List of Symbols

Symbol Definition Description

$E_{max}$ Maximum electric far field density

$I_{dm}$ Differential mode current

Frequency

Surface area, loop area

Distance from Source point to field observation point

Parallel plane separation distance

$\epsilon_{r}$ Relative dielectric constant of the material

$I_{noise}$ Noise current

Width of rectangular parallel planes (y-dimension)

Length of rectangular parallel planes (x-dimension)

Quality factor of a resonator

$\sigma$ Conductivity

$\tan(\delta)$ Loss tangent

$f_{r}$ Resonance frequency

$m,\,n$ Positive integer values

$E_{max}^{res}$ Maximum radiated electric far field from an antenna

$I_{ant}$ Antenna feed point current

$U_{ant}$ Antenna feed point voltage

$R_{1},\,R_{2}$ $\Omega$ Resistances in a circuit

$R_{k}$ $\Omega$ Coupling resistance

$I_{cm}$ Common mode current

$U_{b}$ Supply (battery) voltage

$U_{m}$ Measurement voltage

$C_{k}$ Coupling capacitance

$L_{1},\,L_{2}$ Self inductances of two loops

Mutual inductance of two loops

$L_{k}$ Coupling inductance of two loops

$L_{s1},\,L_{s2}$ Stray inductances of two loops

$\vec{E}$ Electric field density

$\vec{H}$ Magnetic field density

$\vec{J}$ Electric current density

$\epsilon$ Permittivity of a material

$\mu$ Permeability of a material

Imaginary unit: $\sqrt{-1}$ / Integer index

Integer index

$\omega$ Angular frequency: $2\pi f$

$E_{x}$ x-direction component of $\vec{E}$

$E_{y}$ y-direction component of $\vec{E}$

$E_{z}$ z-direction component of $\vec{E}$

$k_{lossless}$ lossless wave number: $\omega\sqrt{\mu\epsilon}$

$L_{c}'$ Parallel plane inductance

$C_{c}'$ Capacitance per parallel plane area

$t_{c_{i}}'$ Plane thickness of metal plane i

$t_{s_{i}}'$ Skin depth of metal plane i

$R_{c}'$ $\Omega$ Parallel plane resistance

$G_{c}'$ Parallel plane conductance

$J_{z}$ z-direction component of $\vec{J}$

$\mathcal{C}$ Boundary curve of a parallel plane structure

$U_{i}(x_{i},y_{i})$ 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

$W_{e}$ Effective width of rectangular parallel planes

$L_{e}$ Effective length of rectangular parallel planes

$k_{m}$ Rectangular parallel plane eigenvalue of x-direction

$k_{n}$ Rectangular parallel plane eigenvalue of y-direction

Wave number

$L_{mn}$ Two dimensional fourier series coefficient

$(x_{i},y_{i})$ Position of port with index i on parallel planes

$(x_{j},y_{j})$ Position of port with index j on parallel planes

$(\Delta x,\Delta y)$ Parallel plane discretization for LCR grid method

$\tilde{U}$ Approximate parallel plane voltage distribution

Residuum (Approximation error) of $\tilde{U}$

$W_{g}$ Finite element weighting function

$\alpha_{k}$ Finite element base functions

$\tilde{u}_{k}$ Nodal approximation of the parallel plane voltage

$\xi_{i},\,\eta_{i},\,\zeta_{i}$ Barycentric (triangular) coordinates of triangle i

$A_{i}$ Area of the triangle with index i

$\textbf{K}$ System matrix

$\tilde{\textbf{U}}$ Approximate solution vector

$\textbf{F}$ Excitation vector

$\tilde{\textbf{U}}_{n}$ Solution vector normal derivation

$\textbf{R}$ Boundary weighting matrix

$K_{kl}$ System matrix elements

$F_{l}$ Excitation vector elements

$\textbf{Z}$ $\Omega$ Parallel plane impedance matrix

$I_{m}$ Measurement port current

$I_{s}$ Trace current at the source position

$I_{l}$ 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

Effective trace length

Horizontal trace length

Trace distance from the ground plane

$C_{13}$ Capacitance of trace and cover

$C_{23}$ Capacitance of cover and ground plane

$C_{12}$ Capacitance of trace and ground plane

$K_{couple\_md}$ Coupling factor from mode decomposition

$\varphi$ Electrostatic potential

$K_{couple\_a}$ Analytical coupling factor

$U_{m_{tr}}$ Plane voltage of model with a trace

$w_{tr}$ Trace width

$\epsilon_{0}$ Permittivity of vacuum

$d_{s}$ Field scan height above the ground plane

$\underline{J}_{s}$ Complex scanned current density

$Jr_{s},\,Ji_{s}$ Components of complex, scanned current density

$K_{s}$ Coupling factor in scan height

$\alpha,\,\beta,\,\gamma,\,\delta$ Coupling constants

$d_{ic}$ Median coupling factor of an IC interconnect path

$d_{i}$ Distance to ground of IC interconnect segment i

$l_{i}$ Length of interconnect segment i

$U_{cm}$ Common mode voltage

$L_{cm}$ Common mode inductance of a trace above ground

$L_{cm\_p}$ Common mode inductance of a trace between two planes

$U_{AB}$ Differential voltage at opposite plane ports

$M_{c}$ Coupling inductance from a trace to parallel planes

$L_{cm\_meas}$ Measured common mode inductance

$Z_{21}$ $\Omega$ Transfer impedance

$\vec{n}$ Surface normal vector

$I_{sp}$ Parallel plane excitation current

$U_{m,n}$ Parallel plane voltage at mode m,n

$K_{s}(x_{sp,y_{sp}})$ Source term of parallel plane voltage

$K_{meas}(x,y)$ Observation point term of parallel plane voltage

$f_{r_{open}}$ Resonances of rectangular power planes

$\vec{A}$ Magnetic vector potential

$\vec{F}$ Electric vector potential

$\vec{r}$ Vector from origin to field observation point

$\vec{r}'$ Vector from origin to source point

$\vec{r}'_{1}$ Vector from source to field observation point

$D_{a}$ Active antenna dimension

$\xi$ Angle between $\vec{r}$ and $\vec{r}'$

$\vec{M}$ Magnetic current density

$\vec{E}_{far}$ Approximate electric far field density

$\vec{H}_{far}$ Approximate magnetic far field density

$\lambda_{0}$ Wavelength in vacuum

$\vec{S}$ Pointing vector

$\vec{S}_{cr}$ Pointing vector of ports with indexes c and r

$Y_{a\_cr}$ Admittance matrix elements

$\textbf{U}_{m}$ Measurement voltage vector

$\textbf{U}{p}$ Interface port voltage vector

$\textbf {Y}_{a}$ Radiation admittance matrix

$\kappa(\textbf{Y}_{a})$ Condition number of matrix $\textbf {Y}_{a}$

$\lambda_{min}(\textbf{Y}_{a})$ Minimum eigenvalue of matrix $\textbf {Y}_{a}$

$\lambda_{max}(\textbf{Y}_{a})$ Maximum eigenvalue of matrix $\textbf {Y}_{a}$

$S_{min}$ Threshold value for matrix condition number

$R_{21}$ Emission reduction estimation for d reduction

$R_{S21}$ Emission reduction estimation for shielding

$Ric_{12}$ Emission reduction estimation for ground under an IC

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