F. L. Neerhoff

Modal factorization of time-varying models for nonlinear circuits by the Riccati transform

In this paper, general linear time-varying systems are addressed. They are considered as small-signal models of nonlinear circuit solutions. The transition matrix is constructed by repeated Riccati transforms. It is shown that each transform factors out a single mode. Key concepts of time-invariant theory are generalized to the time-varying context. They provide a unified mathematical framework, suitable for assessing the local behavior of nonlinear dynamic circuits.

Diagonalization algorithms for linear time-varying dynamic systems

On the basis of a mode-vector representation, we show that its time-varying amplitudes and frequencies can be obtained by diagonalizing the system matrix. Next, we reformulate an explicit diagonalizing algorithm that was earlier proposed by Wu. Then, the missing convergence proof is given. Moreover, we present a new and implicit iteration scheme that is closely related to that given by Wu. In both algorithms, the time-varying system matrix is gradually diagonalized by successive algebraic similarity transformations. It is proved that the convergence conditions are essentially the same. Although the class of systems for which the algorithms are applicable is still not fully known, the results of this paper may be of theoretical and practical interest.

A complementary view on time-varying systems

This contribution is complementary to a previous approach by the authors in that it takes a time-varying mode-vector solution as an a priori assumption. The associated dynamic eigenvalue problem is solved by triangularizing the system equations, again accomplished by successive Riccati transforms. It is explicitly shown that equal eigenvalues do not give rise to the Jordan form. Although the mode-vectors are not uniquely determined, it is demonstrated by example that different representations yield identical transition matrices, as it should.

Dynamic behavior of dynamic translinear circuits: the linear time-varying approximation

Dynamic translinear circuits explore the exponential relation of transistors as a primitive for the synthesis of electronic circuits. In this letter, the linear time-varying approximation is applied to describe the dynamic behavior of a second-order dynamic-translinear oscillator. The Floquet exponents are calculated by the dynamic eigenvalues introduced earlier.

SOME ANALYTIC CALCULATIONS OF THE CHARACTERISTIC EXPONENTS

As is well known, the variational equations of nonlinear dynamic systems are linear time-varying (LTV) by nature. In the modal solutions for these LTV equations, the earlier introduced dynamic eigenvalues play a key role. They are closely related to the Lyapunov- and Floquet-exponents of the corresponding nonlinear systems. In this contribution, we present some simple examples for which analytic solutions exist. It is also demonstrated by example how the classical linear time-invariant (LTI) solutions are related to the equilibrium points of the general LTV solutions.

The linear time-varying approach applied to the design of a negative-feedback class-B output amplifier

The increasing number of battery-operated devices asks for more low-power electronic circuits. A major cause of current consumption are the bias currents. Thus designing circuits having no bias currents may lead to considerable reductions in the power consumption. This inherently implies that we have to deal with the nonlinear behavior of devices. Here the design of a low-voltage low-power negative-feedback amplifiers with a class-B output stage is described. The key issue in the design is the description of the dynamics of the amplifier by the linear time-varying approach. This uses a time-varying small-signal model of nonlinear circuits and enables a generalization of the traditional pole-zero concept. The designed amplifier is capable of driving 1 mA through a piezoelectric load (14 nF). The stand-by current varies from 40 /spl mu/A to 100 /spl mu/A.

The Cauchy-Floquet factorization by successive Riccati transformations

Scalar linear time-varying systems are addressed. In particular, a new factorization method for the associated scalar polynomial system differential operator is presented. It differs from the classical results due to Cauchy and Floquet, in that it is based upon successive Riccati transformations of the Frobenius companion system matrix. As a consequence, the factorization is obtained in terms of the earlier introduced dynamic eigenvalues.