stem and a set of perturbation terms. Example (Voltage-controlled harmonic oscillator): For the system of DAEs (5.16)- (5.20) that model the oscillator circuit in Fig. 5.9, all voltages are normalized with 5.3 OSCILLATOR CIRCUIT EQUATIONS respect to Vosc, an estimate for the amplitude of the resulting differential-mode oscilla- tion. The differential current id(t) is then normalized with respect to = (2w0C)Vosc=osc/(w0L/2) (5.24) The (small) common-mode current icm(t) is best normalized with respect to Iosc/Q. Hereby, (5.25) is the natural frequency of the oscillator's resonant tank, while (5.26) equals its quality factor. With (5.27) being the normalized state variables, we find (5.28) (5.29) (5.30) (5.31) (5.32) Here, the normalized time variable T is as defined in (5.23). The reason for some terms to be enclosed in square brackets is explained in the next section. 5.3.2 Partitioning the normalized circuit equations Having normalized the oscillator's circuit equations, we partition them into a core sys- tem and a set of perturbations. The core system corresponds to the terms of the (nor- malized) circuit equations that are large in magnitude while the perturbation terms cor- respond to the small ones. As such, the core system captures that part of the oscillator that is the main responsible for the oscillations to occur. The perturbation terms cause the core system to change behavior over time over time. Formally, the normalized and partitioned set of oscillator circuit equations can be writ- ten as -(((c(x)+g(x)+ef(x,t)=0 (5.33) 5.3.2 Partitioning the normalized circuit equations Itank noise Rloss feedback Figure 5.10: Two possible ways to partition an harmonic oscillator topology with the parts encircled by the dashed lines corresponding to the perturbation terms. (a) If one is mainly interested in the oscillator's phase noise behavior, only the noise source is treated as a perturbation term. (b) If one targets the oscillator's settling behavior, all currents due to the feedback transistors and tank losses ar