PCB Sample Size Calculator for Relative Dielectric Constant Measurements

   The optimum size of a circular PCB sample for the measurement of ε_r can be calculated by entering the known and predicted data into the text boxes. The predicted data could be from the manufacturers/suppliers specification or just left at the default of 4.2, but if it is too far out it may mean repeating the process and performing the measurement on a second sample once the  approximate value is know.

  This calculator is intended to be used with the Accurate Method for Measuring Relative Dielectric Constant of a PCB Using a Vector Network Analyser The default values in the text boxes relate to the samples used in the article Measuring Relative Dielectric Constant using parallel plate method and a Vector Network Analyser

     Please give us feedback on your use of this calculator, good or bad. It works fine with  FR4, but is not yet proven for all available materials.

       Once the relative dielectric is known the required track width can be found by reference to a handbook, or by using a calculator. There are three calculators for Microstrip including the  Microstrip Transmission Line Characteristic Impedance Calculator Using a Formula by Brian C Wadell. and one Coplanar Waveguide With Ground Calculator

   Many more calculators can be found on the excellent RF Cafe website

http://rfcafe.com/references/spreadsheets/calculator-list.htm

1) The minimum frequency for proximity effect to become established is found first.

From Transmission Lines and Networks by Walter C. Johnson, McGraw-Hill 1963 p58.

Nominal depth of penetration for a copper conductor (δ)  = 6.64 / √f  cm

Where       f = frequency in Hertz

From this a formula is derived to find the frequency that has a depth of penetration of one half of the copper thickness i.e. the lowest frequency that allows the proximity effect to become well established.

                 Formula 1.

2) The minimum frequency is used to calculate the maximum capacitance value at a phase shift of 70^o (Xc = 71.4 Ω). Thus ensuring that the result achieved at 90^o is measured well above the minimum frequency.

              Formula 2.

3) The maximum capacitance is used to find the approximate maximum radius.

From Kirchhoff's formula for a circular capacitor

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

Where ε_r is the Relative Dielectric Constant

r is the radius in cm              

                     d is the thickness of the Dielectric in cm

            Formula 3.

4) Using the approximate radius, an approximation is then found for the Logarithmic part of Kirchhoff's formula

                                  Formula 4

5) A quadratic equation for maximum radius is then derived and solved using

       Formula 5.

The resulting radius is then entered again into  Formula 4 and Formula 5 and this process is repeated five times in order to reduce the small error caused by the approximation to a negligible amount.

6) The maximum area is found and then halved to find the optimum area.

       Formula 6

7) The optimum radius becomes.

         Formula 7

W J Highton Updated 3/4/2008

This calculator is provided free by Chemandy Electronics in order to promote the FLEXI-BOX

Return to Calculator Index