Estimating unstable poles in simulations of microwave circuits

International audienceThe impedance of a microwave circuit has an infinite number of poles due to the distributed elements. This complicates locating those poles with a rational approximation. In this paper, we propose an algorithm to locate the unstable poles of a circuit with distributed elements. The proposed method exploits the fact that a realistic circuit can only have a finite number of unstable poles. We first determine the unstable part whose poles coincide with the unstable poles of the circuit. A rational approximation of the unstable part is used to estimate the unstable poles. Having the ability to trace a circuit's poles as a function of the circuit parameters is a useful design tool. Pole tracking techniques have been used for the design of oscillators [1], [2], to stabilise power amplifiers [3]-[5] and during the design of frequency dividers [1], [6]. The core of a pole tracking tool is a robust automatic algorithm to estimate the poles of a circuit. To determine the poles of a circuit, a two-step procedure is followed. First, an impedance Z(jω) of the circuit is determined at discrete frequency points between 0 and f max with an AC simulation (Fig. 1). Then, the poles of Z(jω) are determined in a post-processing step. In lumped circuits, Z(jω) is a rational function, so a rational approximation can be used to determine the circuit poles. The impedance presented by a circuit with distributed elements, like transmission lines, is non-rational but still meromorphic 1 [7]. This makes estimating the poles of a microwave circuit more difficult. A good fit of Z(jω) can be obtained with a high-order rational function, but the rational approximation will contain spurious poles that do not correspond to poles of the underlying function [8]. To circumvent this problem, another approach proposed in [9], is to compute low-order rational approximants of the circuit's response restricted to small frequency intervals. This local version of the rational approximation scheme, yields precise estimates of poles when these are close enough to the imaginary axis. In this paper, we propose an algorithm that can estimate the unstable poles of a circuit without performing a rational approximation of a non-rational function. The proposed technique exploits the fact that the equilibrium solution of a realistic circuit 2 can only have a finite amount of poles in the right half-plane [10]

Linearized Active Circuits: Transfer Functions and Stability

International audienceWe study the properties of electronic circuits after linearization around a fixed operating point in the context of closed-loop stability analysis. When distributed elements, like transmission lines, are present in the circuit it is known that unstable circuits can be created without poles in the complex right half-plane. This undermines existing closed-loop stability analysis techniques that determine stability by looking for right half-plane poles. We observed that the problematic circuits rely on unrealistic elements with an infinite bandwidth. In this paper, we therefore define a class of realistic linearized components and show that a circuit composed of realistic elements is only unstable with poles in the complex right half-plane. Furthermore, we show that the amount of right half-plane poles in a realistic circuit is finite, even when distributed elements are present. In the second part of the paper, we provide examples of component models that are realistic and show that the class includes many existing models, including ones for passive devices, active devices and transmission lines

Stability analysis of high frequency nonlinear amplifiers via harmonic identification

Nonlinear hyper-frequency amplifiers contain nonlinear active components and lines, that can be seen as linear infinite dimensional systems inducing delays that cannot be neglected at high frequencies. Computer assisted design tools are extensively used. They provide frequency responses but fail to provide a reliable estimation of their stability, and this stability is crucial because an unstable response will not be observed in practice and the engineer needs to have this information between building the actual device. We shall present the models of such devices, and the current methods to compute the response to a given periodic signal to be amplified (this is a periodic solution of a periodically forced infinite dimensional dynamical system) as well as the frequency response of an input-output system associated to the linearization around this periodic solution. The goal of the talk is to present the ideas and preliminary results that on the one hand allow to deduce stability of this time-varying linear system from that frequency response and on the other hand provide a relationship between this stability and the internal stability of the actual nonlinear circuit. The first point resorts from harmonic analysis and perturbation of linear operators. The second one from nonlinear infinite dimensional dynamics and ad'hoc linearization

Obtaining the preinverse of a power amplifier using iterative learning control

\u3cp\u3eTelecommunication networks make extensive use of power amplifiers (PAs) to broaden the coverage from transmitter to receiver. Achieving high power efficiency is challenging and comes at a price: the wanted linear performance is degraded due to nonlinear effects. To compensate for these nonlinear disturbances, existing techniques compute the preinverse of the PA by the estimation of a nonlinear model. However, the extraction of this nonlinear model is involved and requires advanced system identification techniques. The plant inversion iterative learning control (ILC) algorithm is used here in combination with the best linear approximation to investigate whether the nonlinear modeling step can be simplified. This paper introduces the ILC framework for the preinverse estimation and predistortion of PAs. The ILC algorithm is used to obtain a high quality predistorted input for the PA under study without requiring a nonlinear model of the PA. In a second step, a nonlinear preinverse model of the amplifier is obtained. Both the nonlinear and memory effects of a PA can be compensated by this approach. The convergence of the iterative approach and the predistortion results are illustrated on a simulation of a Motorola LDMOS transistor-based PA and a measurement example using the Chalmers RF WebLab measurement setup.\u3c/p\u3

Model-Free Closed-Loop Stability Analysis: A Linear Functional Approach

International audiencePerforming a stability analysis during the design of any electronic circuit is critical to guarantee its correct operation. A closed-loop stability analysis can be performed by analysing the impedance presented by the circuit at a well-chosen node without internal access to the simulator. If any of the poles of this impedance lie in the complex right half-plane, the circuit is unstable. The classic way to detect unstable poles is to fit a rational model on the impedance. This rational approximation has to deal with model order selection, which is difficult in circuits with transmission lines. In this paper, a projection-based method is proposed which splits the impedance into a stable and an unstable part by projecting on an orthogonal basis of stable and unstable functions. Working with a projection instead of a rational approximation greatly simplifies the stability analysis. When the projection is mapped from the complex plane to the unit disc, the projection boils down to calculating a Fourier series. If a significant part of the impedance is projected on the unstable part, a low-order rational approximation is fitted on this unstable part to find the location of the unstable poles

Optimal bounds and matching networks of fixed degree for frequency varying impedances

International audienceIn this paper, matching networks of finite degree are computed. Additionally the presented results are compared with the lower fundamental bounds available in the literature. These bounds are used to certify the optimality of the provided matching networks in function of the attained matching tolerance. To illustrate the presented results, two different examples of matching problems are presented