
Qeff ´ (R1jjR2jj ¢ ¢ ¢ jjRM )C!0

D = kBT ¢ 1 (10) is not a conventional quality factor. Note, in conjunction
| {mz } |{°z}
with (11), that Qeff de¯ned above is a direct measure of the
sensitivity friction amount of noise in the oscillator as it includes every noise

where kB is the Boltzmann constant. The kBT =m factor source in the circuit. If every noise source is of thermal ori-
represents the sensitivity of the Brownian particle to pertur-
bations and becomes smaller for a heavier particle, agreeing gin, every corresponding e®ective equivalent resistance, Rk,
with our intuition. Fundamentally, this sensitivity factor is a loss element and Qeff becomes a conventional quality
derives from the equipartition theorem stating that each in- factor. If there exist non-thermal noise sources such as shot
dependent degree of freedom of a system in equilibrium at
temperature T has a mean energy of kBT =2 [20], that is, noise, the corresponding e®ective equivalent resistances are
hmv2=2i = kBT =2, or hv2i = kBT =m where v is the velocity
of the Brownian particle. On the other hand, for a given not loss elements, making Qeff di®erent from the conven-
Brownian particle (¯xed m), a medium with higher friction tional quality factor.
will exhibit a slower di®usion as we can guess intuitively,
and hence the second factor, 1=°, in (10), originating from Now we will explain D in (12) using the Einstein rela-
the friction (energy loss) of the environment. Summarizing, tion. In (12), the kBT =C factor represents the sensitivity
the Einstein relation suggests that di®usion constant can be of the tank, analogous to the kBT =m factor of the Brown-
determined only when both sensitivity and friction (energy ian particle in (10). This sensitivity factor can be obtained
loss) elements are known.
resorting to the equipartition theorem [20] again, that is,
The phase di®usion in the oscillator is analogous to the hCV 2=2i = kBT =2 or, hV 2i = kBT =C. At the same time,
di®usion of the Brownian particle and hence the phase dif- the !0=Qeff factor in (12) is analogous to the 1=° factor
fusion constant can be also explained using the Einstein re- in (10) for the Brownian motion. If all the noise sources
lation. To this end, we recast the expression for the phase
di®usion constant in (9) into a di®erent form by expressing are of thermal origin, Qeff is a measure of energy loss in
the circuit due to the connection between thermal noise and

loss (°uctuation-dissipation theorem). If some of the noise

sources are of non-thermal origin, Qeff can be interpreted
as a measure of the amount of noise in the circuit. In either

case, !0=Qeff in (12) consists of noise and/or loss elements
corresponding to friction while kBT =C is the sensitivity fac-
tor, hence demonstrating the direct correspondence between

the Einstein relation and the overall phase di®usion con-
stant. The additional factor, 1=V02, in (12) is simply to
convert the di®usion on the limit cycle to the di®usion in

the phase angle.

2.6. Derivation of Leeson's Model These losses account for the low Q of the monolithic spiral
inductor.
Leeson's phase noise model [6] is by far the most cited phase
There are several techniques available for mitigating the
noise model, and hence, it will be worthwhile to see that the losses in the spiral inductor to improve the inductor Q. At
the fabrication level, relatively thick metal layers (typically
phase noise model reviewed in this section is commensurate around 3 ¹m) can be used to reduce the skin e®ect. Larger
distance between the lossy silicon substrate and the metal
with Leeson's model. By using the de¯nition of Q, that is, layer comprising the spiral inductor is preferred to reduce
Q = !0Etank=Ps = !0CV02=(2Ps) where Ps is the power the substrate e®ect. Change of substrate material [24] or
dissipation in the resistive part of the resonator, we can removal of the underlying lossy silicon substrate [25] reduce
the substrate loss, but they are costly or unavailable with
rewrite (12) as standard CMOS processes.

D= Q ¢ kB T ¢ !02 (14) At the design level for a given process technology, place-
Qef f 2Ps Q2 ment of metal strips or patterned ground shields under the
spiral inductor is one way of reducing the substrate loss
By using this in (5), one obtains where the strips prevent to a reasonable degree the inter-
action between the inductor and the lossy substrate [26].
2Q µ !0 ¶2 Strips are used in lieu of a solid plane to prevent Eddy cur-
¢ kBT ¢ rent circulation. While the substrate shielding using metal
Lf¢!g = Qeff 2Ps Q¢! (15) strips often o®ers an attractive solution to reduce the sub-
strate loss, this technique has a limitation. The underly-
which is Leeson's model [6] where we 2Q=Qeff accounts for ing metal strips decrease the self-resonance frequency of the
the notorious ¯tting factor, F , in Leeson's model. inductor due to the increased parasitic capacitance, corre-
sponding to lower energy stored in the spiral inductor. Con-
3. PHASE NOISE REDUCTION TECHNIQUES sidering that Q » Estored=Pdissipated, the metal strips de-
crease the power dissipation but decrease the stored energy
As can be seen from the previous section, the phase di®usion as well. So overall, Q cannot be enhanced substantially [26].
constant is determined by various factors. The phase noise
reduction can be done by in°uencing these factors. This It is worthwhile also to mention MEMS inductors re-
section reviews some of the currently available phase noise ported recently, e.g., [27] - [29]. By using micromachining
reduction techniques. Speci¯cally, Subsec. 3.1 discusses the and/or other 3-D assembly processes as a post-fabrication
enhancement of the resonator quality factor, Q, the most step, one can tilt the planar inductor away from the sub-
general technique for phase noise reduction while Subsec. strate surface (vertical coil) or substantially increase the dis-
3.2 presents the minimization of the noise-to-carrier ratio tance between the planar inductor and the substrate (over-
(NCR) as another method of lowering phase noise. Subsec- hang coil), leading to a much reduced substrate loss and
tion 3.3 presents a case where the reduction of active device hence greater Q in the range of 30 to 70. But again this
noise can signi¯cantly lower phase noise while Subsec. 3.4 improvement comes with a technology which is not cur-
discusses a case where certain exploitation of time variance rently available with standard CMOS processes, requiring
of oscillators can reduce phase noise. Finally Subsec. 3.5 post processing steps.
discusses coupled oscillators and resultant phase noise re-
duction. 3.1.2. E®ective Q enhancement in wave-based resonators

3.1. Q Enhancement Our latest development [30] shows that while the intrinsic
Q of a resonator is limited by the material's loss charac-
Enhancement of Q is the most general technique for phase teristics, it is possible to raise the e®ective Q beyond the
noise reduction and has been one of the primary focuses intrinsic Q in the case of transmission-line or waveguide res-
in oscillator design. In this subsection, we brie°y discuss onators via exploitation of wave properties. Two of the au-
intrinsic Q optimization of a spiral inductor and e®ective thors have recently demonstrated in [30] that e®ective Q of
Q optimization in wave-based resonators via exploitation of a transmission line hosting standing waves can be enhanced
wave properties, where the latter is the most recent devel- by tapering the transmission line such that it is adapted
opment by two of the authors at Harvard University [30]. to the position-dependent standing wave amplitudes. This
tapered structure may be used in standing wave oscillators
3.1.1. Intrinsic Q enhancement in spiral inductors to lower phase noise. A detailed treatment can be found in
[30].
The Q of monolithic planar spiral inductors in silicon tends
to be quite low, typically in the range of 5 to 20, limit-
ing the overall Q of monolithic LC tanks. This is because
the physical constraints in a standard CMOS planar induc-
tor construction introduce two major sources of loss in the
spiral inductor [23]. The thin metal layers comprising the
spiral inductors introduce loss through skin e®ects. The
interaction between ¯elds generated by the spiral inductor
and the underlying lossy substrate is another source of loss.

3.2. NCR Minimization resonator oscillator
LC R
The noise-to-carrier ratio (NCR) in the LC oscillator of Fig. LC R active
1 is de¯ned as [21] device

Ethermal kB T in,1 in,M
Etank C V02
NCR = = (16)

where the thermal and tank energies are given by Ethermal = damping rate : 1/(2RC) V(t) virtual damping rate : D
kBT =2 and Etank = CV02=2, respectively. The minimization t t
of this NCR o®ers another way to reduce oscillator phase
noise. To see this, we introduce a concept of linewidth com- energy phase noise
pression based on the virtual damping concept discussed in spectrum
Subsec. 2.3. Dosc
D res
The left-hand side of Fig. 10 shows a parallel LC res- w0
onator. The energy loss due to the parasitic resistance, R, w0 w w
causes the voltage across the tank to damp exponentially
from a given initial value as shown in the ¯gure. This damp- D res ~ w0 linewidth
ing in the time-domain corresponds to the line broadening Q compression by
in the energy spectrum in the frequency-domain. ~ 1 kB T Q
V02 C Qeff
The right-hand side of Fig. 10 shows an oscillator de-
rived from the same LC resonator by placing it in a positive NCR
feedback loop. The phase di®usion due to the active and
passive device noise is responsible for the line broadening of Figure 10: Linewidth compression.
oscillator's power spectrum (phase noise) as discussed ear-
lier. Since the virtual damping is another manifestation of to high-Q resonator selection discussed in Subsec. 3.1. The
the phase di®usion, the line broadening in oscillator's power second optimization step is to achieve the highest possible
spectrum can be alternatively thought of as the result of linewidth compression by minimizing r in (17). As can be
the exponential virtual damping, and henceforth, the vir- seen in (17), r strongly depends on the NCR, and hence the
tual damping rate in the time-domain corresponds to the maximum linewidth compression in the second step may be
spectrum linewidth in the frequency domain. The virtual achieved through the minimization of the NCR. This is the
damping is extremely slow relative to the oscillation fre- rationale for the NCR minimization to achieve low phase
quency as discussed earlier and hence it is almost always noise.
much slower than the damping in the resonator. Corre-
spondingly, the linewidth of the oscillator's output spectrum This NCR minimization is typically exercised through
is much smaller than the linewidth of the resonator's energy the oscillation amplitude (V0) maximization but there have
spectrum, as shown hypothetically in Fig. 10. recently been emphases on the more proper tank energy
maximization (e.g., [21]). As can be seen from (16), what
In conclusion, placing a resonator in a positive feedback actually matters to reduce the NCR is not V0 alone but
loop to make a self-sustained oscillator results in linewidth Etank determined by both C and V0. If C is already ¯xed,
compression. The ratio of the oscillator's virtual damping every e®ort should be made to maximize V0 in the design. If
rate, D, to the resonator's damping rate, 1=(2RC), is the both C and V0 are the design variables, for a given tank en-
measure of this linewidth compression ratio: ergy, increasing C and thus decreasing V0 does not worsen
the NCR. Therefore, if increasing C can somehow bene¯t
r ´ ¢osc = D = 2Q ¢D factors other than the compression ratio such as original
¢res 1=(2RC) !0 tank Q, C must be increased (and hence V0 must be de-
creased) to improve the phase noise [21].
» 1 ¢ kB T ¢ Q
V02 C Qef f 3.3. Active Device Noise Reduction

= [NCR] ¢ Q (17) Phase noise can be decreased by reducing device noise in
Qef f oscillators as can be seen from (9). Due to the °uctuation-
dissipation theorem, passive device noise reduction is di-
where we have used (12) to obtain the second line and (16) rectly related to the Q maximization already discussed in
to obtain the third line while Q = RC!0 is the quality factor Subsec. 3.1. This subsection brie°y discusses an interest-
of the LC tank.

The linewidth compression concept with the aid of Fig.
10 elucidates that the phase noise of the resonator-based
oscillators can be minimized by a two-step procedure. The
¯rst step is to select a resonator with the narrowest pos-
sible linewidth. Since the resonator's linewidth is ¢res »
1=(2RC) = !0=2Q, this ¯rst optimization step corresponds

Vsupply absolute phase noise and not phase noise per mW supplied
Vcont power this may provide a signi¯cant bene¯t.

Cload Cload 4. CONCLUSION

passive LC This paper reviewed phase noise fundamentals paying spe-
filter cial attention to a simple, yet essential physical picture of
oscillator phase noise based on the phase di®usion concept.
Vbias Several insightful concepts such as virtual damping, Ein-
stein relation in phase noise, and linewidth compression were
Figure 11: A ¯ltering technique to lower LC oscillator phase noise. emphasized to elucidate the fundamental mechanism of the
phase noise phenomenon. The paper also summarized sev-
[31] eral recently developed phase noise reduction techniques,
providing a link between the physical phase noise theory
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tion (hence active device noise reduction) lowers phase noise.
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