IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 33, NO. 2, FEBRUARY 1998 179
A General Theory of Phase Noise
in Electrical Oscillators
Ali Hajimiri, Student Member, IEEE, and Thomas H. Lee, Member, IEEE
Abstract— A general model is introduced which is capable
of making accurate, quantitative predictions about the phase
noise of different types of electrical oscillators by acknowledging
the true periodically time-varying nature of all oscillators. This
new approach also elucidates several previously unknown design
criteria for reducing close-in phase noise by identifying the mech-
anisms by which intrinsic device noise and external noise sources
contribute to the total phase noise. In particular, it explains the
details of how
1
=f
noise in a device upconverts into close-in
phase noise and identifies methods to suppress this upconversion.
The theory also naturally accommodates cyclostationary noise
sources, leading to additional important design insights. The
model reduces to previously available phase noise models as
special cases. Excellent agreement among theory, simulations, and
measurements is observed.
Index Terms—Jitter, oscillator noise, oscillators, oscillator sta-
bility, phase jitter, phase locked loops, phase noise, voltage
controlled oscillators.
I. INTRODUCTION
T
HE recent exponential growth in wireless communication
has increased the demand for more available channels in
mobile communication applications. In turn, this demand has
imposed more stringent requirements on the phase noise of
local oscillators. Even in the digital world, phase noise in the
guise of jitter is important. Clock jitter directly affects timing
margins and hence limits system performance.
Phase and frequency fluctuations have therefore been the
subject of numerous studies [1]–[9]. Although many models
have been developed for different types of oscillators, each
of these models makes restrictive assumptions applicable only
to a limited class of oscillators. Most of these models are
based on a linear time invariant (LTI) system assumption
and suffer from not considering the complete mechanism by
which electrical noise sources, such as device noise, become
phase noise. In particular, they take an empirical approach in
describing the upconversion of low frequency noise sources,
such as
noise, into close-in phase noise. These models
are also reduced-order models and are therefore incapable of
making accurate predictions about phase noise in long ring
oscillators, or in oscillators that contain essential singularities,
such as delay elements.
Manuscript received December 17, 1996; revised July 9, 1997.
The authors are with the Center for Integrated Systems, Stanford University,
Stanford, CA 94305-4070 USA.
Publisher Item Identifier S 0018-9200(98)00716-1.
Since any oscillator is a periodically time-varying system,
its time-varying nature must be taken into account to permit
accurate modeling of phase noise. Unlike models that assume
linearity and time-invariance, the time-variant model presented
here is capable of proper assessment of the effects on phase
noise of both stationary and even of cyclostationary noise
sources.
Noise sources in the circuit can be divided into two groups,
namely, device noise and interference. Thermal, shot, and
flicker noise are examples of the former, while substrate and
supply noise are in the latter group. This model explains
the exact mechanism by which spurious sources, random
or deterministic, are converted into phase and amplitude
variations, and includes previous models as special limiting
cases.
This time-variant model makes explicit predictions of the
relationship between waveform shape and
noise upcon-
version. Contrary to widely held beliefs, it will be shown
that the
corner in the phase noise spectrum is smaller
than
noise corner of the oscillator’s components by a
factor determined by the symmetry properties of the waveform.
This result is particularly important in CMOS RF applications
because it shows that the effect of inferior
device noise
can be reduced by proper design.
Section II is a brief introduction to some of the existing
phase noise models. Section III introduces the time-variant
model through an impulse response approach for the excess
phase of an oscillator. It also shows the mechanism by which
noise at different frequencies can become phase noise and
expresses with a simple relation the sideband power due to
an arbitrary source (random or deterministic). It continues
with explaining how this approach naturally lends itself to the
analysis of cyclostationary noise sources. It also introduces
a general method to calculate the total phase noise of an
oscillator with multiple nodes and multiple noise sources, and
how this method can help designers to spot the dominant
source of phase noise degradation in the circuit. It concludes
with a demonstration of how the presented model reduces
to existing models as special cases. Section IV gives new
design implications arising from this theory in the form of
guidelines for low phase noise design. Section V concludes
with experimental results supporting the theory.
II. B
RIEF REVIEW OF EXISTING MODELS AND DEFINITIONS
The output of an ideal sinusoidal oscillator may be ex-
pressed as
, where is the amplitude,
0018–9200/98$10.00 1998 IEEE
180 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 33, NO. 2, FEBRUARY 1998
Fig. 1. Typical plot of the phase noise of an oscillator versus offset from
carrier.
is the frequency, and is an arbitrary, fixed phase refer-
ence. Therefore, the spectrum of an ideal oscillator with no
random fluctuations is a pair of impulses at
. In a practical
oscillator, however, the output is more generally given by
(1)
where
and are now functions of time and is a
periodic function with period 2
. As a consequence of the
fluctuations represented by
and , the spectrum of a
practical oscillator has sidebands close to the frequency of
oscillation,
.
There are many ways of quantifying these fluctuations (a
comprehensive review of different standards and measurement
methods is given in [4]). A signal’s short-term instabilities are
usually characterized in terms of the single sideband noise
spectral density. It has units of decibels below the carrier per
hertz (dBc/Hz) and is defined as
1Hz
(2)
where
1Hz represents the single side-
band power at a frequency offset of
from the carrier with a
measurement bandwidth of 1 Hz. Note that the above definition
includes the effect of both amplitude and phase fluctuations,
and .
The advantage of this parameter is its ease of measurement.
Its disadvantage is that it shows the sum of both amplitude and
phase variations; it does not show them separately. However, it
is important to know the amplitude and phase noise separately
because they behave differently in the circuit. For instance,
the effect of amplitude noise is reduced by amplitude limiting
mechanism and can be practically eliminated by the applica-
tion of a limiter to the output signal, while the phase noise
cannot be reduced in the same manner. Therefore, in most
applications,
is dominated by its phase portion,
, known as phase noise, which we will simply
denote as
.
Fig. 2. A typical RLC oscillator.
The semi-empirical model proposed in [1]–[3], known also
as the Leeson–Cutler phase noise model, is based on an LTI
assumption for tuned tank oscillators. It predicts the following
behavior for
:
(3)
where
is an empirical parameter (often called the “device
excess noise number”),
is Boltzmann’s constant, is the
absolute temperature,
is the average power dissipated in
the resistive part of the tank,
is the oscillation frequency,
is the effective quality factor of the tank with all the
loadings in place (also known as loaded
), is the offset
from the carrier and
is the frequency of the corner
between the
and regions, as shown in the sideband
spectrum of Fig. 1. The behavior in the
region can be
obtained by applying a transfer function approach as follows.
The impedance of a parallel RLC, for
, is easily
calculated to be
(4)
where
is the parallel parasitic conductance of the tank.
For steady-state oscillation, the equation
should
be satisfied. Therefore, for a parallel current source, the closed-
loop transfer function of the oscillator shown in Fig. 2 is given
by the imaginary part of the impedance
(5)
The total equivalent parallel resistance of the tank has an
equivalent mean square noise current density of
. In addition, active device noise usually contributes
a significant portion of the total noise in the oscillator. It is
traditional to combine all the noise sources into one effective
noise source, expressed in terms of the resistor noise with
HAJIMIRI AND LEE: GENERAL THEORY OF PHASE NOISE IN ELECTRICAL OSCILLATORS 181
Fig. 3. Phase and amplitude impulse response model.
a multiplicative factor, , known as the device excess noise
number. The equivalent mean square noise current density can
therefore be expressed as
. Unfortunately,
it is generally difficult to calculate
a priori. One important
reason is that much of the noise in a practical oscillator
arises from periodically varying processes and is therefore
cyclostationary. Hence, as mentioned in [3],
and are
usually used as a posteriori fitting parameters on measured
data.
Using the above effective noise current power, the phase
noise in the
region of the spectrum can be calculated as
(6)
Note that the factor of 1/2 arises from neglecting the con-
tribution of amplitude noise. Although the expression for the
noise in the
region is thus easily obtained, the expression
for the
portion of the phase noise is completely empirical.
As such, the common assumption that the
corner of the
phase noise is the same as the
corner of device flicker
noise has no theoretical basis.
The above approach may be extended by identifying the
individual noise sources in the tuned tank oscillator of Fig. 2
[8]. An LTI approach is used and there is an embedded
assumption of no amplitude limiting, contrary to most practical
cases. For the RLC circuit of Fig. 2, [8] predicts the following:
(7)
where
is yet another empirical fitting parameter, and
is the effective series resistance, given by
(8)
where
, , , and are shown in Fig. 2. Note that it
is still not clear how to calculate
from circuit parameters.
Hence, this approach represents no fundamental improvement
over the method outlined in [3].
(a) (b)
(c)
Fig. 4. (a) Impulse injected at the peak, (b) impulse injected at the zero
crossing, and (c) effect of nonlinearity on amplitude and phase of the oscillator
in state-space.
III. MODELING OF PHASE NOISE
A. Impulse Response Model for Excess Phase
An oscillator can be modeled as a system with
inputs
(each associated with one noise source) and two outputs
that are the instantaneous amplitude and excess phase of the
oscillator,
and , as defined by (1). Noise inputs to this
system are in the form of current sources injecting into circuit
nodes and voltage sources in series with circuit branches. For
each input source, both systems can be viewed as single-
input, single-output systems. The time and frequency-domain
fluctuations of
and can be studied by characterizing
the behavior of two equivalent systems shown in Fig. 3.
Note that both systems shown in Fig. 3 are time variant.
Consider the specific example of an ideal parallel LC oscillator
shown in Fig. 4. If we inject a current impulse
as shown,
the amplitude and phase of the oscillator will have responses
similar to that shown in Fig. 4(a) and (b). The instantaneous
voltage change
is given by
(9)
where
is the total injected charge due to the current
impulse and
is the total capacitance at that node. Note
that the current impulse will change only the voltage across the
182 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 33, NO. 2, FEBRUARY 1998
(a) (b)
Fig. 5. (a) A typical Colpitts oscillator and (b) a five-stage minimum size
ring oscillator.
capacitor and will not affect the current through the inductor.
It can be seen from Fig. 4 that the resultant change in
and
is time dependent. In particular, if the impulse is applied
at the peak of the voltage across the capacitor, there will be no
phase shift and only an amplitude change will result, as shown
in Fig. 4(a). On the other hand, if this impulse is applied at the
zero crossing, it has the maximum effect on the excess phase
and the minimum effect on the amplitude, as depicted in
Fig. 4(b). This time dependence can also be observed in the
state-space trajectory shown in Fig. 4(c). Applying an impulse
at the peak is equivalent to a sudden jump in voltage at point
, which results in no phase change and changes only the
amplitude, while applying an impulse at point
results only
in a phase change without affecting the amplitude. An impulse
applied sometime between these two extremes will result in
both amplitude and phase changes.
There is an important difference between the phase and
amplitude responses of any real oscillator, because some
form of amplitude limiting mechanism is essential for stable
oscillatory action. The effect of this limiting mechanism is
pictured as a closed trajectory in the state-space portrait of
the oscillator shown in Fig. 4(c). The system state will finally
approach this trajectory, called a limit cycle, irrespective of
its starting point [10]–[12]. Both an explicit automatic gain
control (AGC) and the intrinsic nonlinearity of the devices
act similarly to produce a stable limit cycle. However, any
fluctuation in the phase of the oscillation persists indefinitely,
with a current noise impulse resulting in a step change in
phase, as shown in Fig. 3. It is important to note that regardless
of how small the injected charge, the oscillator remains time
variant.
Having established the essential time-variant nature of the
systems of Fig. 3, we now show that they may be treated as
linear for all practical purposes, so that their impulse responses
and will characterize them completely.
The linearity assumption can be verified by injecting im-
pulses with different areas (charges) and measuring the resul-
tant phase change. This is done in the SPICE simulations of
the 62-MHz Colpitts oscillator shown in Fig. 5(a) and the five-
stage 1.01-GHz, 0.8-
m CMOS inverter chain ring oscillator
shown in Fig. 5(b). The results are shown in Fig. 6(a) and (b),
respectively. The impulse is applied close to a zero crossing,
(a) (b)
Fig. 6. Phase shift versus injected charge for oscillators of Fig. 5(a) and (b).
where it has the maximum effect on phase. As can be seen, the
current-phase relation is linear for values of charge up to 10%
of the total charge on the effective capacitance of the node
of interest. Also note that the effective injected charges due
to actual noise and interference sources in practical circuits
are several orders of magnitude smaller than the amounts of
charge injected in Fig. 6. Thus, the assumption of linearity is
well satisfied in all practical oscillators.
It is critical to note that the current-to-phase transfer func-
tion is practically linear even though the active elements may
have strongly nonlinear voltage-current behavior. However,
the nonlinearity of the circuit elements defines the shape of
the limit cycle and has an important influence on phase noise
that will be accounted for shortly.
We have thus far demonstrated linearity, with the amount
of excess phase proportional to the ratio of the injected charge
to the maximum charge swing across the capacitor on the
node, i.e.,
. Furthermore, as discussed earlier, the
impulse response for the first system of Fig. 3 is a step whose
amplitude depends periodically on the time
when the impulse
is injected. Therefore, the unit impulse response for excess
phase can be expressed as
(10)
where
is the maximum charge displacement across the
capacitor on the node and
is the unit step. We call
the impulse sensitivity function (ISF). It is a dimensionless,
frequency- and amplitude-independent periodic function with
period 2
which describes how much phase shift results from
applying a unit impulse at time
. To illustrate its
significance, the ISF’s together with the oscillation waveforms
for a typical LC and ring oscillator are shown in Fig. 7. As is
shown in the Appendix,
is a function of the waveform
or, equivalently, the shape of the limit cycle which, in turn, is
governed by the nonlinearity and the topology of the oscillator.
Given the ISF, the output excess phase
can be calcu-
lated using the superposition integral
(11)
HAJIMIRI AND LEE: GENERAL THEORY OF PHASE NOISE IN ELECTRICAL OSCILLATORS 183
(a) (b)
Fig. 7. Waveforms and ISF’s for (a) a typical LC oscillator and (b) a typical
ring oscillator.
where represents the input noise current injected into the
node of interest. Since the ISF is periodic, it can be expanded
in a Fourier series
(12)
where the coefficients
are real-valued coefficients, and
is the phase of the th harmonic. As will be seen later,
is not important for random input noise and is thus
neglected here. Using the above expansion for
in the
superposition integral, and exchanging the order of summation
and integration, we obtain
(13)
Equation (13) allows computation of
for an arbitrary input
current
injected into any circuit node, once the various
Fourier coefficients of the ISF have been found.
As an illustrative special case, suppose that we inject a low
frequency sinusoidal perturbation current
into the node of
interest at a frequency of
(14)
where
is the maximum amplitude of . The arguments
of all the integrals in (13) are at frequencies higher than
and are significantly attenuated by the averaging nature of
the integration, except the term arising from the first integral,
which involves
. Therefore, the only significant term in
will be
(15)
As a result, there will be two impulses at
in the power
spectral density of
, denoted as .
As an important second special case, consider a current at a
frequency close to the carrier injected into the node of interest,
given by
. A process similar to that
of the previous case occurs except that the spectrum of
Fig. 8. Conversion of the noise around integer multiples of the oscillation
frequency into phase noise.
consists of two impulses at as shown in Fig. 8.
This time the only integral in (13) which will have a low
frequency argument is for
. Therefore is given by
(16)
which again results in two equal sidebands at
in .
More generally, (13) suggests that applying a current
close to any integer multiple of the
oscillation frequency will result in two equal sidebands at
in . Hence, in the general case is given by
(17)
B. Phase-to-Voltage Transformation
So far, we have presented a method for determining how
much phase error results from a given current
using (13).
Computing the power spectral density (PSD) of the oscillator
output voltage
requires knowledge of how the output
voltage relates to the excess phase variations. As shown in
Fig. 8, the conversion of device noise current to output voltage
may be treated as the result of a cascade of two processes.
The first corresponds to a linear time variant (LTV) current-
to-phase converter discussed above, while the second is a
nonlinear system that represents a phase modulation (PM),
which transforms phase to voltage. To obtain the sideband
power around the fundamental frequency, the fundamental
harmonic of the oscillator output
can be used
as the transfer function for the second system in Fig. 8. Note
this is a nonlinear transfer function with
as the input.
Substituting
from (17) into (1) results in a single-tone
phase modulation for output voltage, with
given by (17).
Therefore, an injected current at
results in a pair
of equal sidebands at
with a sideband power relative
to the carrier given by
(18)
