312
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 2, FEBRUARY 2011
Theory of Injection-Locked Oscillator Phase Noise
Torsten Djurhuus and Viktor Krozer, Senior Member, IEEE
Abstract—The paper describes the development of a model
for the calculation of noise-driven phase response of an injection-locked oscillator perturbed by Gaussian white sources. Being
based on the state space formalism the framework is unified
encompassing all circuit topologies and methods of unilateral
coupling. We thus avoid reverting to the kind of simplified
block-diagram description that one finds in previously published
works on the topic and our approach furthermore allows for all
the main results and model parameters to be derived numerically
based on the netlist description of the circuit. To our knowledge
this constitutes the first attempt at an ILO phase-noise description
not relying on block diagrams or other such phenomenological
modelling strategies.
Index Terms—Injection-locked oscillator, injection locking, oscillator, phase noise, noise, phase macro model.
I. INTRODUCTION
T
HE term injection-locked oscillator (ILO) refers to a
circuit where two asymptotically stable oscillators are
connected unilaterally, with the coupling signal flowing from
the master/reference oscillator (M-OSC) to the slave/local
oscillator (S-OSC). Then, if the difference in operating frequency of the two oscillators is on the order of the coupling,
a so-called saddle-node bifurcation occurs on the slave limit
cycle entraining or locking it to the master. The synchronized
state is hence reached through very weak interaction making
the approach a power efficient alternative to more elaborate
phase-locked systems.
Injection-locking techniques are found frequently in RF and
optical circuit architectures where they are used to implement
low-power frequency multipliers/dividers [1], [2] or as an alternative to a full PLL structure [3], which is often a costly and
power-expensive way to realize synthesizers at RF frequencies.
Injection locking is also used in phase-noise measuring equipment [4]. Furthermore, injection of a low noise reference can be
used as a method of cleaning the phase of a noisy carrier [5]–[9].
This property is explained heuristically as the master imprinting
its pattern on the slave which then inherits all of its frequency
related properties including jitter.
Earlier attempts at an ILO noise description have all been restricted to a specific oscillator class or architecture [5], [8], [10],
[1], [7], [11] with the circuit often modelled via a block-diagram
Manuscript received January 22, 2010; revised April 28, 2010; accepted June
12, 2010. Date of publication October 18, 2010; date of current version January
28, 2011. This paper was recommended by Associate Editor A. Tasic
T. Djurhuus is with the Goethe University of Frankfurt am Main, Max-vonLaue-Strasse 1, 60438 Frankfurt am Main, Germany.
V. Krozer is with the Goethe University of Frankfurt am Main,
Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany. (e-mail:
td_djurhuus@hotmail.com).
Digital Object Identifier 10.1109/TCSI.2010.2071770
structure or some other kind of phenomenological description.
Accordingly, these models will contain a considerable number
of parameters such as, e.g., resonator quality factor, frequency/
phase offset, amplitude saturation coefficients etc. Being inherent to the model these parameters are not always easily calculated, numerically or otherwise, as this would imply a reasonably good fit of a possibly complex circuit topology into the
rather limited framework constituted by a block-diagram type
model.
In this paper we develop a Floquet decomposition of the ILO
linear response starting from a state-space description of the oscillators and coupling circuits with no restriction on the equations except that they are smooth. This approach encompasses
oscillators belonging to all classes (i.e., harmonic, relaxation,
ring etc.) and allows for any circuit topology. Furthermore, the
coupling can take any form and the coupling signal can flow in
any way between the two oscillators as long the interaction is
unilateral. Unlike the phenomenological block-diagram models
there is no fitting of the circuit into a certain template and we
are ensured that all parameters can be derived numerically once
the steady-state has been calculated. The state space methodology presented here builds upon the earlier works on single
oscillator phase noise developed in [12] and [13]. In the single
oscillator case the division between phase and amplitude was
clear-cut; however, in the case of coupled oscillators the line is
more blurred. A basic understanding of the state-space geometry is needed in order to identify the modes representing the
phase dynamics of the circuit. Like its single oscillator predecessors, the model discussed here is set in the time domain but
we note that it is possible to transform the Floquet formalism
into the frequency-domain (see, e.g., [13]).
The paper [5] is an example of a frequency domain conversion
matrix approach to the problem of ILO noise response where
the oscillators are modelled as quasi-sinusoidal signal sources.
It should be clear that these types of models are limited to highly
sinusoidal oscillators. Also, it is worth noting that a numerical
scheme based on the conversion matrix methodology will result
in a singular spectrum close to the carrier proportional to
, as this algorithm fails to capture the neutrally stable
nature of the limit cycle dynamics. Our model, being formulated
in the time domain, does not suffer from this deficiency. In [14]
and [15] a time-domain noise characterization of phase locked
loops (PLL) is discussed. The models described in these papers,
fix the topology of the circuit from the outset by demanding that
it fit into a certain PLL template (i.e., filter, VCO, etc.). Again,
we propose a model free of any templates or block-diagrams.
The aim of the text is to document the development of a unified theory for predicting the noise forced phase dynamics of an
injection-locked oscillator and to show how this general framework relates to the earlier models on the subject. The model is
developed to the point where it should be possible to translate
1549-8328/$26.00 © 2010 IEEE
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
DJURHUUS AND KROZER: THEORY OF INJECTION-LOCKED OSCILLATOR PHASE NOISE
the equations into a useful CAD algorithm/subroutine. However, several minor issues still need to be worked out and while
this subject is interesting to us we do not intend to discuss it further here.
The following text introduces the ILO state-space description
which serves as our underlying model of the circuit. We then
proceed in Section III to introduce and justify our novel definition of ILO phase noise. The text in that section will also serve
as a detailed introduction to the rest of the paper.
II. THE STATE SPACE MODEL
This paper considers a state-space model of an autonomous
circuit where time evolution is described in terms of a smooth
-dimensional first-order ordinary differential equation (ODE)
1 with
referred to as the state-vector.
The diagram in Fig. 1 illustrates in symbolic terms the
main constituents of any injection-locked oscillator circuit
implementation. Two oscillators, the master (M-OSC) and the
slave (S-OSC), are coupled through some-kind of (close-to)
unilateral intermediary. This could be a direct coupling approach entailing an actual buffer/amplifier circuit [7], [16], a
directional coupler/combiner [3], a passive circulator [17], [18]
or an indirect approach such as, e.g., coupling via a bias point
[19], [1], [20]2 . The important point here is that somewhere in
the circuit isolation has to be introduced such that signal energy
is restricted to flow in one direction only (to a good approximation). If this was not true the circuit would not work properly as
an ILO. Note that the theory described here allows for coupling
to be introduced at more than one point and the buffer symbol
in Fig. 1 represents a close-to unilateral multiport.
We choose a coupling point, or points, where the bilateral path
is weak (at the operating frequency). The state space is divided
into M-OSC and S-OSC states and the ILO steady-state solution
is written3
(1)
and
being the
with
and
refer to the
M-OSC and S-OSC state vectors where
state dimension of the M-OSC and S-OSC subcircuits, respec. In order to reach (1) all states
tively; implying that
belonging to the bilateral coupling paths were included on the
S-OSC side. Using the split in (1) the dynamical equations are
written
(2)
(3)
1Note that, except for very special cases, there exist a 1-1 map from the
differential-algebraic equation (DAE) formulation used in CAD programs like
SPICE, ADS and SPECTRE to the ODE formulation employed here.
2In the case where isolation is performed by the S-OSC bias circuit the buffer
in Fig. 1 should not be seen as physical circuit but instead simply as a representation of the signal isolation which exist between the two circuit blocks.
3Throughout the paper we shall use the notation
to denote the transpose
of the real vector/matrix . If
is complex then
refers to the transpose
and complex conjugated vector/matrix. Throughout, an expression of the form
[
] , with
and being column vectors [as, e.g., in (1)], should be understood as the column vector constructed by concatenating the two vectors.
XY
X
Y
X X
X
X
313
with
and
representing
the level of coupling in each direction. It should follow from the
.
above discussion that we have
As a first approximation we simply remove the reverse coupling, that is, we force the circuit to be unilateral
(4)
(5)
where now
and with
in the following
referred to as the coupling parameter. It is important to note that
(4) is no longer an equation but an approximation with an error
on the order of the reverse coupling parameter . Our analysis
will be based on (4)–(5) and the results will hence suffer from
a built-in error term. Luckily, this error will not be that serious,
in fact it is negligible; if it were not, the circuit in Fig. 1 would
not work as an ILO.
The concept of weak coupling is inherent to the design of
any ILO as the purpose of the construction is to realize a synchronous oscillator/divider in a power efficient way. Furthermore, it is well known that strong coupling can lead to bifurcations of the attractor and even chaos. A first-order model in the
coupling parameter is easily derived from (4)–(5)
(6)
(7)
where now
and is the
Jacobian matrix
. Equations (6)–(7) constitute the model to be discussed here and it follows the true dynamics of the circuit in Fig. 1 up to second order in the weak
coupling parameter and first order in the even weaker bilateral
coupling . It is a faithful model of the circuit to the extent that
can be ignored, which is a safe
second-order terms in
assumption.
III. A NEW DEFINITION OF ILO PHASE NOISE
The purpose here is to give an introduction to the main ideas
on which this paper is based. The text will lay out a strategy for
the analysis to follow leading up to the main result of this paper;
that being the calculation of the ILO spectrum in Section VII.
Specifically we include a detailed discussion of our novel definition of ILO phase noise. To our knowledge we are the first to
suggest a completely generalized and unified definition.
Our model is based on the notion of a two-dimensional phase
manifold (PHM).
Lemma 3.1 [The Phase Manifold (PHM)]: There exists a twodimensional, closed and invariant4 manifold holding all asymptotic orbits of the ILO. The ILO steady-state (i.e., the limit cycle)
is a one-dimensional submanifold embedded in this set.
Proof: This follows directly from the smoothness of the
state equations as will be explained in Section IV. For more
formal proofs we refer to the literature [21], [22].
Having established the existence of the PHM we propose the
following novel definition of ILO phase and phase noise.
4By closed we mean that the set is without boundary and by invariant we
imply that orbits initially on the PHM remain there forever.
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
314
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 2, FEBRUARY 2011
Fig. 1. ILO circuit block diagram. A close-to unilateral buffer separates the
two asymptotically stable oscillators ensuring that the coupling signal flows in
one direction only. The buffer symbol can refer to an actual physical circuit or
to a virtual construction representing the extraction of signal isolation offered
by, e.g., a bias point coupling node.
Definition 3.1 [ILO Phase and Phase Noise]: The term ILO
phase refers to the coordinates which index the PHM. The term
ILO phase noise refers to those perturbations of the ILO steadystate (1) which remain tangent to the PHM.
The justification behind the first part of this definition is straight
forward since the PHM is closed and hence must be parameterized by two modulus- coordinates. The two phase variables,
corresponding to the two oscillators in the ILO, are the only two
modulus- coordinates in our model and hence must be defined
as the coordinates which index this set. This discussion mirrors
what is known from the free-running oscillator case where the
limit cycle is indexed by a phase variable [12], [13].
Considering the issue of phase noise measurement, which is
the subject of this paper, we again refer back to the well-known
scenario of the free-running oscillator. There one would isolate
the phase noise perturbations by following the oscillator by an
amplitude limiter or alternatively by considering an oscillator
with sufficiently strong contraction of the amplitude modes. In
this case only the phase noise perturbations would survive and
a measurement would reveal the familiar Lorentzian spectrum
[13], [23], [12]. Consider following the S-OSC (see Fig. 1) by an
amplitude limiter or, alternatively, employing a design with inherently strong S-OSC amplitude contraction. Would we see the
same Lorentzian spectrum as in the single oscillator case? The answer is of course no. Even with the amplitude perturbations completely cut off the spectrum would not be the simple Lorentzian
but would have a more complex characteristic [8], [18], [7]. This
spectrum cannot be modelled by the single-phase/limit-cycle approach as a one-dimensional model would simply lack the sufficient degrees of freedom to capture all the characteristics; the
experiment clearly illustrates that we need a new definition. The
two-dimensional PHM of Theorem 2.1 will persist any type amplitude control of the S-OSC as long as synchronization is maintained [21], [22] and with all amplitude perturbations cutoff the
noise perturbed orbits are constrained to lie in this manifold. The
PHM description is hence the sought after model for the amplitude cut-off scenario, ensuring that we have sufficient degrees of
freedom (i.e., 2) to capture the characteristic of the phase noise
spectrum. By extension the PHM description must therefore also
give the correct definition of the term ILO phase noise for any
other setting of the amplitude control mechanism.
In the single oscillator case the analysis proceed from this
point by projecting the linear response onto the one-dimensional
limit cycle thus isolating the phase from the amplitude response
[13], [12]. We follow the exact same approach here, simply
substituting the two-dimensional PHM for the one-dimensional
limit cycle and we state:
Lemma 3.2 [ILO Phase Noise Calculation]: The ILO phase
noise response is isolated by projecting the noise-driven linear
response onto the two-dimensional PHM.
Proof: Follows directly from Definition 3.1.
In the single oscillator case a projection operator, constructed
using a Floquet decomposition of the oscillator linear response,
was used to project onto the limit cycle [12], [13]. Again we
follow this approach and hence consider constructing a Floquet
decomposition of the linear response around the ILO steadystate/limit-cycle on the PHM (see Lemma 3.1). The following
theorem shows that this is indeed possible.
Theorem 3.3 [ILO PHM Floquet Decomposition]: There
exist two sets of Floquet eigenvectors which together span the
PHM subspace of the ILO linear response. These vectors are
linearly independent and it follows that the ILO phase noise
response can be decomposed in terms of this basis.
Proof: See Section IV.
Once we have found the relevant Floquet vectors we can
easily derive the dual Floquet vectors which we seek in order
to build the projection formalism described in Lemma 3.2. The
detailed calculation of the PHM Floquet vectors and their dual
counterparts is discussed in Section V. With this framework in
place we then proceed to project the noise forced ILO linear response onto the PHM and thus isolate the phase noise dynamics.
This is described in Section VI where we calculate the relevant
correlation matrices. Finally in Section VII we derive ILO phase
noise spectrum.
IV. GEOMETRY OF AN ILO SOLUTION
The domain of the ILO circuit, i.e., the set of initial conditions
in the basin of attraction for the limit set produced by (6)–(7),
is referred to here as the ILO stable manifold5 and denoted ;
a subset of the entire state space
.
We start by considering a scenario where the two oscillators
in (7). Denote
in Fig. 1 are uncoupled corresponding to
the M-OSC and S-SOC limit cycles and , with periods
and , respectively. There exists a two-dimensional, closed and
embedded in . The manifold,
invariant manifold
which is diffeomorphic to the canonical manifold
the so-called two-torus, will in the following be referred to as the
phase manifold (PHM). Orbits with initial conditions on are
now written as in (1) and these trajectories then represent the
asymptotic dynamics or steady-state of the uncoupled system.
We now introduce the concept of manifold coordinate description in our analysis. The M-OSC and S-OSC phase coordiindex and we define the map
nates6
, where the split defined in (1) is used here
to represent an orbit on the PHM. This will be a coordinate map
if we interpret modulus the period .
However, usually we shall prefer to interpret as real number
will no longer be a coordinate map,
and
5From
=
2
the split in (1) it could seem that one could then write
, with
and
denoting the M-OSC and S-OSC domains, respectively.
However, as will be discussed below, this is only true for the case where the two
oscillators are uncoupled.
6Although the coordinate functions
to them throughout as phases.
have dimension of time we shall refer
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
DJURHUUS AND KROZER: THEORY OF INJECTION-LOCKED OSCILLATOR PHASE NOISE
Fig. 2. The asymptotic orbits of the circuit in Fig. 1 and (6)–(7) form a twodimensional closed manifold embedded in
and diffeomorphic to the two. We shall refer to this set as the phase manifold (PHM)
torus
. The tangent space of the PHM at a point p 2 is written
. The PHM
is then defined as the disjoint union of all tangent spaces
tangent bundle
over .
=
315
2
in the strict sense of the word, as the coordinate lines will wrap
around the closed manifold . The important point to keep in
and
produce a coormind is that the lines of constant
dinate grid on which can be used to index the points on the
manifold. Using the (coordinate) map we can now describe
on through its coordinates
; a noan orbit
tation we shall make use of in our further analysis. At a given
, the tangent space (see Fig. 2)
is then spanned
point
by the vectors tangent to the coordinate lines
.
The dynamics of the uncoupled scenario is now briefly discussed. Consider the M-OSC solution to be periodic with period
while the S-OSC period is -close
. We then choose an initial condition for the M-OSC
on , corresponding to
, and map the iterated
dynamics in the S-OSC domain by applying the time-T return
, derived by integrating (6)–(7), to the set
map
. Here
represents the S-OSC stable manifold
and we assume a projection onto the S-OSC domain is implied.
In Fig. 3(a) the orbits of the iterated map, in the S-OSC domain , are drawn in a qualitative manner. Since the two peand
are not rationally related the orbit on
will
riods
not be closed. This is illustrated in the figure where after iterations the orbit overlaps the initial point. Asymptotically with
time the orbit will thus cover the entire PHM, , a characteristic
of the dynamics pertaining to this class of systems. The uncoupled asymptotic dynamics hence consists of orbits which cover
a two-dimensional set . This concludes our description of the
uncoupled circuit dynamics.
in (7)) is now introA small unilateral interaction (
duced between the two oscillators. It follows from the smoothness of the state equations that the PHM will persist the perturbation [21], [22]. Hence, a two-dimensional, closed and in, exists embedded in
variant manifold, the perturbed PHM,
. It furthermore follows from the smoothness of the equaand the perturbed tangent
tions that both the manifold itself
(see Fig. 2) will be -close to their unperturbed
bundle
counterparts [22], [21]. Assuming a periodic synchronized
, is created on
solution of (6)–(7) the ILO limit cycle set,
in a saddle-node bifurcation. More specifically, the bifurcation creates two one-dimensional, compact and invariant limit
sets. One is attracting, a so-called -limit set; this is the limit
cycle
. The other set is repelling, a so-called -limit set,
which we denote
. All trajectories in the ILO stable manifold, including those on
, will approach
asymptotically
Fig. 3. The figure shows the iterated dynamics of the return map (x ) projected onto the S-OSC domain with the M-OSC state fixed. (a) [uncoupled oscillators]: the periods T and T are not rationally related (in the general case)
will eventually return close to starting point but not
and the orbit starting on
exactly on it. The iterated orbit is represented using dots inside circles. After n
iterations the orbit overlaps the initial state shown as dot inside a broken-line
circle. It follows that the orbits will cover the PHM, M , asymptotically with
time. Initial conditions away from , will approach
and hence
asymptotically with time. (b) [locked oscillators]: The figure illustrates the dynamics
in the S-OSC domain (t ) under the mapping of the time-T return-map ,
with the index t referencing the fixed M-OSC state x (t ). A saddle-node
bifurcation takes place on the perturbed PHM,
, and orbits of the iterated
time-T return-map will approach the stable node asymptotically with time. The
(t ) by two orbits 0 (t )
node (t ) and saddle (t ) are connected in
and 0 (t ), the stable manifolds of (t ) and unstable manifolds of (t ).
The union 0(t ) (t ) [ 0 (t ) [ (t ) [ 0 (t ) is a closed set which is
invariant under the dynamics of (x ), where x = x (t ) 2 (t ). The
set 0(t ) then defines the coordinate lines of the S-OSC phase .
with time. In the following the periodic ILO steady-state, i.e.,
a solution of (6)–(7) with initial conditions in
, is written
is then introduced as a coordias in (1). The diagonal phase
nate index of this one dimensional submanifold.
Repeating the procedure from the uncoupled case we now
proceed to investigate the qualitative dynamics of the locked
. We
scenario. The M-OSC state is hence again fixed at
prothen observe the orbits of the iterated return-map
jected onto the S-OSC domain
. Note that we can no
since the S-OSC is enslaved to
longer write
the M-OSC and the state space split can only be done by refreferencing to the M-OSC state; hence the notation
erencing
. The set
is invariant
, by construction, since
lies on the
under the map
where
acts as the identity (as the M-OSC
M-OSC part of
, prois free-running). The orbits of the iterated map
jected onto the S-OSC domain
, are sketched in Fig. 3(b)
and it is seen that fixed points now appear. More specifically, a
and a saddle
are created in the above mennode
tioned saddle-node bifurcation. Also shown in the figure are the
and
connecting the two fixed points,
branches
and the unstable manifolds
i.e., the stable manifolds of
, respectively. We then define the set
of
as the union of the fixed points and the stable/unstable manifolds,
. The set
is invariant under
as it is constructed from
the union of fixed points and their stable/unstable manifolds. We
proceed to state
Lemma 4.1 [Parameterizing the Perturbed PHM]: The PHM,
, can be parameterized by two phase coordinates
through the map
, where ,
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
316
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 2, FEBRUARY 2011
referred to as the S-OSC phase, parameterizes the one-dimenfor each , with
mapping to the stable
sional set
(see Fig. 3(b)).
fixed node
Proof: The fixed points
and
must lie on the PHM, , as this set holds all asymptotic
.
orbits of the system and hence all the fixed points of
invariant it must hold a set of stable/unFurthermore, since
stable manifolds of these fixed points. Hence the sets
and
must also lie on
. The union of
is closed (see
these sets must together form a closed set since
, with
Fig. 3(b)). It then follows that
, lies on the PHM. We can
by introducing
hence parameterize the two-dimensional set
. Substituting
for and leta new coordinate to index
index
we have constructed the map
.
ting
From Lemma 4.1 it follows that the ILO steady-state orbit,
(1), on , can be described by the coordinates
and that the state-space representation is written
,
is the node created in the S-OSC domain as a result of
where
the saddle-node bifurcation [see Fig. 3(b)]. From (1) we get the
and we see that the S-OSC steady-state
identity
orbit can be interpreted as the trajectory of the node fixed point
in the S-OSC domain. This very clearly illustrates the fact that
the S-OSC is no-longer a free running oscillator. As seen from
, we
Fig. 3(b), besides the stable orbit
have also created an unstable orbit
which is the above mentioned -limit set on
.
At this stage we have concluded that the PHM,
, is paramvia a (coordinate)
eterized by the two phase variables
of the set
, into the perturbed
map
PHM. We are, however, ultimately interested in investigating
, and we hence
noise perturbations of the steady-state,
where
seek a coordinate description of the tangent bundle
the subscript refers to the base-space of the bundle which in this
7 . In order to achieve this we procase is the ILO limit cycle
ceed to define the phase vectors
. Here
is a scaled copy of
since , as explained above, indexes
and
is a tangent
is then defined as the tangent
vector to this set. The vector
vector to the set
at the point
[see Lemma
4.1 and Fig. 3(b)]. We now state:
Lemma 4.2 [The Phase Modes]: The vectors
and
are eigenvectors of the ILO Monodromy matrix
corresponding to the characteristic multipliers
and
. These two vectors together
.
span the tangent space
is derived as
Proof: The Monodromy map
the linearization of the time-T return map
. It hence follows
will map tangent vectors of sets, invariant
that
under , into themselves. The ILO Monodromy matrix there, this being a tangent vector to
fore has the eigenvector
the invariant set
at each index . Furthermore, the dynamics
.
in this set is neutrally stable implying the eigenvalue
is also invariant under the time-T reThe set
turn map and
, defined as the tangent to
based at
, is hence also eigenvector of
.
is attracting and it follows that the eigenvalue
The node
must obey
. Furthermore, since the equations are smooth
was equal to one before the bifurcation which led to the
and
locked dynamics it follows directly that the eigenvalue will be
-close, i.e.,
. From Lemma 4.1 we have that
acts as coordinates for the manifold
. By definition
, as defined above, must
it then follows that the set
, where
is defined as the disjoint union of all
span
.
the tangent spaces along the ILO limit cycle
In the following we shall use the Floquet decomposition of
the ILO linear response and we now restate Theorem 3.3 from
Section III:
Theorem 4.3 (Theorem 3.3 ReStated): Among the modes
representing the Floquet decomposition of the ILO
, there exlinear response around the locked steady-state
, also referred to as the ILO phase
ists two sets
, of the
modes, which together span the tangent bundle,
PHM
.
the FloProof: At each point of the solution
are found as eigenvectors of the
quet vectors
Monodromy matrix. From Lemma 4.2 we know that two of
these must be scaled versions of the PHM tangent vectors i.e.,
. Since
and
, by
the proof follows.
construction, span
Theorem 4.3 thus proves that we can always find 2 sets of
which will span the PHM tangent space
Floquet vectors
and hence form a basis for the phase noise response of the ILO.
This means that we now have a way of filtering out the phase
noise response of the noise forced ILO circuit and the rest of
this text will focus on developing this formalism further.
is written
In the following
(8)
, for all [12], [13]. The second vector set/
that is
bundle is only specified above using geometric qualitative argulies entirely in the S-OSC
ments. However, we know that
and we can write
domain
(9)
. A more detailed discussion of this
where
vector is the topic of the following section.
V. A FLOQUET REPRESENTATION OF THE ILO LINEAR
RESPONSE
The ILO steady-state Jacobian matrix is derived by differentiating the state equations (6)–(7) around the periodic solution
in (1)
(10)
7The tangent bundle
can be seen as the disjoint union of tangent spaces
along the set
. That is, a collection of tangent vectors where vectors based at
are not related i.e., they cannot be added together to
two different points of
form a vector etc. See also Fig. 2 for a qualitative illustration of these concepts.
with
and
.
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
DJURHUUS AND KROZER: THEORY OF INJECTION-LOCKED OSCILLATOR PHASE NOISE
Lemma 5.1 [Calculating the ILO STM]: The state-transition
matrix (STM) resulting from integrating (10) is calculated as
317
where time is the parameter of the curve. The linear response
, corresponding to
, is then calculated
around
(11)
is the M-OSC state tranwhere
the locked S-OSC
sition matrix,
represents the
transition matrix and
coupling transition matrix which has the form
(17)
is a small deviation from the curve
where
here refers to the instantaneous time/phase variable
and
(18)
(12)
Proof: See Appendix A.
From the block form of the STM in (11) it should be clear
, will have eigenthat the Monodromy matrix,
vectors of the form
(13)
the matrix
will have
and we collect these in
. In addition
eigenvectors of the form
(14)
, where now the index runs from
to the state dimension
and we collect these as
.
The complete ILO Floquet vector set is then written
(15)
being
and
with
matrices holding the vectors defined in (13)–(14).
In the previous chapter it was proven that two vectors, one
and one from the set
, will span the tanfrom the set
. Fixing these as
gent space of the PHM at the point
the first members of
and
was defined in (8) while
was defined partially in (9) in terms of a
. Using the expression for the ILO STM devector field
rived in Lemma 5.1 we find
Theorem 5.2 [The Second PHM Floquet Mode]: The second
, as defined through (9), is
Floquet phase mode
given, to a first order in , by
The derivation in (17) follows by noting the steady-state in
, is asymptotically
(1), which can also be written simply
stable. Any phase perturbation will hence stay in small region
around this orbit and approach it asymptotically. The variable
in (17) can hence be considered of the order of the perturbation, which in the case of electronic circuit noise, can always
be considered small compared to the solution itself. From this
discussion it should follow that the first-order variation in (17) is
a faithful approximation of the true solution up to second order
on the other hand represent neuin the noise. The phase
trally stable dynamics and hence cannot be considered bounded.
Subsequently, it cannot be taken outside the phase argument of
(17) [12], [13].
From the discussion in Section IV and Theorem 4.3 it follows
and since we consider a
that
Floquet decomposition we have
. Using (9) the
in (17) is written
deviation
(19)
The dual Floquet vectors are defined as the eigenvectors of
. From the
the transposed Monodromy matrix
block form of the ILO STM in (11) it should be clear that this
eigenvectors of the form
matrix will have
(20)
with
more be
and these are collected as
. There will furthereigenvectors of the form
(21)
and
and we colwith
lect these as
. The
complete set of ILO dual Floquet vectors is then written
(16)
(22)
where the modes are defined in (14),
is the S-OSC part
are functions
of the steady-state tangent vector in (1) and
, with
being the
on the order
characteristic Floquet exponent belonging to the mode vector
.
Proof: See Appendix B.
From Lemma 4.1 we know that curves on the PHM are paand we can hence write
rameterized by the coordinates
,
a PHM orbit in the S-OSC domain as
with
being
and
matrices, respectively, holding the vectors defined in (20)–(21).
as
We refer to the dual vectors of the phase modes
which implies that
and
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
(23)
(24)
318
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 2, FEBRUARY 2011
where
,
.
, that is they are
The dual vectors operate on vectors in
functionals, and by definition they obey the bilinear relationship
, for all 8 . Using the expressions
we
(8), (9), and (16) for the PHM Floquet phase modes
then find
further in Section VIII, these simplified expressions will closely
approximate the actual situation.
VI. SECOND-ORDER PHASE STATISTICS OF THE
NOISE-DRIVEN ILO
The Fourier series of the steady-state solution (1) is written
(25)
(30)
(26)
(27)
Inspecting (25) we see that a perturbation in the direction of
the M-OSC phase increase translates into a diagonal phase shift
. This characteristic follows from unilateral coupling of the
circuit and we also say that the M-OSC distributes its phase to
the S-OSC. Inserting (27) into (26) gives
(28)
in (27), as
where we have used the fact that the functions
defined in theorem 5.2, are functions on the order
, with
. Note that
has the
in (25) and so it must correspond to
negative sense of the
(see Footnote 8), at least
the negative M-OSC gradient
to within a small bound
. Again from (27) we get
(29)
and we can begin to understand how to interpret the S-OSC dual
Floquet vector in (24). If both the M-OSC and S-OSC phases
applied
are shifted an equal amount then
to this vector will cancel and the S-OSC phase will not change.
represent a displacement along the neutrally
This shift on
. Then if we shift the M-OSC
stable direction
and S-OSC phases an equal but opposite amount we see from
will be shifted as the M-OSC
(25) that the diagonal phase
phase is shifted; however, this time, as the phase increments
is no longer orthogonal to
have opposite sign, the gradient
the perturbation but instead colinear and no cancellation will
take place. It follows that a step in the direction of the error
will translate to a S-OSC phase shift. The
phase
and
above description is only valid to within a small order
is only meant to give a qualitative understanding of the general
situation. However, under special circumstances, to be discussed
8The functionals v thus are seen to pick out the i’th Floquet component of
a general state-vector and should be interpreted as the gradient vector of the
coordinate used to index the integral curves of which u is tangent [24].
the steady-state frequency in radians,
with
is the complex amplitude vector of the ’th harmonic in
and
reprethe Fourier series with
senting the usual division into M-OSC and S-OSC parts. The
(see (9)) is expanded
S-OSC part of the second phase mode
as,
(31)
Noise is introduced into the model by adding the stochastic
perturbation vector
on the righthand side
of the ILO state equations in (6)–(7), where
models the modulation of the noise by the state variables
is a column vector of delta-correlated,
and
unit variance, Gaussian noise sources [13]. Due to the topology
of the ILO unilateral circuit (6)–(7) this perturbation vector is
written as
(32)
where
model the noise modulation of the ILO subcircuits,
,
and
represent the
while
split of the original dimensional vector into M-OSC and
S-OSC noise sources.
The stationary second-order statistics of S-OSC phase is capdescribing
tured by the correlation matrix
the auto- and cross-correlations of the S-OSC PHM solution.
From (17) this matrix is written
(33)
where all the terms,
, are matrices of the same form
and we consider the statistics in the limit
to
as
ensure stationary statistics. The ILO S-OSC PHM correlation
matrix, as given by (33), is calculated in (87), (100), and
(102) of Appendix C. In the following all terms describing
cross-correlation between the state vector components in (87),
(100), and (102) will be neglected since these do not contribute
to the S-OSC phase noise spectrum which is calculated from
the auto-correlation matrix. This means that we will only need
matrices in
the terms in the main diagonal of the
(33). Also, in order to keep the expressions simple we are only
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
DJURHUUS AND KROZER: THEORY OF INJECTION-LOCKED OSCILLATOR PHASE NOISE
concerned with the components of the correlation functions
which contribute to the spectrum at the first harmonic.
is given by (87)
The steady-state auto-correlation
where the first harmonic term is found by inserting
into
the expression
319
Finally, consider the auto-correlation of the second phase
calculated in (102) of Appendix C. The first
mode
and
harmonic is found by choosing the indices
we write the ’th diagonal component as
(42)
where the constant
is given
(34)
refers to the ’th
diagonal elewhere
denotes the
ment of the matrix inside the square braces,
’th component of the ’th harmonic S-OSC Fourier coefficient
with
being the ’th
vector defined in (30) and
as defined in (83) of
harmonic of the diffusion constant
Appendix C. Next we consider the two terms in (33)
and
representing the cross-correlation between the
two PHM modes. The second of these is calculated in (99) of
Appendix C while the first follows via the identity
. Note that
is only nonzero for
while
is nonzero for
. The sum of these two
functions in (33), as calculated in (100) of Appendix C, is then
defined for all . Inspecting (100) it is seen that several combiwill lead to first harmonic componations of the indices
,
nents. However, as is a small parameter on the order
9 will be
(see Appendix B), the combination
dominant and we can write
(43)
and with
being the ’th harmonic of the S-OSC diffusion
defined in (85) of Appendix C.
constant
Inserting (34), (35), and (42) into (33) the ILO PHM autocorrelation for the ’th S-OSC state variable
is written
(44)
VII. THE ILO PHASE NOISE SPECTRUM
(35)
where from (100) and (101) in Appendix C the scalar parameter
is written
Fourier transforming (44) the phase noise spectral density
around the S-OSC first harmonic (carrier), for the ’th component of the state-vector in (17), is written in (46) at the bottom
is the offset from the
of the next page, where
carrier and
(45)
(36)
is the ’th harmonic of the cross-phase diffusion
where
constant
defined in (84) of Appendix C with the new parameters given as
The first expression in (46) contains second-order terms like,
e.g.,
which are then neglected subsequently.
The phase noise spectrum (in natural numbers) is defined as
, where
denotes the single
and
sided spectrum which is found as
using (46) this is written
(47)
(37)
(38)
where the poles and zeroes follow from (46) as
and
where
(39)
(48)
(40)
(49)
From (100) and using (36)
(41)
9Here n acts as a index and not the state dimension as defined in connection
with (1).
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
(50)
320
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 2, FEBRUARY 2011
VIII. STRONG NORMAL HYPERBOLICITY, SYMMETRY, AND
AMPLITUDE LIMITING
In order to simplify the expression for ILO phase noise, obtained in the previous section, we here consider the noise response of an system restrained by conditions of strong normal
hyperbolicity (SNH), symmetry and/or amplitude limiting with
each of these terms being defined in order.
By the strong normal hyperbolic (SNH) condition we mean
, is very sharply defined implying
that the invariant PHM,
that the normal contraction on the manifold is much stronger
than the tangential part. If the amplitude modes are strongly
contracting this means that the Floquet characteristic expo(see discussion in Section V), except for
nents of the set
, will approach minus
the phase mode exponent
, which implies that terms on the order
infinity
in (16), i.e., the functions
,
can be neglected. We note that the SNH condition can be
interpreted, in a heuristic sense, as a low-Q condition for the
S-OSC since it is well know that the amplitude relaxation time
of a harmonic oscillator is proportional to the quality factor of
the circuit [25].
The term symmetry is used here to define a case where the
evaluate to zero independently of
. From the
discussion in Appendix B we see that this would imply that the
coefficients defined in (77) would all evaluate to zero for
.
Both of these two special conditions would give us a very
simple result for the second phase manifold vector (see (16))
(51)
where we have renormalized
in (16), that is, we have sub. In the frequency domain (51) reads
stituted
(52)
, while (37)
With (52), (38) now evaluates to
. Furthermore, from (25)–(29) in Section V we find
gives
and
.10 Inserting these identities into (83)–(85) of Appendix C will produce the following result
(53)
(54)
10Since we have renormalized u
in (51) by multiplying by a factor jx_ j we
have to divide the dual Floquet vector v in (24) to maintain bilinearity [13].
The expression in (28) then becomes v x_ = 01, since we have set order
terms equal to zero, and from this it follows that v (t) = 0v (t).
where
and
are the dc components of
the diffusion constants (83)–(85),
was introis defined here as
duced in the previous section while
with
and
defined in (24) and
(32), respectively. Assuming strong normal hyperbolicity we
which in turn
can hence write
. From (36), (40), (43), (53), and
implies from (40) that
(54) we can now write the two defining noise parameters
and
as
while (48)–(50) is written
(55)
(56)
and we see that the zero of the spectrum (47) becomes real.
We can then write the phase noise spectrum for SNH (or low-Q)
and symmetry conditions as
(57)
given as in (47) and given by (56).
with the poles
Now consider connecting the output of one of the S-OSC
nodes to an ideal (i.e., resistive) amplitude limiter. Since S-OSC
amplitude response at the output of the limiter is cut-off we must
have that the subspace spanned by the S-OSC amplitude mode
exclude this node of the circuit. Let the
vectors
output node of the amplitude limiter be referenced by the S-OSC
mode vecindex, i.e., not full state vector index, . The
tors, defined through (14), would then have their index components set to zero. From (16) it then follows that (51) would
and we
again be true for the ’th component of the vector
would rediscover all the above results including the spectrum
(57).
The above text discussed three situations: SNH/low-Q, symmetry and amplitude limiting, where the equations reduce significantly in complexity and the rest of this paper will deal exclusively with this simplified model.
A. Locking Versus Tracking
Of the three singularities in (57) it should be noted that
for all parameter values as
for all practical purposes. The real parts of
and are then the only adjustable parameters and the value of
these will determine whether the S-OSC phase noise spectrum
(57) displays a locked or a tracking characteristic.
(46)
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
DJURHUUS AND KROZER: THEORY OF INJECTION-LOCKED OSCILLATOR PHASE NOISE
Fig. 4. The spectrum (57), plotted as 10 log( (! )) [dBc] (dB below carrier),
for three values of D with the parameters = 1; ! = 2 [s ]; 1! =
j ln(0:95)j [s
]; D = 1 2 10 . [thin solid line]: D = 1 2 10 . From
(55) < 0 and the zero lies to the left of the pole p in the complex plane corresponding to a tracking characteristic. [bold solid line]: D = 1 2 10 . From
(55) = 0 and the zero lies lies exactly at p leading to mutual cancellation and
the resulting characteristic is the free-running M-OSC Lorentzian. [broken line]:
D = 1 2 10 . From (55) > 0 and the zero lies to the right of the pole p
which gives rise to a locked characteristic.
The transition from locked to tracking mode occurs at
which from (56) implies
321
identical to the SDE derived in [8, p. 239] and all the rest of the
results of that paper follow from this result.
Model 2: The ISF Model—Verma et al. 2003: The authors of
[1] formulate a model based on a block-diagram interpretation of
the ILO circuit. The noise response is formulated using the impulse sensitivity function (ISF) approach which was developed in
[26]. Following this method the authors develop a SDE description equivalent to the Kurokawa equation (59).
Model 3: A Harmonic Balance Model—Zhang et al.
1992: In [5] the authors analyze a circuit block diagram
of a standard harmonic feedback oscillator with a subharmonic tone injected at a summing point. This is the same
circuit originally studied by Adler [27]. An expression for
the phase noise spectrum of a subharmonic ILO perturbed
by white noise is derived using a harmonic balance type
analysis. The phase noise of the M-OSC is now written
where
. Using this
the last approximation holds for
approximation (57) is written
(58)
meaning that the S-OSC is locked when
and otherwise
tracking. As noted above, represents the signal to noise ratio of
is the equivalent number for the S-OSC and
the M-OSC while
the relation then only says that the S-OSC is locked if it is at least as
noisy as the M-OSC otherwise it is tracking. Fig. 4 gives a qualitative illustration of how the characteristic shape of the power spectrum in (57) depends on the position of the poles and zeros.
B. Comparison With Earlier Models
In this section we shall compare our result for the SNH/symmetry/amplitude limited phase noise spectrum in (57) with earlier
published ILO phase noise models. We have picked out three contributions which together give a good representation of the current
state of this topic.
Model 1: The Classical Model—Kurokawa 1968, 1973: In [8]
and[18]K. Kurokawastudiedharmoniclockingof LCoscillators.
In Section V it was explained how under the SNH/symmetry/amplitude limited conditions the S-OSC phase was advanced by a
step in the error-phase direction with the error phase being given
. The S-OSC stochastic differential equation
as
(SDE) for is thus derived by adding to the first-order ODE
the S-OSC noise and minus the M-OSC domain noise.
This last minus is very important in order to obtain the correct end
result and we note that this comes out of our equations very naturally. From this description we can then write
(60)
After inserting the approximation given in Footnote 11 and
(60) then
using the definition
equals the expression ([5], (3.21)).
IX. CONCLUSION
The paper documents a phase noise model of an injectionlocked oscillator. We make no assumptions regarding circuit
topology, class or method of unilateral coupling and the paper is
hence an attempt at a unified framework. The theory builds on
top of a numerically derived steady-state allowing for a future
implementation of the derived formulaes in any CAD program
with a steady-state solver.
APPENDIX A
PROOF OF LEMMA 5.1
Consider the matrix ODE equation
(61)
representing the linear response of the ILO ODE with
and
given in (10). The solution of
with
(61) is now split as
(59)
(62)
where we have used that the noise-forced dynamics of the freerunning M-OSC is defined through
[13]. When
into (59) it is
inserting the first harmonic approximation11 of
and
are defined in (10). Since this matrix is block
where
diagonal the solution is given as
11The parameter is periodic in the phase argument defined in (38) and the
first harmonic Fourier coefficient is approximated as 1! cos() where
1! is known as the locking bandwidth and = + =2. This expression
was derived by noting that, for the first-harmonic, we must have = 0 for
= (0; ). At this point the M-OSC and S-OSC are phase shifted 90 and it is
well know that this leads to loss of synchronization.
(63)
where
Writing
and
.
the full expression in (61) is found
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
322
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 2, FEBRUARY 2011
as
the Floquet eigenvectors and dual eigenvectors [13], and using
that
(69)
(64)
which means that we can write
and using (63)
(70)
where
and
(65)
where
was defined in (10) and the relationship
was employed. Because of the block
form of
in (65) we find that
for all
. Specifically, this implies that the matrix in (65) commutes
,
with any copy of itself at any time i.e.,
. This is important as this means that the solution to
for all
can be written in the simple form
(66)
Using the results in (63) and (66) we hence find (67), the
expression for the ILO STM, shown at the bottom of the page.
APPENDIX B
PROOF OF THEOREM 5.2
The tangent vector to the steady state solution,
, is an eigenvector of the monodromy matrix
with eigenvalue 1. Multiplying this vector with
(67) we find
and
(71)
where
The matrix product
according to
.
in (68) is now decomposed
(72)
where are the S-OSC parts of the Floquet eigenvectors/modes
defined in (14). Equation (72) follows from the observation that
(see discussion in Section V) spans the S-OSC
the set
domain at each index and there must then exist a unique expansion of the left-hand side of (72) in terms of this set. Furthermore, since the vector on the left-hand side of (72) is periodic,
as are the vectors in , the expansion coefficients must be periodic with the same period.
Using (72) we can write the term being integrated in (68) as
(73)
and inserting this into (68) using (71), we get
(74)
(68)
where we have used that
, where
The periodic functions
are expanded by a Fourier series
. Writing
and hold
(75)
(67)
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
DJURHUUS AND KROZER: THEORY OF INJECTION-LOCKED OSCILLATOR PHASE NOISE
Inserting (75), the integral in (74) is evaluated as
323
where
. The approximation sign in (81) signifies
that the expression is correct up to second order in the noise. The
S-OSC phase mode (see (17) and (19)) linear response following
from the noise perturbation is
(76)
(82)
with
(77)
Inserted into (74) this gives
where again the approximation sign signifies that the expression
is only correct up to second order in the noise.
We define the phase manifold diffusion constants
(83)
(78)
As a first approximation we know that
since
this Floquet exponent was zero before the bifurcation which led
term domto synchronization and we hence find that the
mode
inates for the
(79)
From Lemma 4.2 and Theorem 4.3 we know that the Floquet
, corresponding to the second Floeigenvalue
, must be of the order
, which then
quet mode
must be on the order of
in turn implies that
. Furthermore, in the uncoupled limit
we must have
since this mode then corresponds to the zero Floquet
characteristic exponent. This discussion then leads to the result
. The conclusion is hence that
,
that
to a first order in
, is equal to
. Next let us inspect the
second term on the righthand side of (79). We can safely assume
for the S-OSC amplitude modes.
that we have
term dominates in (79) and this term is
Clearly then the
of the order
. This follows since the coeffi, as discussed above. Colcients in (77) are on the order
lectively, the above discussion leads to the following first-order
approximation to (79):
(84)
(85)
which are periodic functions with Fourier series
(86)
It is well known [23], [12], [13] that, asymptotically with
becomes a regular
time, the process
, where
is given as
Wiener process with power
the dc component of (83). From this description the autocorreis easily calculated [23], [12], [13]
lation matrix
(87)
The matrix describing the cross-correlation between the two
phase manifold modes is written
(80)
(88)
where we have normalized the left-hand side of (79) as the mode
vectors in
(see discussion in Section V) are assumed to be
unit length. In (80) are functions on the order
.
where we have used the Fourier transform of the ILO steadystate defined in (30). Using the identity
APPENDIX C
CORRELATION MATRICES
(89)
is of the order
and represents the correlation
where
time of the second S-OSC phase mode, the exponential in (88)
can be written
The statistics of the phase increment variable
is characterized through [12], [13], [23]
(90)
(81)
where the different variables are defined in Sections V and VI,
and
In the above expression a Taylor expansion of the exponential
was used
which, as explained in [23], is a valid approximation as long
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
324
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 2, FEBRUARY 2011
as the condition
holds; which we shall assume.
Using (89), (90) in (88) and remembering that nonoverlapping
become uncorrelated asymptotically
intervals of the process
with time
(91)
where
(92)
(93)
It can be shown easily that
implying that
and we get
with
given in
(92). Consider the first factor of this expression, if we exchange
with
the value of the expression will stay approximately the same
is much longer than that of
since the correlation time of
and the exponential is hence to a good approximation constant
. Making this exchange the first ensemble
in the interval
average is found from the same calculations which lead to
(87)12 . Secondly, if we exchange
with
in the second factor of (92)
the expression is also not going to change significantly. This
is because, to a good approximation, the factor is zero outside
as this is the correlation time of the S-OSC phase mode
. The above approximations allow us to evaluate (88) as
(96)
As is well known, asymptotically with time, the phase will
become uniformly distributed due to the neutral stability of the
limit cycle dynamics. Hence, in all periodic functions where this
, we can simply
function occurs in the phase, i.e.,
with 14 . Note specifically,
exchange the phase
in (95).
that this implies that
as deUsing the Fourier series of the diffusion constant
fined in (84), (86) together with the Fourier series (31) we can
write the integral in (95) as
(97)
Inserting the above expression into (95) and (95) into (94) and
using
(98)
(94)
In (96)13 we calculate the ensemble average in (94) with the
result
we can write (94)
(95)
where
.
(99)
The other cross-correlation matrix
by employing the identity
(99) we find
is then found
and from
12The ensemble average hexp(jk! 1 )i = exp(0(1=2)k ! D j j) is
(t + ) 0
(t) a
simply the characteristic function of the process 1 =
Gaussian variable with power D j j and mean zero [13].
13The approximation sign in (96) implies that the expression is correct to
first order in the noise. Also, the full calculation is not included for notational
purposes as this would contain an exponential exp(0 [ ( ) 0 (t)]) (t)) ( is a small parameter on the order =T (see Ap1 0 ( ( ) 0
pendix B)). Including the second factor into (96) would imply an ensemble average h ( ) ( ) ( )i. However, as is a Gaussian stochastic variable this
evaluates to zero. The sum total of including exp(0 [ ( ) 0 (t)]) in
(96) would therefore be to multiply by one.
(100)
14Technically this follows from averaging over
but we shall not perform
this calculation explicitly but instead simply change the phase.
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.
DJURHUUS AND KROZER: THEORY OF INJECTION-LOCKED OSCILLATOR PHASE NOISE
where we have defined the matrices
(101)
Following the same approach which led to (99) we calculate
with the result
the correlation matrix
(102)
REFERENCES
[1] S. Verma, H. R. Rategh, and T. H. Lee, “A unified model for injectionlocked frequency dividers,” IEEE J. Solid-State Circuits, vol. 38, no. 6,
pp. 1015–1027, Jun. 2003.
[2] F. Ramírez, M. Pontón, S. Sancho, and A. Suarez, “Phase-noise analysis of injection-locked oscillators and analog frequency dividers,”
IEEE Trans. Microw. Theory Tech., vol. 56, no. 2, pp. 393–407, Feb.
2008.
[3] K. Kamogawa, T. Tokumitsu, and M. Aikawa, “Injection-locked
oscillator chain: A possible solution to milimeter-wave MMIC synthesizers,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 9, pp.
1578–1584, Sep. 1997.
[4] K. F. Tsang and C. M. Yuen, “Phase noise measurement of free-running microwave oscillators at 5.8 GHz using 1/3-subharmonic injection locking,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4,
pp. 217–219, Apr. 2005.
[5] X. Zhang, X. Zhou, and A. S. Daryoush, “A theoretically and experimantal study of the noise behaviour of subharmonically injection
locked local oscillators,” IEEE Trans. Microw. Theory Tech., vol. 40,
no. 5, pp. 895–902, May 1992.
[6] E. Shumakher and G. Eisenstein, “On the noise properties of injectionlocked oscillators,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 5,
pp. 1523–1537, May 2004.
[7] B. Razavi, “A study of injection locking and pulling in oscillators,”
IEEE J. Solid-State Circuits, vol. 39, no. 9, pp. 1415–1424, Sep. 2004.
[8] K. Kurokawa, “Noise in synchronized oscillators,” IEEE Trans. Microw. Theory Tech., vol. MTT-16, no. 4, pp. 234–240, Apr. 1968.
[9] A. Mazzanti and F. Svelto, “A 1.8-GHz injection-locked quadrature
CMOS VCO with low phase noise and high phase accuracy,” IEEE
Trans. Circuits Syst. I, Reg. Papers, vol. 53, no. 3, pp. 554–560, Mar.
2006.
[10] R. Adler, “A study of locking phenomena in oscillators,” Proc. IEEE,
vol. 61, pp. 1380–1385, Oct. 1973.
[11] G. R. Gangasani and P. R. Kinget, “Time-domain model for injection
locking in nonharmonic oscillators,” IEEE Trans. Circuits Syst. I, Reg.
Papers, vol. 55, no. 6, pp. 1648–1658, Jul. 2008.
[12] F. X. Kaertner, “Analysis of white and f-a noise in oscillators,” Int. J.
Circuit Theory Appl., vol. 18, no. 5, pp. 485–519, Sep. 1990.
[13] A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: A unifying theory and numerical methods for characterization,”
IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 5, pp.
655–674, May 2000.
[14] A. Mehrotra, “Noise analysis of phase-locked loops,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 9, pp. 1309–1316, Sep.
2002.
325
[15] A. Demir, “Computing timing jitter from phase noise spectra for oscillators and phase-locked loops with white and 1/f noise,” IEEE Trans.
Circuits Syst. I, Reg. Papers, vol. 53, no. 9, pp. 1869–1884, Sep. 2006.
[16] M. Tiebout, “A 50 GHz direct injection locked oscillator topology as
low power frequency divider in 13 m CMOS,” in Proc. 29th Eur.
Solid-State Circuits Conf. 2003. (ESSCIRC ’03), 2003, pp. 73–76.
[17] M. Hines, J.-C. R. Collinet, and J. G. Ondria, “Fm noise suppression
of an injection phase-locked oscillator,” IEEE Trans. Microw. Theory
Tech., vol. MTT-16, pp. 738–742, Sep. 1968.
[18] K. Kurokawa, “Injection locking of microwave solid-state oscillator,”
Proc. IEEE, vol. 61, no. 10, pp. 1386–1410, Oct. 1973.
[19] C. van den Bos and C. J. Verhoeven, “Frequency division using an injection-locked relaxation oscillator,” in Proc. IEEE Int. Symp. Circuits
Syst., 2002, pp. 517–520.
[20] C. Poole, “Subharmonic injection-locking phenomena in synchrous oscillators,” Electron. Lett., vol. 26, no. 21, Oct. 1990.
[21] N. Fenichel, “Persistence and smoothness of invariant manifolds for
flows,” Indiana Univ. Math. J., vol. 21, no. 3, pp. 193–226, 1971.
[22] F. C. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural
Networks. New York: Springer-Verlag, 1997.
[23] F. X. Kaertner, “Determination of the correlation spectrum of oscillators with low noise,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 1,
pp. 90–101, Jan. 1989.
[24] T. Djurhuus, V. Krozer, J. Vidkjær, and T. Johansen, “Oscillator phase
noise: A geometrical approach,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 7, pp. 1373–1382, Jul. 2008.
[25] T. Djurhuus, V. Krozer, J. Vidkjær, and T. Johansen, “Nonlinear analysis of a cross-coupled quadrature harmonic oscillator,” IEEE Trans.
Circuits Syst. I, Fundam. Theory Appl., vol. 52, no. 11, pp. 2276–2285,
Nov. 2005.
[26] A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical
oscillators,” IEEE J. Solid-State Circuits, vol. 33, no. 2, pp. 179–194,
Feb. 1998.
[27] R. Adler, “A study of locking phenomena in oscillators,” Proc. IEEE,
vol. 61, no. 10, pp. 1380–1385, Oct. 1973.
Torsten Djurhuus received the M.S. and his Ph.D.
degrees in electrical engineering from the Technical
University of Denmark in 2003 and 2007, respectively.
From 2007 to 2009 he was a Research Assistant
at DTU Elektro, Technical University of Denmark.
Since October 2010 he has been working as a
Postdoc at the Johann Wolfgang Goethe University,
Frankfurt, Germany. His research interests include
nonlinear circuit analysis, circuit noise analysis,
MMIC design, and RF oscillator design.
Viktor Krozer (M’91–SM’03) received the
Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering at the Technical University Darmstadt in
1984 and 1991, respectively.
In 1991 he became Senior Scientist at the TU
Darmstadt working on high-temperature microwave
devices and circuits and submillimeter-wave electronics. From 1996 to 2002 he was a Professor at the
Technical University of Chemnitz, Germany. From
2002 to 2009 he was a Professor at Electromagnetic
Systems, DTU Elektro, Technical University of
Denmark, and was heading the Microwave Technology Group. Since 2009 Dr.
Krozer has been the endowed Oerlikon-Leibniz-Goethe Professor for Terahertz
Photonics at the Johann Wolfgang Goethe University, Frankfurt, Germany. His
research areas include terahertz electronics, MMIC, nonlinear circuit analysis
and design, device modeling, and remote sensing instrumentation.
Authorized licensed use limited to: National Tsing Hua Univ.. Downloaded on April 19,2023 at 08:22:28 UTC from IEEE Xplore. Restrictions apply.