List of Symbols

Symbol	Definition	Description
$E_{max}$	$V/m$	Maximum electric far field density
$I_{dm}$	$A$	Differential mode current
$f$	$Hz$	Frequency
$A$	$m^2$	Surface area, loop area
$r$	$m$	Distance from Source point to field observation point
$h$	$m$	Parallel plane separation distance
$\epsilon_{r}$	$-$	Relative dielectric constant of the material
$I_{noise}$	$A$	Noise current
$W$	$m$	Width of rectangular parallel planes (y-dimension)
$L$	$m$	Length of rectangular parallel planes (x-dimension)
$Q$	$-$	Quality factor of a resonator
$\sigma$	$S/m$	Conductivity
$\tan(\delta)$	$-$	Loss tangent
$f_{r}$	$Hz$	Resonance frequency
$m,\,n$	$-$	Positive integer values
$E_{max}^{res}$	$V/m$	Maximum radiated electric far field from an antenna
$I_{ant}$	$A$	Antenna feed point current
$U_{ant}$	$V$	Antenna feed point voltage
$R_{1},\,R_{2}$	$\Omega$	Resistances in a circuit
$R_{k}$	$\Omega$	Coupling resistance
$I_{cm}$	$A$	Common mode current
$U_{b}$	$V$	Supply (battery) voltage
$U_{m}$	$V$	Measurement voltage
$C_{k}$	$F$	Coupling capacitance
$L_{1},\,L_{2}$	$H$	Self inductances of two loops
$M$	$H$	Mutual inductance of two loops
$L_{k}$	$H$	Coupling inductance of two loops
$L_{s1},\,L_{s2}$	$H$	Stray inductances of two loops
$\vec{E}$	$V/m$	Electric field density
$\vec{H}$	$A/m$	Magnetic field density
$\vec{J}$	$A/m^2$	Electric current density
$\epsilon$	$F/m$	Permittivity of a material
$\mu$	$H/m$	Permeability of a material
$j$	$-$	Imaginary unit: $\sqrt{-1}$ / Integer index
$i$	$-$	Integer index
$\omega$	$1/s$	Angular frequency: $2\pi f$
$E_{x}$	$V/m$	x-direction component of $\vec{E}$
$E_{y}$	$V/m$	y-direction component of $\vec{E}$
$E_{z}$	$V/m$	z-direction component of $\vec{E}$
$k_{lossless}$	$1/m$	lossless wave number: $\omega\sqrt{\mu\epsilon}$
$L_{c}'$	$H$	Parallel plane inductance
$C_{c}'$	$F/m^2$	Capacitance per parallel plane area
$t_{c_{i}}'$	$m$	Plane thickness of metal plane i
$t_{s_{i}}'$	$m$	Skin depth of metal plane i
$R_{c}'$	$\Omega$	Parallel plane resistance
$G_{c}'$	$S$	Parallel plane conductance
$J_{z}$	$A/m^2$	z-direction component of $\vec{J}$
$\mathcal{C}$	$m$	Boundary curve of a parallel plane structure
$U_{i}(x_{i},y_{i})$	$V$	Parallel plane voltage at position $(x_{i},y_{i})$
$Z_{ij}(x_{i},y_{i},x_{j},y_{j})$	$\Omega$	Mutual plane impedance matrix element
$W_{e}$	$m$	Effective width of rectangular parallel planes
$L_{e}$	$m$	Effective length of rectangular parallel planes
$k_{m}$	$1/m$	Rectangular parallel plane eigenvalue of x-direction
$k_{n}$	$1/m$	Rectangular parallel plane eigenvalue of y-direction
$k$	$1/m$	Wave number
$L_{mn}$	$-$	Two dimensional fourier series coefficient
$(x_{i},y_{i})$	$(m,m)$	Position of port with index i on parallel planes
$(x_{j},y_{j})$	$(m,m)$	Position of port with index j on parallel planes
$(\Delta x,\Delta y)$	$(m,m)$	Parallel plane discretization for LCR grid method
$\tilde{U}$	$V$	Approximate parallel plane voltage distribution
$Res$	$V/m^2$	Residuum (Approximation error) of $\tilde{U}$
$W_{g}$	$-$	Finite element weighting function
$\alpha_{k}$	$-$	Finite element base functions
$\tilde{u}_{k}$	$V$	Nodal approximation of the parallel plane voltage
$\xi_{i},\,\eta_{i},\,\zeta_{i}$	$-$	Barycentric (triangular) coordinates of triangle i
$A_{i}$	$m^2$	Area of the triangle with index i
$\textbf{K}$	$-$	System matrix
$\tilde{\textbf{U}}$	$V$	Approximate solution vector
$\textbf{F}$	$V$	Excitation vector
$\tilde{\textbf{U}}_{n}$	$V/m$	Solution vector normal derivation
$\textbf{R}$	$m$	Boundary weighting matrix
$K_{kl}$	$-$	System matrix elements
$F_{l}$	$V$	Excitation vector elements
$\textbf{Z}$	$\Omega$	Parallel plane impedance matrix
$I_{m}$	$A$	Measurement port current
$I_{s}$	$A$	Trace current at the source position
$I_{l}$	$A$	Trace current at the load position
$K_{couple}$	$-$	Trace to cavity coupling factor
$Z_{w}$	$\Omega$	Characteristic trace impedance
$\gamma$	$-$	Trace propagation constant
$Z_{l}$	$\Omega$	Trace load impedance
$l$	$m$	Effective trace length
$ll$	$m$	Horizontal trace length
$d$	$m$	Trace distance from the ground plane
$C_{13}$	$F$	Capacitance of trace and cover
$C_{23}$	$F$	Capacitance of cover and ground plane
$C_{12}$	$F$	Capacitance of trace and ground plane
$K_{couple\_md}$	$-$	Coupling factor from mode decomposition
$\varphi$	$V$	Electrostatic potential
$K_{couple\_a}$	$-$	Analytical coupling factor
$U_{m_{tr}}$	$V$	Plane voltage of model with a trace
$w_{tr}$	$m$	Trace width
$\epsilon_{0}$	$F/m$	Permittivity of vacuum
$d_{s}$	$m$	Field scan height above the ground plane
$\underline{J}_{s}$	$A/m^2$	Complex scanned current density
$Jr_{s},\,Ji_{s}$	$A/m^2$	Components of complex, scanned current density
$K_{s}$	$-$	Coupling factor in scan height
$\alpha,\,\beta,\,\gamma,\,\delta$	$-$	Coupling constants
$d_{ic}$	$m$	Median coupling factor of an IC interconnect path
$d_{i}$	$m$	Distance to ground of IC interconnect segment i
$l_{i}$	$m$	Length of interconnect segment i
$U_{cm}$	$V$	Common mode voltage
$L_{cm}$	$H$	Common mode inductance of a trace above ground
$L_{cm\_p}$	$H$	Common mode inductance of a trace between two planes
$U_{AB}$	$V$	Differential voltage at opposite plane ports
$M_{c}$	$H$	Coupling inductance from a trace to parallel planes
$L_{cm\_meas}$	$H$	Measured common mode inductance
$Z_{21}$	$\Omega$	Transfer impedance
$\vec{n}$	$-$	Surface normal vector
$I_{sp}$	$A$	Parallel plane excitation current
$U_{m,n}$	$V$	Parallel plane voltage at mode m,n
$K_{s}(x_{sp,y_{sp}})$	$A$	Source term of parallel plane voltage
$K_{meas}(x,y)$	$-$	Observation point term of parallel plane voltage
$f_{r_{open}}$	$Hz$	Resonances of rectangular power planes
$\vec{A}$	$A$	Magnetic vector potential
$\vec{F}$	$V$	Electric vector potential
$\vec{r}$	$m$	Vector from origin to field observation point
$\vec{r}'$	$m$	Vector from origin to source point
$\vec{r}'_{1}$	$m$	Vector from source to field observation point
$D_{a}$	$m$	Active antenna dimension
$\xi$	$rad$	Angle between $\vec{r}$ and $\vec{r}'$
$\vec{M}$	$V/m^2$	Magnetic current density
$\vec{E}_{far}$	$V/m$	Approximate electric far field density
$\vec{H}_{far}$	$A/m$	Approximate magnetic far field density
$\lambda_{0}$	$m$	Wavelength in vacuum
$\vec{S}$	$W/m^2$	Pointing vector
$\vec{S}_{cr}$	$W/m^2$	Pointing vector of ports with indexes c and r
$Y_{a\_cr}$	$S$	Admittance matrix elements
$\textbf{U}_{m}$	$V$	Measurement voltage vector
$\textbf{U}{p}$	$V$	Interface port voltage vector
$\textbf {Y}_{a}$	$S$	Radiation admittance matrix
$\kappa(\textbf{Y}_{a})$	$-$	Condition number of matrix $\textbf {Y}_{a}$
$\lambda_{min}(\textbf{Y}_{a})$	$S$	Minimum eigenvalue of matrix $\textbf {Y}_{a}$
$\lambda_{max}(\textbf{Y}_{a})$	$S$	Maximum eigenvalue of matrix $\textbf {Y}_{a}$
$S_{min}$	$-$	Threshold value for matrix condition number
$R_{21}$	$dB$	Emission reduction estimation for d reduction
$R_{S21}$	$dB$	Emission reduction estimation for shielding
$Ric_{12}$	$dB$	Emission reduction estimation for ground under an IC

C. Poschalko: The Simulation of Emission from Printed Circuit Boards under a Metallic Cover