## About the Wire loop probe

15.12.2016 21:08

Recently I was writing about how my father and I were checking a HiFiBerry board for a source of Wi-Fi interference. For want of better equipment we used a crude near-field probe that consisted of a loop of stripped coaxial cable and a trimmer capacitor. We attempted to tune this probe to around 2.4 GHz using the trimmer to get more sensitivity. However we didn't see any effect of capacitance changes on the response in that band.

The probe was made very much by gut feeling, so it wasn't that surprising that it didn't work as expected. We got some interesting results nonetheless. Still, I thought I might do some follow-up calculations to see how far off we were in our estimates of the resonance frequency.

Our probe looked approximately like the following schematic (photograph). The loop diameter was around 25 mm and the wire diameter was around 1 mm. Trimmer capacitor was around 10 pF:

Inductance of a single, circular loop of wire in air is:

L = \mu_0 \frac{D}{2} \left( \ln \frac{8D}{d} - 2 \right) \approx 50 \mathrm{nH}

The wire loop and the capacitor form a series LC circuit. If we ignore the effect of the coaxial cable connection, the resonant frequency of this circuit is:

f = \frac{1}{2 \pi \sqrt{LC}} \approx 200 \mathrm{MHz}

So it appears that we were off by an order of magnitude. In fact, this result is close to the low frequency peak we saw on the spectrum analyzer at around 360 MHz:

Working backwards from the equations above, we would need capacitance below 1 pF or loop diameter on the order of millimeters to get resonance at 2.4 GHz. These are very small values. Below 1 pF, stray capacitance of the loop itself would start to become significant and a millimeter-sized loop seems too small to be approximated with lumped elements.

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