Gas station without pumps

2013 June 27

Inductance of large inductor summarized

Filed under: home school — gasstationwithoutpumps @ 11:25
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This post summarizes my inductance measurements, some of which I did with my son, some independently.

The goal was to determine the inductance of the large inductor that my wife found lying in the street and that I’ve used for a few things in the physics class with my son (like the Speed of sound lab).

Our first measurement used the Arduino data logger to measure an L/R time constant. Using a Schmitt trigger to clean up a mechanical switch output, we measured the voltage across a series resistor (hence the current) for step inputs (both upward and downward).  The upward steps were not useful, as internal resistance of the Schmitt-trigger inverter meant that the step was not clean—the voltage drooped as the current went up. The downward step did not have this problem. The inductance and resistance of the coil were determined by making measurements with both a 10Ω and a 100Ω external resistor, and fitting exponential time constants to the downward steps.  This gave a pair of linear equations in L and R to solve: L=(R+10\Omega)T_{10} and L=(R+100\Omega)T_{100}.  From these equations I got estimates of 78.5Ω and 0.410H. Note that the resistance includes all the wiring resistance, which was substantial, as I was using flexible jumper wires that are not made from copper and have a high resistance.

The next successful measurement again used a series connection of the unknown inductor and a 100Ω resistor, but the stimulus was a sine wave, rather than a step.  I measured the RMS voltage across the resistor and across the inductor for several different frequencies, and fit an L+R model to the magnitude of the impedance as a function of frequency: |R + j 2 \pi f L| = V_{L}/V_{100\Omega} 100\Omega.  With this method, I measured the inductance and resistance as 0.369H and 73Ω.  I also tested an AIUR-06-221 inductor (nominally 220µH and 0.252Ω) with a 1Ω current-measuring resistor and got 229µH and 0.258Ω.

Once I finally figured out that DC bias changes capacitance of ceramic capacitors enormously, I managed to get a Colpitts oscillator to work. I should be able to use the frequencies of oscillation with the small inductor and with the large inductor to get an estimate of the inductance of the large inductor.  With the center of the Colpitts tank at virtual ground, the AIUR-06-221 inductor oscillates at 7691Hz and the large one at 180.04 Hz  (both after a few minutes of warmup, as the frequency changes initially).  The ratio of the inductances should be the square of the ratio of the frequencies, that is L = \mbox{220E-6 H} (7691/180.04)^2 = 0.401 \mbox{H}, or 0.418H, if we use 229µH as the inductance of the AIUR-06-221. Note that this measurement depends on knowing the inductance of the small inductor, but does not require knowing the capacitance of the ceramic capacitors—just that the capacitance is the same for either inductor.

These oscillator measurements and the step-response measurements are consistent, but the modeling of the magnitude of impedance from the voltage measurements at different frequencies seems a bit out of line with them.  The problem may have been that the current-measuring resistor was small, so that the voltage measurements were small.  Perhaps I need another series of measurements with a 10kΩ current-measuring resistor, which should allow a better estimate of L (though not a good estimate of R). I tried again using a 10kΩ resistor and a frequency range from 320Hz to 15kHz, and got a fit for L=0.349H, the lowest estimate so far!

I’m not sure what the problem is with the measurements using the external sine wave.  Perhaps I should do another set using a different function generator, as the Bitscope Pocket analyzer produces pretty bad harmonic distortion at the higher frequencies.  I tried with the Elenco FG-500 function generator (which also has bad harmonic distortion, but different) from about 70Hz to about 16kHz, and got a fit for 0.356H.  This is similar to the measurements I got with the Bitscope function generator.

So, I’m left with a discrepancy that I can’t explain:  measurements of  0.35H for the sine-wave excitation and 0.41H for the L/R step and the Colpitts LC oscillator.

 

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