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I have a piece of single sided FR4 PCB substrate that I need to build a 2.4 GHz patch antenna on using copper tape.

Other than the dimensions and it being FR4 I know nothing about the substrate.

Is there a way I can determine the dielectric constant experimentally with a standard set of lab equipment (scopes, multimeters, VNA, signal gens, etc. possibly more that I'm not yet aware of us having at university)?

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    \$\begingroup\$ better to start with a substrate for which there is published data ... \$\endgroup\$
    – citizen
    Commented Mar 14 at 11:51

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There are many standard methods. For example, you can get a proper test jig from Keysight or etc., cut a coupon of the material according to the instructions, and follow the measurement procedure. This is probably most economical when the utmost precision is needed and testing is done regularly; such instruments are not cheap. They detail that here:
https://www.keysight.com/us/en/assets/7018-01284/application-notes/5989-2589.pdf
which amusingly is locked behind a login, but is currently downloadable for free from here:
Basics of Measuring the Dielectric Properties of Materials | Keysight Technologies

On a more basic level, consider a calculator such as:
https://www.chemandy.com/calculators/sample-size-calculator.htm
and the related pages on the same site.

If nothing else, simply make a patch of known geometry, and apply a suitable capacitance formula. Give or take whether you've cut out a cylinder exactly the size of the round electrodes, or left it oversize (fringing in air vs. fringing in dielectric), or what the diameter to thickness ratio is, or choice of frequency, this may have various errors, but you can at least get close; and such a measurement can be done with equipment as simple as an oscilloscope and signal generator, or even just an AC voltmeter. For these, I would suggest using the calculations here,
LC Resonance | Calculators | Seven Transistor Labs (links to my site)
particularly "L or C by Resistor Divider" (scalar measurement, can use an AC (high frequency capable) voltmeter) and "Frequency and Q Factor" (resonate with a known inductor, solve for parameters).

Keep in mind that FR-4 is a material which varies significantly with temperature, and frequency (dispersion, loss); which also factors into antenna gain / efficiency, but more to the point, you should probably measure its properties near the frequency of interest, and this will require high-frequency rather than lumped-element methods.

There are also transmission line and resonator methods, for which I do not have citations offhand, but in which the dielectric constant appears as a parameter. These for example, consider it an introduction; you will have to find exact methods based on the keywords:
Measuring Dielectric Constant | Microwaves101
One of the simplest cases is merely laying down a copper strip of constant width, between a pair of SMA or whatever connectors, with the housings well bonded to the backside ground plane. Run this on a VNA or TDR to find the transmission line impedance, and work backwards from dimensions to find κ.

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One way would be to build an antenna on the substrate, using a design with known behavior, and trim the copper until the antenna resonates at the desired frequency. Then back-calculate what the dielectric constant must be for the antenna to work with the final copper dimensions.

An added bonus of this procedure is that if what you really want is a working antenna, you can just skip the final step and not worry about the exact value of the dielectric constant.

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I used a very simple method for guestimating it - take an un-etched 2 sided PCB piece, measure dimensions, then use an LCR meter to measure capacitance. From there, it's simple to calculate Dk at the measurement frequency. For me, measuring at 100 kHz was close enough for opearation at 433 MHz, but my design was a wilkinson divider, not an antenna, so it was more forgiving there.

Another method that comes into mind is to build a test coupon and measure with a VNA. I would do a simple beatty resonator scaled for lower frequencies (possibly from meandered transmission line), but I have no real-world experience with those, only touched them in simulations.

The third way is the simplest way of tuning any single-resonant component: design it with a guestimate Dk, build it, measure parameters, than scale phisical dimensions to move the resonant peak to it's intended frequency.

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You'd etch through lines and antennas onto the substrate, add connectors that you've already de-embedded in your VNA, and characterize the material properties of the substrate. With the rest, you can then build patch antennas. If you have many boards, you probably want to sample a few of them, because it's not known how consistent they are.

In other words: more testing cost than any stack of FR4 that you could carry alone would be worth. Use the FR4 for something that is not RF, and build your antennas on something that comes with a datasheet.

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