Circuits and Systems, 2011, 2, 311-319
doi:10.4236/cs.2011.24043 Published Online October 2011 (http://www.scirp.org/journal/cs)
Derivation of Floquet Eigenvectors Displacement for
Optimal Design of LC Tank Pulsed Bias Oscillators
Stefano Perticaroli, Nikend Luli, Fabrizio Palma
Department of Information Engineering, Electronics and Telecommunications,
Sapienza Università di Roma, Rome, Italy
Received June 13, 2011; revised July 3, 2011; accepted July 22, 2011
Abstract
The paper presents an approximated and compact derivation of the mutual displacement of Floquet eigen-
vectors in a class of LC tank oscillators with time varying bias. In particular it refers to parallel tank oscilla-
tors of which the energy restoring can be modeled through a train of current pulses. Since Floquet eigenvec-
tors are acknowledged to give a correct decomposition of noise perturbations along the stable orbit in oscil-
lator's space state, an analytical and compact model of their displacement can provide useful criteria for de-
signers. The goal is to show, in a simplified case, the achievement of oscillators design oriented by eigen-
vectors. To this aim, minimization conditions of the effect of stationary and time varying noise as well as the
contribution of jitter noise introduced by driving electronics are deduced from analytical expression of ei-
genvectors displacement.
Keywords: Floquet Eigenvectors Noise Decomposition, Pulsed Bias Oscillator, Oscillator Phase Noise
1. Introduction
Noise decomposition through Floquet eigenvectors is
widely acknowledged to be a correct methodology for the
analytical treatment of phase noise in electronic oscilla-
tors [1]. For this reason, numerical algorithms which rely
also on the Floquet eigenvectors have been developed in
order to obtain accurate prediction of power density spec-
trum (PDS) around the fundamental oscillation frequency.
A well known example is found in the commercial simu-
lator SpectreRF that computes Floquet eigenvectors from
the shooting matrix and use them as a theoretical based
correction for the PNoise analysis [2]. Although elec-
tronic simulators offer results in good agreement with
effective measurements, they cannot be used to infer gen-
eral properties of the underlying system. Despite the great
amount of work presented even recently in literature [3,
4], since the pioneeristic work of Kaertner [5] and the
latter efforts accomplished by Hajimiri-Lee [6], Demir [7,
8] and Buonomo [9] not many significant theoretical con-
tributions have really extended the capability to develop
innovative techniques in the design of oscillators systems.
In authors opinion the innovation should be founded on
derivation of architecture-related expressions of Floquet
eigenvectors [10], leading to a proper design methodol-
ogy. This approach has been limited in the past by the fact
that, also in relatively simple cases, Floquet eigenvectors
cannot be obtained in analytical closed form.
Mutual displacement among eigenvectors along the pe-
riod of oscillation regulates indeed how noise perturba-
tions affect the spectral purity of oscillator system. In
particular projections of noise on the eigenvectors deter-
mine the PDS of the oscillator [11]. We notice that mu-
tual displacement depends in general on the quality factor
of tank and on the way the lost energy is restored, i.e. on
the chosen architecture. Even if in recent years several
new architectural solutions were proposed for the reduc-
tion of phase noise in CMOS technologies [12-15], none
of them was justified on eigenvectors based considera-
tions. In particular, to knowledge of the authors, in litera-
ture there is no attempt to establish a direct relationship
between Floquet eigenvectors and circuit parameters for a
certain class of oscillators.
In this paper we present an analytical derivation of
mutual displacement of Floquet eigenvectors and of their
relationship with design parameters in the class of the LC
tank oscillators with pulsed bias. First, we propose to in-
troduce a parametrical model for the pulsed bias concept.
Then we extend Floquet theory for the class in study in
order to analyze noise dynamics and thus to optimize

S. PERTICAROLI ET AL. 312
oscillators implemented through architectures with time
varying bias. Finally we validate the proposed analysis
and optimization criteria through the comparison with
numerical results from a dedicated Matlab simulator of
introduced model.
2. Simplified Model of LC Tank Pulsed Bias
Oscillator
In order to characterize the class of LC tank oscillators
with pulsed current bias we adopt the simplified parallel
RLC model in Figure 1: it presents only two state vari-
ables, corresponding to the capacitor voltage VC and the
inductor current IL of tank. This assumption can be seen
as a rather drastic simplification for a model of a real
oscillator, nevertheless, if pulsed bias circuitry does not
introduce parasitic comparable to those in tank, the addi-
tional state variables can be neglected.
In this model the energy refilling is demanded to
pulsed current generator i(t) which is controlled, in par-
ticular, by the capacitor voltage VC. Crossing of a thresh-
old |Vth| by the capacitor voltage, occurring at time de-
fined as Tth, triggers the accumulation of a fixed delay T1.
After this delay the current generator is turned on for a
time T2 at fixed amplitude Imax. The turning on and off
are assumed to be instantaneous, thus application of the
ideal pulses gives rise to non derivable points of the ca-
pacitor voltage. In order to clarify the aforementioned
timings and parameters of the model, a sketch of the
generator current and of the resulting capacitor voltage
waveform is reported in Figure 2.
The differential algebraic equations (DAE) describing
dynamic of the proposed model is reported in Equation (1).

 

11 1
()
10
0
C C
L L
Vt Vt RC Cit
C
It
It
L

 

 

 
 

   
 





(1)
We choose to not explicitly express the large signal
dependence of i(t) as a function of VC, since in this paper
we are interested in a variational analysis for the study of
noise only. However the existence of a stable orbit can be
easily proved by means of mathematical approach for the
analysis of nonlinear systems (e.g. the describing function
Figure 1. Model of the simplified LC tank pulsed bias oscil-
lator.
Figure 2. Sketch of capacitor voltage and current pulse of
the generator in time.
technique) and will be not reported here.
In Figure 3(a) sketch of the stable orbit and the projec-
tions onto eigenvectors of superimposed noise perturba-
tion b at certain time instant are reported. Due to the
adopted symmetrical model, every half of the orbit can be
subdivided into two different portion delimited by dashed
lines in Figure 3. A first one corresponds to the RLC
dumped evolution only and the second one corresponds to
the evolution when the refilling current pulse is also
active.
3. Floquet Theory for the Study of Noise
Floquet theory describes the periodical linear time vary-
ing (LPTV) response of oscillators systems to small per-
turbations superimposed on the stable oscillation orbit.
Floquet eigenvectors are usually extracted from Mono-
Figure 3. Sketch of the stable orbit and of the projections
onto eigenvectors of superimposed noise perturbation b at
certain time instant. Dashed lines indicate phase portion
when pulsed bias current is active in a half of period.

S. PERTICAROLI ET AL.313
dromy matrix. In general Monodromy matrix is obtained
as the product of sequence of eventually different transi-
tion matrices along one oscillation period. In our treat-
ment Monodromy matrix is calculated by following the
evolution of the state variation vector [vC iL]T in the sys-
tem along the orbit, apart from non derivable points.
Evolution is derived from (1) imposing i(t) = 0 and is
given by

  
11
10
C
L
vt vt
RC C
it it
L



 
 
 
  
 




C
L





, (2)
The state transition matrix Φ(t) of dumped RLC system
(2) results in
  
 
11 12
21 22
tt
ttt







(3)
Every component of (3) can be obtained by direct in-
tegration of (2) and hence assumes the form of dumped
sinusoidal evolution as expressed in Equation (4)
 
  
 
  
22
11
22
12
21
22
cos
sin
cos
sin
cos
cos
t
nn
t
nn
t
n
t
n
e
t
Le
t
e
tt
L
e
tt




t
t














 



 






 


(4)
where
2
00
0
00
0
0
12π
,1
1
,
222
1
2












 
  

  

 
n
n
n
LC T
R
QLRCQ
arctgarctgarctg Q
(5)
It has to be noticed that real period T differs in general
from the nominal period Tn as a main effect of the pulsed
bias.
The above described model holds until current pulse
edges are reached. Since at this time instants the capaci-
tor voltage is not derivable, we will need to introduce
Interface matrices [16,17] to properly describe state evo-
lution.
In the simplified adopted model only two Floquet
eigenvectors exist, with two corresponding eigenvalues.
We call “first eigenvector”, u1(t), the one which is tan-
gent to the orbit with unitary eigenvalue and the “second
eigenvector”, u2(t), the one with corresponding eigen-
value lower than 1. Once the evolution of the eigenvec-
tors is available, we may project the perturbation b(t)
multiplying it by inverse eigenvectors. The effect of
noise projection is a state deviation which evolves as the
eigenvector itself and sum in square to further noise pro-
jections along the orbit. Projections depend on the instant
when b(t) is applied, so that the system appears as peri-
odic time variant. In Figure 4(a) sketch of power density
spectrum of the eigenvectors is reported [11]. Contribu-
tion to the PDS of noise projected on the first eigenvec-
tor, can be described by a transfer function with square
modulus 1/ω2, where is the offset with respect to the
fundamental. We notice that at very low offset a cut-off
must be assumed, not reported in the figure, due to
nonlinear behaviour of the system for large values of
phase deviation [7]. Noise projected on the second ei-
genvector can be instead described by a transfer function
with square modulus 1/(γ2
2+ω2), where γ2 is the pole
pulsation related to the second Floquet multiplier.
As a result the main contribution to the overall power
density spectrum at frequencies close to the fundamental
arises from projection on the first eigenvector. On the
contrary contribution arising from the second eigenvec-
tor becomes relevant only at high frequency offsets
(ω>>γ2) due to its low-pass shape with respect to the
fundamental.
Only noise contributions parallel to u2 produce null
contributions on u1 and thus do not increase the phase
noise [1]. Floquet decomposition points out that eigen-
vectors, in general, are not orthogonal. As a result, as-
suming a white noise current generator in parallel to the
RLC resonator, as it will be discussed in next section, a
noise contribution in the direction [1 0]T (the voltage
Figure 4. Sketch of power density spectrum contributions
due to the eigenvectors.

S. PERTICAROLI ET AL. 314
variation axis) applied in correspondence of the voltage
maximum may not yield to a null projection on u1.
These considerations lead to investigate a pulsed bias
strategy, designed to activate the bias only when the pro-
jection of noise related to the biasing electronics on the
first eigenvector is at least around a minimum. Never-
theless a complete evaluation of the proposed architectc-
ture must take into account also the effect of eigenvec-
tors displacement due to the pulsed bias on all the other
noise sources present in the circuit.
As formerly stated, the eigenvectors evolutions are
almost everywhere derived from state transition matrix
of the dumped RLC system except for time instants when
the generator switches on/off. Since an oscillator has no
time reference, in the adopted model we may assume that
at t = 0 the first eigenvector must be equal to u1(0) =
[Vu1(0) Iu1(0)] = k[1 0]T, with null current com-
ponent (an eigenvector can always be defined by means
of a proportionality constant). Until the first discontinu-
ity, occurring at t = Tth + T1, eigenvectors evolutions can
then be described by the following equations normalized
to amplitude of superimposed perturbation
k
 



 



1
2
2
2
cos
sin
cos
sin
μt
n
u1 μt
u1 n
μt
n
uμt
un
et
Vt
ut== e
It t
F
et
β
Vt
ut== e
It t+β
F































(60)
where coefficient F depends on the quality of the tank
and on resonating elements as in (7) and β represents the
phase displacement between eigenvectors.

2
1
1
2

L
F= +
CQ
(7)
4. Phase Displacement between Eigenvectors
as a Function of Pulse Paramaters
Figures 5 and 6 show a sketch of the evolutions respec-
tively of first and second eigenvector. The effect of turn-
ing on and off of the bias current, taking place at t = Tth
+ T1 and t = Tth + T1 + T2 respectively, corresponds to
discontinuities of eigenvectors states, in direction [1 0]T
(the V voltage variation axis). Due to the introduction of
the Interface matrix theory [16], and as shown in Figure
7, we have to consider that any state variation along each
eigenvector, evolved from t = 0 until Tth, produces a de-
lay t with respect to the crossing of the threshold |Vth|.
The same delay is reported at the time of the turning on
and off of the current generator. The delay is calculated as
Figures 5. Sketch of evolution of the first eigenvector for a
half of the period.
Figures 6. Ske tch of evolution of the second eigenvector for
a half of the period.
Figure 7. Current of the generator in time with delay due to
the variation (a) and additional contributions at the pulse
edges (b).
the ratio of voltage variation to the time derivative of the
capacitor voltage at threshold crossing. Using a smart
notation with subscript J = 1,2 where XJ=1 = 0 and XJ=2 =
 , delay for both eigenvectors results in

S. PERTICAROLI ET AL.315

cos
th
th
T
J
nth J
CtT
e
tT
V




 
X
(8)
Implicit voltage variation at numerator ensures dimen-
sional balance in Equation (8). As a result of the delay, at
the pulse edges two additional contributions opposite in
sign sum to the former state variation. Since we are
dealing with a pulsed bias for hypothesis, we now as-
sume T2<<T and then we neglect the relative phase rota-
tion between the pulse edges in both eigenvectors evolu-
tions. As depicted in Figure 5, this allows us to ap-
proximate the effect of the bias current pulse as a unique
discontinuity of the state in direction [0 1]T (the I current
variation axis). We hence calculate discontinuity ampli-
tude for both eigenvectors using (6) and (8) as the sum of
opposite contributions approximating sinusoidal function
for small T2 in

max
2
Icos




 

th
th
T

J
nnth
CtT
J
I
eTT X
CV F
(9)
The overall state variation, including the effect due to
discontinuities, evolves along a new orbit. The second
part of the evolution can be obtained for both the eigen-
vectors as
 




Acos
Asin
 




 









μt
nJ
J
Jμt
JnJ J
tX
Vt
ut= =
It et+X
F






J

(10)
where A and

J are unknown general solution parame-
ters.
We notice that at t = Tth + T1 the voltage components
remain unaltered due to the assumption T2<<T, whereas
the current components of state variations are affected by
the  
J drops. This observation leads to define a system
of two non-linear equations in the two unknown A and

J:




1
1
1
1
()
1
()
1
()
1
()
1
Acos
cos
Asin
1sin .

 

 
















th
th
th
th
μTT
nth nJJ
μTT
nth nJ
μTT
nth nJJ
μTT
nth nJJ
eTTX
eTTX
eTT+X
F
eTT+X
F
I


(11)
In order to achieve an approximate solution of system
(11) we expand the trigonometric function containing

J in Mac Laurin series, limited to the first order and
obtain:


1
1
cos .
cos( )sin



 
Jnthn J
J
Jnthn
IFT TX
=IFT TX
Since in most of practical cases tank quality factor Q
is at least greater than 5 we further observe that

11
cos coscos1
22
1
J
arctg QQ
IF


 



 

 



(13)
thus we may reformulate (12) neglecting terms due to
pulse in the sum at denominator to solve the general pa-
rameter 
J for both the eigenvector as





11
1
21
2
cos
cos
cos
cos
nth n
nth n
IFT T
IFTT
β





 

 

2
(14)
where  
1 and 

2 account the phase discontinuities due
to the bias current pulse, respectively, for the first and
the second eigenvector.
Since, by the definition, eigenvectors are periodic, and,
due to the symmetry of the adopted model in the two
halves of period they cover the same phase angle, we
may infer that phase discontinuities caused by the bias
current pulse must be equal, i.e.
1


 (15).
From these observations, substituting the condition (15)
in Equation (14) and expanding  
J drops, it is straight-
forward to obtain the value of the phase displacement
between the eigenvectors u1 and u2, at least out of the
orbit portion when pulsed current is active as


1
2
nth n
TT2

   . (16)
We want to remark that in Equation (16)  is only a
function of the threshold crossing time, of the delay of
current pulse application, and of the tank quality factor
(related to parameter  ).
5. Optimal Phase Displacement between
Eigenvectors
Phase displacement between eigenvectors becomes here
the main tool toward the paper goal to offer design in-
sights for the reduction of phase noise which, in our
treatment, means to minimize projection of noise on the
first eigenvector. To this aim we are now going to dem-
onstrate optimum conditions on phase displacement be-
tween eigenvectors, in order to reduce noise due to bias,
parasitic resistance and jitter on accumulation of delay
time T1.
We assume perturbations to be realizations of white
Gaussian noise processes with zero time average. This is
not a restrictive assumption. In fact, also noise sources
J
(12)

S. PERTICAROLI ET AL. 316
with non-delta autocorrelations, i.e. 1

f
 with 

,
may be taken into account through the projections on the
eigenvectors. Referring to [5], in particular Sections 6
and 7, the power density of a flicker noise source is ob-
tained as an infinite sum of autocorrelation spectra of
statistically independent Ornstein-Uhlenbeck processes
(that are also Gaussian processes). Since we are dealing
with a parallel RLC tank, perturbations are introduced as
parallel noise current sources. Such noise current sources
cause a variation of the capacitor voltage, hence we
model the normalized noise perturbation through a con-
stant vector
1
0
b
b
V
bI








  . (17)
In order to determine the PDS we need to calculate the
projection of vector b on the base formed by the two ei-
genvectors. Integration along the orbit of the square value
of the projection on u1 leads to the c1 coefficient of Demir
[7], while projection on u2 must be multiplied before in-
tegration by a properly evaluated exponential factor [11].
We first search for the condition which zeroes the pro-
jection on u1 of noise due to the bias current, usually the
largest source of noise in integrated technologies. Bias
current noise is cyclostationary because it arises only
during the bias pulse. Assuming again T2<<T we may
consider constant the projection angle during the pulse,
then integration reduces to the multiplication of the pro-
jection by time T2. We notice that noise projection has to
be performed after the turning on of the bias current, after
the eigenvector discontinuity in Tth + T1 has occurred. As
depicted in Figures 5 and 6, the additional component is
in the direction [1 0]T, thus the necessary and sufficient
condition which ensures that once added the Interface
matrix component the eigenvector has null current com-
ponent is
 
21 21
1
0
0

 


th th
uT TbuT T0
. (18)
Following the definition of u2 in Equation (6) and sub-
stituting in its expression the phase displacement of
Equation (16), condition (18) is equivalent to:

1
11
()
()2 2
2.
nth
nthnth n
nth
TT
TTT T
T




 



0 
(19)
Equation (19) states that, in possible implementations
of pulsed bias oscillators, the |Vth| threshold should be
chosen around the zero crossing of the oscillation voltage.
We may further search for the minimization of the
contribution arising from parasitic resistance, which re-
mains active all along the orbit. We recall that, from
Equation (16) and following condition T2<<T which led
to (15) phase displacement between eigenvectors remains
constant along the entire evolution except for region
where the pulsed current bias is active.
However, since we assumed bias active for a short time
compared to period, the projections of noise on the two
eigenvectors, respectively u1p and u2p, are given by
1
2
sin( )
() sin( )
sin( )
() sin( )
n
p
n
p
t
ut
t
ut












(20)
then we obtain the integral of the square of the projec-
tions
22
122
00
22
222
00
11
() dsin ()d2
sin()sin()
11
() dsin ()d2
sin ()sin ()













TT
pn
TT
pn
T
ut ttt
T
ut ttt
(21)
As expected the two terms in Equation (21) are equal.
The first one is the Demir c1 coefficient [7], while the
second is only a majorant term in the expression of phase
noise due to u2.
The actual period T depends on the angle β between
the eigenvectors, however in hypothesis Q > 5 it results
very close to Tn (the nominal period). This leads to state
that factor sin2(β) predominates the result of integration
and allows us to infer that minimization of phase noise
distribution due to a stationary noise source occurs if the
condition β = ±/2 holds.
Further considerations can be derived from this result
and from condition expressed in (19). For example, we
notice that a jitter noise contribution arises from the elec-
tronic circuit when we introduce the delay T1. This jitter
is proportional to the delay itself and inversely propor-
tional to the power used to accumulate the delay [18].
The choice to avoid any delay may be of interest in order
to save an additional source of noise and to reduce en-
ergy dissipation [19]. Imposing T1 = 0 in combination
with condition (21) we obtain that the optimum choice is
to set

10
ππ
224  







nth nth
T
TT
(22)
It results that condition on minimization of stationary
noise (21) together with condition on minimization of
accumulated jitter can be simultaneously satisfied only
for a very low quality factor (Q = 0.5). Then in any prac-

S. PERTICAROLI ET AL.317
tical case a suboptimal condition must be searched, e.g.
accepting a non zero delay time T1.
Hence the last optimum condition we can search for is
derived from minimization of both resistance noise and
bias current noise for a given quality factor Q and in pres-
ence of a delay time of the current pulse T1. Using results
from Equations (19) and (21) in Equation (16) we obtain
1
ππ 1
2 2arctg
222


 

nTQ
(23)
6. Numerical Comparison and Discussion
In order to evaluate the accuracy of the proposed analyti-
cal expressions we need to compare them with numerical
simulations. In fact, at the knowledge of the authors, there
is no measurement setup and/or post-processing that can
extract eigenvectors from measure of physical quantities
in a real circuital implementation.
Moreover there are no commercial circuit simulator
which can compute Floquet eigenvectors in presence of
discontinuities in space state. Then we developed a dedi-
cated Matlab simulator for the model defined in Section
III. The simulator derives the Monodromy matrix through
a shooting algorithm which use Interface matrices for the
treatment of discontinuities.
Even if the proposed analysis is not dependent on the
oscillation period, in the following simulations we fixed
frequency at fn = 
n/(2
 5 GHz. Along with frequency,
we choose to keep constant quality factor Q = 7.5, pulse
duration T2 = T/15 and amplitude Imax = 60 mA ensuring
the oscillator to be refilled by a fixed amount of energy
and to be perturbed by the same amount of noise power.
Moreover in case Vth ≠ 0 V we keep constant also the sum
Tth+T1, allowing to reach the same limit cycle of Vth = 0
V simulations set. The two case of study, respectively Vth
≠ 0 V and Vth ≠ 0 V, differ only in the starting point with
respect to period of T1 delay accumulation and in the
weight of jitter contribution which is proportional to T1.
All kind of noise sources are modeled as parallel current
generators that adds to the model in Figure 1. Stationary
noise is introduced as thermal noise of the parasitic resis-
tance depending on quality factor Q through 4kBTamb/R
[A2/Hz] relation (kB is Boltzmann constant Tamb is ambi-
ent temperature in Kelvin) whereas the cyclostationary
noise is introduced as shot noise (modeling eventual de-
vices) related to current Imax through 2qImax [A2/Hz] rela-
tion (q is electron charge). Jitter noise source is instead
considered as an additional time shift due to the accumu-
lation of both T1 and T2 delay and it is added to the one
due to evolution of the initial variation until time of thre-
shold crossing Tth (8). Jitter source is the result of the
charging of a capacitor through a dissipative media (again
assume a MOS device) so it has been modeled through
8kBTamb  T
1(KMOS/ID
3)0.5 [s] relation where is
 transistor
noise constant, KMOS and ID are respectively the large sig-
nal current gain and drain current of MOS device. We
choose to fix drain current to ID = Imax/10 = 6 mA. In fact
the eventual auxiliary circuit suited for the introduction of
desired delay must necessarily have a lower power con-
sumption with respect the refilling process of the tank in a
real design.
In Figure 8 the phase displacement β is reported in
function of T1 normalized to period T in two cases Vth = 0
V and Vth = 0.75 V.
Amplitude and period of oscillation, as stated before,
vary in function of pulse position between [2.45 V, 3.11
V] and [4.8 GHz, 5.2 GHz] respectively.
It can be observed a good match between simulated
and calculated (16) trend with a maximum absolute error
of about 13 in the evaluated range for case Vth = 0.75 V.
Former considerations on factor sin2(β) in case Vth = 0
V suggest that in correspondence of T1/T ≈ 0.223 when
 ≈ /2 a minimization of noise projection on the first
eigenvector should occur. Moreover for T1/T ≈ 0.223 the
maximum of amplitude and the nominal frequency are
obtained. In case V
th = 0.75 V the enhancement of time
shift (8) due to late starting point Tth of delay accumula-
tion overwhelm any reduction of jitter weight due to
shorter T1.
In Figure 9 the simulated PDS of the oscillator is re-
ported for three typical frequency offsets (100 KHz,
1 MHz and 10 MHz) from the fundamental in function of
Figure 8. Eigenvectors phase displacement out of the fast
region simulated (dotted trace) and calculated (continue
trace) in function of normalized pulse position T1 for T2 =
T/15, Q = 7.5 in cases Vth = 0 V and |Vth| = 0.75 V.

S. PERTICAROLI ET AL. 318
Figure 9. Simulated PDS evaluated at three different offset
from fundamental frequency in function of normali zed pulse
position T1 for T2 = T/15, Q = 7 .5 in cases Vth = 0 V (continue
trace) and |Vth| = 0.75 V (dotted trace).
T1 in case of Vth = 0 V and Vth = 0.75V.
PDS has been computed following [11]. It can be ob-
served that the continue trace related to the lowest offset,
where the major contribution to PDS arises from projec-
tion on first eigenvector, exhibits the minimum exactly in
correspondence of the prediction made by means of  ≈
/2 criterion for Vth = 0 V. At higher offsets contributions
arising from projections on cross-correlations between
eigenvectors and on second eigenvector become prepon-
derant and the minima shift toward earlier T1/T ratio for
both Vth cases. However it can be noticed that the Vth = 0
V case achieves the better performance at the lowest off-
set whereas the Vth = 0.75 V case reaches the best results
at large offsets for quite all T1/T ratio.
In order to justify this observed trend we propose to
monitor noise introduced through first eigenvector (refer-
ring to u1p(t) noise projection) in term of injected level of
energy from all noise sources on first eigenvector.
In Figure 10 the energy injected along the normalized
period on first eigenvector by perturbation vector b is
reported in cases Vth = 0 V and Vth = 0.75 V for T1/T =
0.24, T2 = T/15, Q = 7.5. It can be noticed that the zero
of injected energy occurs during the current bias pulse
only in case Vth = 0 V. For any Vth ≠ 0 V the zero shifts
toward later time instant. Moreover, as a result of non
orthogonal phase displacement between eigenvectors in
case Vth ≠ 0 V the maxima of injected energy on first
eigenvector (when only stationary and jitter noise
sources are present) can be greatly increased, thus de-
grading phase noise especially at low offsets. This ob-
servation is congruent with former defined design crite-
rion (19) and validates the proposed analysis.
Figure 10. Simulated energy injected on first eigenvector by
perturbation vector b in cases Vth = 0 V (continue trace) and
|Vth| = 0.75 V (dotted trace) on left y-axis and capacitor volt-
age VC right y-axis along oscillation period for T1/T = 0.24,
T2 = T/15, Q = 7.5.
7. Concluding Remarks
The paper presented an approximated and compact deri-
vation of the mutual displacement of Floquet eigenvec-
tors in the class of parallel RLC tank oscillators with
pulsed current bias. Mutual displacement was proved to
be strongly connected with the chosen architecture and
consequently to determine phase noise distribution. As
appreciable result, the derived expression of displace-
ment is compact and straightforward, thus it can suggest
primary guide lines for designers in the field of oscilla-
tors architectures with time varying bias.
In particular we demonstrated conditions for minimi-
zation of stationary as well as cyclostationary noise. Also
the jitter noise introduced in the positioning of the pulsed
bias is taken into account and its relation with noise pro-
jections on the eigenvectors is determined. Minimization
conditions were formulated using parameters of the pro-
posed model for the pulsed bias class, allowing to di-
rectly infer the circuit design. Finally the analytical re-
sults are compared with a dedicated simulator, showing
that proposed criteria for noise reduction are congruent
with observed trend in the simulated PDS.
Future works will provide developments and exten-
sions of the proposed analytical noise treatment to VCOs
and PLLs systems.
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