and the quality factor Q=ω0RC, the time evolution of a signal is given by:1 Introduction
QuickWave-3D is a time-domain simulator. This implies that the software calculates the frequency domain characteristics of a circuit by analysing the time evolution of signals at all its ports. In the initial phase of the simulation, the signal energy is transmitted from the source to the structure. When the source is switched off the build up of energy in the structure also stops and the structure starts to leak the accumulated energy to the load. The rate of decay of the energy depends on the quality factor Q of the structure.
In a simple RLC circuit with the resonance angular frequency and the quality factor Q=ω0RC, the time evolution of a signal is given by:
(P 1-1)
From the above expression it is clearly seen that the higher the Q-factor, the slower the signal decays. This simple example of a damped oscillator evidently shows that all time domain simulation techniques, including the one used by QuickWave-3D may involve exceedingly long computing time when applied to modelling of structures with high and moderate quality factors.
Recall that the Q-factor determines the frequency spread of a system response. For instance, the Fourier transform of signal (P 1-1) leads to a well-known Lorentzian curve of a damped oscillator and the width of this curve defines the quality factor. In more complex systems the frequency response may be shaped by several Lorentzians. In any case, however, the narrower and sharper the peaks or nulls in the frequency response of the circuit to be modelled, the higher the Q-factors of the Lorentzians. In other words, circuits with sharp features in the frequency domain, such as narrow band pass filters, require considerably longer simulation times than broadband structures devoid of internal resonances.
Obviously, the simulation in QuickWave-3D can be stopped at any point. However, if the time domain signature is curtailed before the signal has sufficiently decayed, the nulls in the frequency domain are shallower and less "sharp". Consequently, the results of the modelling may be unacceptable.
A very efficient way to reduce the simulation time is to use digital signal processing techniques. Signal processing allows one to extract the essential features of time signature based on the analysis of time samples obtained at the initial simulation phase. This is achieved by creating an appropriate model of each signal and using this model to compute the frequency domain response. Among various signal processing techniques which have been investigated in the technical literature in context of time domain electromagnetic simulators are nonlinear exponential autoregressive models [7], all pole auto regressive (AR) linear predictors [7, 8], pole-zero (ARMA) [8] models and various methods leading to Prony's models [1, 2, 3]. Unfortunately, a successful application of any of these techniques requires considerable knowledge about the modelling procedures. In order to point out the main difficulties, a few issues essential for good quality model creation should be mentioned:
· Parametric models should be used to describe slowly decaying components of the signal. For this reason initial time samples, which contain transients, should be discarded.
· One has to determine the number of samples required to construct a good model or, in other words, decide on the moment for terminating the simulation. Without a termination condition the time domain simulation is either too short or unnecessarily long.
· For each time sequence one has to select the model order. Several authors pointed out the difficulties in the proper model order selection [2, 3, 6]. When the model order is too low the results are inaccurate, but when the model order is too high, spurious modes or instabilities of the postprocessing technique may appear. In general, however, overmodelling is less dangerous than undermodelling.
In order to assist the users of QuickWave-3D in the analysis of high Q structures, a special software module QProny was developed. QProny uses one of the most robust signal processing techniques known as the Generalised Pencil of Function Method (GPOF) [5] and employs unique and innovative methods for automatic selection of the most important parameters such as:
· the number of initial samples to be skipped,
· the number of samples required for model construction,
· the model order.
As a result, high quality models may be created without any user's intervention. Inexperienced user can construct a correct model without knowing anything about intricacies of the applied signal processing techniques.
The following section gives a general overview of the methods used in the QProny module and can be skipped by the users who are not interested in technicalities. It should be noted, however, that reading these sections may be helpful in better understanding of various warning messages generated by the module. An even more detailed description of the techniques used in the QProny module can be found in journal papers [10,11]