Measurement of atmospheric ion concentrations

Research Group: Emily MacDonald, Katherine Ratcliff, and Chasen Shaw

Launch: Whitworth Fall 2014

''A capacitor was hand-rolled and placed within a pod that was to be sent into the troposphere via a weather balloon. As the capacitor ascended and descended, the capacitance would change based off the differing ionization levels. The ion concentration would change the dielectric, and the capacitance would change in a measurable way. Our purpose was to measure the radiation density in the lower atmosphere over regions near Spokane and contrast it with average radiation levels. This was achieved by measuring changes in atmospheric ion concentration by calculating the changing capacitance (and so, dielectric) from the output voltage of the hand-rolled capacitor. The pod ascended to the stratosphere and data was collected in its ascent and descent. The data retrieved shows no significant trend until it is converted to capacitance, and then averaged throughout the flight. A synopsis of the experiment, the theory behind it, and the analysis of the data follows.''

Background
The basic objective of the experiment is to measure the radiation density over areas around Spokane in order to contrast it with average radiation levels in the lower atmosphere. In the Earth's atmosphere, cosmic rays collide with the ionosphere at 85-600 km altitude, avoiding lower layers and the altitudes reached by our weather balloon. Given this information, it is fair to assume that the radiation density in lower layers of the atmosphere, like the troposphere, should be lower than in the ionosphere. Any spikes or irregularities in radiation levels down must be due to sources of radiation on the ground, specifically the ground of areas around Spokane. Radiation density in an area is proportional to its ion concentration because radiation like x-rays and gamma rays are just very energetic photons that strike valence electrons in atoms, freeing them, and thus ionizing the atom.

A Gerdien cylinder (specifically, capacitor) can be used to study ionization of the atmosphere by cosmic rays. Since ionized air has a greater conductivity, the amount of ionized air in a region can be measured by finding the change in conductivity. The Gerdien cylinder does this with two electrodes of the same material, one an outer cylinder and another a small rod inside the outer one. This essentially acts as a capacitor with the incoming ions accumulating on oppositely charged electrodes, effectively acting as a linear dielectric. The degree of ionization is proportional to the change in capacitance in the Gerdien cylinder. It is also worth noting that a Gerdien cylinder of stainless steel and one of aluminum yielded different graphs in a function of conductivity vs. altitude. This is due to electrochemical potentials in the atmosphere turning the Gerdien cylinder into an electrochemical cell. Specifically, stainless steel and aluminum have to be biased at different positive potentials to achieve zero ion current in the cylinder.

There were two methods to using a Gerdien cylinder to measure conductivity: voltage drop across the capacitor and current measurement. In terms of current, ions would flow into the cylinder and accumulate on surfaces of opposite voltage. When enough ions accumulate on the surfaces, a small current is created which is proportional to the ion concentration and conductivity of the air. The dimensions of the Gerdien cylinder had to be increased in order to increase the current to an amount that could be observed. The cylinder's dimensions were usually 8 cm by 50 cm. Given the constraints of current measurement, it was decided instead that voltage decay through air within the cylinder would be measured.

The relationship between conductivity $$\sigma$$ and voltage is illustrated by the equations

$$\sigma = \frac{\epsilon_{0}}{\tau} $$ and $$V = V_{0} e^{\frac{-t}{\tau}} $$

The first equation can be derived from Ohm's Law $$J=\sigma E $$. J can be isolated by realizing the electric field is proportional to the charge per unit of surface area.

$$J = \frac{\sigma Q}{S_{A} \epsilon_{0}} $$

This equation is easily solved since $$J=\frac{I}{A}$$, but instead of I we will use i, which is defined here as the flow of negative ions from a positively charged body.

$$i = -\frac{\sigma Q}{\epsilon_{0}} = \frac{dQ}{dt} $$ whose solution is $$Q = Q e^{\frac{-\sigma t}{\epsilon_{0}}} $$

Seeing as $$Q = Q e^{\frac{-\sigma t}{\epsilon_{0}}} = Q e^{\frac{-t}{RC}} $$ where $$\tau=RC$$.

Given this relationship between Q and $$\sigma$$, it is evident that Q is exponentially proportional to C. A graph of the change in capacitance with altitude will show the ion concentration as a function  of altitude, which will in turn show any changes in radiation density in the upper atmosphere.

Circuit background
The circuit used to measure the change in capacitance is basically just two capacitors in series, with one known and one unknown capacitor. While it may seem that there is only one capacitor in the final circuit, in reality there are two. Here it is assumed that the handmade capacitor is the test capacitance and the stray capacitance of the Arduino is the known capacitance, and thus stays constant. Stray capacitance is actually a small amount of capacitance that arises in any real circuit.

Given this information, it is worthwhile to explain how the circuit works. Because the two capacitors are in series with the same source, the sum of the two capacitors is

$$C_{T}= \frac{ C_{1} C_{2} }{ C_{1} + C_{2} }$$

With this in mind, the voltage across one of the capacitors is

$$ V_{CX}=V_{S} \frac{ C_{T} }{ C_{X} } $$

The voltage across one of the capacitors, in this instance C1, is

$$ V_{C1}=\frac{ C_{1} }{ C_{1}+C_{2} } V_{T}$$

$$ V_{out} = \frac{(V_{in}) C_T}{C_1+C_T} $$

By solving this equation for $$C_{T}$$, the test capacitance can be isolated given a $$C_{2}$$ and $$V_{T}$$ as constants and measuring the $$V_{out}$$ of the test capacitor.

Materials

 * Pod, 20x15x15 cm
 * Two copper plates 30.5x10x0.075 cm
 * Two foam pieces, approximately 9 mm thick, cut in the shape of our capacitor bottom
 * Four 1-cm cube pieces of foam
 * Brushless mini-fan
 * Arduino
 * Alligator clips and wires
 * Radio transmitter #5
 * 5-volt battery

Description
The contents of the pod consisted of two sheets of copper hand-coiled together to form a capacitor, two foam platforms, four foam cubes, and a brushless mini-fan. Each plate was 0.075 cm thick, 30.5 cm long, and 10 cm wide. Once coiled together lengthwise, the capacitor at its maximum width was 71.78 cm, where the two plates came to an end on the outside of the coil. The height of the capacitor remained 10 cm, the width of each plate. Two pieces of foam were placed on the top and the bottom of the coil, the two plates wedged into pre-made ridges that aligned with the curves, so that the plates would remain untouching as the pod moved. Two holes, one in the top piece of foam located at the middle of the coil between the two plates, and one located in the bottom piece of foam located at the end of the coil between the two plates existed to force air to travel throughout the entire capacitor in a consistent manner. In each corner, a one-centimeter-tall piece of foam supported the bottom foam platform, to allow a maximum circulation of air inside the pod. A computer fan, of dimensions 40x40x9.5 mm, was attached to the top of the inside lid of the pod. Air was pulled through the capacitor by this fan, in order to keep the readings related to the altitude. The two plates of the capacitor were connected to the arduino to read the voltage through the plates, connected by alligator clips and secured with electrical tape. The radio transmitter was wedged on the side of the capacitor. The battery was plugged into the Arduino, used to power it. The battery was also wedged into the side, next to the capacitor itself. The pod lid was not fully sealed, to allow airflow.

Electrical
The capacitor was connected by alligator clips to wires which led into the A0 and A4 pins of the Arduino, so that the Arduino could read the output voltage. Soldering would have been preferred and more reliable; however, the wires could not be directly soldered to the copper plates due to the heat dissipating properties of copper. The fan was connected to the 5V output and ground pin of the Arduino. The radio transmitter was connected to digital pin 5 and the ground pin of the Arduino.

Software


The Arduino was coded to record the output voltage of the capacitor as it changed altitude and, presumably, ionization levels. The code was created by Jonathan Nethercott on PIC Tutorials.

Program Flow
The code takes advantage of the fact that the voltage of a circuit with two capacitors will settle to within 1% of its final voltage. This results from the equation:

$$ V_{C} = V_{0} e^\frac{-t}{R C} $$

This equation calculates the capacitance using the circuit in Figure 4, where one capacitor of known capacitance is placed in series with an unknown test capacitance. An initial voltage is then introduced to the circuit and the voltage across the test capacitor is used to find its capacitance, furthering the equation to:

$$ V_{out} = \frac{(V_{in}) C_T}{C_1+C_T} $$

where C1 is a known capacitance used in the circuit to test for the value of CT. Mr. Nethercott assumed here that the stray capacitance of the Arduino itself can be used to find the test capacitance, removing the need for a second capacitor and significantly simplifying the circuit. It should be noted that the stray capacitance is assumed to only be 30pF, which leads to a range of 3.5-225 pF. with an accuracy of 1%. Given the small value of the capacitor in the pod, this seemed sufficient and the most convenient in terms of space since it hardly requires a circuit to function. Mr. Nethercott experimentally verified the stray capacitance in the code to be 24.4pF.

The code itself loops once a second and records an output voltage each time. The output voltage is then run through the equation:

$$ C_{out} = \frac{(V_{out}) C_1}{V_{in}-V_{out}} $$

This equation yields the capacitance, which is output to the serial monitor on the computer. A set of analog pins were all that was available. Unfortunately, the analog pins can only transmit voltages, so it was decided that transmitting the output voltage directly would be the best option. To do this, the Arduino was programmed to output a voltage to the analog pins that was equal to the output voltage of the circuit. Since the Arduino can only output a voltage between 0 and 5 volts, a duty cycle was used to decrease the 5V down to the output voltage. One of the pulse with modulation (PWM) pins had to be used, as the duty cycle was a necessary addition. This could then be converted into the resulting capacitance in analysis.

Code
FinalProject.ino

Capacitor
The capacitance of the capacitor was tested frequently, using two different methods. The first method was by using a digital multimeter. The capacitance was measured before the plates were wedged in the foam, and after to ensure the plates were not actually touching. Each testing read an average of 270 pF. The second method used to measure the capacitance of the hand-curled capacitor was by using the code provided in the Electrical section. This value oscillated a lot, but remained relatively around the value measured by the multimeter. This was to be expected from the capacitor as it was a very low value of capacitance, and capacitors are noisy in general.

Code
Our code was originally made to measure the capacitance of the capacitor directly. This code was tested for three different capacitors whose values were known: 98 nF,1 nF (actually a fraction of this, measured in pF, but the multimeter cannot read this magnitude), and 89 nF. These three capacitors were first measured with the digital multimeter and then used to validate our program.

Fan
The fan was tested to make sure it worked, first with a battery, and then plugged into the Arduino 5V output.

Data and Analysis
The Arduino gave a voltage for several points between 843.2 ft and 95000 ft. The output voltage from the capacitor was directly measured, transmitted, and recorded. The output voltage was converted to capacitance by multiplying the difference between the input and output voltages by a constant and dividing by the output voltage:

$$ C_{out} = \frac{(V_{out}) C_1}{V_{in}-V_{out}} $$

$$C_1$$ was determined to be 24.4 pF and $$V_{in}$$ was 5V.

The data retrieved from the command pod showed no significant trend, even after being converted to capacitance, implying there was little if any relationship between the dielectric function of the atmosphere and the altitude. The majority of data points were distributed seemingly at random between 0 and 94 pF, with several outliers of around 2500 pF between 60443 ft and 78514 ft. However, the average value of the data points was 83.30068 pF, 69% less than the value found in testing. The uniformity of the graph shows that the capacitance stayed constant across the different altitudes, implying that capacitance may have changed between the initial tests and the actual use of the capacitor in the pod. A possible source of the discrepancy may have been the decreased flow of air within the pod, as compared to in the lab where the capacitance was first tested.

The 69% change in capacitance seems reasonable when the magnitude of the capacitance is taken into account. Seeing as it is in the picometer range, a change in air could lead to a 187 pF change in the capacitance. Since the dielectric constant of air is reported to be 1.00059 at standard atmospheric pressure, then air can lead to a maximum of 0.6 mF increase in the capacitance.

The constant trend of the graph argues that the ionic content of the atmosphere is uniform, at least up until 95,000 ft. The uniformity of the capacitance as a function of altitude suggests that the radiation from the ground is consistent throughout the troposphere and the stratosphere. While it is possible the radiation from the ground is giving a constant ionization of the air over areas in Washington, ideally that would not be the case all the way up to 95000 ft. There would be some net decrease in ion concentration. Given our original assumption that the radiation would lead to spikes or irregularities in ion concentration, it is fair to say that any excess radiation near or in Spokane is not high enough to detect by using information from the change in ion concentration alone.

Future Improvements
A more sensitive capacitor could better detect the change in ionization with increasing altitude. The original assumption in the design of the capacitor was that the inverse relationship between capacitance and separation distance means that minimizing the distance between the plates can maximize the potential; however, it would also minimize the flow of air through the capacitor. An alternate way to increase sensitivity would be to increase surface area of the capacitor plates, or to be able to more precisely roll the plates together, to make sure the plates were uniformly equidistant.