# PCB Sample Size Calculator for Relative Dielectric Constant Measurements

The optimum size of a circular PCB sample for the measurement of εr 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.

 INPUT DATA Enter the predicted εr of the PCB: Enter the thickness of the copper: mm Enter the thickness of the dielectric: mm

Note: 1 oz/ft2 of copper = 0.035 mm thickness

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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 an equation by Brian C Wadell. and one Coplanar Waveguide With Ground Calculator

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

http://www.rfcafe.com/references/calculators/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 an equation 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. Equation 1.

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

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

From Kirchhoff's equation for a circular capacitor

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

εr is the Relative Dielectric Constant

r is the radius in cm

d is the thickness of the Dielectric in cm  Equation 3.

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

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

The resulting radius is then entered again into Equation 4 and Equation 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. Equation 6. Equation 7.