shielding / cables & connectors
N u m e r i c a l S o l u t i o n o f C o m p l e x EMC P r o b l e m s
Figure 4. Selection of some of the cable types supported in FEKO.
FEM Solver [pF/m]
Relative Error [%]
C11
14.9395
14.9718
0.22
C12
-6.0894
-5.5062
9.58
C22
18.8111
18.7565
0.29
C13
-1.6117
-1.4734
8.58
C23
-6.0894
-5.5069
9.56
C33
14.9395
14.9710
0.21
the cross sectional dimension information about a specific line is contained in these parameters. Under the fundamental transverse electromagnetic field structure assumption, the per unit length parameters of inductance, capacitance, and conductance are determined as a static solution to Laplace’s equation 2 (x,y)=0 in the two-dimensional cross sectional (x,y) plane of the line. The determination of the per unit length parameters can be simple or very difficult depending on whether the cross sections of the conductors are circular or rectangular, and whether the conductors are surrounded by a homogeneous or an inhomogeneous dielectric medium. In fact, there are very few transmission lines for which the cross sectional fields can be solved analytically to give simple formulas for the per unit length parameters. To illustrate, the self-inductance and mutual inductance terms of n widely spaced cores above an infinite ground (see Figure 5) can be derived as [9]
Figure 6. Field excitation of a transmission line using only voltage sources (Agrawal method).
where r wi is the wire radius, hi is the wire height above ground and sij is the center to center spacing between wires. If however these cores are surrounded by an insulation medium or the separation is not wide, one has no alternative but to employ approximate, numerical methods. In FEKO a 2D static FEM solver to Laplace’s equation is used. Table 1 compares the analytic to the numerical per unit length capacitance matrix entries for 3 wires above ground (r w1=r w3 =1.0 mm, r w2 = 1.5 mm, h1=h3 =52 mm, h2 =50 mm, s12=s23=15.13 mm, s13=30 mm. The 2D FEM solution is based on minimizing the stored field energy per unit length. The self-capacitance matrix entries are a direct function of the total energy in the system, and hence these terms agree very well with those of the analytic prediction. The mutual capacitance matrix entries are derived from the FEM solution (integration over -) and as such use a lower order approximation, also explaining the larger differences between the analytic and numerical solutions.
where x denotes the longitudinal direction and parameters Z and Y are the per unit length impedance and admittance parameters of the line Z = jωL + R Y = jωC + G. As a generalization to the two-conductor system, a multiconductor transmission line model is simply a distributed parameter network for an arbitrary cable cross section (see Figure 4) where the voltages and currents can vary in magnitude and in phase over its length. Per Unit Length Cable Parameters The per unit length parameters of inductance, capacitance, resistance and conductance are essential ingredients in the determination of transmission line voltages and currents from the solution of the transmission line equations. All of interference technology
Analytical [pF/m]
Table 1. Transmission line per unit length capacitance matrix entries for three widely spaced conductors above ground: Comparison of analytical values with the numerical static FEM solution.
Figure 5. Parameters for the computation of the per unit length inductances of widely separated wires above a ground plane.
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Entry
emc Directory & design guide 2011