Abstract
The term pole-zero identification refers to obtaining the poles and zeros of a linear (or linearized) system described by its frequency response. This is usually done using optimization techniques (such as least squares, maximum likelihood estimation, or vector fitting) that fit a given frequency response of the linear system to a transfer function defined as the ratio of two polynomials [1], [2]. This kind of linear system identification in the frequency domain has numerous applications in a wide variety of engineering fields, such as mechanical systems, power systems, and electromagnetic compatibility. In the microwave domain, rational approximation is increasingly used to obtain black-box models of complex passive structures for model order reduction and efficient transient simulation. An extensive bibliography on the matter can be found in [3]-[6]. In this article, we focus on a different application of pole-zero identification. We review the different ways in which pole-zero identification can be applied to nonlinear circuit design, for power-amplifier stability analysis, and more. We provide a comprehensive view of recent approaches through illustrative application examples. Other uses for rational-approximation techniques are beyond the scope of this article.
Highlights
Pole‐zero identification refers to the obtaining of the poles and zeros of a linear system described by its frequency response
It provided a simple and intuitive way to solve the generalized eigenvalue problem without the need to access the Jacobian of the system [8] or to the internal nodes of the non‐linear models of active devices [9]. It is perfectly fitted for use in combination with conventional microwave simulators as Advanced Design System (ADS) or Microwave Office (MWO)
Perhaps the most attractive features of pole‐zero identification for stability are its simplicity and the graphical and intuitive nature of the results: whenever a pair of complex conjugate poles is found lying on the Right‐Half Plane (RHP), i.e. poles with positive real part, an oscillation will begin to grow from the steady state
Summary
Controlling the position of unstable poles on the complex plane can be used to improve the transient characteristics for applications in which switching times are relevant. Pole‐ zero identification is used to obtain the pair of complex‐conjugate poles σ ± jω of the unstable solution from which the oscillation builds up. This pair of poles will dominate the initial transient of the circuit (at least while the amplitude of the growing oscillation is still small). The technique in [17] carries out a tuning of the oscillator element so as to increase the positive real part σ of the unstable poles while simultaneously the required oscillation frequency and first harmonic amplitude are maintained. The magnitude of the first harmonic is similar, the two measured waveforms have a very different harmonic content, as can be seen in the detailed plot of Fig. 12b
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