# Using the Transmission Line Matrix Method to Solve Electromagnetic Problems

### Key Takeaways

• The transmission line matrix method is a numerical technique that can give an approximate solution to electromagnetic wave propagation through complex geometries.

• The transmission line matrix method models the two-dimensional space where wave propagation and scattering occur by a cartesian matrix of nodes.

• The transmission line matrix method discretizes both the time and space of electromagnetic wave propagation. The transmission line matrix method is an iterative method with two main processes of scattering and connecting. Electromagnetic waves are scattered in conducting and dielectric mediums

Electromagnetic waves get scattered in the conducting and dielectric mediums, and Maxwell’s equations and boundary conditions govern the incidence, reflections, and scattering of these electromagnetic waves. Numerical methods such as the time-domain finite difference, finite volume, and finite element methods are commonly used to solve integral and differential formulations of Maxwell’s equations.

The transmission line matrix method is another method that can solve problems involving Maxwell's equations. This method is a numerical technique that can give an approximate solution to electromagnetic wave propagation through complex geometries. The transmission line matrix method analyzes electromagnetic scattering from conducting and dielectric materials and provides the solution to wave propagation in the time domain.

## Modeling Two-Dimensional Space Using the Transmission Line Matrix Method

The transmission line matrix method is a numerical simulation method used to solve two-dimensional scattering problems in electromagnetic wave propagation. This method models the two-dimensional space where wave propagation and scattering occur by a cartesian matrix of nodes. Each node is separated by a mesh parameter of length Δl. The time taken for an electromagnetic pulse to travel from one node to another is represented by Δt. The Δl,  Δt, and the velocity of the electromagnetic wave (c) are related as follows: The upcoming section introduces the cartesian matrix model used in the transmission line matrix method.

### The Cartesian Matrix Model

In the Cartesian matrix model, the energy of an incident wave hitting a node is scattered in all four directions isotropically. The energy of the pulses scattering in four directions is one-fourth of the incident wave. The field quantities are reduced by a factor of ½. A negative sign accompanies the field quantities and energy that gets reflected in the direction of incidence.

The electric and magnetic fields of Maxwell’s equations guiding the wave propagation in two-dimensional space can be directly equated to voltages and currents on the mesh. The voltages and currents represent the field quantities in the transmission line matrix method. In this way, the transmission line matrix method develops the electrical network for solving Maxwell’s equations associated with electromagnetic field problems.

In the two-dimensional transmission line matrix method, the mesh of two transmission lines forms the electrical network model. The nodes are the interconnection of transmission lines in the two-dimensional transmission line matrix method. Transmission line matrix nodes are characterized by the way the transmission lines are connected and the properties of the medium. Series transmission line matrix nodes simulate transverse electric modes of wave propagation and shunt transmission line nodes simulate transverse magnetic modes of wave propagation.

## The Transmission Line Matrix Method Algorithm

The transmission line matrix method discretizes both the time and space of electromagnetic wave propagation. The transmission line matrix method is an iterative method with two main processes of scattering and connecting:

• The scattering process determines the reflected impulse voltage on the nth node at (k+1)Δt, provided the voltage incident at the node at kΔt instant is known.

• The connecting process finds out the incident pulse at a node, given the information of the reflected impulses at neighboring nodes.

The processes of scattering and connecting can be mathematically represented as the following equations, where S and C are the scattering and connection matrices:

rVn=S iVn                       (2)

iVn+1=C rVn               (3)

The iteration visualizes the spread of incident energy over the two-dimensional space in which the wave propagates.

## Uses for the Transmission Line Matrix Method

The transmission line matrix method is a significant numerical technique in computational electromagnetics. The transmission line matrix method can be applied to the analysis of scattering and radiation problems. It is used for understanding and solving electromagnetic field problems in optical waveguiding structures. PCB crosstalk can also be modeled using the transmission line matrix method.