Equation 185
Equation 186
Equation 187
Equation 188
It is hard to do measurements and susceptibility tests of telecommunication systems since they usually are of very large size and the frequencies are relatively low. It could be cheaper and more convenient to carry out the measurements on a small scaled model of the system instead. Another advantage is that equipment calibrated for frequencies up to 30MHz (used by xDSL modems) are less common than equipment calibrated for frequencies above 30MHz. A test site for open area measurements is also easier and cheaper to build.
The theory is based on the fact that Maxwell’s equations are linear and on the law of similarity. Non-linear materials like ferromagnetic materials and time-dependent materials are excluded to get a linear model. Following is a short theory of how different quantities can be scaled.
Consider a system with the electric field E(r,t) and the magnetic field H(r,t) at the point r at time t. E, H and r are vectors. Scaling these quantities with different scale factors and representing the scaled variables with a prime we get:
|
Equation 185 |
|
Equation 186 |
|
Equation 187 |
|
Equation 188 |
where p is a mechanical scale factor, t is a time scale factor, g is an electric scale factor and h is a magnetic scale factor. Then applying that Maxwell’s equations shall be satisfied in both systems we get the following requirements:
 |
Equation 189 |
 |
Equation 190 |
We can see from these equations that if the media is air we get:
 |
Equation 191 |
Where the time and mechanical factors are equal and also the electric and magnetic factors are equal.
One can build two kinds of models. The first is a geometric model for simulating field lines, but not field levels, where the mechanical and the time factors are fixed and the electric and magnetic scale factors are arbitrary. The other is an absolute model where even the field levels can be obtained. An absolute model is built with fixed mechanical and electrical scale factors.
Following is a table showing the correspondences between the main quantities:
Table 5 Real system quantities and corresponding model quantities
| Quantity | Real system |
Model | Geom. |
Absolute |
| Length | l | l´ = l/p | Ä |
Ä |
| Time | t | t´ = t/p | Ä |
Ä |
| Frequency | f | f´ = pf | Ä |
Ä |
| Wavelength | l | l ´ = l /p | Ä |
Ä |
| Phase velocity | v | v´ = v | Ä |
Ä |
| Permittivity | Î | Î ´ =Î | Ä |
Ä |
| Permeability | m | m ´ = m | Ä |
Ä |
| Conductivity | s | s ´ = ps | Ä |
Ä |
| Impedance | Z | Z´ = Z | Ä |
Ä |
| Capacitance | C | C´ = C/p | Ä |
Ä |
| Inductance | L | L´ = L/p | Ä |
Ä |
| Electric field | E | E´ = E/h | Ä |
|
| Magnetic field | H | H´ = H/h | Ä |
|
| Surface current density | Js | Js´ = Js/h | Ä |
|
| Volumetric current density | J | J´ = Jp/h | Ä |
|
| Current | I | I´ = I/hp | Ä |
|
| Surface charge density | r | r s´ = r s/h | Ä |
|
| Volumetric charge density | r | r ´ = r p/h | Ä |
|
| Charge | Q | Q´ = Q/hp2 | Ä |
The first eleven quantities depends only on the mechanical scale factor p, while the remaining, also controlled in an absolute model, in addition depends on the magnetic scale factor. We can see that if we increase the frequency, the size of the system can be reduced by the same factor. It could however be hard to scale the conductivity, s , of the material with the same factor since you have to find another material for the conductors. Thus the attenuation of the current along the line, for example, can be hard to simulate in a realistic way.
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EMC of Telecommunication Lines
A Master Thesis from the Fieldbusters © 1997
Joachim Johansson and Urban Lundgren