Comments on “ Oscillator with Ogd-Symmetry Characteristics Elirknates Low-Frequency Noise Sidebands ”

The subject paper [1] concluded that the elimination of even-ordered nonlinearity in an oscillating system would prevent the transfer, of low-frequency flicker noise up to the oscillation frequency zone. This communication is a response prompted by the question: if such a simple expedient as achieving odd symmetry in an oscillator can eliminate 1/f phase noise, why is it that all oscillators are to some degree corrupted by flicker noise? The significance of the experimental method employed in [1] to demonstrate the desirability of achieving odd symmetry is questioned. Finally, some additional factors relating to oscillator flicker phase noise generation are considered.

2) The condition K = L, which has been discussed in [2] and [3], constitutes a restrictive sufficient condition for the Lyapunqv equation (1) to hold.' 3) Closed-form modified Lyapunov equations for K and W in the special case of minor interest of separable-denominator lower quality 2-D digital filters may be directly deped from the general equations (6a) and ( 23).Specifically, according to the notation used here, the unit pulse response sequence f( i, j) of the state vector due to a unit pulse input sequence is generally given for A, = 0 by the formula 1 form,n>O   f(m,n)  =A(m,n)b= for m > 0, n = 0 Equation ( 33) may be easily proved by induction.The substitution of ( 33) into (6) leads to the modified Lyapunov equation in K that appeared in [4] and [5].The modified Lyapunov equation in W is similarly derived.
4) The general forms of matrices K and W given by ( 6a) and ( 23), which were first reported in [l], constitute the beginning of all the efforts for the minimization of the roundoff noise in 2-D state-space filtering.Therefore, other closed-form expressions may be directly derived from (6a) and ( 23) in special cases, where the system matrices have a simplified form.With respect to the general case, the associated synthesis problem has been successfully solved in [6] by using (6a) and ( 23 Comments on "Oscillator with Ogd-Symmetry Characteristics Elirknates Low-Frequency Noise Sidebands" CHASE F. HEARN ,qmYtct -The subject paper [lj concluded that the elimination of even-ordered nonlinearity 'in an oscillating system would prevent the transfer, of low-frequency flicker noise up to the oscillation frequency zone.This communication is a response prompted by the question: if such a simple expedient as achieving odd symmetry in an oscillator can eliminate I/f phase noise, why is it that all oscillators are to some degree corrupted by flicker noise?Jhe significance of the experimental method employed in [I] to demonstrate the desirability of achieving odd symmetry is questioned.Finally, some additional factors relating to oscillator flicker phase noise generation are considered.

A. Experimental Methodology
The method employed in [l] to demonstrate the advantage of eliminating even-ordered nonlinearity in a feedback oscillator was to inject a low-frequency perturbing signal into the oscillating system and observe the power spectrum around the oscillation fundamental frequency, or first spectral zone.From the measurement theory viewpoint, this approach has a shortcoming in that is does not distinguish between spectral cqntributions arising from amplitude fluctuations and those due to phase fluctuations.
-' 4 complete characterization of the output signal should make that distinction smce amplitude fluctuations do not necessarily affect the instantaneous phase of the process.Therefore, the spectral contributions of amplitude and phase fluctuations are needed individually rather than collectively.Commercial stability analyzers are available, which perform those two measurements.Another shortcoming of the experimental approach is that the injected perturbation signal does not accurately simulate circuit noise associated with "synthesized" nonlinearities, as discussed next.A major contention of this paper is that the conclusion of [l] is correct only when odd symmetry exists with respect to both the loop oscillation and the flicker noise, or the perturbing signal; that is, only if those two signals were additive prior to symmetrical limiting.This condition was satisfied in the experimental circuits; however, in practice, it may not be satisfied.Odd-symmetry nonlinearity is usually synthesized by suitably combining nonlinear elements which individually lack such symmetry as, for example, by shunt-connecting opposite polarity semiconductor diodes.Associated with each diode is a physical source of flicker noise which is independent of the source associated with the other diode.When used as the limiting element in an oscillator, the diodes would conduct on alternate half-cycles of the oscillationf the translated (product) noise processes would not cancel since they were generated by uncorrelated noise voltages interacting in nonlinearities lacking odd symmetry.Synthesis of odd symmetry is therefore advantageous only if the major source of flicker noise is external to the nonlinearities.However, in such a circumstance, frequency-selective coupling between the flicker noise source and the limiting function could reduce upconversion by reducing the flicker noise amplitude with respect to that of the oscillation signal-subject to limitations discussed in the next section.One of the experimental circuits described in [l] used source-coupled FET's as (synthesized) odd-symmetry amplifiers.As with the diode example just discussed, the translated device noises would not cancel since the FET's are alternately driven into cutoff by the loop oscillation.Therefore, the extent to which the circuit of Fig. 3 in [l] would reduce the translation of FET-generated flicker noise is believed to be conjectural.Riddle and Trew [5] reported an impressive 20-dB reduction in phase noise by using source-coupled FET's instead of a single FET in a microwave oscillator.Before accepting this as a substantiation of the conclusions of [l], it should be appreciated that the models in [l] and microwave oscillator models differ significantly.One major difference is that the voltage-dependent capacitances associated with FET's and BJT's became very significant at microwave frequencies where they become sizable relative to fixed (linear) circuit capacitances.It is this author's opinion that the improvement cited in [S] was due primarily to the reduction of the linear term in the capacitance-voltage relationship that results with a source-coupled negative resistance amplifier configuration.This occurs because the gate-to-source capacitances (of the FET's) are effectively paralleled with opposing polarities, thereby (ideally) eliminating the linear dependency between capacitance and voltage which is responsible for direct-frequency modulation of the oscillator by flicker noise.In addition, the large-signal input and output conductance varies much less in a source-coupled pair than in a single-stage amplifier [5].

C. Envelope-to-Phase Conversion Mechanisms
Nonlinear capacitive effects were not considered in [l] since it dealt with very low frequency oscillators in which they would be swamped by lumped fixed capacity.Consider now the question of how the envelope fluctuations produced by resistive upconversion of flicker noise (or a perturbation signal) produces phase fluctuations.To the extent that the amplitude and phase conditions necessary for steady-state oscillation can be independently satisfied, there is no mechanism to allow AM to PM (or FM) conversion.Obviously, this is no small accomplishment since most oscillators do exhibit some dependency between the amplitude and frequency of self oscillation.This dependency is a consequence of at least two distinct mechanisms which will be discussed.Groszkowski [6] analyzed one such mechanism which results from multiple nonlinear interactions in an oscillator.Harmonics generated in the nonlinearity'are phase shifted by the resonator, which is highly reactive to harmonics.These phase-shifted harmonics then interact with each other and the fundamental in the nonlinearity to produce a phase-shifted fundamental component which produces a change in the closed-loop phase shift.The oscillation frequency must change to produce a compensating phase shift.The amplitude-frequency dependency is due to the fact that the harmonic amplitudes are dependent upon the amplitude of the loop oscillation.Having identified and quantified the effect, Groszkowski showed that it could be greatly reduced by clever circuit design; however, his recommendations have been largely ignored by oscillator designers.Another mechanism which can convert amplitude fluctuations into phase fluctuations is created when a resistive nonlinearity is reactively loaded [7]-[9].This effect was first analyzed .byKochenburger [lo] in his application of describing function analysis to nonlinear control systems.He derived the magnitude and phase of the describing function for a capacitively loaded, symmetrical shunt limiter using ideal (piecewise-linear) diodes, as depicted in Fig. 1.This circuit has the property that both the magnitude and phase of the describing function decrease monotonically when A > V,,.Fig. 2 is an adaptation of Kochenburger's results which more clearly demonstrate the dependency between the input amplitude and fundamental frequency input-output phase shift at a fixed frequency.When this circuit is enclosed within a feedback oscillator to perform the limiting function, amplitude-frequency dependency is directly proportional to its operating point phaseamplitude slope [9].The phase-amplitude slope is in turn proportional to the nonlimiting, or linear, phase shift (when A < V,) of the circuit, which is the reason reactive loading of nonlinearity should be strictly avoided.
The maximum phase-amplitude slope for Kochenburger's model was 0.118 rad per unit normalized amplitude for a nonlimiting phase shift of n/4 rad.This corresponds to an amplitude-to-phase conversion efficiency of -18.5 dB.For comparison purposes, this author [9] evaluated a similar circuit using Schottkey-barrier diodes and found the corresponding values to function approach which simplifies the formulation of the system equations be 0.181 rad per unit normalized amplitude, or -14.8 dB, which for the network.Our application of the procedure is in computer-aided attests to the numerical significance of this amplitude-to-phase noise analysis of switched-capacitor circuits.conversion mechanism.These two conversion mechanisms and resistive upconversion are independent of frequency as opposed I. INTRODUCTION to nonlinear capacitance effects which increase with frequency, An active RC network driven by independent zero mean white and are therefore believed to be the dominant causes of flicker noise sources is considered.Our interest is in numerically comphase noise in low-frequency oscillators.
[l]  concluded that the elimination of evenordered nonlinearity in an oscillator will prevent (device) flicker noise from being translated up to the oscillation frequency.This opinion has been expressed by other authors [2]-[4].The purposes of this communication are to: (a) note the limitations of the experimental technique used in [l]; (b) discuss nontrivial circuit conditions for which the conclusion of [l] 'is incorrect; (c) note that some form of amplitude-to-phase (or frequency) conversion mechanism must be present in an oscillator to convert the envelope modulation produced by resistive upconversion of flicker noise into phase (or frequency) fluctuations.II.DISCUSSION Manuscript received September 2, 1986.The author is with NASA, Langley Research Center, Hampton, YA 23665.IEEE Log Number 8612946.U.S. Government work not protected by U.S. copyright IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL.CAS-34, NO. 3, MARCH 1987 B. Circuit Modeling
[l], [2].If one analyzes noise subject of flicker phase noise in oscillators than is addressed in generated in SC networks, one has to consider the continuous-[l].More specifically, it was argued that the synthesis of time circuits which realize the various configurations of capaciodd-symmetry may not defeat the translation of low-frequency tor/amplifier interconnections given by the switch positions of device noise up to the oscillation frequency when the noise is the different SC phases.We assume that switches in their ON associated with the nonlinear elements.The role of active device state have finite ON conductances which are noisy.Since the set nonlinear reactance was cited as a major consideration in micro-of all switch ON conductances together with the capacitors have a wave oscillators with regard to flicker noise transfer.Finally, the bandlimiting effect, the conductance noise can mathematically be role of AM to PM conversion associated with reactively loaded modeled as white noise; the filtering effect of the capacitor-connonlinearities was shown to be a significant factor in low-ductance combinations assures finite variances of the observable frequency oscillators of the sort discussed in [l].The substantial noises.The amplifier model often features two noise components, amount of recent interest in oscillator noise theory attests to the a white noise and a l/f noise component.Here, only the white fact that this is far from a closed subject.noise part is taken into account.The amplifier model is assumed to be bandlimiting in order to be self consistent with the white noise component.Due to the periodically switching-on of the