Various capacitors, charged to various initial voltages, are discharged through a filament lamp. The resulting graphs are analysed and an attempt is made at finding a trend linking the initial voltage to the rate of decay of the voltage across the capacitor during the discharge.
The aim of this experiment is to study the discharge of capacitors through filament lamps as a function of the initial voltage across the capacitor, and, if possible, model this process.
The initial brief (Bath, 1998) explained that the voltage across a charged capacitor of capacitance C, when discharged through a resistor of resistance R, varies exponentially according to the following equation.
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Q = Q_{0} e |
This is covered in most introductory text books, for example Bell, 1978.
If R is not a constant but a function of the voltage to which the capacitor is initially charged (i.e., R = f(V_{0})), then the relationship is no longer simply exponential. This can occur, for instance, if the resistor is in the form of a filament lamp that rapidly increases temperature when a current flows through it.
In this case, one would expect that the graph would start non-exponentially, since at voltages roughly > 1V the lamp is lit, and thus non-ohmic, and then when the voltage reaches (and goes below) 1V, the curve would become exponential as for an ohmic resistor.
At the centre of circuit lay the capacitor being tested. The other components in the circuit were the voltage source, a digital multimeter (DMM), a chart recorder, the filament lamp, and a two-way switch.
The DMM was wired across the voltage source and was used to set the initial voltage across the capacitor. The chart recorder was similarly wired across the capacitor.
The switch was used to select whether the capacitor should charge or discharge by being connected to the capacitor on the common terminal, the voltage source on one of the two other terminals, and the lamp on the final terminal.
Readings were taken for each capacitor/initial voltage pair several times. This was done by recording the voltage drop across the capacitor with the chart recorder. (The graphical output was then measured by hand before being analysed electronically).
Furthermore, to reduce the error in the data collected, the voltages selected were staggered. Instead of reading 1V, 2V, 3V, 4V and 5V consecutively in series, the voltages were set in a shuttling fashion: 1V, 5V, 2V, 4V and finally 3V.
The independent variables in this set-up are the initial voltage V_{0}, the capacitance C, and the properties of the bulb. For purposes of consistency, the same bulb was used throughout the experiment. To provide for some control data and to allow for comparison, a 10 W resistor was used to collect one set of 1V results.
The dependent variables include the voltage across the capacitor, the temperature of the filament, the resistance of the filament, and the current passing through each part of the circuit.
One finds exactly the expected curves.
In particular, the resistor data is exponential (figure 1), as is the 1V lamp data (figure 2). The lamp data at higher voltages moves away from the ideal exponential curve (figure 3), but becomes exponential as the voltage decreases (figure 4).
Figure 5 demonstrates that the exponential fit is already poor at 2V. Close examination of the data indicates that 1V is roughly the limit at which the lamp behaved exponentially; while the voltage is greater than 1V the lamp would be emitting light (that is, it would be hot).
Note that in all of the above graphs, the data is plotted using linear axes, and the best fit lines were Excel-fitted exponential trendlines. These lines were forced through the correct (as set with the help of the DMM) initial voltage.
The random error in each reading is notably small. For each capacitor/voltage combination three readings were taken, and when superimposed most sets were almost indistinguishable from each other.
One source of random errors is the limit of accuracy of the chart recorder. Accordinging to the documentation the pointer can fluctuate within a distance of ±2mm. This is an error of about 0.5% of the range, but for each experiment the error should be constant over the duration of the discharge. In order that this error not become very significant, we did not examine any of the data beyond 1% of the range of the chart recorder.
Any data points that are obviously outliers are almost certainly transcription errors. To transfer the data from the graphs output by the chart recorder to the Excel spreadsheet required the excruciatingly (over seven hour) long process of carefully measuring the graph every two millimetres and writing the measurement into Excel. On initial graphing, many errors came to light and were immediately found to be transcription errors.
All of the original graphs that were selected for analysis were perfectly smooth. (Some graphs were collected that had steps or other dramatic imperfections, these were ignored as the variations were due to faults in the equipment. For example, in one case a connecting wire was jarred out of position and the capacitor recharged — this would obviously in no way help the analysis required by this project and so the graph was discarded.)
A brief analysis of the graphs gives ± 0.1V and ± 0.1s for the data that is plotted. This error was mainly introduced from the manual conversion of the analogue graphs to the digital data.
There were many potential sources of systematic errors.
The first induces a systematic error within the readings of a chart. This is the fluctuation in the mains power supply that was very apparent during times when the lab was being used by many people. (For example, on Friday afternoons, when the lab is busy, the power supply would only stay constant to the first decimal place over a few seconds, while on Wednesday afternoon, when the lab was virtually empty, the power supply provided a constant voltage output to the fifth decimal place over a period of many minutes).
The result of this fluctuation is an uncertainty in the initial voltage.
While this error is systematic over a single chart, it is a random error when several charts are taken into consideration for each capacitor/voltage pair. Therefore, the plotting of best fit lines that take into consideration all the data for each capacitor/voltage pair reduces the effect of this error a great deal.
Another source of systematic error is the graph paper upon which the chart recorder recorded the charts. Initially we assumed that the separation was 1cm per major grid line, however, later, more accurate measurements indicated that the actual separation was about 0.995mm. This should not affect the majority of results to a great extent, however on longer runs (for example, the 24mF capacitor discharged through the resistor) the error adds up to about 2% in the final value. We have tried to compensate for this by using an acetate overlay of a more accurate grid to measure the graphs, instead of relying on the lines printed on the chart paper.
A third important source of systematic error is the occasionally poor quality of the capacitors themselves.
We noticed that some of the capacitors leaked current to varying degrees. One particular capacitor leaked current so fast that it would lose around 10% of the voltage it was charged up to between the switch being in the "charge" position and the switch being in the "discharge" position. Another capacitor showed very clear symptoms of dielectric absorption, gaining 0.55V volts of charge in about 2 minutes! (We recorded this data for possible future analysis.)
These defects are common to dielectric capacitors such as the ones we used, especially after long periods of inactivity.
In the majority of cases, slightly faulty capacitors were replaced with more stable ones. In cases where this was not possible, a slight drift in the data may be apparent, particularly at the start of the discharge and at the very end.
Different sets of readings were taken on different days, and the ambient temperature thus probably varied a few degrees between different parts of the experiment. This may have introduced some variation of the resistance of the wires. This error is likely to be extremely minor, however, in comparison to some of the more important errors mentioned above.
The data indicates quite clearly that the theory mentioned in the introduction is sound. However, an exact analysis of the data to obtain an equation modelling the non-ohmic behaviour of the filament lamp and its effect on the discharge of the capacitors is currently beyond me.
For the 6V data set, which was the least exponential, the pre-1V data points are approximated to quadratics by Excel as follows:
V = 0.63t^{2} - 3.6324t + V_{0}
V = 0.6818t^{2} - 3.9506t + V_{0}
V = 0.1564t^{2} - 1.8001t + V_{0}
V = 0.1343t^{2} - 1.6965t + V_{0}
V = at^{2} - bt + V_{0}
Plotting a and b against C and fitting various trendlines suggests that a is linked to C logarithmically and b is linked to C by a power law. In particular, for this lamp and the initial voltage of 6V, calculations suggested that a and b were roughly related to C by the following equations:
a = 33.212 C^{-1.7183}
b = -2.6714 ln(C) + 10.107
This would suggest the following general equation for V in terms of t, C and some numeric constants:
V = (33.212 × C^{-1.7183})t^{2} - (-2.6714 × ln(C) + 10.107)t + V_{0}
Had collecting the data been less time consuming, more samples could have been taken and this relationship could have been extended further, possibly removing the four numeric constants of uncertain origin.
The objectives were fully met, in that the data confirmed the breakdown of the exponential relationship when a capacitor is discharged through a (hot) filament lamp. The process was modelled, although the extent to which this was achieved is not ideal.
The change from an analogue straight-to-paper chart recorder to a digital straight-to-computer probe would have greatly facilitated the analysis of the data. A great deal of time was spent merely on measuring the graphs, when it could have been spent on writing the report.
I would also suggest that before further experiments using the electrolytic capacitors are carried out, the capacitors be held at a constant voltage for a few hours. This would help them rebuild their internal structure so that the degeneration of the capacitors does not affect the results to such a degree.
At low voltages, lamps act just like ohmic resistors and thus the discharge of a capacitor at low voltage through a filament lamp is exponential. However, lamps do not act as ohmic resistors when they are emitting large numbers of visible-light photons. An exact numerical analysis provided no great insight into the mathematical relationship between the variables involved, however it did suggest a direction for future research.
A filament lamp consists of a very thin tungsten wire (the filament) which has a relatively high resistance (as compared to thicker copper wires). The tungsten rapidly increases temperature when a current of any consequence is passed through it, and rapidly reaches the white-hot stage. At this point, the lamp is emitting large numbers of photons all the way across the visible-light spectrum, and so appears to be emitting white light. The temperatures involved are between 1000K and 2000K.
Research established that the relationship between the resistances of a metal at two different temperature was given by the following equation.
R = R_{0} (1 + a × DT)
In equation 10, the constant a is the temperature coefficient of the metal (in this case tungsten), and DT is the difference in temperature between the that at which R_{0} was measured and that at which R is wanted.
Unfortunately, even the most careful research did not discover a recognised value for a. Because that leaves us with two unknowns in this equation (a and R), this equation could not be put to much use.
I would like to thank Mike Harriman, who proved very helpful during the construction and troubleshooting stages of our circuit design. Thanks should also be extended to Dr Andrews who gave some initial pointers, Dr Squire who suggested suitable values for the filament lamp and the capacitors used, and Dr Sullivan who had some ideas on where to begin in the data analysis.
Bell, D A (1978). Fundamentals of electric circuits. ISBN: 0879093188
University of Bath (1998). Discharge of a capacitor through a filament lamp. EXPERIMENT 50.