A study of amplitude‐to‐phase noise conversion in planar oscillators

This paper explores the many interesting implications for oscillator design, with optimized phase‐noise performance, deriving from a newly proposed model based on the concept of oscillator conjugacy. For the case of 2‐D (planar) oscillators, the model prominently predicts that only circuits producing a perfectly symmetric steady‐state can have zero amplitude‐to‐phase (AM‐PM) noise conversion, a so‐called zero‐state. Simulations on standard industry oscillator circuits verify all model predictions and, however, also show that these circuit classes cannot attain zero‐states except in special limit‐cases which are not practically relevant. Guided by the newly acquired design rules, we describe the synthesis of a novel 2‐D reduced‐order LC oscillator circuit which achieves several zero‐states while operating at realistic output power levels. The potential future application of this developed theoretical framework for implementation of numerical algorithms aimed at optimizing oscillator phase‐noise performance is briefly discussed.


| A BRIEF REVIEW OF ESTABLISHED THEORY
We consider an n-dimensional autonomous vector-field f : R n ! R n generating a set of n-coupled ordinary differential equations (ODEs) with xðtÞ : R ! R n being the n-dimensional time-dependent state-vector. The dynamical system in Equation (1) produces a stable limit set γ known as the oscillator limit-cycle. A steady-state solution x s ðt + TÞ = x s ðTÞ is a T-periodic solution orbit with an initial condition in γ. It then follows that x s (τ) 2 γ for all τ. The system in Equation (1) is timenormalized using the timescale τ = 2πðt=TÞ which implies a 2π periodic solution x s ðτ + 2πÞ = x s ðτÞ and corresponding oscillator frequency ω 0 = 2π=T = 1.

| Linear response map Φ and the oscillator tangent bundle TR n
The oscillator response, δx(τ), to small perturbations around the steady-state, γ(τ), is governed by the oscillator linearresponse map Φðq, sÞ : T s R n ! T q R n . This map produces the oscillator output δxðqÞ = Φðq, sÞζðsÞ , at time τ = q , in response to a small (deterministic or random/noise) perturbation vector ζ(s), at time τ = s. Here, the oscillator tangent space T η R n represents the usual real n-dimensional vector-space R n with the origin translated to the point γ(η) in statespace (an affine vector-space). The entire domain and co-domain (image) of the map Φ, TR n , is known as the oscillator tangent-bundle, is then given as union of the individual tangent spaces

| The Floquet decomposition
For hyperbolic stable oscillators, the following unique decomposition of Φ will always exist: where {μ i } are the n characteristic Floquet exponents, {u i (τ)} is the complete set of Floquet vectors, and {v i (τ)} is the complete set of dual * Floquet vectors. [10][11][12] The Floquet vectors and dual vectors support the following defining relationship: v † i ðτ a Þu j ðτ b Þ = δðτ a −τ b Þδ ij fori = 1,2,…n, ð4Þ where v † i denotes the transposed of the (dual) vector v i , δ ij is the Kroenecker delta-function, whereas τ a , τ b are two normalized time-points. The delta-function component of Equation (4) signifies that the bi-orthogonality property only holds for equal time arguments. The set {u i (τ)} is complete, and we can hence always decompose any perturbation vector function as ζðτÞ = a 1 ðτÞu 1 ðτÞ + a 2 ðτÞu 2 ðτÞ + … + a n ðτÞu n ðτÞ, ð5Þ where {a i (τ)} are n expansion coefficient functions. From Equation (4), one then finds directly v † k ðτÞζðτÞ = a k ðτÞ, ð6Þ and the dual vector v k hence picks out the component of ζ proportional to u k . For an autonomous oscillator solution, the special exponent μ 1 = 0 will always occur. The corresponding Floquet vector u 1 , known as the phase-mode, is a scaled version of _ x s , where x s is the steady-state solution of the system in *The difference between vectors and dual vectors is that they transform as contra-variant and co-variant vectors, respectively. 11 Henceforth, the terms vector and vector-bundles are thus connected to contra-variant vectors and vector-functions, whereas the terms dual vector and dual vector-bundles are associated with co-variant vectors and vector-functions.
Equation (1) and hence tangent to the limit-cycle γ. 10,11 Setting k = 1 in Equation (6), it follows that the dual-vector v k = v 1 collects the part of the perturbation attributed to phase-mode u k = u 1 as seen from Equation (5). When ζ is a noise perturbation, it thus collects the oscillator phase-noise contribution. The dual vector, v 1 , hence has special status as it relates to the oscillator phase-response and is often referred to in the literature as the perturbation-projection vector (PPV). 10 For an autonomous oscillator, all that is not phase must, by definition, be amplitude. Hence, the remaining n − 1 Floquet vectors fu i ðτÞg n i = 2 are referred to as amplitude-modes and stability requires ℜ{μ (i > 1) } < 0, where ℜ{μ i } denotes the real part of the complex coefficient μ i .
The n Floquet vector functions {u i (τ)} are now grouped in two so-called vector-bundles 11 TI = spanfu 2 ðτÞ,u 3 ðτÞ, …,u n ðτÞg ð8Þ with TM containing the single phase-mode, whereas the bundle TI holds the n − 1 amplitude-modes. The oscillator tangent-bundle in Equation (2) can then be decomposed into phase and amplitude contributions as The bundle definitions in Equation (7)- (8) then prompt the definition of the equivalent dual bundles (see footnote *) The geometric interpretation behind the concepts discussed above are illustrated qualitatively in Figure 1A. For a more thorough discussion of these topics, we refer to Djurhuus et al. 11 2.3 | The orthogonal decomposition of TR n We consider the scenario where the phase-mode, u 1 , and the PPV, v 1 , are colinear u 1 ðτÞ = av 1 ðτÞ, a 2 R. From Equations (7) and (10), we then have † TM ' T * M. From the rule in Equation (4) and the definition of TI in Equation (8), it follows that v 1 is always orthogonal to TI and since u 1 and v 1 are colinear, it follows that u 1 now also must be orthogonal to TI. By definition, u 1 is orthogonal to the dual space NM defined in Equation (11), and we therefore must have TI ' NM. We sum up the above discussion as where Q ⊥ W implies that the bundles Q and W are orthogonal. This orthogonal decomposition scenario is illustrated qualitatively in Figure 1B.

| The oscillator phase-noise spectrum
In this section, we consider the noise-perturbed version of the oscillator defined in Equation (1) _ x = f ðxÞ + bðxÞzðτÞ, ð13Þ † Here, we use the notation X ' Y to indicate that the vector-bundle (contra-variant bundle) X and the dual bundle (co-variant bundle) Y describe the same subspace of TR n (see footnote *).
where b : R n ! R n × p is the noise modulation matrix 10 and z : R ! R p is a p-dimensional vector of uncorrelated, zeromean, unit-power Gaussian white noise sources ‡ . Note that since the noise analysis herein is based on the linearresponse (see discussion in Sections 2.1-2.2), the model implicitly assumes that the power of the noise sources z k are weak compared to the deterministic signals x = ðx 1 , x 2 ,…x n Þ † . Specifically, this is the situation for the important case of electrical oscillators operating at room temperature. The noise perturbation in Equation (13), evaluated on the oscillator steady-state solution x s (τ), can now be written where b k (τ) is the kth column of the modulation matrix b(x(τ)), whereas z k (τ) is the kth component of the noise-vector z.
As the p noise sources {z k (τ)} in Equation (14) are uncorrelated, and equally distributed (i.e., white Gaussian noise), the total response of any ensemble average function (such as diffusion constants) can found by adding the individual contributions using the superposition principle. We can hence simplify the notation of the following analysis significantly by only considering the response of the oscillator to a single noise source. Hence, let 1 ≤ q ≤ p and consider the qth term in Equation (14). We then define the simplified perturbation source as where κðτÞ = b q ðτÞ and sðτÞ = z q ðτÞ . In the following analysis, the various calculations are performed using the source in Equation (15) as the oscillator noise perturbation. Later in the analysis phase, the superposition principle will then be applied to show how this partial, single-source, response actually directly implies the total oscillator response produced by the full perturbation vector in Equation (14).
From the discussion in Section 2.2, and the expression of the Φ in Equation (3), we know that the dual Floquet vector v 1 (PPV) collects all noise perturbations attributed to the phase-mode u 1 , i.e., phase-noise. Specifically, given the noise perturbation vector ξ, the so-called oscillator phase-diffusion D is calculated as The oscillator phase-noise spectrum, around the first harmonic (carrier), then takes the (time-normalized) form [10][11][12] Lðf m Þ = 10log 4D where f m is the normalized offset-frequency from the carrier. The model leading to spectrum in Equation (17) assumes a stable carrier which is an acceptable approximation for most practical electrical oscillator circuits, at least for reasonable time-scales. However, it should be mentioned that this approach does neglect thermal and noise-induced drift of the carrier frequency. For a more detailed description of the stochastic oscillator phase process, including drift, we refer to the literature. [14][15][16][17] We now decompose the phase-diffusion in Equation (16) into two separate contributions where D ϕϕ represents the part of the phase-diffusion due to PM noise (noise tangent to γ) and D ϕa represent the component due to AM noise (noise orthogonal to γ), i.e., the AM-PM noise conversion. The calculation of these parameters is the topic of the following section.

| AM-PM noise conversion
We consider the vector-bundle N * M ' NM which is simply just the contra-variant version of the dual vector-bundle NM in Equations (11) (see footnotes * and †).
Together with the unit vector bundle ϕðτÞ 2 TM, we can construct the orthonormal frame ðφ,fâ i gÞ shown in Figure 2. From this frame, the noise vector ξ(τ) in Equation (15) is decomposed into phase (PM) and amplitude (AM) components where α 0 ðτÞ = κ † ðτÞφðτÞ, α i > 0 ðτÞ = κ † ðτÞâ i ðτÞ. The following projections are now defined where v 1 is the dual phase-mode Floquet vector (PPV) discussed in Section 2.2. From Equation (6), it follows that the TM = spanðu 1 ðτÞÞ component of any arbitrary vector x is calculated from the projections v † 1 ðτÞx . Furthermore, if we define y = x −v † 1 ðτÞx, then, it must follow that y 2 TI; as v † 1 ðτÞy = 0 (see discussion in Section 2.2). Using the definition of TI in Equation (8) where {ν k (τ)} are a set of n − 1 unspecified functions representing the oscillator amplitude-noise response, whereas ρ, λ : R ! R are two real functions, representing the PM/AM-noise contributions to the phase mode u 1 . From this description, it follows that ρ holds the phase-mode (u 1 ) noise component due to PM noise (PM-PM), whereas λ holds the phasemode noise component due to AM noise (AM-PM). From Equations (20) and (21)- (22), these two functions are written The following statement explains under what circumstances the AM-PM contribution in Equation (23), λ(τ), can be zero (25) will be zero if, and only if, the Floquet decomposition is orthogonal (see Section 2.3 and Figure 1) Proof. From Equations (25) and (22) where wðτÞ = P n− 1 i = 1 α i ðτÞâ i ðτÞ. The frame fâ i g, shown in Figure 2, spans the normal-space N * M ' NM which implies wðτÞ 2 N * M. This frame is orthogonal to the limit cycle γ (see Figure 2). As wðτÞ 2 N * M, Equation (27) then implies that v 1 is orthogonal to N * M ' NM which means that it must be tangent to γ (see Figure 1). The vectors v 1 and u 1 are hence colinear, i.e., TM ' T * M (see Section 2.3). But then it follows directly from Equation (12) that the Floquet decomposition must be orthogonal.
We proceed with the statement Corollary 2.2. Given the decomposition of ξ in Equation (23) and the definition of the AM-PM phase-diffusion constant D ϕa in Equation (18), we have where the function λ : R↦R is defined in Equation (23).
At this point, we must remind ourselves that the above results were derived using the single source κðτÞsðτÞ = b q ðτÞz q ðτÞ in Equation (15) which represented the qth component of the sum in Equation (14). Hence, from this point on, the notation D ðqÞ ϕa is used to refer to the partial AM-PM phase-diffusion constant due to the perturbation b q (τ)z q (τ), whereas D ϕa is now reserved for the response due to the full noise perturbation in Equation (14). Combining the results in corollaries 2.1 and 2.2, then gives the result As mentioned above, the superposition principle can be applied to tie the partial result in Equation (29) to full response. For this purpose, we now state Corollary 2.3. The partial result in Equation (29), calculated using the single source in Equation (15), holds equally for the response due to the full perturbation vector in Equation (14) D where, as mentioned above, D ϕa now refers to the full AM-PM phase-diffusion constant calculated using ξ(τ) in Equation (14).
Proof. Repeating the above calculation for each term b k (τ)z k (τ) in Equation (14), we would again arrive at Equation (29) for q = k. Since the sources z k (τ) are uncorrelated, the ensemble averages will not contain any cross-terms meaning that the response of each of the terms are mutually independent. It then follows directly from the superposition principle that the full AM-PM phase-diffusion constant is given as the sum D ϕa = P p k = 1 D ðkÞ ϕa . Furthermore, as a diffusion constant cannot be negative, we have which says that D ϕa is zero if, and only if, all the partial terms fD ðqÞ ϕa g are zero. Each term in this sum obeys Equation (29), and combining this result with Equation (31), one arrives directly at Equation (30).
As discussed in Section 1, we refer to D ϕa = 0 (zero AM-PM noise conversion) as zero-states. Equation (30) then says that zero-states are only possible if the Floquet decomposition is orthogonal, i.e., TI ⊥ TM. Using the definitions of TM and TI in Equations (7) and (8), this translates directly to the following condition for the special two-dimensional (n = 2) case Hence, in-order for a two-dimensional oscillator to attain a zero-state, we must demand that the corresponding Floquet phase and amplitude-modes, u 1 (τ) and u 2 (τ), are orthogonal for all values of τ.

| THE SYM-AM/PM MODEL
In 13 a new methodology, based on the concept of oscillator conjugacy, was developed with the aim of analyzing oscillator noise response. In this text, we refer to this theoretical framework as the SYM-AM/PM model.
One prominent and wide-reaching prediction which emerges from the SYM-AM/PM model is represented by the following statement 13 : where S a is the collection of rotationally symmetric limit sets with W s being the 2-D oscillator stable manifold § which is indexed here using the complex coordinate w. Note that S a simply holds all limit-sets which have the form of concentric circles centered at the point a. In words, Equations § As explained in Djurhuus and Krozer, 13  (33)-(34) say that an orthogonal decomposition is achieved if, and only if, the underlying steady-state oscillator solution γ(τ) is perfectly symmetric. This connection between γ-symmetry and an orthogonal Floquet decomposition was first discovered in Djurhuus and Krozer. 13 Combining the result in Equation (33) with Equation (32), calculated in the previous section, we arrive at the novel statement which says essentially that for 2-D oscillators, zero AM-PM noise conversion follow from the symmetry of the steadystate (deterministic) oscillator solution.
In-order to numerically test and verify the novel hypothesis in Equation (35), we will need to quantify the levels of γ-symmetry and AM-PM noise conversion. For this purpose, we introduce the measure where ρðτÞ = jγðτÞj 2 = γ 1 ðτÞ 2 + γ 2 ðτÞ 2 , with γ 1, 2 being the two components of the 2-D limit-cycle solution (γðτÞ = ðγ 1 ðτÞ, γ 2 ðτÞÞ), whereas ρ max representing the maximal value of the function ρ on the interval τ 2 [0; 2π]. From the definition of S a in Equation (34) and the expression above, it should be clear that Λ = 0 , γ 2 S a and Λ hence measures the level of limit-cycle symmetry, with Λ = 0 for a perfectly symmetric solution. Next, we consider the function where u 1, 2 (τ) are the first and second Floquet vector functions, respectively. It follows directly from Equation (32) that Υ measures the level of AM-PM in the solution, with Υ = 0 representing the zero-states discussed above. Using the definitions in Equation (36)-(37), it follows that in order for Equation (33), and consequently Equation (35), to hold true, we must observe in our numerical simulation results. The predictions which emerge from the SYM-AM/PM model directs us to seek out oscillator topologies and parameter settings which produce the symmetric (Λ = 0) steady-state solution, contained in the set S a in Equation (34), in order to achieve oscillators operating at the zero-states (Υ = 0). Unfortunately, as will be discussed below, these Λ = 0 states seem to generally not occur in industry-standard RF and μ-wave oscillator circuit topologies. In fact, as will be made evident in subsequent simulation trials, these circuit-classes only attain zero-states in special limit-cases which are unworkable from a practical design perspective, the so-called A-STATES. The issues discussed here will become more accessible in the following section where we consider a reduced-order model of a specific MOSFET transistor differential-pair cross-coupled oscillator which is a unit found in many modern communication and remote-sensing applications. This state-of-affairs will prompt us in Section 5 to demonstrate the synthesis of a novel 2-D LC oscillator circuit, constructed using the design rules newly acquired from the SYM-AM/PM model, which in fact does present several practically useful zero-states, the so-called P-STATES.

| A STANDARD MOSFET TRANSISTOR OSCILLATOR CIRCUIT-AMBIGUOUS ZERO-STATES (A-STATES)
We consider the MOSFET differential pair LC oscillator, shown in Figure 3A. Here, we consider both the series resonator load (red) and parallel resonator load (blue) options. The circuit operates in the current-limited regime 18 implying that the NMOS differential pair behaves as an ideal switch. From Figure 3A, the circuit is biased by the DC current source I B and the small-signal transconductance is then written where k n = μ n C ox W n =L n with μ n , C ox , W n , and L n being the charge mobility, oxide capacitance, gate width, and length, respectively. Using the result in Equation (39), the full nonlinear transconductance function of the MOSFET differential pair in Figure 3A is then approximated 19 where v C is the resonator differential capacitor voltage (see Figure 3). In Figure 3B, the differential-mode equivalent of the full transistor circuit, referred to as the THETA-OSC, is shown. We consider both the series resonator load THETA-OSC/SER (red) and parallel resonator load THETA-OSC/PAR (blue) options. In the following, we normalize the time variable as τ = ω n t where ω n = 1:0= ffiffiffiffiffiffi LC p is the natural frequency of the resonator. The dynamical equations of the 2-D differential-mode THETA-OSC reduced-order circuit are then written as with x = v C =z 0 , y = i L z 0 , where v C , i L are the voltage across the capacitor C and current through inductor L, respectively, whereas z 0 = ffiffiffiffiffiffiffiffi ffi L=C p is the resonator characteristic impedance and Θ was defined above in Equations (39)-(40) ¶ . Here, R p , R s ≠ 0 corresponds to the THETA-OSC/[PAR/SER] circuits in Figure 3 with corresponding color codes (blue/red). The capacitance C and inductance L do not occur explicitly in Equation (41) but are easily calculated from the given parameters # . Figure 4A shows a series of numerical calculations for the THETA-OSC/PAR circuit in Figure 3B as modeled by Equation (41) (blue). We consider a constant Q circuit with Q = 10 corresponding to a resonator load R p = Q=ðω n CÞ = Qz 0 . The figure plots the two measures Υ, Λ in Equations (37) and (36) as function of the bias current I B (see Figure 3A) and for four values of the resonator characteristic impedance z 0 . In Figure 4A, the concept of an F I G U R E 3 (A) A cross-coupled MOSFET (NMOS) LC oscillator operating in the current-limited regime. (B) The equivalent 2-D circuit for the differential-mode oscillation, THETA-OSC, where the nonlinear transconductance Θ models the MOSFET pair in figure A). The circuit includes either a series (THETA-OSC/SER) or parallel (THETA-OSC/PAR) resonator load [Colour figure can be viewed at wileyonlinelibrary.com] ¶ In the following, we fix k n from Equation (39), in time-scaled form, as k n = 2AV −2 . # As the resonator natural frequency ω n = 1:0= ffiffiffiffiffiffi LC p is fixed (time normalization), we can express the circuit capacitance and inductance in terms of parameters ω n , z 0 as C = 1:0=ðω n z 0 Þ and L = z 0 =ω n . oscillator start-up boundary (SB) is introduced. Herein, the term SB represents the point in parameter space where the oscillator solution is born, i.e., starts-up, from the DC operating point in some-kind of bifurcation (Hopf-bifurcation in our case). The THETA-OSC/PAR circuit starts-up (turns-on) at G m > G m0 = 1:0=R p = 1:0=ðQz 0 Þ . Using the relation in Equation (39), we can then find the bias current I B0 at start-up Here, I B0 describes the SB as for I B < I B0 , the system is at rest (i.e., DC rest point) and for I B > I B0 , the dynamics of the system is described by a periodic limit-cycle solution. The boundary I B = I B0 hence represents the transition between these two states (i.e., the bifurcation point). The four SBs, one for each value of z 0 , in Equation (42), are indicated in the figure as dotted vertical lines. We note that Λ, Υ ! 0 as I B ! I B0 exclusively for the parameter value z 0 = 1:0. Each curve in Figure 4A corresponds to a unique set of circuit resonator parameters (C, L, R p ) as explained in footnote # and the above derivation of the parallel load as R p = Qz 0 . Figure 4B repeats the experiments displayed in Figure 4A above but this time for the THETA-OSC/SER circuit (see Figure 3B [red]) modeled by Equation (41) (red). Here, the resonator quality factor Q is again fixed as Q = 10 corresponding to a resonator load R s = ω n L=Q = z 0 =Q. A simple series-to-parallel transformation then gives the following approximation for the equivalent parallel loadR p ≈R s ð1 + Q 2 Þ. From the discussion above, we know that the oscillator circuit turns on for G m > G m0 = 1:0=R p = ðQ=z 0 Þ=ð1 + Q 2 Þ. Using Equation (39) results in the following approximate values for the SBs for the TANH-OSC/SER circuit in Figure 3B, which depend on the value of the characteristic impedance z 0 , and are indicated in Figure 4B using vertical dotted lines.
Unlike what was observed in Figure 4A, no singularities, for either measure, occur for the TANH-OSC/SER at the SBs. Each curve in Figure 4B corresponds to a unique set of circuit resonator parameters (C, L, R s ) as explained in footnote # and the above derivation of the series load as R s = z 0 =Q. Figure 4A, which represents simulation results for the THETA-OSC/PAR circuit, featured a singularity in both measures Λ and Υ, at the SB, for the parameter setting z 0 = 1. Furthermore, this singularity occurred strictly pairwise, by which is meant that is was never the case that Υ ! 0 while Λ ! ̸ 0 for any parameter setting. From the above description, it follows that the THETA-OSC/PAR circuit has a zero-state (i.e., Υ = 0) at the SB. In contrast, the numerical experiments in Figure 4B, for the THETA-OSC/SER circuit, showed no evidence of singularities. While the THETA-OSC/SER again attains minimum AM-PM, as measured by Υ, for z 0 = 1:0, it never achieves any zero-states at the SB (or anywhere else) exactly because it never attains a perfectly symmetric limit-cycle solution (Λ = 0 ) at the SB (or anywhere else). The above numerical experiments, detailed in Figures 4A and 4B, hence, follow the rule in Equation (38) and thus fully corroborate and validate the SYM-AM/PM model and specifically the prediction in Equation (35).
In Figure 4A, we see that the THETA-OSC/PAR circuit exhibits a zero-state (Υ = 0) directly at the SB for z 0 = 1. As mentioned in the introduction, we refer this type of zero-state on the SB as an ambiguous zero-state or A-STATE. The A-STATE in Figure 4A is very important for the purpose of verifying the predictions of the SYM-AM/PM model, and this issue was discussed in detail above. However, as we approach the SB, the steady-state oscillator solution, γ, approaches the quiescent DC operating point, with the corresponding oscillation amplitude approaching zero. It hence follows that an A-STATE, such as the one shown in Figure 4A, holds no significant relevance for practical oscillator design.
In contrast, a proper zero-state, or P-STATE, refers to a zero-state where the oscillator operates at nonzero power output. The prospect of designing oscillators operating at P-STATES is indeed very interesting as these solution modes represent a local, if not global, minima of the total phase-noise response. Fortunately, with the emergence of the novel SYM-AM/PM model, we, for the first time, have the tools available in order to formulate rules which can be used to guide our design in terms of topology and parameter settings. In the following section, we proceed to apply these newly attained design rules in order to synthesize a novel electrical oscillator circuit which achieves several viable P-STATES. Figure 5 plots the THETA-OSC oscillator limit-cycle γ and corresponding Floquet vector-bundles (u 1 (τ), u 2 (τ)) for various circuit parameter configurations. The figure contains simulations of the THETA-OSC/[PAR/SER] circuits given two different parameter configurations, labeled ➀ and ➁ (see Figure 5 caption and footnote k ). The parameter configuration ➀ corresponds to the SB for the specific parameter choice z 0 = 1:0. For the THETA-OSC/PAR circuit, this operating point corresponds to the A-STATE shown in Figure 4A. The existence of this special state is clearly detected in Figure 5A. Here, the perfectly symmetric limit-cycle and orthogonal Floquet vector-bundle directly corresponds to the singularities in the measures Λ, Υ, defined in Equations (36) and (37), respectively, observed in Figure 4A (blue curves). Figure 5C shows simulations for the same configuration but for the THETA-OSC/SER circuit. The figure clearly illustrates that no such symmetry property holds here and furthermore that the bundle is no longer orthogonal. This again directly corresponds to the lack of an A-STATE (singularity) at the SB for the THETA-OSC/SER circuit in Figure 4B (z 0 = 1:0, blue curves). Finally, Figures 5B and 5D shows the equivalent simulations for the THETA-OSC/ [PAR/SER] for the parameter configuration ➁. Clearly, neither circuit produces a symmetric limit-set in this configuration, and the bundles are also not orthogonal. These results again correlate directly with the curves in Figure 4A-B (z 0 = 0:7, turquoise curves). The plots in Figure 5 hence correlate directly with the equivalent curves in Figure 4A-B and hence simply convey a different, perhaps more intuitive, representation of the same results.

| SYNTHESIS OF AN OSCILLATOR WITH PROPER ZERO-STATES (P-STATES)
We consider the electrical circuit displayed in Figure 6 which we shall refer to throughout as the SYM-OSC circuit. In order to model this circuit, we proceed to introduce the following two Q-factors: where ω n = 1:0= ffiffiffiffiffiffi LC p is the natural frequency of the circuit resonator. In Figure 6, the controlled sources i N and v N have the form where the values and units of the various parameters in the above expressions are listed in Table 1.
Continuous nonlinear functions such as those in Equation (45) can be realized electronically using various different function synthesizer circuit topologies. The perhaps simplest option involves the use of standard op-amp log/anti-log 20,21 and summer circuits which allows for a bare-bone implementation of parametric polynomial functions. Alternatively, operational-transcondutor-amplifier (OTA) circuits can be worked to synthesize any mathematical function characteristic (see, e.g., Sánchez-Sinencio et al 22 ). Finally, a dedicated chip such as the AD538 23 can be used to synthesize any polynomial function.
In the following, we normalize the time variable as τ = ω n t . The dynamical equations modeling the circuit in Figure 6 can then be written as F I G U R E 6 The novel SYM-OSC oscillator circuit, synthesized using design guidance attained from the novel SYM-AM/PM model. The circuit contains both a nonlinear transconductance i N and transimpedance v N which are defined in Equation (45) k Here, 5 mA refers to a value close to, but a little above I B = 5mA which roughly represent the SB for both PAR/SER circuits, for z 0 = 1:0, as seen from Equations (42)-(43) and footnote ¶. We hence choose the of value of I B closest to the SB which still produces a converging limit-cycle solution.
where x = v C =z 0 , y = i L z 0 with z 0 = ffiffiffiffiffiffiffiffi ffi L=C p being the resonator characteristic impedance, whereas v C , i L are the voltage across the capacitor C and current through inductor L, respectively. Note that each set of circuit parameters in Equation (46) corresponds to a unique ** component set (C, L, R p , R s ) in Figure 6. Figure 7A plots the two measures Λ and Υ, defined in Equations (36) and (37), for the SYM-OSC in Figure 6 with Q p , Q s = 10:0. The measures are plotted as a function of the transconductance parameter a 1 for four values of the resonator characteristic impedance z 0 . Here, the parameter a 1 describes the linear part of the transconductance i N defined in Equation (45) with the value of the remaining transconductance and transimpedance parameters being kept fixed to the values displayed in Table 1. Each curve represents a unique set of the SYM-OSC circuit parameters introduced in Figure 6 as explained in footnote **. The figure displays a clear zero-state at the point a 1 = 6,z 0 = 1. As discussed in the introduction, since this operating point is located away from the SB, we are dealing with a so-called proper zero-state also referred to here as a P-STATE. Figure 7B again plots the two measures Λ and Υ for the SYM-OSC in Figure 6 with Q p , Q s = 10:0. This time as a function of the resonator characteristic impedance z 0 = ffiffiffiffiffiffiffiffi ffi L=C p and for three values of the transimpedance parameter b 2 . Here, the parameter b 2 describes the cubic part of the transimpedance v N defined in Equation (45)  # Value Units **Throughout, we shall assume that the Q-factors in Equation (44) are fixed, and the corresponding parallel and series loads in Figure 6 are then derived as R p = Q p =ðω n CÞ = Q p z 0 and R s = ω n L=Q s = z 0 =Q s . Likewise, the capacitance C and inductance L are easily derived from the circuit parameters as explained in footnote #.
remaining transconductance and transimpedance parameters being kept fixed to the values displayed in Table 1. Each plot point in this figure corresponds to a unique set of SYM-OSC circuit parameters as explained in footnote **. The figure reveals a clear zero-state at the point z 0 = 1,b 2 = 0:3, and again, we are dealing with a P-STATE. The curves in Figure 7A,B all fully corroborate and validate the SYM-AM/PM model discussed in Section 3. Both zero-states in two above figures follow the prediction in Equation (35) as formulated in terms of Υ and Λ in Equation (38). Importantly, we never observe behavior diverging from this rule (e.g., we never observe Υ ! 0^Λ ! ̸ 0 ). Furthermore, as was noted, both of the zero-states in the above figures are so-called P-STATES which implies an operating point away from the SB. Hence, in both cases, we are able to achieve a zero-state while operating the oscillator at nonzero output-power. This is in stark contrast to the MOSFET oscillator example detailed in the previous section which could only attain an ambiguous zero-state (A-STATE). We are still in the early stages of exploring and discovering all the implications which follow from the novel SYM-AM/PM framework developed in Djurhuus and Krozer. 13 The above results for the SYM-OSC circuit, coupled with an upcoming expansion to higher dimensions currently being prepared for publication, clearly demonstrate the promise of this framework for the purpose of synthesis of practical oscillator designs with the optimized phase-noise response.
The ideas discussed herein open the door on a new class of numerical algorithms aimed at optimizing the phasenoise performance of free-running oscillators. Our work details a novel link between the noise performance of an autonomous oscillator system and symmetry properties of the underlying nonlinear steady-state oscillator solution. This insight allows us to understand which directions in oscillator parameter-space will lead to improved noise performance. Our work shows that these are the directions which lead to an increase in symmetry of the steady-state where full symmetry correspond to a fully optimized state (i.e., a zero-state). Knowing the zeros of the optimizing measure is a necessary component in the formulation of these types of routines. Work is currently ongoing in this area, and we plan to publish our results in the near future.

| CONCLUSION
The paper investigates the many interesting practical implications of the newly proposed SYM-AM/PM model framework. A prominent prediction which has emerged from this work is that, for the case of 2-D (planar) oscillators, only circuit topologies generating perfectly symmetric steady-states can have zero AM-PM noise conversion, the so-called zero-states. The concept of zero-states is an interesting topic for practical oscillator design as they represent a minimum which is at least local, if not global, for the total oscillator phase-noise response.
The paper includes simulation results for the important example of a MOSFET differential-pair LC oscillator circuit which is a unit found in most modern RF and μ-wave systems. The various simulations all unequivocally verified the correctness of every prediction made by the SYM-AM/PM model. In addition, these simulations showed that zerostates, for this class of transistor circuits, were only possible at the start-up boundaries (SB) where the oscillator solution bifurcates from the DC operating point, so-called A-STATES. These zero-states are only interesting from a verification point-of-view and do not translate into practical oscillator design solutions.
In an effort to remedy this situation, and to illustrate the power of the newly acquired design rules which follow from the SYM-AM/PM framework, we demonstrated the synthesis of a novel reduced-order LC oscillator circuit, known as the SYM-OSC circuit, which attained these zero-states while operating at nonzero output power, so-called P-STATES. This circuit was synthesized entirely by following the instructions, rules, and predictions derived from the novel SYM-AM/PM model framework. The developed SYM-OSC example clearly illustrates how this new theory can be used the synthesize practical oscillator solutions which attain zero-states while operating at realistic output-power levels (P-STATES). The work detailed herein furthermore clearly demonstrate a possible path toward the eventually development of novel numerical algorithms, based on the SYM-AM/PM model, aimed optimizing oscillator phase-noise performance.