Periodic noise analysis of electric circuits: Artifacts, singularities and a numerical method

Abstract

SUMMARYIn this paper it is shown that a numerical method largely adopted for the simulation of noise in autonomous circuits is affected by singularities that manifest when the frequency at which the noise analysis is carried out approaches a harmonic of the autonomous circuit. The resulting noise power spectral density (PSD) is thus characterized by spurious spikes. The presence of these singularities is for the first time justified from an analytical standpoint and their effects are shown by simulating some oscillators, employed as benchmarks. Furthermore, the presented approach justifies the 1/( f s − f ) 2 shape of the PSD of noise at the output when the f s frequency approaches the f fundamental of a stable oscillator and the 1/| f s − f | 3 shape when the effects of flicker noise are manifest.

Stability analysis of voltage regulators versus different digital control strategies by analog-mixed-signal circuit simulation

“…To optimise the size of the capacitor bank while ensuring stability and keeping the performance of the VRM acceptable, we performed efficient steady state analyses and computed the small signal R Ao (f ) impedance as detailed in [1] by exploiting the periodic small signal (PAC) analysis [15], [16] extended to AMS circuits through saltation matrix [17]. R Ao (f ) can be considered as the periodic time varying transfer function between the current 4 injected by A o and the variation of the V Ao voltage (CPU land).…”

Periodic Small Signal Analysis of a Wide Class of Type-II Phase Locked Loops Through an Exhaustive Variational Model

“…Equation (16) can be rewritten as (18) The matrix can be expressed through the Fourier decomposition of each one of its entries as (19) where is an complex elements matrix, and are elements diagonal matrices. As a consequence, (18) can be finally transformed into (20) where are diagonal matrices whose -element ( ) is (21) In practice, the summations in (19) and (20) are truncated to terms, and, in the proposed example, is fixed to 10. Assuming that the periodic input to system (15) is an elements vector whose th entry is , the integral in (21) can be solved in close form as shown by (22), shown at the top of the next page.…”

“…As a consequence, (18) can be finally transformed into (20) where are diagonal matrices whose -element ( ) is (21) In practice, the summations in (19) and (20) are truncated to terms, and, in the proposed example, is fixed to 10. Assuming that the periodic input to system (15) is an elements vector whose th entry is , the integral in (21) can be solved in close form as shown by (22), shown at the top of the next page. (22) The evaluation of in (18) can then be done, for any of the aforementioned time instants at which is known, by computing at first according to (22) for and , and then evaluating the summation in (20) for .…”

Steady State Computation and Noise Analysis of Analog Mixed Signal Circuits

“…The generalized saltation matrix expression is (10) It is important to notice that the denominator of the second term in the r.h.s of (10) does not depend on the expression assumed by after the switching event. This term is related only to the system characteristics for .…”