t tanks with a sufficiently high Q-factor, they can be treated as perturbations. Hence, the (coupled) harmonic oscillator's core system is chosen to coincide with the K lossless resonant tanks. 5.8.1 Model extraction theory 5.8.1.2 Characterizing the oscillator's unperturbed core. The oscillator's core system corresponds to K independent lossless resonant tanks. It is described by (5.151)-(5.152) for E set to zero. The steady-state solutions to this system are given by (5.153) Xsk(T)=Akcos(T+Ok) (5.154) Each tank resonates with an arbitrary but fixed amplitude Ak and phase Ok. These resonant tank amplitudes and phases are combined in the core system's state vector O2 ... AK (5.155) Each particular value of the state vector identifies a possible state (steady-state solution) of the oscillator's core. 5.8.1.3 Oscillator perturbation analysis For small but nonzero perturbations, i.e. for 0 < E << 1, the state vector p slowly changes with time. Up to first order in E, the behavior of p is governed by (5.156) Here, g(t,p) (5.157) -S1A1COS(T+01) is the vector in R2K containing the perturbation terms. The projector VT(t,p) E R2KX2K can be shown to equal sin(t+01) cos(t+01) V(t,p) = (5.158) a block-diagonal square matrix. Since in most practical applications phase differences are of greater interest than ab- solute phase values, we again change variables. Equation (5.156) is reformulated in 5.8 HARMONIC OSCILLATOR BEHAVIORAL MODELING terms of one common-mode phase 0 = 01 and the phase differences = Ok - 01 between the k-th resonant tank and the first one. In stable oscillators, these phase dif- ferences converge to a constant operating-point value. This property is exploited later on in our model extraction procedure. Introducing the vector AQK 0] (5.159) = Tqp (5.160) where ER2KX2K (5.161) the ODE (5.156) can be recast into (5.162) Equation (5.162) captures the dynamic behavior of the oscillation amplitudes, the phase differences and the common-mode phase. It is accurate up to first order in E. 5.8.1.4 Averaging The pre