Measuring Relative Dielectric Constant of PCB's Using Parallel Plate Method and a Vector Network Analyser page 2

PCB-B

  This has four square samples of 1, 4, 9 and 16 cm² 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_rstill 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.

  It was decided that the measurement problem was related to the perimeter length to area ratio. i.e.

Sample 1 )    1 square centimetre, perimeter length = 4 x 1 = 4 centimetre.

Ratio = 1 : 4

Sample 2 )    4 square centimetre, perimeter length = 4 x 2 = 8 centimetre.

Ratio = 4 : 8

or     Ratio = 1 : 2     etc

  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 cm² 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_oC_cgs    Formula 2.

C_cgs = (r²/4d) + 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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