dicates that a bad job has been done in partitioning the original set of circuit equations. CHARACTERIZING THE OSCILLATOR'S UNPERTURBED CORE Figure 5.12: The set of all points that lie on one of the steady-state solutions of the harmonic oscillator's core system make up a 2-dimensional manifold (surface). All other solutions that solve the unperturbed core system's equa- tions are quickly attracted towards this manifold. arbitrary amplitude and phase. The algebraic equation (5.40) makes the third state variable, the common-mode voltage vcm(t), "track" the differential voltage vd(t). In in Fig. 5.11(c), these solutions are pictured as points in a 2-dimensional state-vector space. Fig. 5.12 plots the 2-dimensional manifold (surface) M -defined by (5.37)- consisting of all points that lie on a steady-state solution. This manifold is stable as it quickly attracts all other solutions. Example (Voltage-controlled harmonic oscillator): The (normalized) unperturbed core system associated with the voltage-controlled oscillator topology in Fig. 5.9 is described by the system of equations (5.41) (5.42) (5.43) -Xcm Xn (5.44) (5.45) These equations are obtained by neglecting the perturbation terms in (5.28)-(5.32), i.e. the terms in the square brackets. The periodic steady-state solutions (states) of the core system are given by xd(T) Acos(t+0) (5.46) ya(t) Asin(t+0) (5.47) xcm(T) = (5.48) ycm(T) (5.49) (5.50) All steady-state solutions taken together yield a 3-dimensional manifold. Hereby p=[A02]`ER (5.51) is the corresponding state vector. Note that, compared to the previous example, the dimension of the state vector p has increased by one by means of the addition of the common-mode variable Z. This increase is due to the presence of the control voltage Vcontrol in Fig. 5.9. This control voltage induces the common-mode charging and dis- charging of the resonant tank capacitors. This, in turn, introduces an extra slow-varying (common-mode) process in the oscillator's transient behavior. Typical analyses, e.g. [Ham01], do not