Improved Small-Signal Analysis for Circuits Working in Periodic Steady State

Abstract

This paper considers the formulation of the variational model (VM) of autonomous circuits (oscillators) working in periodic steady-state conditions. The shooting method, which is largely used to compute the solution in the time domain when the VM is forced by a small-signal perturbation, is studied. The proposed analytical approach can be exploited to improve accuracy in the simulation of the effects of noise sources. In particular, we justify from an analytical standpoint the adoption of a suitable periodicity constraint in the shooting method. We exploit the properties of block circulant matrices that naturally arise in the description of the problem. We prove that the frequency of the small-signal perturbation must be different from that of the unperturbed oscillator to avoid inaccuracy of the shooting method due to the existence of singularities in the VM formulation, and derive a method that allows us to get closer to the singularity

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

“…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 .…”

“…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 .…”

“…These effects are determined through an approach often referred to as PAC, for what concerns periodic small signal analysis, and PNOISE for what concerns periodic linear noise analysis [17]- [22]. In both analyses it is assumed that the superposition theorem holds and thus the effects of the small signal perturbations are added to the large signal result, that, in our case, is the large unperturbed solution of the PLL.…”

On the Cyclostationary Noise Analysis in Large RF Integrated Circuits

“…The harmonic balance formulation is better suited to RFIC noise analysis than is discrete time-domain formulation [4], [6], [8], [9], [18], since a frequency-domain formulation accurately handles multitone large-signals as well all arbitrary frequency-defined characteristics which are common case in RF applications.…”

“…Derivation of transfer function is itself a multiple steps process as follows-first, solve for the (quasi) periodic noiseless equilibrium of the circuit-second, linearize the system equation around the noiseless equilibrium to get the perturbation equation-third, solve the perturbation equation. The solution of the perturbation equation can be performed either in the frequency-domain which results in harmonic-balance based noise analysis [3], [5], [10] or in finite-difference time-domain which results in time-shooting based noise analysis [4], [6], [8], [9], [18]. When dealing with large circuit size analysis, derivation of the transfer function is a numerically critical point to address efficiently in terms of arithmetic operations count and memory storage.…”