While building a filter for a WSPR beacon, I noticed that the primary inductor in the circuit behaved as if it had more inductance than expected.
To learn more about what was happening, I built a test jig made up of a single 180pf SMD capacitor with two SMA connectors and used a vector network analyzer (VNA) to measure the resonant frequency of test coils to determine their effective inductance. This gave me enough data to derive an empirical formula for estimating inductance given the number of turns used.
SPOILER 1: the standard formula doesn't account for connecting the coil to the rest of the circuit. My core also seems to have a bit higher than expected relative permeance.
SPOILER 2: a second experiment where the coil leads are twisted to minimize the effect of the last turn gives results much closer to specs, but still requires an offset.
The following figure shows a schematic of the test jig
And the picture of the test jig as built below.
This shows how a coil can be laid in place for measurements without permanent connections. This picture also shows a common problem in that the turns of the coil have been pushed so that they are not evenly spaced. Uneven spacing like this an increase the inductance of a toroidal coil by as much as 20%. This uneven spacing caused data from coils with 12 through 16 turns to be unusable.
This picture also illustrates a common problem with toroidal coils as mounted on a printed circuit board. The distance between the holes where the leads for the coil is attached to the circuit is about 17mm. These leads together with the path through the resonant capacitor form a very significant inductive loop. Because one leg of the toriodal core is inside this loop, the total inductance of the coil appears to be significantly larger than would be expected from the number of turns in the toroid itself. The inductance of this loop appears as an offset term in the model derived below.
For the second experiment, the leads were given a half twist at the base of the coil to minimize the area of this last turn. This twist is visible in the following picture:
In both experiments, the test procedure consisted of measuring the resonant frequency for coils with different numbers of turns. The measurements were made on the same coil as one turn at a time was removed. The leads from the coils were inserted into the holes in the test jig after stripping insulation and the parallel resonant frequency was recorded. One turn was then removed, insulation removed and the remaining turns were spread evenly around the toroid. This process was repeated until only four turns were left on the coil. The first experiment had no twist and the second included a twist of the leads at the coil.
As an example, resonant frequency for the 4-turn coil is clearly seen in the following figure. In this picture, the marker is not exactly at the resonant minimum which is why the displayed value is not quite the same as in the results given below.
The resonant frequency and the value of the parallel capacitor (180pF) determine the inductance of the coil using
This inductance for the coil includes the inductance of the leads and test fixture.
These measured values of inductance were then fitted to a linear model with the square of the number of turns as the observed values. The fit was done heuristically to minimize relative error.
The results for the first experiment are shown in the following table
| Turns | f_resonant (MHz) | L_actual (nH) | L_fit (nH) | error (%) |
|---|---|---|---|---|
| 11 | 17.7645 | 445.92 | 450.46 | -1.0% |
| 10 | 18.949625 | 391.89 | 382 | 2.6% |
| 9 | 20.945625 | 320.76 | 320.06 | 0.2% |
| 8 | 22.95 | 267.18 | 264.64 | 1.0% |
| 7 | 25.275 | 220.29 | 215.74 | 2.1% |
| 6 | 28.375 | 174.78 | 173.36 | 0.8% |
| 5 | 31.875 | 138.51 | 137.5 | 0.7% |
| 4 | 36.325 | 106.65 | 108.16 | -1.4% |
The form of the model is
The results for the second experiment are shown in the following table:
| Turns | f_resonant (MHz) | L_actual (nH) | L_fit (nH) | error (%) |
|---|---|---|---|---|
| 15 | 14.3500 | 683.38 | 683.25 | 1.000 |
| 14 | 15.3000 | 601.15 | 600.6 | 1.001 |
| 13 | 16.2750 | 531.28 | 523.65 | 1.015 |
| 12 | 17.6750 | 450.45 | 452.4 | 0.996 |
| 11 | 19.0400 | 388.18 | 386.85 | 1.003 |
| 10 | 20.8400 | 324.02 | 327 | 0.991 |
| 9 | 22.7470 | 271.97 | 272.85 | 0.997 |
| 8 | 25.0575 | 224.13 | 224.4 | 0.999 |
| 7 | 28.1600 | 177.46 | 181.65 | 0.977 |
| 6 | 31.3325 | 143.34 | 144.6 | 0.991 |
| 5 | 36.0000 | 108.58 | 113.25 | 0.959 |
| 4 | 42.0000 | 79.78 | 87.6 | 0.911 |
The model that fits these data is
The coefficient on the turns term is much closer to the 2.7 nH/turn^2 value given in the Micrometals data sheet.
Even with the twist, an offset is still required in the model. This model replicates the observed data within 2% of the actual value and the 5-95% confidence interval for the model is
The results here give significantly different values than would be derived from data sheets for the -6 micrometal.
For instance, RF Microwave quotes the datasheet from Micrometals and
gives the model
Some of the variation measured here may be due to the cores under test being from unknown manufacturer. Some of it is, however, due to the geometry of the circuit.