cm(Xd,xcm) - Ib 0 (E.3) (E.4) (E.5) (E.6) (E.7) 1/VLC (E.8) (E.9) (E.10) 1 Note that, in this appendix, we neglect all transistor capacitances. This introduces only small errors as far as the dynamics induced by u(t) is concerned. However, when interested in the upconversion of the 1/f noise coming from the bias-current sources Ib, it is of great importance to take the transistor capacitances into account. These capacitances induce (a slight) skewing of the common-mode voltage which can be shown the main responsible for the upconversion of 1/f noise coming from the bias currents. With some extra effort, the reasoning outlined in this appendix can be extended to include the effects of the transistor capacitances as well. E.1 DETERMINING THE OSCILLATOR'S AVERAGED DYNAMICS Figure E.1: Harmonic oscillator topology injected with a (small) current u(t). In the above, iL is the current flowing through the inductor while Vosc is an estimate for the oscillation amplitude. Furthermore, IM, is the differential transistor current while IM,cm equals the common-mode transistor current. As usual, the terms of (E.1)-(E.3) in square brackets denote perturbation terms while the others correspond to the oscillators core system. The set of all steady-state solutions to the equations corresponding to the oscillator's core system behavior, i.e. (E.1)-(E.3) with the terms between the square brackets re- moved, is given by Acos(t+0) (s(t,p) Asin(t+0) (E.11) f(Acos(t+0)) the oscillator's state vector. In (E.11), the function Xcm = f (xd) is determined by solving (E.3) for Xcm. Next, using (E.11) we find -Asin(t+0) Acos(T+0) (E.12) Using the theory outlined in section 5.5.2 of chapter 5, we find that the corresponding projector V1(t,p) equals the solution to (E.13) V[(t,p)C(t,p)U(t,p) = I (E.14) where IE R2x2 is the unit matrix and C(t,p) = (E.15) G(t,p) (E.16) This solution is found to be (E.17) Hence, using (5.70) from section 5.5.2 in chapter 5, the oscillator's amplitude and phase behavior is seen to be governed by (E.18) In the above