# PCB Sample Size Calculator for Relative Dielectric Constant Measurements

In order to make an accurate measurement of the relative dielectric constant (εr ) of a circular sample PCB, it is important to ensure that skin effect has become established, that edge effect is taken into account in the calculations and that the measuring instrument (VNA) is reasonably well impedance matched. The optimum size of a circular PCB sample for the measurement of εr can be calculated by entering the known dimensions and approximate εr into the text boxes. The approximate εr could be from the manufacturers/suppliers specification or just left at the default of 4.2, but in extreme cases when the first sample size it is totally inappropriate it may mean repeating the process and performing the measurement on a second sample size.

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 approximate εr of the PCB: Enter the thickness of the copper (t): mm Enter the thickness of the dielectric (d): 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) Minimum Frequency for Well Established Skin Effect

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

Nominal depth of penetration for a copper conductor   cm  Equation 1.

Where f = frequency in Hertz

This equation is transposed to find the measurement frequency that has a depth of penetration of one half of the required copper thickness i.e. a practical frequency that ensures that skin effect has become well established on both sides of the copper even though the distribution is known to be non-symmetric. The units are also changed to a more convenient millimetres.

Hz  Equation 2.

Where t = copper thickness in mm

2) Maximum Capacitance

The minimum frequency for skin effect is used to calculate the maximum capacitance value at a phase shift of 70o (Xc = 71.4Ω in a 50Ω measurement system). Thus ensuring that the final measurement result achieved at 90o phase shift is measured well above the minimum frequency required for skin effect.

First the standard equation used for the reactance of a capacitor is taken (the negative sign for capacitive reactance is not required in these calculations).

Ohms  Equation 3.

The standard equation used for the reactance of a capacitor is transposed to find the capacitance at 71.4 Ohms.

pF  Equation 4.

The maximum capacitance is used to find an approximate maximum radius.

Starting with Kirchhoff's equation for a circular capacitor

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

pF  Equation 5.

Where:

εr is the Relative Dielectric Constant

r is the radius in mm

d is the thickness of the Dielectric in mm

Just the first part of Equation 5 is selected in order avoid the mathematical complexity of calculating the additional value due to edge effect.

pF  Equation 6.

Equation 6 is then transposed to find the first approximate maximum radius.

mm  Equation 7.

Using the first approximate radius, an intermediate value is found for the capacitance (Ci) using Kirchhoff's equation which includes the capacitance due to edge effect (see Equation 5)

A factor is then found using the maximum capacitance (Cmax) in Equation 4 and the initial capacitance (Ci) found using Kirchhoff's equation and the first approximate radius. This is then multiplied by the first approximate radius (rapp) to give a further approximation of the maximum radius (rmax). This calculation is repeated 100 times in a FOR loop and the error in maximum radius iteratively reduced until it is negligible in relation to the required maximum radius (Equation 8).

mm  Equation 8.

6) Optimum Area

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

mm2  Equation 9.