1.1 Fundamentals of the Method of Analysis
The electromagnetic fields are in general characterised by three-dimensional Maxwell equations. In the Finite Difference Time Domain method (FDTD) these equations are directly discretised (in space and time) and solved explicitly. The discretisation in space means that the considered domain should be divided into small subdomains called cells. Since the time variable is also discretised we calculate the field distribution in consecutive moments separated by the time step Dt. The algorithm simulates the natural processes in the considered space. The increments of the E-field components between the time instants tn and tn+1 = tn + Dt are calculated from the space derivatives of the H-components at the instant tn + 0.5 Dt. Next, the increments of the H-field components between the time instants tn+0.5 and tn+1.5 = tn+0.5 +Dt are calculated from the space derivatives of the E-components at the instant tn+1. The process of field calculation for a consecutive time instant advanced by Dt is called iteration.
FDTD was first introduced by Yee in 1966 and since then applied by many authors. It is currently one of the most popular methods among microwave circuit researchers. The version of the FDTD method applied here is based on the research conducted by W. K. Gwarek and his co-workers. The main difference with respect to the classical FDTD method is much more flexibility in the shape of individual cells. This reduces the main disadvantage of the classical FDTD, which is the necessity of stair case approximation of curved boundaries. QW-3D software also contains many other features based on the original research (see User Guide 3D: Bibliography for references).
In most of the FDTD applications we use a pulse excitation at one or more ports of the modelled device. After the wave simulation is accomplished, the comparison of the Fourier transforms of the input and output signals gives the S-matrix parameters of the circuit. Furthermore, the Fourier transforms of tangential fields on a surface enclosing the antenna or a scatterer can be used to extract the radiated or scattered far fields, respectively. Theoretically, the Fourier transform calculations require infinite period of time. However, since the exciting pulse has limited duration and the power entering the circuit is being dissipated at the input and output (which are matched), or at the absorbing boundaries, the signal at the ports becomes negligible after a limited period of time and the Fourier transform calculation can be limited to this period without causing significant errors.
As all numerical methods the FDTD method has its limits of accuracy for given computer resources. There are two basic sources of errors:
Error due to finite discretisation of space (finite cell size)
The division of the circuit into cells (called meshing) is performed in the QW-Editor automatically but it is controlled by the user. The user chooses the basic cell size, which determines the accuracy on one hand, and the computer time and memory needed for the simulations on the other hand. The best way to check the level of the discretisation error for a particular application is to analyse the same circuit with different cell size and to compare the results. In QW-3D reducing the cell size by half brings the space discretisation errors down by a factor of 4, or nearly 4. However, it should also be mentioned that the computing time will rise almost 16 times and memory occupation 8 times.
Error due to finite computing time used for wave simulation and the stop criteria for FDTD simulations
This error results from the fact that calculation of Fourier transforms of the input and output signals is restricted to the number of iterations, in which not all the energy injected into the device by the virtual source is dissipated in the input and output terminations, and on internal losses of the circuit.
There is quite a simple way of predicting the needed number of iterations implemented in Amigo option. It is based on the observation that since the FDTD method simulates natural processes in the device, the necessary number of iterations (and the computing time) directly depends on the loaded Q-factor of the analysed structures. High-Q resonant structures need much longer computing time. On the other hand the Q-factor directly influences the frequency resolution needed to reproduce the characteristic of interest with sufficient accuracy. Thus declaration of the frequency resolution is sufficient to calculate the needed number of iterations. The user is typically well aware of the frequency resolution he requires to make the simulation of the frequency-domain characteristics informative and thus application of this criterion is quite straightforward. A concern may arise that there are structures of a very low-Q but very large size and thus the long simulation time is needed to simulate displacement of the virtual pulse through the structure. However the size of the structure also directly influences the variations of the frequency-domain characteristics and the user can usually easily predict the frequency resolution needed to take into account the “size factor”. To make such a prediction easier Amigo informs the user how many times the pulse would travel across the structure with the assumed number of iterations.
There is also Energy Stop criterion feature, which monitors the level of energy in the circuit and uses it as a criterion for simulation suspension. Resuming, we recommend using one of the above features as a basic way of predicting the needed number of iterations and suspending/terminating the simulation, but we also encourage the users to experiment with other criteria listed below.
There are two general criteria for terminating the simulation during its interactive operation:
In every FDTD iteration during the simulation we can watch the field distribution to find out when the field amplitudes are reduced to negligible values.
In every FDTD iteration we can also watch the S-parameter curves to find out when their shape stabilises.
Two other criteria are available in QW-Simulator for two important categories of problems:
S-parameter analysis of shielded lossless multiports. QW-Simulator integrates power entering and leaving through all the ports. The user can watch the power balance curve, and stop the simulation when power balance becomes equal to unity in the considered frequency band.
Analysis of radiation patterns and return loss of lossless antennas. QW-Simulator calculates radiation efficiency as a ratio of power radiated by the antenna to power delivered to the antenna by the source. The user can stop the simulation when efficiency approaches 100% at the frequencies of interest.
Note that after introducing losses, the simulation time for a particular structure can be reduced. The user sets the number of iterations by application of Amigo or energy stop criterion in the QW-Editor, or in the so-called tasker files. It is understood that an inexperienced user will start from the Amigo option. When the simulation is suspended at the number of iterations calculated by Amigo he will resume the simulation to see if the obtained characteristics really converged adequately to the final solution. This will help in setting in the next runs the frequency resolution giving the best results in reasonable computing time. In most cases an experienced user will be able to set easily a correct number of the frequency resolution in Amigo or directly the needed number of iterations based on previous experience with the particular types of circuits.
To terminate the subject of needed number of iterations let us note that in the case of structures of very high-Q an optional signal post-processing module QProny can be used to decrease the needed number of iterations.