Oscillator Phase Noise: A Geometrical Approach

Abstract

Abstract-We construct a coordinate-independent description of oscillator linear response through a decomposition scheme derived independently of any Floquet theoretic results. Trading matrix algebra for a simpler graphical methodology, the text will present the reader with an opportunity to gain an intuitive understanding of the well-known phase noise macromodel. The topics discussed in this paper include the following: orthogonal decompositions, AM-PM conversion, and nonhyperbolic oscillator noise response.

Eliminating1/fnoise in oscillators

“…This follows because the effect of the noise sources on the oscillator phase noise is given by projecting the noise onto a particular direction, the phase sensitivity vector v ⊥ , in the phase space. This vector is perpendicular to the isochrone at the limit cycle-the surfaces of perturbations to the limit cycle that decay to the same phase point on the cycle [17][18][19][20]-and can be calculated as the eigenvector with zero eigenvalue of the adjoint of the Jacobian of the evolution equations linearized about the no-noise limit cycle [16,21]. To implement this we write the complex amplitude equation as a pair of real equations for (a, ).…”

Selective Determination of Floquet Quantities for the Efficient Assessment of Limit Cycle Stability and Oscillator Noise

“…According to (6), the last (n − m)N t rows ofR are zero (N t = 2N H +1), and this property can be effectively exploited to minimize the size of the generalized eigenvalue problem (11). To this aim, the columns of matrixR need to be reordered by means of the reduced row echelon form (RREF) reduction technique [16]; since for the problem to admit a solution matrixB should be full rank, applying the RREF toB (11) can be cast in the form…”

“…We discuss here the error introduced in the solution of (4) by the approximation on the R matrix performed in (6). The starting point of the analysis is (4), where the approximation leading to R (t) is expressed in perturbative form…”

“…The upper triangular matrix D(t) is characterized by finite elements, and by zero elements in the first m rows and columns [see (6)]. Also note that, since the result of the QR factorization with column pivoting has the largest magnitude element of each row on the diagonal, matrix D(t) can be chosen as normalized with nonzero diagonal element not larger than min t∈]0,T ] R m,m (t) .…”

Derivation of Floquet Eigenvectors Displacement for Optimal Design of LC Tank Pulsed Bias Oscillators

“…Although electronic simulators offer results in good agreement with effective measurements, they cannot be used to infer general 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 contributions have really extended the capability to develop innovative techniques in the design of oscillators systems.…”