Transmission line matrix method

Many problems of modern microwave technology can only be solved satisfactorily by numerical methods. For example, the technical properties of a cell phone antenna, such as its radiation behavior, can no longer be solved with closed mathematical approaches, but only with computer-aided numerical methods. One of these methods is the transmission line matrix method, or TLM method for short .

Basic algorithm

The transmission line matrix method (TLM method) is a differential numerical method in the 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. As with any numerical method for solving electromagnetic field problems, the simulation space is divided into many small basic cells. In the case of the TLM method, through a network of TEM waveguides (transmission lines) that are connected to one another at their entrance gates. A construct of two TEM lines crossing at an angle of 90 ° is called a TLM cell or TLM node.

Two-dimensional TLM nodes according to Johns and Breule

When the network is excited at the entrance gates, Dirac impulses (see delta distribution ) spread on the TEM waveguides , which are scattered in the middle of each cell, run back to the entrance gates and then as new impulses at the entrance gates of neighboring cells issue. The next iteration step again consists of a spreading of the impulses applied to the entrance gates and the further distribution to the neighboring cells. The impulses - hereinafter always referred to as wave amplitudes - can, as will be shown later, be related to individual electromagnetic field components ( electric field strength and magnetic field strength ). Over the years, different types of TLM cells have also been developed and published for the three-dimensional case. The most common node is the (SCN Symmetrical Condensed Node) introduced by Johns in 1987. In its basic version, the SCN has 12 gates. Due to the 12 wave amplitudes, two polarizations that are rotated by 90 ° are reproduced on each cube surface. The SCN is the most common node type. In general, the TLM scheme for the SCN can be specified in the following form:

${\displaystyle \mathbf {b} _{k;l,m,n}=\mathbf {S} \mathbf {a} _{k-1;l,m,n}}$

${\displaystyle \mathbf {a} _{k-1;l,m,n}=\mathbf {\Gamma } \mathbf {b} _{k-1;l,m,n}}$

b and a represent twelve-dimensional (in the basic form of a vacuum) vectors , which combine the incoming ( a ) and reflected ( b ) wave amplitudes of each TLM cell. represents the connection operator that takes over the distribution of the scattered wave amplitudes to neighboring cells after the scattering of the incoming wave amplitudes. The index k denotes the discrete time step, with integer k marking the point in time at which the wave amplitudes are exactly between the interfaces of the TLM cells and n * k ± 1/2 the point in time immediately before or after a scatter. ${\displaystyle \Gamma }$

S denotes the scatter matrix ( matrix ) via which the incoming and reflected wave amplitudes are connected to one another. It has the following shape:

${\displaystyle \mathbf {S} ={\begin{bmatrix}\mathbf {0} &\mathbf {S} _{0}&\mathbf {S} _{0}^{T}\\\mathbf {S} _{0}^{T}&\mathbf {0} &\mathbf {S} _{0}\\\mathbf {S} _{0}&\mathbf {S} _{0}^{T}&\mathbf {0} \\\end{bmatrix}}}$ With ${\displaystyle \mathbf {S} _{0}={\begin{bmatrix}0&0&{\frac {1}{2}}&-{\frac {1}{2}}\\0&0&-{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}&0&0\\{\frac {1}{2}}&{\frac {1}{2}}&0&0\\\end{bmatrix}}}$

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

• Scattering of the incoming wave amplitudes.
• Distribution of the scattered wave amplitudes to neighboring cells.

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

In the basic version, the SCN has twelve gates and thus twelve wave amplitudes that propagate synchronously on the connecting lines and are scattered synchronously. This can only be maintained if all connecting lines have the same propagation properties - i.e. the same wave impedance :

${\displaystyle Z_{0}={\sqrt {\frac {\mu _{0}\mu _{r}}{\varepsilon _{0}\varepsilon _{r}}}}}$

and thus the same speed of propagation:

${\displaystyle c={\frac {1}{\sqrt {\mu _{0}\mu _{r}\varepsilon _{0}\varepsilon _{r}}}}}$.

A comparison of Maxwell's equations and the line equations in the two-dimensional case shows that a TLM grid models a medium with double permittivity compared to the TEM lines. The speed of propagation is therefore lower by a factor of 1/2 than given by the material parameters. This must be taken into account by scaling the real simulated time step . In the case of the SCN, the scaling factor is given by 1/2, with which the time step must be scaled. For a grid of TLM cells of the geometric size, this results in a real simulated time step of: ${\displaystyle \Delta T}$ ${\displaystyle \Delta T}$ ${\displaystyle \Delta T}$

${\displaystyle \Delta T={\frac {\Delta t}{2}}={\frac {\Delta l}{2c}}={\frac {\Delta l}{2}}{\sqrt {\mu _{0}\mu _{r}\varepsilon _{0}\varepsilon _{r}}}}$.

Mapping between the network and field sizes

The mapping of the wave amplitudes, which are used in the processing of the scattering and distribution operations, and the physical field sizes again represent a crucial point in the process of the algorithm, since the 12 wave amplitudes of the SCN must be mapped to 6 field sizes or when the Network 6 field sizes on 18 wave amplitudes. In his original work on the SCN, Johns suggests the following figure, which is based on the idea that only those wave amplitudes can contribute to a field component that are also polarized in the same coordinate direction (in the case of the E-fields) or enclose the corresponding field component ( in the case of the H-fields). The field components are formed exclusively from incoming wave amplitudes of the current time step k of a cell. ${\displaystyle a_{k-1/2;l,m,n}^{i}}$ ${\displaystyle i=1,\ldots ,12}$

${\displaystyle E_{k-{\frac {1}{2}};l,m,n}^{x}={\frac {2}{\Delta l}}\left[{\frac {a^{7}+a^{8}+a^{9}+a^{10}}{4}}\right]_{k-{\frac {1}{2}};l,m,n}}$

${\displaystyle E_{k-{\frac {1}{2}};l,m,n}^{y}={\frac {2}{\Delta l}}\left[{\frac {a^{1}+a^{2}+a^{11}+a^{12}}{4}}\right]_{k-{\frac {1}{2}};l,m,n}}$

${\displaystyle E_{k-{\frac {1}{2}};l,m,n}^{z}={\frac {2}{\Delta l}}\left[{\frac {a^{3}+a^{4}+a^{5}+a^{6}}{4}}\right]_{k-{\frac {1}{2}};l,m,n}}$

${\displaystyle H_{k-{\frac {1}{2}};l,m,n}^{x}={\frac {2}{Z_{0}\Delta l}}\left[{\frac {-a^{11}+a^{5}-a^{6}+a^{12}}{4}}\right]_{k-{\frac {1}{2}};l,m,n}}$

${\displaystyle H_{k-{\frac {1}{2}};l,m,n}^{y}={\frac {2}{Z_{0}\Delta l}}\left[{\frac {a^{9}-a^{3}-a^{10}+a^{4}}{4}}\right]_{k-{\frac {1}{2}};l,m,n}}$

${\displaystyle H_{k-{\frac {1}{2}};l,m,n}^{z}={\frac {2}{Z_{0}\Delta l}}\left[{\frac {a^{1}-a^{7}-a^{2}+a^{8}}{4}}\right]_{k-{\frac {1}{2}};l,m,n}}$

In order to be able to calculate power flows correctly, the signs of the H-fields are inverted to maintain a right-handed coordinate system in contrast to John's original work. For the mapping of the physical fields to the wave amplitudes of the TLM grating, which is required to excite a structure when starting a simulation, Johns gives the following assignment in his original paper:

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{1}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{y}-Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{z})}$

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{2}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{y}+Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{z})}$

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{3}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{z}+Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{y})}$

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{4}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{z}-Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{y})}$

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{5}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{z}-Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{x})}$

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{6}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{z}+Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{x})}$

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{7}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{x}+Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{z})}$

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{8}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{x}-Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{z})}$

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{9}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{x}-Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{y})}$

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{10}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{x}+Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{y})}$

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{11}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{y}+Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{x})}$

${\displaystyle a_{k-{\frac {1}{2}};l,m,n}^{12}={\frac {1}{2}}\Delta l(E_{k-{\frac {1}{2}};l,m,n}^{y}-Z_{0}H_{k-{\frac {1}{2}};l,m,n}^{x})}$

The above-mentioned change of sign to maintain the right-handed coordinate system for correct power flow calculation is also included here. refers to the geometric dimensions of a TLM cell. ${\displaystyle \Delta l}$

See also

• Finite difference method

literature

• P.B. Johns and RL Beurle: Numerical solutions of 2-dimensional scattering problems using a transmission-line matrix, Vol. 118, No. 9, pp. 1203-1208, Proceeding of the IEE, 1971.
• P.B. Johns, A symmetrical condensed node for the TLM method, Vol. 35, No. 4, pp. 370-377, IEEE Transactions on Microwave Theory and Techniques, 1987.
• W. J. R. Hoefer: The transmission-line matrix method - theory and applications, Vol. 33, No. 10, p. 882893, IEEE Transactions on Microwave Theory and Techniques, 1985.
• W. Dressel: Modeling of electromagnetic structures using the TLM method using static sublattices, ISBN 3-89825-981-1 , dissertation.de, 2005.

Freely available program packages

• http://www.petr-lorenz.com/emgine/