^{*}

Using a new Admittance-based model for electrical noise able to handle Fluctuations and Dissipations of electrical energy, we explain the phase noise of oscillators that use feedback around L-C resonators. We show that Fluctuations produce the Line Broadening of their output spectrum around its mean frequency f
_{0} and that the Pedestal of phase noise far from f
_{0} comes from Dissipations modified by the feedback electronics. The charge noise power 4FkT/R C
^{2}/s that disturbs the otherwise periodic fluctuation of charge these oscillators aim to sustain in their L-C-R resonator, is what creates their phase noise proportional to Leeson’s noise figure F and to the charge noise power 4kT/R C
^{2}/s of their capacitance C that today’s modelling would consider as the current noise density in A
^{2}/Hz of their resistance R. Linked with this (A
^{2}/Hz?C
^{2}/s) equivalence, R becomes a random series in time of discrete chances to Dissipate energy in Thermal Equilibrium (TE) giving a similar series of discrete Conversions of electrical energy into heat when the resonator is out of TE due to the Signal power it handles. Therefore, phase noise reflects the way oscillators sense thermal exchanges of energy with their environment.

In a previous paper under this title [_{Ref}, a Pedestal of electrical noise was generated. This Pedestal of 50% the amplitude of the native noise but wider bandwidth was due to the confusing action of the 50% noise the CF samples in quadrature with the carrier whose amplitude it aims to sustain in time. This is so because the CF implicit in the Automatic Level Control (ALC) systems or limiters of actual oscillators is phase-locked to the carrier whose amplitude it has to keep in time. This subtle effect was shown in a convenient resonator of, which was a capacitor of capacitance C shunted by a resistance R to account for its losses. Accordingly to [_{Ref} plus a small amplitude offset that is the error signal driving the CF towards its goal: to counterbalance the excess of PF used during the start of the oscillator.When this counterbalance was achieved, this oscillator of sustained a voltage where v_{ε}(t) was a constant average value plus electrical noise superposed to it. The reference V_{Ref}(t_{n}) that the CF needs at instant n to clamp the output amplitude was available from the reference V_{Ref}(t_{n}_{–}_{1}) at instant n–1 because in this “convenient oscillator”. This availability of V_{Ref}(t_{n}), which is not possible when f_{0} ≠ 0 because it would require having in advance an electrical reference of the signal the oscillator is going to create, allowed us to show the origin of the aforesaid Pedestal of electrical noise of amplitude 2FkTR V^{2}/Hz. Whereas the 50% of noise power sampled in phase by the CF was heavily damped as expected, the 50% noise power sampled in quadrature (e.g. “midway” 0˚ for NF and 180º for PF) was enhanced by the CF and gave the aforesaid Pedestal. As we advanced in [_{0} ≠ 0 would require a sinusoidal reference V_{Ref}(t) whose generation in advance (e.g. just before to be used) didn’t help to explain the noise Pedestal, whence it can be seen the usefulness of the “resonator of” we handled in [

Shunting the R-C parallel circuit of a capacitor with a finite inductance L ≠ 0 one gets an L-C-R parallel resonator whose resonance frequency f_{0} ≠ 0 allows oscillators repeating phase each seconds. Actually, this repetition exactly each T_{0} seconds is impossible, thus meaning that the spectrum of their output signal won’t give a “line” or d(f – f_{0}) function. Instead, it will have a non null width due to the charge noise existing in the resonator at temperature T, as we will show. This noise coming from the charge noise of the lossy resonator and from the noise added by the electronics, both collected by Leeson through an effective noise figure F [^{2}/s [

Before handling resonators with f_{0} ≠ 0, let’s recall that to sustain the amplitude of their output signal v(t) in time endures to sample v(t) each T_{0} seconds and the ALC system or limiter will react from this set of sampled data. Due to the phase noise (jitter in time domain) of the own amplitude thus generated, this sampling won’t be done exactly each T_{0} (or each T_{0}/2 seconds by sampling positive and negative peaks), although we will consider it as fast and accurate enough to allow the ALC system to handle properly amplitude changes with spectral content up to f_{0}/2 or up to f_{0} by sampling each T_{0}/2, accordingly to Nyquist sampling theorem. The high speed this sampling rate could provide to the ALC system is not used in general because amplitude changes endure energy ones in the resonator. Since the quality factor Q_{0} of an L-C-R resonator at its resonance frequency is: “π times the lifetime of its output voltage v(t) measured in periods T_{0}”, amplitude changes during one period in high-Q resonators will be small. Thus, the aforesaid sampling rate will work well for quartz resonators like that of [_{0} factors over 10^{4} are often found. This allows considering that the CF associated to the ALC system or limiter works as expected and since these electromechanical resonators use to be studied by highly selective L-C-R circuits, our results can be applied to them easily.

It’s well known that the sinusoidal voltage and current existing at any frequency f in an electrical Susceptance are in-quadrature. To work in parallel mode let’s use the Admittance function of frequency whose real part is Conductance G(jω) and whose imaginary part is Susceptance B(jω):, where the imaginary unit j means that currents through G(jω) and those through B(jω) are in-quadrature. Considering a sinusoidal voltage existing between the two terminals of Y(jω), currents through B(jω) allow Fluctuations of electrical energy in Y(jω) whereas those through G(jω) lead to Dissipations of electrical energy [

_{S} in series with the inductance L_{S} of an inductor shunting C, the circuit transform leading to the circuit of

A native L_{S}-C-R_{S} series resonator is thus replaced by its parallel equivalent circuit of

mittance formed by C and R in parallel. This noise comes from a random series of Thermal Actions (TA) that occur in C at an average rate l_{T} (TAs per second), given by [

where q is the electronic charge and is the thermal voltage at Temperature T.

Each TA triggers a Device Reaction (DR) aiming to remove the previous Fluctuation of energy in C due to the TA. The use of the parallel circuit of _{0}. Since these random DRs occur in time at the average rate λ_{T} of (1), the spectral content of the noise they will give will be like that of _{0} and with half its bandwidth due to the two times larger time constant (2RC) of

With an output carrier of non null frequency f_{0} ≠ 0 (not the “dc carrier” used to show basic ideas on a CF in [_{0} in the tens of MHz for example. This allows using a narrow-band approach around f_{0} to speed calculations (e.g. the factor Q_{0} of the L-C-R of _{0}, but for Q(f_{0}) = Q_{0} = 100, its value and meaning remains for a sideband frequency 1.001f_{0}). From the above it isn’t difficult to realize that a narrow-band CF working synchronously with the carrier at frequency f_{0} ≠ 0 will damp well the noise 2FkTR V^{2}/Hz it sees in phase, whereas the noise 2FkTR V^{2}/Hz it sees with phase error of –90˚ will mislead it so as to create the Pedestal of 2FkTR V^{2}/Hz around f_{0} shown in [_{0} that we called Technical phase noise in [

It is worth noting that the oscillating voltage of _{0}_{ }≠ 0 has two times larger lifetime () than that of the energy present in the resonator with studied in [_{c}/2 around f_{0}, see

Since the noise power Dissipated by R does not depend on the L shunting C because Equipartition sets the mean square voltage noise in C [^{2}/Hz amplitude for as it will happen in high-Q_{0} resonators where. This reasoning becomes less straightforward in low-Q_{0} resonators where f_{C} ≈ f_{0} and they won’t be considered for simplicity. The case with of [

This reasoning giving directly the spectrum of thermal noise in L-C-R resonators considers that the energy each DR dissipates is no other than the Fluctuation of energy stored by its preceding TA [_{T} of TA-DR pairs only depends on R since λ_{T} defines G by (1). It’s worth noting that

in _{0} where the Pedestal of Phase Noise [3,7,8] meets the Lorentzian line proposed in [

Mimicking what we did in [_{0} in C from its own thermal noise and to sustain it in time once it has reached an amplitude close to a reference V_{Ref}(t). Knowing that the ideal L-C resonator for which _{0} where, _{FB} to overcompensate its losses represented by R. This is done by feeding-back a current by the network of transconductance V/A, the same type of feedback used in _{FB} = R, this PF would compensate exactly the power lost at each instant in R, thus the power lost in R at any f. Although this exact compensation will fail at high f because the finite bandwidth BW_{FB} of the feedback of _{0}, provided a fast enough electronics is used. Given that the effects of any phase error in the loop due to the finite BW_{FB} and its associated phase noise were shown in [_{0}.

To start the oscillator from thermal noise of C, the PF of _{1} of [_{start}, will build the oscillating v(t) whose amplitude will drive the CF of the ALC system or limiter. Mimicking [

of PF once the aimed amplitude is reached. This is a CF whose implementation was discussed in [_{ε}(t) would be the difference between the sinusoidal signal v(t) of amplitude V_{0} and a reference signal of the same frequency and phase, but slightly lower amplitude V_{Ref} to generate this error signal. Thus, v_{ε}(t) would be a sinusoidal signal synchronous with v(t), with amplitude The NF of v_{ε}(t) through the transconductance β_{ALC} would counterbalance at each instant the excess of transconductance ∆β_{Y} used during t_{start}. Using the Clamping Factor of [_{Y} is driven by v(t), which is times larger than the v_{ε}(t) signal driving the CF (see Equation (8) in [

Thus, the transconductance β_{ALC} that feeds back negatively the resonator with the error signal v_{ε}(t) will be much higher than the transconductance that allows the reliable start of the oscillator (e.g. by a loop gain as we used in [

Accepting a 0.1% amplitude error and T_{start} = 2 (e.g. R_{FB}_{ }= R/2) like those values used in [^{3}. Using these values in (2) we find that the NF counterbalancing the excess of PF to clamp the output amplitude at 1.001V_{Ref} will be shunting the resonator by a resistance R_{DIF} = R/1001. This will be so for any signal affecting v_{ε}(t) as the random series of DRs that form the noise of C. Because DRs appear randomly in time, they endure 50% noise in-phase with v_{ε}(t) or with the carrier to which the CF is phase-locked, and 50% noise inquadrature with v_{ε}(t) (recall ^{2}/Hz around f_{0} with bandwidth f_{c}) will be enhanced by the broadening of its spectrum away from ±f_{c}/2 [

As we discuss in [_{0} of the resonator and on the loop gain T_{start} and Clamping Factor CL used in the design of the oscillator. Added to this, the CL of a limiter would have to be taken in an average form, because as the output amplitude approaches its limit, the clamping action becomes harder. This means that the bandwidth of the Pedestal shown in

In the model for electrical noise of [_{T} s^{–}^{1} of (1). Since each DR comes from the integration in time of the impulsive current of each preceding TA, a possible Phase Modulation (PM) of the carrier by DRs will be equivalent to its Frequency Modulation (FM) by TAs. This way, the PM approach to Phase Noise of [_{m}(t) and PM due to the time integral of x_{m}(t). Concerning noise, it’s worth mentioning that when the L-C-R resonator is in Thermal Equilibrium (TE), its noise spectrum is that of _{0}, the noise increases by F and also split into the damped noise and Pedestal of ^{2}/Hz increased by the noise of the electronics and by any small heating effect, both included in F) together with two noises that only exist out of TE when the resonator stores the energy

U_{E} corresponding to the amplitude V_{0} of v(t).

Since v(t) is quite a sinusoid let’s have some figures by using. Therefore, the energy stored in the resonator is: J and the average power converted into heat by R (e.g. the mean power leaving the resonator as heat) is:. This leakage of energy per unit time is the price we pay to store U_{E} in this resonator out of TE. Another price we pay concerns the purity of the voltage v(t) on C because the aimed exchange of energy at 2f_{0} shown in _{0} volts peak (or CV_{0} Coulombs peak) we aim to have in C. Considering the distinction made in [

Considering that the damping of a DR during a period of v(t) is small, _{0}. It’s worth realizing that a null damping of this DR triggered by a TA would mean that this resonator has no losses, but having suffered a TA it must be lossy [_{T}/f_{0}_{ }≈ 1) would be vanishingly small: Using (1) at room T, a Q_{0}_{ }≈ 0.3 C/q would appear. Thus, _{0} > 50 for example. Since each TA is a charge noise of one electron in C, it always gives the same voltage shift ±q/C V no matter the instant α it takes place. However, the Amplitude Modulation

(AM) it produces depends on α [7,8], reaching its maximum for α = T_{0}/4 or α = 3T_{0}/4, when the charge in C has its peak value Q_{p}_{ }= CV_{0} C. From typical circuit values we can say that this non null AM is negligible however, because in an L-C tank with C = 16 pF and V_{0} = 10 V, the peak charge appearing in one of the plates of C is: Q_{p} = 1.6 ´ 10^{–}^{10} C, thus N ≈ 10^{9} electrons. The change of ±1 electron in N at α = T_{0}/4 by a TA would be an AM of 0.001 parts per million (–180 dB) that would be lower for AM due to TAs taking place at other instants. Thus, we won’t consider this AM in this introductory paper on phase noise, although it can play a role in high-frequency, low-power oscillators.

The spectrum of the output signal will contain both the Damped and the Feedback-induced Pedestal of noise shown in _{0} due to the “carrier” whose amplitude is kept by the CF or better said: a broadened line around f_{0}, because _{0}/4 or α = 3T_{0}/4. Since each TA displaces one electron in C, it shifts its voltage by V depending on its sign and the oscillation continues with new amplitude A and phase (ω_{0}t + fD). Thus we have:

Since each TA is an impulsive current unable to change the magnetic energy stored in the resonator (this requires some elapsed time), it only will modify the instantaneous energy of C to pass its charge Q(t) to Q(t) ± q C. The associated energy change is therefore:

thus equal to the Fluctuation J needed to displace one electron between plates of C (or to break charge neutrality separating +q and –q charges in C) plus the energy required to move a charge q in a region where charge neutrality already was broken by opposed charges like the dipolar charge of C that is the source of its voltage between terminals.

The very different scale for ∆U and qV_{0} in actual circuits appears by considering their ratio qV_{0}/∆U = 2N, for N being the peak number of electrons in one of the plates of C. For C = 16 pF and V_{0} = 5 V we have: qV_{0}/∆U = 10^{9}, thus meaning that the addition of one electron to the negative plate of this circuit needs a 10^{9} times higher energy than the Fluctuation of energy ∆U required to displace one electron between the plates of C with V_{0} = 0. This huge value raises this question: Where comes from this huge energy when a TA makes an electron to appear as a charge –q on the negative plate of C? A likely source is [_{0} = 5 V, the electron borrows a small fraction (1%) of the energy U_{f} it had as a free carrier in the CB, thus releasing only 0.99U_{f} J as heat in the negative plate that collects it. For a TA of opposed sign in which the electron appeared as a charge –q at the positive plate of C with V_{0} = 5 V, an energy 1.01U_{f} J would be released as heat on the positive plate on its arrival (e.g. the energy U_{f} it had as a free carrier in the CB plus the energy acquired from the electric field in C by an electron passing from its negative plate to its positive one).

Due to the equal probability for positive and negative TAs, the average energy Converted into heat by each TA is U_{f} and this account well for Joule effect accordingly to [

From (4) and (5), the phase shift ∆f produced by a TA occurring at time t within the period is [

Using N, the peak number of electrons accumulated in the negative plate of C without TAs, (6) becomes:

Because TAs have the same probability to increase N than to decrease it, the average phase shift resulting from (7) for the l_{T} TAs per second given by (1) is null, but this is not so for the mean square phase shift due to the huge amount of TAs taking place within a period of the output signal. Although it’s easy to show that for N values like those found in actual oscillators (e.g. N > 10^{6}) an increment or decrement of one electron gives a similar ∆f shift with opposed sign, we prefer to use the equal number of positive and negative TAs on average to add the square of (7) for a positive TA (e.g. with its ± signs replaced by + signs) to the square of (7) with its ± signs replaced by – ones, thus obtaining twice the mean square phase shift of a TA taking place at time t in C. Representing half this sum in the first quarter of period () we obtain the s-curve shown in

For, this averaging of positive and negative TAs is numerically irrelevant and one can take directly the s-curve of ^{2}. Since this s-curve and fit very well, we can integrate directly ∆f(ω_{0}t) to obtain the same mean value. Therefore, the mean square phase modulation (PM) of each TA is:

Although (8) is the mean square PM expected for TAs

occurring in the first quarter of the period, the mean square PM expected for TAs occurring in other quarters is the same because the s-curve for is the mirror image of ^{2}. Since (1) is the rate of TAs giving the charge noise power 4kT/R C^{2}/s of the capacitance C [^{2}/Hz of the resistance R) the extra noise added by the feedback electronics, collected by the effective Noise _{0} by Fl_{T} TAs per second disturbing the otherwise sinusoidal carrier of amplitude V_{0}, will be:

when a period finishes, a new one starts and the phase modulation accumulated in the finished period is lost. Thus the way the phase of v(t) is degraded by TAs as time passes within each period will be:

This linear dependence with time comes from the cuasi-linear, cuasi-continuous accumulation of TAs as time passes due to its huge rate Fλ_{T}. Considering now the Phase Noise model of [

Thus, this nice model for an ensemble of M identical L-C oscillators subjected to white noise is formally equal to our single L-C-based oscillator subjected to the white charge noise of [

that gives the phase diffusion constant D found in Equation (21) of [^{*} given by (12) that is F/2 times higher (although its physical meaning is similar) in order to use Equation (3) of [^{*} given in rad/s, we obtain:

where ∆ω is 2π times the frequency offset from the central frequency f_{0} taken as carrier frequency (see below (3) of [_{0} as (13) shows. Moreover, given the integration in time done by C of each impulsive displacement current or TA to give a voltage step modulating in phase the “carrier” of static frequency f_{0}, what we have described is the Frequency Modulation (FM) of this carrier at f_{0} by the impulsive current noise of the TAs. To say it bluntly: the displacement currents generating electrical noise [_{0} and this shows vividly that v(t) is not a pure sinusoidal carrier giving a spectral line at f_{0}, but a FM carrier whose frequency f(t) wanders randomly with time around f_{0}, tracking the random wandering around zero of the noise current of spectral density 4FkT/R A^{2}/Hz that reflects the thermal noise of C. This noise includes the Nyquist noise traditionally assigned to R, the noise coming from the feedback electronics and the extra noise coming from the unavoidable heating of the resonator Converting into heat a power W on average when it stores the energy U_{E} fluctuating at 2f_{0}.

Equation (13) gives the spectral dispersion S_{V}(∆ω) of the mean-square carrier voltage of v(t) due to the effect of TAs (Fluctuations) creating electrical noise in C [_{V}(∆ω) is a spectral density with the same units V^{2}/Hz of the noise density S_{NV}(∆ω) coming from energy Dissipations shown in _{0}. It is worth noting that (13) nothing says about the small, but not null AM of v(t) already discussed concerning _{0} itself taken as 0 dB reference, although perhaps the reason is a deeper one because this residual AM disappears when the quantization of charge for each TA is neglected, as it would do a noise model unaware about the discrete nature of the electrical charge. In any case, the power density S_{V}(∆ω) near f_{0} is so high, that the noise sidebands due to this residual AM will be overridden by it in the same way the feedback-induced Pedestal overrides the damped noise in

The random noise added by the TA-DR pairs to the sinusoidal v(t) of _{N}, added to a big phasor of amplitude V_{0} rotating uniformly in time at f_{0} times per second to represent a “carrier” of static frequency f_{0}. This noise vector can be decomposed into a small noise vector A_{AM} along the phasor that represents a random AM of the carrier, and a small noise vector A_{PM} orthogonal to the phasor that represents random PM of the carrier. Due to the random orientation of A_{N}, there is equal probability for A_{AM} and A_{PM} at each instant, thus meaning that the native noise density 4FkTR V^{2}/Hz of ^{2}/Hz density would be extended up to frequencies well above f_{0}/(2Q_{0}), being this the Pedestal of electrical noise shown in

The Pedestal of 2FkTR V^{2}/Hz shown in _{V}(∆ω) given by (13), both normalized by the mean-square carrier voltage in order to have the familiar single-sideband spectral density of Phase Noise L{∆ω} found in Equation (12) of [_{F-D} where S_{V}(∆ω) drops down to 2FkTR V^{2}/Hz or where the phase noise due to Fluctuations of energy in C and the phase noise due to Dissipations of energy in R modified by the feedback are equal, we obtain: as shown in ^{2} to a Pedestal of Phase Noise far from f_{0} [^{3} as we approach more f_{0} [

Recalling what we wrote about (13): that its Lorentzian Line reflects a FM of the carrier of frequency f_{0} by a noise of flat spectrum that is F times the Nyquist noise current usually assigned to R, we can explain easily the region of Phase Noise varying as 1/(∆ω)^{3} that appears in oscillators, especially in those using resonators of high Q_{0} (thus) and with electronics bearing 1/f-like noise below some frequency f_{CN}, no matter if it is excess noise (resistance noise in Solid-State devices [_{0}. From (12), the constant D^{*} for resonators with high Q_{0} values is correspondingly low. This is why the phase noise found in these oscillators uses to be the region of (13) where it is proportional to 1/(∆ω)^{2}, a “skirt” of Phase Noise that drops down to the Pedestal as ∆ω increases for. Let’s consider the Phase Noise of these oscillators that use high Q_{0} resonators when the flat spectrum of noise that modulates in Frequency the carrier is filtered previously by a low-pass filter with cut-off frequency.

_{0} and ^{2} for to for, an effect due to the integration of the modulating signal that precedes the Phase Modulation in a Frequency Modulator. This integration in t leads to a term inversely proportional to the modulating frequency f_{m}, whose effect in the phase noise spectrum appears at. Hence, the flat spectrum of the modulating signal gives phase noise proportional to 1/(∆ω)^{2} whereas the region whose power density drops as 1/(ω_{m})^{2} gives phase noise proportional to 1/(∆ω)^{4}. From this fact, it is easy to understand that a sum of the above low-pass filtered signals will give a sum of Phase Noise spectra like that of

Thus, a sum of low-pass filtered noise spectra like those of ^{3}, as the empirical one reported by Leeson [

Considering that the 1/f noise of Solid-State devices is synthesized in the way shown in

we have shown the origin of Phase Noise varying as 1/(∆f)^{3}. As a way to show the mechanism giving the Line Broadening of the output spectrum of oscillators based in L-C resonators, _{0} in this resonator at temperature T. This CCO models the Line Broadening around f_{0} of these oscillators that will be a Lorentzian line for white current noise and that will have another shape if the current noise modulating the carrier is “coloured” noise. This Phase Noise around f_{0} is the random Phase Modulation of the carrier by DRs (or its random FM by TAs) that one obtains neglecting the decay of each DR within one period of v(t), thus being valid for resonators with negligible Dissipation (e.g. Q_{0} > 50). Since each DR really decays as it dissipates the electrical energy stored in C by its preceding Fluctuation or TA, these random decays also generate electrical noise of bandwidth ±f_{C}/2 around f_{0} whose power in-quadrature with the output signal mislead the CF designed to reduce amplitude noise, thus producing the Pedestal of phase noise 2FkT/P_{0} that accompanies their carrier Line Broadening and that the FM modulator of

Using a new model for electrical noise based on Fluctuation-Dissipation of electrical energy in an Admittance, the Phase Noise of resonator-based oscillators is explained as a simple consequence of thermal noise. The discrete Fluctuations of energy involving single electrons produce the observed Line Broadening whereas the noise associated to subsequent Dissipations modified by the feedback electronics, lead to the Phase Noise Pedestal far from the “carrier”. Therefore, a monochromatic carrier of static frequency f_{0} never is obtained and the oscillator’s output corresponds to a Frequency Modulated carrier of central frequency f_{0} whose instantaneous frequency f(t) wanders randomly with time around f_{0}, tracking the random wandering of the noise current of density 4FkT/R A^{2}/Hz that collects the noise of the resonator, its electronics and the extra noise due to any heating effect due to the Signal power converted into heat in the resonator. In summary: Phase Noise shows the way the oscillator senses the charge noise power 4FkT/R C^{2}/s that exists in its L-C resonator while it stores the energy corresponding to the output signal it sustains in time because these oscillators always are CCOs driven by the charge noise of their capacitance.