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

Abstract

Performing 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. 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. When the unstable part lies significantly above the interpolation error of the method, the circuit is considered unstable. Working with a projection provides one, at small cost, with a first appraisal of the unstable part of the system. Both small-signal and large-signal stability analysis can be performed with this projection-based method. In the small-signal case, a low-order rational approximation can be fitted on the unstable part to find the location of the unstable poles.Comment: Longer version of the paper published in IEEE Transactions on Microwave Theory and Technique

Estimating unstable poles in simulations of microwave circuits

“…In [11], the Fast Fourier Transform (FFT) is used to compute the f k . This introduces an aliasing error in the method.…”

“…Above that frequency, we assume that Z(jω) = 0. This sudden jump to zero at f max will have a strong unwanted influence on the obtained stable and unstable part [11]. To suppress this influence,Z(θ) is multiplied by a low-pass filter with its stop-band starting at f max :…”

“…In [11], an elliptic filter on the complex plane is used. On the unit disc, this elliptic filter has an infinite number of Fourier coefficients, which increases the aliasing error in the method.…”

“…The f filt k are then used in Φ N to compute the unstable poles. With the elliptic filter used in [11], the sum in the convolution is infinite. With the new filter, this sum becomes finite.…”

“…In [11], it was shown that estimates of the unstable part can be obtained by computing the Fourier coefficients of a pre-filtered version of the impedance function. In this paper, the method is extended so as to yield estimates of the unstable poles.…”