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The aim of this article is to give insight into the mechanisms behind the behavior of oscillators from a new angle, introducing the idea of "frozen eigenvalues." This approach is based on piecewise-linear modelling and a study of the eigenvalues of the time varying linearized Jacobian of the nonlinear differential equations describing the oscillator. A multivibrator and a quadrature oscillator are used as test examples. The mechanisms behind the oscillations of the two circuits are compared.