OSCILLATORS, HYSTERESIS AND ”FROZEN EIGENVALUES” Erik Lindberg Ørsted•DTU Department, 348 Technical University of Denmark, Ørsteds Plads, Kgs. Lyngby, DK2800, Denmark e-mail: el@oersted.dtu.dk WWW: http://www.es.oersted.dtu.dk/∼el/ Abstract— The aim of this tutorial is to pro- vide insight in the mechanism behind the be- haviour of oscillators. A 10MHz negative resis- tance oscillator is used as an example. A hys- teresis phenomena in connection with the negative resistance characteristic is found. By means of piece wise linear modeling and the ”frozen eigen- values” approach sinusoidal oscillations are inves- tigated. I. Introduction Amplifiers are normally considered linear cir- cuits with a stable time-independent dc bias point as reference for the signal amplification. Real os- cillators are nonlinear amplifier circuits which have an unstable dc bias point as a startup ref- erence. In the steady state the bias point of the linearized small signal model vary with time. By means of the Barkhausen Criteria a sinu- soidal oscillator is normally designed as a linear amplifier with a frequency determining feed-back circuit so that the poles of the whole circuit are on the imaginary axis of the complex frequency plane, i.e. an ideal harmonic oscillator which may be modelled as an ideal linear inductor in parallel with an ideal linear capacitor. In order to startup the oscillations some parameters of the circuit are adjusted so that the poles (eigenvalues) of the lin- earized circuit are in the right-half plane, RHP, i.e. in the dc bias point the circuit is unstable and signals will start to increase when the power sup- ply is connected. It is obvious that if the poles are very close to the imaginary axis the time constant is very large (days, months, years ?) and the tran- sient time to steady state behaviour is very large, i.e. apparently steady state sinusoidal oscillations take place. The placement of the poles on the imaginary axis is an impossible act of balance. Due to the nonlinearities, the physical nature of the circuit, the signals will always be limited in some way. The crucial point is whether there will be oscillations or just a transition to a new stable dc bias point. Very little is reported about how far out in RHP the poles should be placed ini- tially. Also many authors assume that due to the nonlinearities the poles are fixed on the imaginary axis at a certain constant amplitude of the signals Fig. 1. A negative resistance oscillator. VCC 1= VCC 2 = 5V, RC 1 = RC 2 = 50kΩ, C 1 = 106pF, L1 = 2.39μH, RLS = 1.0Ω,VNL = V 12 = 0V, Q1= Q2=2N 2222. [1], [2]. It is an open question if this assumption is true. The normal classification of oscillators in two groups ”sinusoidal” and ”relaxation” is also questionable because the same oscillator topology may perform both ”kinds” of oscillations depend- ing on the parameter choice [3]. Classification should be based on the mechanism behind the be- haviour. At a certain instant you may investigate the lin- earized small signal model of the oscillator. The eigenvalues of the Jacobian of the linearized differ- ential equations are the poles of the network func- tions. It is obvious that the signals will increase