PCB-B

This has four square samples of 1, 4, 9 and 16 cm^{2 }on the top, and a
mirrored ground plane on the bottom. Measurements were made from 1 MHz to 1 GHz and the resultant ε_{r
}(K) calculated the same as PCB-A. The results are shown below in Figure 2.

Figure 2

Figure 2 shows that the measured ε_{r
}still varies with both frequency and sample size, but the curves are a little closer together and have dropped slightly at the centre
frequencies. There is still no single point where the curves all coincide. The minimum spread has now reduced to 6.9% at 200 MHz, but this is still
too great.

Next, the samples were changed to circles because the circumference to area ratio of a circle is marginally lower than that of the perimeter to area ratio of a square of similar area.

PCB-C

This has four circular samples of 1, 4, 9 and 16 cm^{2 }on the top, and
a mirrored ground plane on the bottom. Measurements were made from 1 MHz to 1 GHz and the resultant ε_{r
}(K) calculated the same as PCB-A. The results are shown below in Figure 3.

Figure 3

Figure 3 shows that the measured ε_{r
}still varies with both frequency and sample size, but the curves are again a little closer together and have dropped even more at the centre
frequencies. There is still no single point where the curves all coincide. The minimum spread is now 4.6% at 200 MHz. Clearly, there is still a
problem with the edge effect that the sample size and shape cannot overcome. It was decided to use a formula by Kirchhoff that takes into account the
edge effect of a circular capacitor, which is what we now have!

From Landau, L.D. & Lifschitz, E.M., (1987). *Electrodynamics of Continuous Media.* Oxford, England: Permagon Press, p. 19.

C_{MKS} = 4πε_{o}C_{cgs }Formula 2.

C_{cgs} = (r^{2}/4d) + r/4π [ln(16πr/d)-1] Formula 3.

Where r = radius

d = thickness of dielectric

ε_{o} = dielectric constant

C_{MKS }= capacitance in metre - kilogram - second units

C_{cgs } = capacitance in centimetre - gram - second units

From Formula 1, 2, & 3 the following formula was derived.

_{}Formula 4.

This formula is used in the Relative Dielectric Constant Calculator.

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