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Estimating unstable poles in simulations of microwave circuits

Abstract : The 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].
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Submitted on : Tuesday, December 18, 2018 - 11:30:42 AM
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  • HAL Id : hal-01958795, version 1

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Adam Cooman, Fabien Seyfert, Smain Amari. Estimating unstable poles in simulations of microwave circuits. IMS 2018, Jun 2018, Philadelphia, United States. ⟨hal-01958795⟩

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