Space cloth

The figure to the right shows a coaxial cable terminated by space cloth. In the case of a closed structure like a coaxial cable, the space cloth may be trimmed to the boundary of the outer conductor. The computation of resistance between the conductors can be computed with 2D electromagnetic field solver methods including the relaxation method and analog methods using resistance paper.

Resistance of an annular ring of material having a surface resistance of η per square. The radius of the outer surface of the inner electrode (yellow) is r₁. The radius of the inner surface of the outer electrode (magenta) is r₂.

In the case of a coaxial cable, there is a closed-form solution. The resistive surface is considered to be a series of infinitesimal annular rings, each having a width of dρ and a resistance of (η/2πρ)dρ. The resistance between the inner electrode and the outer electrode is just the integral over all such rings.

${\displaystyle R=\int _{r_{1}}^{r_{2}}{\frac {\eta }{2\pi \rho }}d\rho ={\frac {\eta }{2\pi }}\ln {\frac {r_{2}}{r_{1}}}.}$

This is exactly the equation for the characteristic impedance of a coaxial cable in free space.[4][lower-alpha 2]

Transmission line composed of two parallel wires terminated by space cloth.

The characteristic impedance of a two parallel wire transmission line is given by[6][lower-alpha 3]

${\displaystyle Z_{0}={\frac {\eta }{\pi }}\ln {\frac {2D}{d}},}$

where d is the diameter of the wire and D is the center to center separation between the wires.

Transmission line composed of two parallel wires cross section

If the second figure is taken to be two round pads on a printed circuit board that has surface contamination resulting in a surface resistivity of R_s (50 MΩ per square, for example) then the resistance between the two pads is given by:

${\displaystyle R={\frac {R_{s}}{\pi }}\ln {\frac {2D}{d}}}$

Cross section of a three conductor transmission line composed of two parallel plates and a rectangular shield.

The figure shows the cross section of a three conductor transmission line. The structure has two transmission eigen-modes which are the differential mode (conductors a and b driven with equal amplitude but opposite phase voltages with respect to conductor c) and the common mode (conductors a and b driven with the same voltages with respect to conductor c). In general, the eigen-modes have different characteristic impedances.

If w ≫ h₁, h₂ ≫ t, then the field in region IV and V and can be ignored.

The resistance of regions I–III are

${\displaystyle R_{\text{I}}=\eta {\frac {h_{1}}{w}}}$

${\displaystyle R_{\text{II}}=R_{\text{III}}=\eta {\frac {h_{2}}{w}}}$

where η = impedance of space cloth in ohms per square

In the common mode, conductors a and b are at the same voltage so there is no effect from region I. The common mode characteristic impedance is the resistance of region II in parallel with region III.

${\displaystyle Z_{CM}={\frac {R_{\text{II}}}{2}}={\frac {\eta h_{2}}{2w}}}$

In the differential mode, the characteristic impedance is the resistance of region I in parallel with the series combination of regions II and III.

${\displaystyle Z_{DM}={\frac {(R_{\text{I}})(2R_{\text{II}})}{R_{\text{I}}+2R_{\text{II}}}}={\frac {\eta }{w}}{\frac {2h_{2}h_{1}}{h_{1}+2h_{2}}}}$