Periodic Motions

Computation of all the Floquet eigenfunctions in autonomous circuits

“…Initially we chose Ct = 0 (this is equivalent to remove the capacitor) and perform the Floquet analysis. The computed working frequency is 985.14 MHz and the first three multipliers are 0 = 0.99985, 1 removed. From Figure 9, we can easily see that v 0 (t)b A (t) and v 1 (t)b A (t) are very different; thus, it is important to identify the Floquet multiplier theoretically equal to 1 and the corresponding v 0 (t)b A (t) eigenfunction.…”

“…In particular, since the columns of U(·) are equal to the rows of V(·), we have v 0 (t) = u 0 (t) and v 1 (t) = u 1 (t) as shown in the IV quadrant of Figure 1. We have in this specific case u 0 (t) orthogonal to u 1 …”

“…The theoretical background of these VMs is closely related to Floquet theory [1]. Since VMs are linear and time varying, they can be represented by linear circuits composed of time-varying elements.…”

“…This analysis is denoted as periodic AC (PAC) in the sequel. In most simulators the PAC analysis constitutes the basis for noise analysis of circuits 719 is the one based on MNA, the relation mapping the two matrices is expressed by the linear transformation B(t) = QA(t)Q T (1) where Q ∈ R M×N with M N , all entries in the set {−1, 0, 1} and rank(Q) = N . The Q matrix transforms voltage state variables in node voltages (leaving untouched state variables that are expressed as currents).…”

“…When the circuit is autonomous, D has (at least) ‡ one entry equal to 0 [1]. Reordering equations and variables, we assume that d 0 = 0 corresponds to the u 0 (t) column of U(t) and to the v 0 (t) row of V(t), having defined u j (t) and v j (t) the columns of U(t) and the rows of V(t), respectively (we assume that rows/columns are numbered starting from 0).…”