on is a non-orthogonal one. The projector VT (t,p) is calculated by solving the homogeneous linear system of DAEs (5.72) Furthermore, its initial value VT(0,p) satisfies V[(0,p) C(0,p)U1(0,p) C(0,p)U2(0,p) G(0,p)U3(0,p) 0 0 (5.73) with I E RPxP the P-dimensional unity matrix. Here, U1(t,p) and U3(t,p) are re- spectively defined by (5.58) and (5.62). The columns of U2(t,p) contain the stable modes of (5.59), i.e. the solutions that satisfy (5.61). The condition VT(0,p)G(0,p) X U3 (0,p) = 0 is imposed in order for VT (0,p) to be a valid initial condition for solving the system of DAEs (5.72). Example (Voltage-controlled harmonic oscillator): For the voltage controlled har- monic oscillator in Fig. 5.9, we find, by means of (5.46)-(5.50) and with p=[Aez]T, (5.74) 5.5.3 In summary cos(t+0) -Asin(t+0) sin(t+0) Acos(t+0) U1(t,p) (5.75) Furthermore, with C(t,p) = G(t,p) (5.76) we obtain U3(t,p) (5.77) while U2(t,p) is empty. Using (5.72) and (5.73), the projector VT(s, can be shown to equal (5.78) Hence, the equations that govern the changes of the state vector p are (5.79) (5.80) (5.81) These equations capture the dynamic behavior of the oscillation amplitude A, the phase 0 and the common-mode level Z. In (5.79)-(5.81), the differential transistor current IM,a(t+0,A,z) is computed by substituting the voltages V1 (T) = and V1 ( T = Vosc(z - Acos(t+0)) (see the equations (5.12)-(5.15), (5.27) and (5.46)- (5.50)) into (5.21). Using a likewise procedure, we can obtain the common mode transistor current IM,cm (T+0,A,z). Note that the phase 0 always appears in conjunction with the time variable T. The equations (5.79)-(5.81) capture that part of the oscillator's dynamic behavior that is, for most practical purposes, of greatest interest. They describe the transient dynam- ics of the oscillator's amplitude, phase and common-mode behavior. Compared to the original system of five equations, i.e. (5.28)-(5.32), the number of equations needed 5.6 AVERAGING Rloss Figure 5.16: Startup behavior of a harmonic oscillator. Here, amplit