rresponding to the circuit's modified nodal equations: (5.6) Here, V E RN is a vector storing the circuit's state variables, for example, the circuit's nodal voltages. Furthermore, q(v) E RN is a vector of capacitor charges and/or inductor fluxes while j(v) ERN is a vector of branch currents. The vector b(t) E RN models the action of some weak (small) input sources, e.g. noise. Example (Voltage-controlled harmonic oscillator): The voltage-controlled harmonic oscillator in Fig. 5.9 is described by the system of DAEs (5.7) (5.8) (5.9) (5.10) (5.11) Here, the resistors with values R/2 model the inductor losses, IM (Vgate, Vdrain) denotes the drain-source current of the MOS transistors and the capacitors C capture some 5.3.1 Normalizing the oscillator circuit equations small (transistor) capacitances. By introducing the differential voltages and currents id = (i1 - i2 ) 2 = (5.12) (5.13) Vd = (v1 - v2) / 2 = (5.14) Vcm = (v1+v2) / 2 = (5.15) the equations (5.7)-(5.11) can be written as (5.16) (5.17) (5.18) (5.19) (5.20) In the equations above, IM,d = - (5.21) (5.22) respectively denote the differential and common-mode transistor currents. Further- more, in order not to overload the example, we neglected the nonlinear characteristic of the resonant tank's capacitors (varactors). Also, in this example, we assumed the components of the vector b(t) to be zero. 5.3.1 Normalizing the oscillator circuit equations Although not strictly necessary, it is good practice to normalize the equations (5.6) such that all quantities involved are, approximately, one in order of magnitude. To do so, we normalize all voltages and currents with respect to some nominal value. These normalization constants can be user-provided. They can also be obtained by means of short SPICE-like trial simulations. Moreover, we also normalize the time variable T=Wot. . (5.23) Here, wo is chosen to make derivatives with respect to T one in order of magnitude. Normalizing the oscillator circuit equations later on facilitates partitioning them into a core sy