3.4 Time-Domain Techniques for Distortion Analysis
3.4.1 Time-Step Integration Basics
Substituting (3.257) into (3.256) and assuming dynamic time-step selection, we have
GvO(tk) + qNL[vO(tk)] −qNL[vO(tk−1)]
hk +iNL[vO(tk)] =iS(tk) (3.258a) or
hkGvO(tk)+qNL[vO(tk)] +hkiNL[vO(tk)] =hkiS(tk)+qNL[vO(tk−1)]
(3.258b)
which shows we can determinevO(t) for any time-sample,tk, from the knowledge of the forcing function at that point,iS(tk), and the past circuit solution, or system state,vO(tk−1).11
So,vO(t) is calculated for all timet0<t <tKbeginning with the knowledge of the initial conditionvO(t0), and solving the nonlinear algebraic equation (3.258) for each time-step:
vO(t0)
h1GvO(t1)+qNL[vO(t1)] +h1iNL[vO(t1)] =h1iS(t1) +qNL[vO(t0)]
⯗
hkGvO(tk) +qNL[vO(tk)]+hkiNL[vO(tk)] =hkiS(tk) +qNL[vO(tk−1)]
⯗
hKGvO(tK) +qNL[vO(tK)] +hKiNL[vO(tK)] =hKiS(tK)+qNL[vO(tK−1)]
(3.259) which results in a set ofK nonlinear algebraic equations that can be successively solved forvO(t1), . . . ,vO(tk), . . . ,vO(tK) using the Newton-Raphson scheme.
This formulation follows directly from our intuitive knowledge of dynamic systems’ operation. So, it should be of no surprise that it was used in the first digital computer programs of circuit analysis and is still nowadays the most widely used numerical method for that purpose. It is present in all SPICE or SPICE-like computer programs.
Time-step integration is a particular implementation of what is usually known in differential equations as aninitial value problem,because it solvesvO(t) for all tk from the knowledge of the initial condition (or state)vO(t0). Therefore, it is clearly tailored for finding the circuit’s transient responses. However, if the objective is the determination of the steady-state, there is no other way than to pass through the painful process of integrating all transients, and expecting them to vanish. In circuits having extremely different time constants, or highQresonances, time-step integration can be very inefficient. And, unfortunately, this is typical of RF and microwave circuits, because, not only they are narrowband tuned, as their bias networks present time constants that are various orders of magnitude higher than the excitation period, or the time constants of the signal networks.
Since distortion problems demand for very high numerical dynamic ranges, and the transition from transient to steady-state behavior is gradual, the response periodicity must be guaranteed with high precision, which contributes to also
11. Because our circuit is of first-order—only first-order time-derivatives are involved—the system state is characterized by onlyvO(tk−1). In a circuit ofnth order, the state would require knowledge ofvO(tk−1), . . . ,vO(tk−n).
exacerbate simulation time. But, this is not the only problem time-step integration poses to distortion simulation. For example, a two-tone excitation involves two different time-scales, which correspond to the tones’ period and the tones’ separa- tion, or envelope period. If the tones are commensurated, but close in frequency, the number of time-steps can be very large. If they are uncommensurated, the period tends to infinity, and the circuit will never reach a periodic steady-state.
Obviously, postprocessing these simulation results by some kind of windowing and FFT computation will always involve a certain amount of error [19]. This error, appearing as a spectrum ‘‘noise floor,’’ can easily mask small amplitude distortion components. Another common origin of this numerical noise floor comes from the dynamic step usually adopted to reduce simulation time. Because FFT algorithms require a uniform time-sampling, this type of postprocessing is always preceded by interpolation. And this introduces an amount of noise which often cannot be tolerated. In such cases, there is no alternative way than to impose a fixed time-step (e.g., by declaring a very conservative time-step ceiling) and cope with the resultant huge amount of data points and simulation time.
Beyond these drawbacks, time-step integration shares with any other time- domain methods a disadvantage that is particularly significant to microwave cir- cuits. It cannot handle directly circuit elements having a frequency-domain represen- tation. Examples of these are dispersive transmission lines, transmission line discontinuities (as microstrip cross-junctions, bends, impedance-steps, coupled- lines, etc.), or, in general, any admittance or scattering matrix coming from labora- torial network analysis. To circumvent that, some time-domain simulators suggest the use of approximated lumped equivalent circuits [20]. Nevertheless, these lumped equivalents are so difficult to extract and involve such a large number of elements that in those cases a frequency-domain simulator, as the ones using HB, is usually preferable.
Time-domain methods present, however, an important advantage over HB for lumped circuits: they are capable of handling much stronger nonlinearities. Facing strong nonlinear regimes, HB requires Fourier expansions with a large number of coefficients and its harmonic-Newton Jacobian matrix loses its characteristic diagonal dominance. As a consequence, harmonic-Newton becomes very inefficient, both in memory storage and simulation time. Time-domain methods do not suffer from these problems as the time-variable can be used as a natural continuation parameter: circuit solution at the previous time-step is always used as the initial estimate for the next Newton-Raphson iteration. In this respect, time-step integra- tion is so good, compared to HB, that there has been a continuous push of that simulation method into the RF domain and a steadily proposal of new time-domain methods that circumvent some of the above-mentioned drawbacks.
The following sections will briefly review some of these alternatives for time- domain simulation. Steady-state sinusoidal excitation is addressed first, and then guidelines for its generalization to multitone will be given.