Coil capacitance calculations example

When me and my brother IN3LBQ finished building our variometer we were eager to check the actual inductance of the coils, in order to be sure it was enough to bring our vertical antenna to resonance on the 137 kHz band. But it was also clear that such a big structure will have also a not-so-small parasitic capacitance, that could affect the measured inductance, so we decided to resonate the coils with several different capacitances and use the previously described method to estimate both the true inductance and distributed capacitance.
The following table shows the measured resonance frequencies with three different external capacitances in parallel for the two variometer coils alone and for the assembled variometer in its minimum and maximum inductance configurations:

Parallel capacitance outer coil inner coil variometer at max. inductance variometer at min. inductance
Cp1 = 18 pF 375 kHz 647 kHz 207 kHz 405 kHz
Cp2 = 518 pF 153 kHz 230 kHz 99 kHz 177 kHz
Cp3 = 1318 pF 100 kHz 149 kHz 67 kHz 117 kHz

With the aid of the Inductor Capacitance and Inductance Estimation form the coils capacitance and inductance were computed for different combinations of the external parallel capacitance; the values are reported in the following table:

Parallel capacitance outer coil inner coil variometer at max. inductance variometer at min. inductance
Cp1, Cp2 1804 µH
82 pF
837 µH
54 pF
3986 µH
130 pF
1308 µH
100 pF
Cp1, Cp3 1810 µH
82 pF
831 µH
55 pF
3886 µH
134 pF
1304 µH
100 pF
Cp2, Cp3 1813 µH
79 pF
828 µH
61 pF
3823 µH
158 pF
1302 µH
103 pF
 
max. error 0.28 %
2.5 %
0.60 %
7.6 %
2.3 %
12.3 %
0.26 %
2.0 %

The last row shows the maximum error between the three computed values of true inductance and distributed capacitance for the four different coils configuration. Note the very good agreement between the inductance values, except for the variometer at max inductance case, where the measured resonance frequencies with Cp2 and Cp3 were quite close; since the calculations are based on the relative frequency differences, a small error in the readings may lead to a large error in the estimated inductance. For example, if a resonant frequency of 66.5 kHz is used in the calculation instead of 67 kHz the error in the estimated inductance will decrease to 1.3 %. The distributed capacitance estimations may be also considered in good agreement, taking into account that these values (differently from the true inductance) depend also on the exact parallel capacitance values, which were not measured but the nominal values were used instead. Also here, if we use a resonant frequency of 66.5 kHz instead of 67 kHz for the variometer at max inductance case the estimated distributed capacitance error decreases, to 4 % .