The simplified model we refer to is reported in 
. Here and in the following all vectors and parameters quantities are in real numbers field if not differently specified. Coupling between resonators in a positive feedback configuration is accomplished by pulsed current generators emulating active devices. This model can be seen as a rather drastic simplification of a real oscillator, however if we assume the parasitic introduced by transconductors to be much smaller than those in resonators they can be merged in resonators themselves, thus avoiding to increment the effective space state dimension.
The model assumes a current pulse of fixed duration injected in the driven resonator and triggered by crossing 0V common mode level of the capacitance voltage of the driving oscillator. The impulsive characteristic of the generators is described by the piecewise linear expressions in (1) where 
and 
represent respectively the limiting current and the pulse width. The 
represents time instant of threshold crossing of the driver oscillator. 
and 
indicate respectively the driver and the driven oscillator. There are two pulses in each period with alternated sign depending on the sign of driver voltage derivative.
Furthermore the “±” takes into account the possible degree of freedom in determination of 
reciprocal phase. Sign choice determines indeed the quadrature relationship between the oscillators. In order to roughly depict the role of the sign one can notice that, assuming in (1) the sign “-” for current pulse in OSC_Q at the zero crossing of OSC_I with positive derivative, a negative
pulse is applied to OSC_Q. Such pulse corresponds to a negative half-wave for driven oscillator OSC_Q that becomes delayed of 
with respect to driver oscillator OSC_I. In the following calculations we shall always assume the sign “-” for current pulse in OSC_Q.
In time intervals when the pulse generators are not active, following RLC system equations, eigenvectors can be described by
refer to the relative eigenvector number. 
and 
are unknown eigenvector amplitudes, 
and 
are the unknown phase displacements between eigenvector components in the two resonators. The 
factor accounts for the ratio between currents and voltages whereas 
factor accounts for relative phase between currents and voltages in a resonator and are given respectively by
as the amplitude of components related to OSC_Q at 
rather than at
. We recall that in a stable oscillator always exists a “first” eigenvector with unitary eigenvalue and with components tangent to the space state orbit. This property makes first eigenvector behavior directly related to the oscillator state. E.g. the threshold crossing which triggers the bias current pulse can be indicated either in term of eigenvector components or of oscillator state. In the two oscillators the threshold is indeed crossed as the capacitance voltage zeroes, i.e. when current component of eigenvector is null
. Since eigenvectors components can always be defined by a proportionality constant, we choose to fix 
with no loss of generality. Similarly we assume the threshold of OSC_I is crossed at 
and for
. We recall once more that the value of the eigenvalue of first eigenvector must be one. This aspect brings to four the number of independent conditions on eigenvector components. We notice that Equations (2) contain only two unknowns in front of four conditions. The two exceeding conditions indeed will lead to determination of both oscillation period and amplitude.
is short enough compared to actual oscillation period T, the overall effect of the current pulse can be approximated with a drop in current components of eigenvectors. In [
currents and voltages components in resonators are not exactly in quadrature, then current drops may produce both amplitude variations and phase shifts of the orbit. In 
and 
for the two oscillators.
the phase evolution in (2) between application of the bias pulse to OSC_Q resonator and the time instant when its current component, 
, zeroes (0 V threshold crossing by OSC_Q voltage component). We define as 
the phase change between the application of the bias pulse to OSC_I resonator and the instant when its current component, 
, zeroes (0V threshold crossing by OSC_I voltage component).
but with a different amplitude and an additional phase shift.
to OSC_Q resonator giving rise to phase shift
phase, 
(the OSC_Q threshold) is crossed and a second current pulse is applied this time to OSC_I. The second induced phase shift 
retains memory of former phase shift 
resulting in
is dependent on the amplitude 
which is still an unknown. In order not to run through an intricate calculation we now attempt a by inspection solution (requiring a later verification). Basing again on symmetry we assume
phase shift is obtained
.
is well approximated with resonator quality factor just greater then few unities. In the simplified case of 
expression (11) is verified only for
. This assumption implies 
, i.e. the oscillation period is equal to the natural period of damped RLC resonator. The assumption would also lead to state that
. In sections 3.2 and 3.3 we are going to perform further calculations on remaining eigenvectors under this particular assumption. Then in section 5 this approximation will be verified through comparison with a dedicated Matlab simulator.
we may assume 
in (2) and this ensures that OSC_Q is delayed with respect to OSC_I, thus confirming the effect of signs choice in (1).
by (9), we may impose the condition only to one component of eigenvector. If we consider 
its value must be equal to current state component at time origin changed in sign
obtaining
and
) it must be solved in conjunction with evolution of OSC_I components. We may calculate the evolution of 
at 
just after the current drop is added by the bias pulse (superscript ‘+’ indicates in the limit from the right). In particular 
can be expressed through (5) at a time instant 
when the OSC_Q component can be easily calculated using reflection condition as
. We hence obtain
the 
zeroes. Reflection conditions can now be imposed on the voltage component 
. A straightforward consequence is
. Again 
and 
are unknown eigenvectors amplitudes, 
and 
are the unknown phase displacements between eigenvector components in the two resonators.
. Since second eigenvector cannot be related to the oscillator state evolution, the only additional condition we may impose is the independence from the first one. We recall we are going again to perform the calculation under the particular assumption that 
(which corresponds to assume
), i.e. a damped sinusoidal evolution must cover a phase of 
in a semiperiod.
a first current pulse is applied to OSC_Q whereas at 
the second current pulse (in first half of period) is applied to OSC_I.
, thus at 
the first eigenvector has the OSC_Q components in the {I} current axis direction and the OSC_I in the {V} voltage axis direction.
for OSC_Q at least a finite component in the {V} direction and for OSC_I a finite component in the {I} direction.
a total phase evolution 
is achieved. Damped sinusoidal evolution (which is equal in all the eigenvectors) is already 
and no additional phase can be added.
and if the OSC_Q voltage components is null at
.
and
.
damping of the system. There is no further condition to be applied in order to calculate the unknown amplitude
. This means that reflection conditions on second and third eigenvectors are satisfied for any vectors laying in the plane 
where
, i.e. that there are two independent eigenvectors with the same eigenvalue:
and
.
and 
are unknown eigenvectors amplitudes, 
and 
are the unknown phase displacements between eigenvector components in the two resonators.
, 
, and
.
the phase propagation of both OSC_I and OSC_Q amounts in total to
. Thus also 
must have null values of phase steps. This condition occurs if one of the following two conditions is satisfied:
and
. Null values of voltage displacement components at threshold crossing corresponding to null current drops, but this condition is completely dependent on eigenvectors 
and
;
and
. Eigenvector is parallel to current drops at the time of thresholds crossings thus giving rise to null phase steps. The condition with sign “+” is that of
.
then we write at 
the independence condition among the four eigenvectors
and 
we obtain
The fourth eigenvector evolution is reported in 
and 
for (9) we may impose the reflection conditions only to one component: after a semi-period evolution any eigenvector component must reach a value which is the square root of 
times its values at 
changed in sign. We follow the calculation performed for the first eigenvector. For OSC_Q at 
we have
at 
where the current drop is added due to the bias pulse. ΔI1_I can be expressed through (5). Pulse on OSC_I depends on the amplitude 
The reflection condition on OSC_I is written as
h4_Q = 1/h4_Q = 1.
. Recalling that the reflection condition can be written as
we know that
component of phase noise close to the fundamental. We can thus try to evaluate the contribution of projection of the two cited main noise sources.
and 
current pulses are found on OSC_Q. We model a normalized white Gaussian process as two additional parallel and independent current sources (one source per resonator). Such noisy process generates a state variation in the direction 
for OSC_Q. This variation is in the plane of eigenvectors 
and 
and orthogonal to
. Thus it implies a null projection onto the first eigenvector.
and 
the pulse is in OSC_I. In this case the process generates a state variation in the direction
. Eigenvectors 
and 
after a phase evolution of 
still span a plane which contains this vector and thus remains orthogonal to
. This implies again a null projection onto the first eigenvector.
is not decreased by the exponential damping due to the eigenvalue
, non orthogonal eigenvectors reflect unavoidably in a phase noise enhancement.
generated by the parasitic resistance in one of the two resonant circuit and verify the condition of minimum projection in a period. We extract in (33) the left eigenvectors v at 
inverting the right eigenvectors matrix u derived from former results. The components of first eigenvector are normalized to one in order to make more explicit the result of projection even if a normalization is not necessary. Being uncorrelated, noise contributions from the two resonators can be considered independently. Noise has a fixed direction while each component of the eigenvectors rotates with pulsation ωn as given in Equation (2). For the sake of simplicity we may change the reference assuming new axes solid with the eigenvectors, so that noise components appear as a rotating 
vector. At every time instant the total energy from the two noise components is maintained constant.
and 
. It is worth to notice that the projections onto first and fourth eigenvectors are identical with coefficient 1/2 while there is a projection on eigenvector 
with coefficient 1.
and
. Projections onto the first and fourth eigenvectors are again identical with coefficient 1/2 while this time we have projection on eigenvector 
with coefficient 1.
and 
producing ideal resonance at 
even if the developed theory is not frequency dependent. We fix also the total charge of a single current pulse in
. In 
(error always <1%) whereas calculated 
is underestimated with respect to the simulated one (error <5% for
). Underestimation is due to 
assumption which leads to higher error in determination of expression (32).
and 
noise perturbation vectors onto the first eigenvector in case
. Such noise vectors describe both the effect of bias current generator and parasitic resistance noise respectively of
capacitor. Absence of any increase above value 1/2 of projection is the effect of the eigenvectors orthogonality.