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Registered Member #543
Joined: Tue Feb 20 2007, 04:26PM
Location: UK
Posts: 4992
Case question: a parallel plate capacitor has a dielectric consisting of fine weave glass sheet saturated with a liquid dielectric. How can we best express the effective relative static permittivity of the mixed insulator, given that we have adequate data on the glass and the liquid dielectric?
Registered Member #72
Joined: Thu Feb 09 2006, 08:29AM
Location: UK St. Albans
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With no approximation, we can say the dielectric constant is bounded above and below by the constants of the two materials, but glass and (maybe) oil are fairly different constants, so that's still leaves quite a large uncertainty.
You'll get a lot nearer with the mean of the two constants, if we guess that in practice, fine weave glass sheet occupies around 50% of the total volume.
However, the dielectric constant is *not* er1*rv1 + er2*rv2, where ern is the dielctric constant of dielectric n, and rvn is the relative volume of dielectric n. Consider the two extreme ways of splitting two dielectrics, shown below, each occupying 50% of the capacitor volume. Each is fairly easy to calculate, one as the parallel sum of two smaller caps, the other as the series sum of two larger caps. These provide tighter upper and lower bounds on the actual capacitance when they are mixed in some intermediate way. An average of these two bounds is as close as you are going to get without some serious work.
For instance, if the two materials are "average soda glass" of er 7.5, and mineral oil of er 2.2, and we assume 50% volume fraction each, then the first parallel configuration gives the upper bound of their arithmetic mean = 4.85, the second series configuration the lower bound of their harmonic mean = 3.4, with the truth for a mix being somewhere between these limits. The geometric mean of the dielectrics is 4.06, which might be somewhere close.
To get closer, you do need to perform a 3D integration of charge and voltage throughout a mixed medium for the particular fibre size, though finite element will probably be more practical. Once fibres are much smaller than the thickness of the total dielectric, the overall constant will tend to a limit. Whether the fibres are random or regular, and how each weave lines up with the one below it will affect the sum slightly.
I use FR4 epoxy-glass board (which is a glass fibre in epoxy resin) and need to know the dielectric constant for sizing transmission lines, and the er that the manuafactuerr quotes varies from 3.7 to 4.2, depending on the thickness and whether it's core or pre-preg which controls the glass/resin ratio. Interesting, the possible figure for a 50% mix of glass and oil (which is not far off epoxy) is contained within that range.
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Query: relative static permittivity calculation for a mixed dielectric.
