Transmission Lines

Transmission Lines

Background
Benchmark ID	transmissionLines_n1600m14q14 transmissionLines_n2624m30q30 transmissionLines_n5248m62q62
Category	misc
System-Class	LTI-FOS
Parameters
nstates	1600 2624 5248
ninputs	14 30 62
noutputs	14 30 62
nparameters	0
components	A, B, C, E
Copyright
License	NA
Creator	Giovanni De Luca
Editor	Giovanni De Luca Christian Himpe
Location
NA

Definition

In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, and computer network connections.

In many electric circuits, the length of the wires connecting the components can often be ignored. That is, the voltage on the wire at a given time can be assumed to be the same at all points. However, when the voltage changes as fast as the signal travels through the wire, the length becomes important and the wire must be treated as a transmission line, with distributed parameters. Stated in another way, the length of the wire is important when the signal includes frequency components with corresponding wavelengths comparable to or less than the length of the wire.

A common rule of thumb is that the cable or wire should be treated as a transmission line if its length is greater than $1/10$ of the wavelength, and the interconnect is called "electrically long". At this length the phase delay and the interference of any reflections on the line (as well as other undesired effects) become important and can lead to unpredictable behavior in systems which have not been carefully designed using transmission line theory.

Description

The decrease of integrated circuit feature size and the increase of operating frequencies require 3-D electromagnetic methods, such as the Partial Element Equivalent Circuit (PEEC) method; it stems from the integral equation form of Maxwell's equations. The main difference of the PEEC method with other integral-Equation-based techniques, such as the method of moments, resides in the fact that it provides a circuit interpretation of the Electric Field Integral Equation (EFIE) in terms of partial elements, namely resistances, partial inductances, and coefficients of potential. In the standard approach, volumes and surfaces are discretized into elementary regions, hexahedra, and patches respectively over which the current and charge densities are expanded into a series of basis functions. Pulse basis functions are usually adopted as expansion and weight functions. Such choice of pulse basis functions corresponds to assuming constant current density and charge density over the elementary volume (inductive) and surface (capacitive) cells, respectively. Following the standard Galerkin's testing procedure, topological elements, namely nodes and branches, are generated and electrical lumped elements are identified modeling both the magnetic and electric field coupling. Conductors are modeled by their ohmic resistance, while dielectrics requires modeling the excess charge due to the dielectric polarization. Magnetic and electric field coupling are modeled by partial inductances and coefficients of potential, respectively.

The magnetic field coupling between two inductive volume cells $\alpha$ and $\beta$ is described by the partial inductance

$L_{p_{\alpha\beta}}=\frac{\mu}{4\pi}\frac{1}{a_{\alpha}a_{\beta}}\int_{u_{\alpha}}\int_{u_{\beta}}\frac{1}{R_{\alpha\beta}}\,du_{\alpha}\,du_{\beta}$

where $R_{\alpha\beta}$ is the distance between any two points in the volumes $u_{\alpha}$ and $u_{\beta}$ with $a_{\alpha}$ and $a_{\beta}$ their cross section. The electric field coupling between two capacitive surface cells $\delta$ and $\gamma$ is modeled by the coefficient of the potential

$P_{\delta\gamma}=\frac{1}{4\pi\epsilon}\frac{1}{S_{\delta}S_{\gamma}}\int_{S_{\delta}}\int_{S_{\gamma}}\frac{1}{R_{\delta\gamma}}\,dS_{\delta}\,dS_{\gamma}$

where $R_{\delta\gamma}$ is the distance between any two points on the surfaces $\delta$ and $\gamma$, while $S_{\delta}$ and $S_{\gamma}$ denote the area of their respective surfaces .

Generalized Kirchhoff's laws for conductors, when dielectrics are considered, can be rewritten as

$\textbf{P}^{-1}\frac{d\textbf{v}(t)}{dt}-\textbf{A}^T\textbf{i}(t)+\textbf{i}_e(t)=0,$

Figure 1: Illustration of PEEC circuit electrical quantities for a conductor elementary cell (Figure from ).

$-\textbf{A}\textbf{v}(t)-\textbf{L}_p\frac{d\textbf{i}(t)}{dt}-\textbf{v}_d(t)=0,$

$\textbf{i}(t)=\textbf{C}_d\frac{d\textbf{v}_d(t)}{dt}$

where $\textbf{A}$ is the connectivity matrix, $\textbf{v}(t)$ denotes the node potentials to infinity, $\textbf{i}(t)$ and $\textbf{i}_e(t)$ represent the currents flowing in volume cells and the external currents, respectively, $\textbf{v}_d(t)$ is the excess capacitance voltage drop, which is related to the excess charge by $\textbf{v}_d(t)=\textbf{C}_d^{-1}\textbf{q}_d(t)$. A selection matrix $\textbf{K}$ is introduced to define the port voltages by selecting node potentials. The same matrix is used to obtain the external currents $\textbf{i}_e(t)$ by the currents $\textbf{i}_s(t)$, which are of opposite sign with respect to the $n_p$ port currents $\textbf{i}_p(t)$,

$\textbf{v}_p(t)=\textbf{K}\textbf{v}(t),$

$\textbf{i}_e(t)=\textbf{K}^T\textbf{i}_s(t).$

An example of PEEC circuit electrical quantities for a conductor elementary cell is illustrated, in the Laplace domain, in Fig. 1, where the current-controlled voltage sources $sL_{p,ij}I_j$ and the current-controlled current sources $I_{cci}$ model the magnetic and electric coupling, respectively.

Thus, assuming that we are interested in generating an admittance representation having $n_p$ output currents under voltage excitation, and let us denote with $n_n$ the number of nodes, $n_i$ the number of branches where currents flow, $n_c$ the number of branches of conductors, $n_d$ the number of dielectrics, $n_d$ the additional unknowns since dielectrics require the excess capacitance to model the polarization charge, and $n_u=n_i+n_d+n_n+n_p$ the global number of unknowns, and if the Modified Nodal Analysis (MNA) approach is used, we have:

$\left[ \begin{array}{cccc} \textbf{P} & \textbf{0}_{n_n,n_i} & \textbf{0}_{n_n,n_d} & \textbf{0}_{n_n,n_p} \\ \textbf{0}_{n_i,n_n} & \textbf{L}_p & \textbf{0}_{n_i,n_d} & \textbf{0}_{n_i,n_p} \\ \textbf{0}_{n_d,n_n} & \textbf{0}_{n_d,n_i} & \textbf{C}_d & \textbf{0}_{n_d,n_p} \\ \textbf{0}_{n_p,n_n} & \textbf{0}_{n_p,n_i} & \textbf{0}_{n_p,n_d} & \textbf{0}_{n_p,n_p} \end{array}\right]\frac{d}{dt}\left[ \begin{array}{c}\textbf{q}(t) \\ \textbf{i}(t) \\ \textbf{v}_d(t) \\ \textbf{i}_s(t) \end{array}\right]=$

$= - \left[ \begin{array}{cccc}\textbf{0}_{n_n,n_n} & -\textbf{P}\textbf{A}^T & \textbf{0}_{n_n,n_d} & \textbf{P}\textbf{K}^T \\ \textbf{AP} & \textbf{R} & \Phi & \textbf{0}_{n_i,n_p} \\ \textbf{0}_{n_d,n_n} & -\Phi^T & \textbf{0}_{n_d,n_d} & \textbf{0}_{n_d,n_p} \\ -\textbf{K}\textbf{P} & \textbf{0}_{n_p,n_i} & \textbf{0}_{n_p,n_d} & \textbf{0}_{n_p,n_p} \end{array}\right]\cdot\left[ \begin{array}{c} \textbf{q}(t) \\ \textbf{i}(t) \\ \textbf{v}_d(t) \\ \textbf{i}_s(t) \end{array}\right]+ \left[ \begin{array}{c}\textbf{0}_{n_n+n_i+n_d,n_p} \\ -\textbf{I}_{n_p,n_p} \end{array}\right] \cdot [ \textbf{v}_p(t) ].$

Here $\textbf{0}$ is a matrix of zeros, $\textbf{I}$ is the identity matrix, both are with appropriate dimensions, and $\Phi=\left[ \begin{array}{c} \textbf{0}_{n_c,n_d} \\ \textbf{I}_{n_d,n_d} \end{array}\right]$. Then, in a more compact form, the above equation can be written as:

$\left\{ \begin{array}{c} \textbf{C}\frac{d\textbf{x}(t)}{dt}=-\textbf{G}\textbf{x}(t)+\textbf{B}\textbf{u}(t)\\ \textbf{i}_p(t)=\textbf{L}^T\textbf{x}(t) \end{array}\right . \qquad (1)$

with $\textbf{x}(t)=\left[ \begin{array}{cccc} \textbf{q}(t)\quad\textbf{i}(t)\quad\textbf{v}_d(t)\quad\textbf{i}_s(t) \end{array}\right]^T$. Since this is an $n_p$-port formulation, whereby the only sources are the voltage sources at the $n_p$-ports nodes, $\textbf{B}=\textbf{L}$ where $\textbf{B}\in\mathbb R^{n_u\times n_p}$ (for more details on this model, refer to ).

Motivation of MOR

Since the number of equations produced by 3-D electromagnetic method PEEC is usually very large, the inclusion of the PEEC model directly into a circuit simulator (like SPICE) is computationally intractable for complex structures, where the number of circuit elements can be tens of thousands.

Data

All data sets (in a MATLAB formatted data, downloadable in TransmissionLines.rar) in Fig. 1 are referred to as the multiconductor transmission lines in a MNA form, coming from the PEEC method (then, with dense matrices since they are obtained from the integral formulation of Maxwell's equation). The LTI descriptor systems have the form of, equation $(1)$, where $C\in\mathbb R^{n\times n}$ (with $C=C^T\ge0$), $G\in\mathbb R^{n\times n}$ (with $G+G^T\ge0$), $B\in\mathbb R^{n\times m}$ and $L=B^T$, $x(t)\in\mathbb R^n$ is the vector of variables (charges, currents and node potential), the input signal $u(t)\in\mathbb R^m$ are the sources (current or voltage generators depending on what one wants to analyze: the impedances or the admittances) linked to some node, the output $y(t)\in\mathbb R^m$ are the observation across the node where the sources are inserted. An accurate model of the dynamics of these data sets is generated between 10 kHz and 20 GHz.

Name of the data set	Matrices	Dimension	Number of inputs
dsysPEEC-MTLn1600m14	$G$, $B$, $C$ ($L=B^T$)	1600	14
dsysPEEC-MTLn2624m30	dss object (*)	2624	30
dsysPEEC-MTLn5248m62	dss object (*)	5248	62

one can extract the matrices with Matlab command:

   [G,B,L,D,C] = dssdata(dssObjectName);

e.g., if one wants to work on one of the last two data sets of this table, just load it into the Matlab Workspace and type the command aforementioned on the Command Windows; for the first example, once one loads the data, the Workspace shows directly the matrices. Note that $D = 0$.

Dimensions

System structure:

$\begin{array}{rcl} E\dot{x}(t) &=& Ax(t) + Bu(t) \\ y(t) &=& C x(t) \end{array}$

System dimensions:

$E \in \mathbb{R}^{N \times N}$, $A \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times M}$, $C \in \mathbb{R}^{M \times N}$.

System variants:

dsysPEEC-MTLn1600m14: $N = 1600$, $M = 14$, dsysPEEC-MTLn2624m30: $N = 2624$, $M = 30$, dsysPEEC-MTLn5248m62: $N = 5248$, $M = 62$

Contact

Giovanni De Luca

Lihong Feng