Transmission-line matrix method

Many problems of modern microwave technology can be solved satisfactorily only by numerical methods. For example, let the technical features of a mobile phone antenna as not solve radiation with closed mathematical approaches, but only with computer-based numerical methods. One of these methods is the transmission -line matrix method, in short: TLM method.

Basic algorithm

In the transmission -line matrix method (TLM method) is a differential numerical method in time domain for solving hyperbolic differential equations. It was first introduced by P. B. Johns and R. L. Beurle in 1971 to solve two-dimensional electromagnetic field problems. The simulation space is thereby divided as with any numerical method for solving electromagnetic field problems in many small base cells. In the case of the TLM method by a network of TEM waveguides (English transmission lines ), which are interconnected at their input gates. A construct of two at an angle of 90 degrees intersecting lines referred to as TEM TLM TLM cell or node.

Wide on the TEM waveguides at the suggestion of the network at the entrance gates Dirac pulses ( see delta distribution ), which are scattered in the nodal center of each cell, run back to the entrance gates and then as a new impetus at the entrance gates of adjacent cells applied. The next iteration in turn consists of a scattering of the voltage applied to the gates pulses and the further distribution to the neighboring cells. The pulses - hereafter always referred to as wave amplitudes - can, as will be shown later, set with single electromagnetic field components ( electric field strength and magnetic field strength) in relation. Over the years, different types of TLM cells were developed and published for the three-dimensional case. The most widespread is the node ( SCN Symmetrical Condensed Node), which was presented by Johns in 1987. In its basic configuration, the SCN has 12 goals. 12 by the wave amplitudes are two polarizations are rotated by 90 °, reproduced on each face of the cube. The SCN is the most common type of node dar. general, the TLM scheme for the SCN in the following form to be specified:

B and a set zwölfdimensionale (which affects the basic shape of the vacuum chamber ) vectors here, summarize the incoming (a) and reflected ( b ) wave amplitudes of each TLM cell. since the connection is operator which handles the distribution of the scattered wave amplitude on neighboring cells according to the scatter of the incident wave amplitude. The index k denotes the discrete time step, integer k mark the time at which the wave amplitudes are located exactly between the interfaces of the TLM - cells and n * k ± 1/2 the time immediately before and after scattering.

S is the scattering matrix (matrix) on the incoming and the reflected wave amplitudes are connected. It has the following form:

In summary, the TLM algorithm are divided into the following two steps:

• Scattering of the incident wave amplitude.
• Distribution of the scattered wave amplitudes at adjacent cells.

These relationships can be very easily implemented in a numerical machine code to be executed by a computer.

In the basic version of the SCN has twelve gates, and thus twelve wave amplitudes, synchronously spread on the connecting lines and are scattered in sync. This can only be maintained if all connecting lines have the same propagation characteristics - ie the same characteristic impedance:

And thus the same propagation velocity:

A comparison of the Maxwell equations and the two-dimensional case in the line equations shows that a TLM grid models a double medium permittivity compared with the TEM lines. Thus, the propagation velocity is accordingly by a factor of 1/2 smaller than dictated by the material parameters. This must be taken into account by a scaling of the real simulated time step. In the case of the SCN, the scaling factor is given by 1/2, the time step should be scaled. For a grid of TLM cells of the geometric size, this results in a simulated real time step of:

Mapping between the network and field sizes

The mapping of the wave amplitudes, which are used during the execution of the scattering and distribution operations, and the physical field variables is again a crucial point in the execution of the algorithm, since the 12 wave amplitudes of the SCN must be mapped to six field variables or at the suggestion of the network field sizes 6 to 18 wave amplitudes. John suggests in his original work on the SCN following illustration before, which is based on the consideration that only the wave amplitudes can contribute to a field component (in the case of the E-fields ) are polarized in the same coordinate direction or the corresponding field component enclose ( in the case of the H- fields). The field components are thereby formed exclusively of incoming wave amplitudes of the current time step k ​​of a cell.

In order to calculate power flows correctly, the sign of the H fields to maintain a right-handed coordinate system, in contrast to John's original work are inverted. For the imaging of the physical fields on the amplitudes of the TLM grid that is required for the excitation of a structure at the start of a simulation, John is in its original work on the following assignment:

Again, the above-mentioned change of sign for the maintenance of the right-handed coordinate system to the correct power flow calculation is already included. denotes the geometric dimension of a TLM cell.