# Gas station without pumps

## 2013 June 18

### L/R time constant lab

Filed under: Data acquisition,home school — gasstationwithoutpumps @ 00:28
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In our home-school physics today, we tried modifying the RC time constant lab to look at inductors instead of capacitors.

The basic idea was simple: put an inductor and resistor in series, switch the voltage between two levels, and monitor the current.

Circuit for looking at current through inductor after step-wise change at Vin.

The equations for Vout are simple:
$V_{\mbox{out}}(t) = R I(t)$
$V_{\mbox{in}}(t) - V_{\mbox{out}}(t) = L \frac{dI(t)}{dt}$

If we start out with 0 current and have an upward step of Vin to Vmax, we have the solution
$V_{\mbox{out}}(t) = V_{\mbox{max}} (1 - e^{-R/L t})$

Initially, we wanted to use one of the 220µH inductors that I had bought for the circuits class, but the L/R time constant is too short to be easily measured with the Arduino data logger—with a resistor large enough to avoid shorting the power supply in the steady state (say, 47Ω) the time constant would be only about 5µs, and we need a time constant of 5msec or more, so we’re a factor of 1000 too low.

Coil used for speed of sound lab and as inductor for L/R time constant.

I do have one larger inductor: the one that my wife found on the street and which I used previously as a sensor for the speed of sound lab. I had previously estimated that inductor as about 0.5H, but I’d forgotten that estimate when we did the measurements today, so we did a little trial and error to get reasonable resistor sizes (47Ω, 10Ω, and 100Ω were all eventually used).

Rise in current (as seen by rise in output voltage across 47Ω resistor).

The maximum Arduino output of 413 is compatible with a DC resistance of (1024-413)* R/413 = 69.5Ω for the inductor, close to the 68.3Ω we measured in Dec 2011. The inductance of 0.177H also seems fairly reasonable, though lower than my previous estimates.

I also realized that I did the analysis of the L/R time constant wrong, since I did not include the DC resistance of the coil, which is substantial (measured, along with the wires to it, at 70.31Ω). This changes the equations to

$V_{\mbox{out}}(t) = R I(t)$
$V_{\mbox{in}}(t) - V_{\mbox{out}}(t) = L \frac{dI(t)}{dt} + R_{\mbox{ind}} I(t)$

If we start out with 0 current and have an upward step of Vin to Vmax, we have the solution
$V_{\mbox{out}}(t) = V_{\mbox{max}} (1 - e^{-t (R+R_{\mbox{ind}})/L})$

With this correction, the estimated value of the inductance is 0.445H.

We did not get a reliable fall time measurement in the first set of measurements, probably because the time it took for the mechanical switch to go from 5V to 0V was too long.  We tried again with the output of Schmitt trigger as Vout, instead of a mechanical switch. Since the 74HC14N chips only deliver about 24mA, I put 5 of the Schmitt triggers in parallel.  Furthermore, to avoid problems with switch bounce, I put a low-pass RC filter between the mechanical switch and the Schmitt-trigger input (R=1kΩ, C=4.7µF, for a 4.7msec time constant). Without the low-pass filter, I sometimes got a bounce visible on the Schmitt-trigger output.

With a smaller resistance (10Ω) to get better time resolution, we took another set of measurements, getting good curves for both rising and falling edges:

Rising edge with a 10Ω series resistance. Note that the input voltage is not a simple step—there is apparently another 10Ω or so of source impedance when pulling up Vin, resulting in an IR drop at Vin, though no such effect was visible when pulling Vin down.

The falling edge resulted in a similar estimated value for L, even though the transition of Vin was very sharp on the downward edge (<1ms), so there was no evidence for extra source resistance on downward transitions.

I trust the falling edges more than the rising ones (because of the non-step nature of the input rising edge), so 0.372H is currently my best estimate of the inductance of the coil.  I think we want to make some more measurements with larger resistance values, so that the resistance of the wiring has less effect.  I think that we can get a sufficiently good estimate of the time constant even if it is only 1/3 as long as the current one, which should allow a total resistance around 240Ω, or a load resistance up to 170Ω.

We tried again with a 100Ω load, then tried finding a resistance for the source, wiring, and inductor that gave a consistent value for the inductance with either load.  I found this resistance by fitting the R/L time constants to the falling edges, then solving the pair of linear equations:

$10+R_{\mbox{ind}} = L / \tau_{10}$

$100+R_{\mbox{ind}} = L / \tau_{100}$

The solution turned out to be 78.5Ω and 0.41H.  I believe that to be a fairly good estimate of the inductance of the coil, but the resistance includes source and wiring resistance other than just the 70.31Ω DC resistance measured with my multimeter (and even that included some wiring resistance, since it was measured at the breadboard, not at the coil).

Fit for falling edge with 100Ω load, using the larger estimate of 78.5Ω for the series resistance of the inductor.

We’ll want to do more experiments with inductors, to validate this inductance estimate, and to measure the inductance of the 220µH inductor using other means.

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