Chaos in an LRD Circuit

In this lab, non-linear behavior was investigated using a diode circuit.  The three main passive components of electronic circuits are resistors R, capacitors C, and inductors L.  In an LRC circuit, the voltage across them depends on charge q.  If a single sine wave voltage at frequency υ is applied, the output voltage across any of the three devices is also a single sine wave at frequency υ.  If the capacitor C is replaced by a diode D, then many frequencies can be generated simultaneously.  The voltage across the diode has non-linear terms in the charge q (eg q2, etc) and its derivatives.  I investigated the non-linear electronic response of an LRD circuit.

The setup of the circuit involves an Agilent DSO-X2014A Oscilloscope and Agilent 3320-A Waveform Generator.  The waveform generator can generate a wave up to a frequency of 10 MHz.  By keeping the frequency at 1 kHz, the amplitude of the sine curve can be varied.

First I explored an LRC circuit, since that was simpler and much easier to understand.  There are three components: a resistor, a capacitor, and an inductor.  The input and output signals are viewed simultaneously in y-t mode.  When switching to the x-y mode on the oscilloscope, the image on the screen shows a perfect circle, because the input and output are both sine waves.  The Fast Fourier Transform (FFT) mode shows the Fourier transform of the output curve.  In an LRC circuit, there is only one peak in FFT mode since it is only a sine curve.  By varying the frequency, the amplitude of the output curve and the phase difference between the input and output curves can be measured and the following graphs resulted:

phi vs f

Phase Difference (phi) vs Frequency (kHz).  This graph shows how as the frequency changes, the phase difference also varies.

A vs f

Amplitude vs frequency graph.  At around 60 kHz, the amplitude hit its peak and then began to decrease.  This means that at around this frequency, the circuit hit resonance.

Next I explored the LRD circuit, which replaced the capacitor with a diode.  DC offset is now a variable because the diode is asymmetric.  A diode is a two-terminal electronic component with an asymmetric transfer characteristic.  A diode has a low resistance to current flow in one direction and high resistance in the other.  The most common function of a diode is to allow an electric current to pass in one direction while blocking current in the opposite direction.  The input amplitude also becomes a variable because the degree of non-linear response depends on it.  The frequency response may or may not be different depending on whether frequency is increased or decreased and it may or may not depend on the starting frequency.  Because the diode only lets current pass in one direction, that is why there is a flat plane where the positive part of the sine curve should be.

Now, the interesting part is that the FFT mode shows lots of peaks instead of one peak.  The general trend seems to be

freqand f0 is the input frequency.  At low amplitudes, these peaks are the only ones in the FFT.  However, at 1 Vpp, and f = 3 kHz, other peaks suddenly popped up at 1 kHz, 2 kHz, 4 kHz, 5 kHz, etc.  There is really no discernible pattern with this, and that is why this circuit is chaotic.  However, what stays the same are the peaks that occur at nf0, which are islands of stability.  What is cool is that at a frequency of 5 kHz and 10 kHz, no extra peaks turn up even when the amplitude is increased up to the maximum.  Unfortunately, I did not get a chance to finish the lab, and this was the furthest I got, but what I explored so far in the lab was fascinating.

Electron Spin Resonance

In this lab, I explored how an electron interacts with an external magnetic field.  When this happens, the electron gives up energy to or takes energy from its surroundings.  The interaction energy between spin and field is quantized into “spin up” and “spin down” energy levels.  A photon of specific energy is emitted or absorbed when the spin makes a transition from one state or another.  This lab explores the technique of electron spin resonance and the physics of “spin.”  The magnetic moment μ of the electron can be calculated by measuring the magnetic fields used.

In a magnetic field B, two different energy levels differ by

ΔE = 2μB

where μ is the magnetic moment of the electron.  With photons of E = hf,

hf = 2μB

In this experiment, frequency f is the independent variable.  I measured the B that produces resonance at each frequency.  The slope of the data will give the magnetic moment of the electron.

The unpaired electrons used in this experiment come from diphenylpierylhydrazil, or DPPH.  This sample is placed in a coil that acts as an inductor in an LRC circuit.  The magnetic field B is produced by Helmholtz coils.  This determines the energy difference ΔE as well as the frequency of the photons that must be provided to produce resonance.  The current is measured, and when the spin is changed, the signal changes.

The Helmholtz coil configuration involves two identical circular coils that are placed symmetrically on each side of the experimental area.  The distance h which separates the two coils should be equal to the radius R of the coil.  By setting h = R, the nonuniformity of the field at the center of the coils is reduced.  The pair of Helmholtz coils for this experiment have N = 320 turns.  The current is limited to 2 amperes, and the radius of the coils is 6.8 cm.  The magnetic field at the center of a circular loop of wire is

B

where μ0 = 4π x 10-7 Tesla*m/A = 4π x 10-5 Tesla*cm/A, and R is the radius of the loop.  If the loop has N turns

BNIf the object is distance z from the center of the loop along the axis which is perpendicular to the plane of a loop of N turns, then

BNz

which simplifies back to the previous equation when z = 0.  If the two loops are 2z apart, then the equation is reduced to

BN2zIn our case, 2z is 6.8 cm, and plugging in all the other values, we come up with

BCalcSo B and I have a linear relationship B = DI, where D = 4.231 x 10-3 Tesla/A.

Data for the voltage of resonance at different frequencies were taken, and graphed on a voltage vs frequency graph, shown below.

Vvsf

The slope of the line, G, was 8.5 ± 0.2 mV/MHz = (8.5 ± 0.2) x 10-9 AΩs.  Since V = IR as well, Gf = IR.  We just calculated B = DI, so I = B/D.  This means

fSince hf = 2μB,

mu

Since I know the values of h, R, G, and D, I plugged in all the values and got

μ = (9.22 ± 0.04) x 10-24 J/Tesla.

The accepted value is μ = 9.28 x 10-24 J/Tesla.  So my calculated value is within 2σ of the accepted value!

PVT in Carbon Dioxide

In this lab, Orsola and I worked together. We used a CO2 isotherm apparatus to measure the pressure P and volume V of a CO2 sample at various temperatures T for a fixed amount n of gas.  Using the Van der Waals equation

VanderWaals

where R is the gas constant, T is the temperature, a and b are parameters to be investigated and V is the molar volume given by:

Vbar

where V is the volume and n is the number of moles in the volume. Long-range intermolecular potential is a and b is short range intermolecular potential.  At high T or low density, the equation behaves like the ideal gas law, PV = nRT.

The apparatus we used was really old, and hadn’t been cleaned since 1999. Therefore, we expected our measurements to have large uncertainties. The apparatus consisted of a capillary tube containing a fixed amount of CO2, which was compressed with an oil-mercury pump. The pump compressed the oil, which then moved the mercury and in turn compressed the CO2. Since the pump controlled the compression of the CO2 sample, it also controlled the pressure the CO2 was subject to. We were able to measure this pressure with a pressure gauge.

We measured the temperature with a mercury thermometer placed beside the capillary tube. We wanted to change the temperature of the CO2 sample and keep it constant. Therefore we pumped water with a certain temperature into the outer tube that contained the capillary tube, which was called a cooling jacket. The water came from a large vat, where we could heat it with three A.C. electric heaters supplied by a variable voltage source. But since this took forever, we just boiled water in an electric teapot and dumped it into the vat. We still had to wait for the temperature of the water in the cooling jacket to stabilize and equal the temperature of the water in the vat, but this would only take about five to ten minutes instead of three hours.

The temperature of the water had to be constant in order for us to reduce the uncertainty in our measurements, but this was difficult to achieve. The temperature of the water in the vat after some time would start decreasing because it was exposed to room temperature. Also, the cooling jacket leaked, especially when the temperature of the water was high. We stopped the leaking with play-dough and a sponge from the physics kitchen (we had to get a new sponge afterwards).

We first calibrated the apparatus by measuring the height of the capillary tube, which we found to be 40.5 cm.  We then took seven runs of data at seven different temperatures: 28.6°C, 33.3°C, 35.9°C, 39.0°C, 41.9°C, 44.2°C, and 50.2°C.  We took data by varying the pressure inside the capillary tube and measuring the height of the mercury.  The height of CO2 was the difference between the height of the capillary tube and the mercury.  Afterwards, we plotted the P/P0 vs V/V0.  P0 and V0 are given constants for C02, where P0 = 7.38 × 106 Pa and V0 = 94.0 cm3 mol-1.  We came up with the following graph:

PvsV

P/P_0 vs V/V_0 graph. This shows all the data we gathered at seven temperatures.

We knew that the behavior of real gases can be modeled with the following virial equation, derived from the Van der Waal’s expression:

Virialwhere B, C, and D are virial coefficients that become progressively smaller for higher order terms of V.  We then discarded all higher order terms because they were too small to affect the curve fit.  The final equation, after variable manipulation, was:

PEqWe knew the value of all of the variables except for n and B.  The n value is constant for all temperatures, but B is temperature dependent.  In order to calculate for n and B, we fit the data to a curve and solved for n and B.   We then found the average number of moles, which was 5.8×10-3, and recalculated all the B values using this value.  All the data we calculated is shown on this table:

CalculationsWe graphed the recalculated values for B vs Temperature (K), which was somewhat close to what we expected.

B vs Temperature

B vs Temperature graph. The general shape of the graph is similar to the example graph that was shown to us.

 

e/k Ratio and the Band Gap

In this lab, the relationship between e, the magnitude of the charge of an electron, and k, Boltzmann’s constant, is explored.  The current-voltage relationship of a p-n junction is measured at different temperatures to determine the e/k ratio.  These measurements show the physical properties of semiconductor junctions and a large range of currents can be measured.

First, I learned some theory.  The measured current I can be calculated by the following equation:

I eq1where V is the applied voltage and I and V are positive quantities.  When eV is much greater than kT, the equation simplifies to

I eq2The charge of an electron is q = -e, where e is a positive constant, and so eV/kT is a positive quantity.  I0 and e/kT can be determined from an I vs V graph.

The experiment uses a l kΩ heliport and a 1.5V battery, a picoammeter, a transistor, and a multimeter.  The transistor is an n-p-n transistor.  Inside the transistor, a single piece of silicon is doped with electron contributing impurities in the left and right regions, such as arsenic.  The center region is doped with hole contributing impurities, such as Gallium.  The n-type region on the left is the emitter (e), and on the right the collector (c).  The p-type region is the base (b).  An image is shown below

npn

The set up of an npn transistor.

The circuit is connected as shown:

circuit

The circuit created by the equipment for the experiment.

The same semiconductor should be used for the entire experiment.  In the beginning, the semiconductor should be in the test tube filled with oil that has been at room temperature for a while.

The first run was at room temperature of 297.15 Kelvin.  Putting a mercury thermometer in the oil with the semiconductor, quick measurements of I and V were made.  After plotting the data by hand on semilog graph paper, an rough experimental e/k value of 11088 k/V was found.  This was compared to the value of 11609 K/V, which was calculated using values from the NIST website, where e = 1.60 x 10-19 C and k = 1.380 x 10-23 J/K.

The linear relationship between ln(I) and V was determined using a graphing program.  The slope of the line is e/(kT); in order to determine e/k, the slope must be multiplied by the temperature of the environment the transistor is in.  I0 is found by using the formula ey-intercept.

This experiment was repeated for four different environments: a water/ice mix at 273 K, dry ice/isopropyl alcohol at 197.5 K, liquid nitrogen at 80 K, and boiling water at 369.4 K.  e/k values and I0 were determined after each run. After plotting all five sets of data on one graph, a general trend appeared:

ln(I)vsV(All)

All five sets of ln(I) vs V data on one graph.

The colder the temperature was, the higher the initial voltages were, and the steeper the slope of the line was.  This makes sense, because temperature is inversely related to e/(kT).

Finally, the e/(kT) values with error derived from all five sets of data were plotted against 1/T to calculate the final e/k value of 11500 ± 200 K/V.  To find the electron band gap, the slope of the ln(I0) vs 1/T graph was used.  This slope represented Egap/k, so to calculate Egap, the slope was multiplied by the k value to give a final band gap energy of 1.195 ± 0.008 eV.  Both graphs are displayed below.

e_kTvs1_T

e/kT vs 1/T graph. The slope of this graph is the final averaged e/k value for this experiment.

ln(I_0)vs1_T

ln(I_0) vs 1/T graph. The slope of this line is the band-gap energy in silicon over k value.

Millikan Oil Drop

The purpose of this lab is to calculate the electric charge carried by a particle, and determine how many electrons are on the drop.  This is done by measuring the force experienced by the particle in an electric field of known strength.  In this lab, I will study the behavior of small charged droplets of oil.  They will have masses of 10-12 g or less, and are observed in a gravitational and electric field.  If I use a different drop to measure each charge, the effect of the drop on the charge is questioned, which causes an uncertainty in the experiment.  This can be corrected for by charging a single drop with different charges several times.

I began by learning some theory behind the experiment.  The drop in air has two forces: Ff, the force of friction, and Fg, the force of gravity.  Ff = -kv0, where v0 is the initial velocity of the fall, and k is the coefficient of friction between air and the drop.  Fg = -mg, where m is the mass of the drop and g is the acceleration of gravity.  The sum of the two forces is zero, and k = -mg/v0.  The drop rising in the electric field has three forces: FE, the force of the electric field, Ff, and Fg.  FE = qE, where q is the charge carried by the drop, and E is the electric field.  Ff = -kv, where v is the velocity of the fall.  The sum of the forces equal zero, and through some manipulation, we find

The volume of the sphere is m = (4/3)πa3ρ, where a is the radius of the droplet, and ρ is the density of the oil.  We are then given

where η is the coefficient of viscosity.  ηeff is the corrected η, ηeff = η/(1 + b/(pa)), where b is a constant, p is the atmospheric pressure.  The final equation for a is

To find the charge q, I use a plot of v vs E, where s is the slope of the plot: s = -qv0/(mg), and q = -smg/(qv0).  The final equation for q is

where

            q = charge carried by droplet

            g = gravity = 9.80 m/s2

            s = slope of v vs E

            ρ = density of oil = 886 kg/m3

            b = constant = 8.22 x 10-3 Pa*m

            p = barometric pressure = 101.3 x 103 Pa

            η = viscosity of dry air

            v0 = terminal velocity of fall

The droplet viewing chamber has a lid and housing cylinder.  Inside, there was a droplet hold cover, an upper capacitor, a plastic spacer, and a lower capacitor.  The width of the plastic spacer was 0.7 cm.  Using the focusing wire provided, the reticle was brought into focus using the focusing rings, and the light from the halogen lamp was adjusted to illuminate the viewing area.  The atomizer is the apparatus that delivers the oil droplets into the viewing chamber by squeezing the bulb of the atomizer.  The high voltage power supply was connected to the capacitor, and the settings adjusted to deliver about 500V.  The thermistor measures the temperature inside the capacitor, which can be used to measure η.

To collect data, the plate voltage was changed so the droplet was “driven” to the top of the viewing area.  To find v0, set the plate voltage to neutral and time droplet as it falls one major division, which is 0.5 mm.  Record polarity required to drive droplet (±500V) upward.  Record the time it takes to travel the one division, and repeat the process while driving the droplet downward.  Repeat this several times, and then change the voltage delivered to the capacitors to 400V, then 300V.  After this run, I lost sight of the droplet.  I then graphed the velocity vs voltage.

This graph shows the relationship between velocity and voltage. Using the slope of this graph, I can determine the amount of charge that the droplet carried.

Using the previous equation, I was able to determine that there this droplet carried 4.59 x 10-17 C and carried 287 electrons.

The second part of this experiment involved hitting the droplet with more charge and measuring the amount of time it took the charged droplet to travel up and down one division.  I picked one droplet and recorded the time it took to drive the droplet up and down several times.  Then I hit the ionization switch for a few seconds and repeated the process.  I did this twice before the droplet ran out of my sight.  The equation I used to determine the charge of the droplet was

Using this equation, I calculated that the original droplet had four electrons.  After one ionization, the droplet had 10 electrons, and after a second ionization, there were 26 electrons on the droplet.

Fourier Optics

The purpose of this lab is to explore the world of lasers and lenses, using different slits to quarantine specific parts of images.  The setup of the experiment involved a Neon-Helium laser, three lenses, and a camera connected to a computer with the uEye program. There were three planes where images could be seen.  Plane 1 (P1) is where the original image is placed.  Plane 2 (P2) is where the slit is placed for the Fourier transform to occur.  Plane 3 (P3) is the final Fourier transformed image.  The setup is diagrammed in the image below.

What happens to the image as it travels through the experimental setup.

First, I began with a simple image: a grid.  Two shots were taken: one with the camera at P3, and one at P2.  These images are displayed below:

Image of unfiltered grid.

The Fourier image of the unfiltered grid.

Using the 0.16 mm vertical slit from a set of diffraction slits, I was able to block out all the horizontal lines on the grid, and came up with this result

Horizontal lines of filtered grid.

Fourier image of vertically filtered grid.

After this, I changed the orientation of the slit to horizontal, and then only the vertical lines remained on the grid.

Vertical lines of filtered grid.

Fourier image of horizontally filtered grid.

After the grids, I moved on to more complex things.  I put a fingerprint on a glass slide and tried to enhance the contrast of the print.  This involved using a wire to try to block out the center dot in the Fourier plane.  This was really hard, and I tried this two times and then the third print is from when I used the diffraction slide with a point in the center.

Unfiltered image of fingerprint.

Fourier image of fingerprint.

Image of first attempt using a wire.

Image of fingerprint after second attempt using wire.

Image of fingerprint after using a diffraction slide.

For the final part of the lab, there was an image that said, “Here is some text.” with vertical lines running through it.  I tried to get rid of the lines and simply have the filtered image of the text, but that proved to be impossible with the setup I had.  Because the text and the lines intersected at places, there was no way to get rid of the lines without the text.

The unfiltered image of the text.

Fourier image of the unfiltered text.

Filtered image of the text.

 

Alpha Spectroscopy

The lab that I just finished was about alpha spectroscopy.  The purpose of this lab is to study alpha particles, which are helium-4 nuclei, with two protons and two neutrons.  The source is Americium-241.  Since alpha particles can only travel a few centimeters in air at atmospheric pressure, various degrees of vacuum will be used to allow the particles to travel from the source to the silicon surface-barrier detector.  First I explored the setup of the experiment, which included a vacuum system, a laptop, and various electrical connections.  I learned how to properly evacuate the chamber.  There are three handles, and the tubes are open when the handle is parallel to the tube.  The handle controlling the tube letting air in should always be closed when the pump is on.  After the chamber was at zero Torr, Channel A on the Ortec 428 was turned up to 125 V.  The analyzer program allowed me to pick the number of bins and the run time, which was how long the program collected data.  After a quick-and-dirty run with the minimum number of bins, 256, and only 30 seconds of data collection, the first actual experiment involved a five-minute run with the maximum amount of bins, 8192.  The important data collected was the channel number with the highest peak, number of counts at the peak, the full width at half maximum, which involved finding the channel with half the counts at the peak and then multiply the difference by two.  There are three main peaks in the data, but they were clustered very closely together, so it is not very easy to distinguish among the three peaks.

The second experiment involved changing the detector bias supply voltage.  Data was taken at three different voltages: 25V, 60V, and 95V.  In the end, after comparing the data, I found that running the detector at 125V made the graph a lot cleaner.

The third experiment involved letting the program run for ten hours (36,000 seconds), and then fitting a sum of three Gaussian equations to the graph using the program Kaleidagraph.  The process took a substantial amount of time, because Kaleidagraph is very picky about the starting values of the variables.  The final fit is displayed in the figure below.

Number of Counts per Channel vs Channel Number

 

The fit yielded the channels for which the three peaks occurred, and using this along with the data provided about the energy of these three peaks, I then graphed a channel vs energy (MeV) graph, displayed below.  Using the fit derived from this graph, I will now be able to convert channel numbers into energy values.

 

Channel Number vs Energy with error bars

 

The final experiment of the lab had me changing the pressure in the chamber and measuring peak energy, total counts, and half-width.  As expected, the higher the pressure, the lower the peak energy and total counts.  After 600 Torr, there was no peak anymore; instead, it was a few channels with one count per channel.