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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.

Enter the predicted εr of the PCB:  Maximum sample radius:  cm
Enter the thickness of the copper:   mm Optimum sample radius:  cm
Enter the thickness of the dielectric:   mm
Note: 1 oz/ft2 of copper = 0.035 mm

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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 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

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.

[Minimum frequency for proximity effect formula]Formula 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.

[Maximum capacitance formula]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.

 [Circular capacitor formula]

Where εr is the Relative Dielectric Constant

r is the radius in cm

d is the thickness of the Dielectric in cm

[First part of circular caapacitor formula] 

[Approximate radius formula]Formula 3.

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

[Logarithmic part of circular capacitor formula]Formula 4.

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

[Quadratic equation formula]

[r max formula]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.

[Optimum area formula]Formula 6.

7) The optimum radius becomes.

[Optimum radius formula]Formula 7.

W J Highton Updated 3/4/2008

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

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