Physics 4BL Experiment 2

Laboratory 2

Uncertainty and statistics

Max Woo
10/16/14
Lab 11, Michael Ip
Partners: Ivy Wang, Zoe Zhu

Introduction

The overall purpose of our experiment was to observe the responses of different combinations of circuit components to constant and varying voltage sources. For the first part, we will investigate the current-voltage relationships of a resistor and a diode, making use of Ohm’s law:

Ohm’s Law: $V = IR$ (Eq. 1)

For the second part, we will investigate an RC circuit, and LC circuit, and an RLC circuit, verifying the formulas for calculating time constant and resonating frequency.

To aid us with calculating the uncertainty of certain calculations, we derive the propagation of error for a few fundamental relationships. Starting with the general equation for propagation of uncertainty:

$\sigma_{\overline{F}} = \sqrt{(\frac{\delta f}{\delta x} \sigma_{\overline{x}})^2 + (\frac{\delta f}{\delta y} \sigma_{\overline{y}})^2}$

We can derive the expression for propagation of error for:

F = kx : $\sigma_{\overline{F}} = k \sigma_{\overline{x}}$ (Eq. i)
F = x + y: $\sigma_{\overline{F}} = \sqrt{\sigma_{\overline{x}}^2 + \sigma_{\overline{y}}^2}$ (Eq. ii)
F = x * y: $\sigma_{\overline{F}} = \overline{F} \sqrt{\sigma_{\overline{x}}^2 + \sigma_{\overline{y}}^2}$ (Eq. iii)
F = 1/x: $\sigma_{\overline{F}} = \overline{F} \sigma_{\overline{x}}$ (Eq. iv)

Experimental Results

Part 1

We started by setting the function generator to a triangular wave with a 10 Hz frequency and amplitude of a few volts. We then created a circuit with 2 resistors in series, a resistor $R$ with a resistance of $998 \pm 0.5 \Omega$ and a resistor $r$ with a resistance of $98.5 \pm 0.05 \Omega$. We then connected channel 0 of the myDAQ to measure the voltage across $R$ and connected channel 1 to measure the voltage across $r$. We then used a sampling rate of 5000 samples/sec and a sample size of 1000 points. The full setup is shown through the circuit diagram below:

Fig. 1: The setup used the measure the voltage across 2 resistors in series

A graph of the resulting current-voltage relationship is shown below in Graph 1. The relationship is a line with positive slope, verifying Ohm’s law, shown in Eq. 1 to have be a linear relationship with positive slope.

Graph 1: The current-voltage relationship of resistor $R$

Next, we used the same setup as shown in Fig. 1, but replaced the resistor $R$ with a diode. Using the same settings on the myDAQ, we measured and recorded a data ensemble of the voltages across the diode and resistor over time. The graph of the current-voltage relationship is shown below in Graph 2:

Graph 2: The current-voltage relationship of the diode

This graph has an exponential curve, which is what we expect from the equation for the diode, which is also an exponential relationship:

$I = I_0 (e^{|e|V/(nk_B T)} - 1)$ (Eq. 2)

Part 2

The second part of our experiment involved measuring the response of certain electrical components to a time-varying voltage source. We first chose a time constant $]tau$ by choosing a resistor $R1$ and a capacitor $C$ so that $\tau = R_1C \approx 10^{-3}$. We then set the function generator to output a square wave with a frequency smaller than $1/(2\pi\tau)$, and an amplitude of about 2V. We also offset the voltage by the same amount as the amplitude, so the waveform never goes below 0 volts. We then used the resistor and capacitor chosen earlier to build the circuit shown in Fig. 2:

Fig. 2 The setup used to measure the response of an RC circuit to a varying voltage source

We adjusted the myDAQ sample rate until one to two periods of the square wave fit within the display, and saved one ensemble of data. A graph of that data is shown below in Graph 3:

Graph 3: The voltage across the capacitor of the RC circuit over time

The graph has a noticeable pattern of sharply approaching a high voltage value, then asymptotically getting closer to that value, then sharply going towards a low voltage value, and asymptotically getting closer, and so on. This function makes sense because the capacitor resists change to voltage, so when the voltage provided by the source jumps, the capacitor takes time to reach that voltage. In addition, as the charge on the plates build up, incoming charge is met with more and more repulsion from the charge already on the plates, slowing down the build-up of voltage the closer the capacitor voltage gets to the source voltage. In addition, the equation for the capacitor voltage is given by an inverted exponential equation:

$V_c (t) = V_b (1 - e^{-t/RC})$ (Eq. 3)

This inverted exponential relationship is repeated in the graph every time the voltage changes.

For the next part of the experiment, we chose an inductor $L$ such that we again set the time constant $\tau = L/R_1 \approx 10^{-3}$. We also find a resistor $R2$ with a resistance of approximately 5 $\Omega$, and use the components to build the circuit in Fig. 3:

Fig 3: Setup used to measure the response of an RL circuit to a varying voltage source

We again adjusted the myDAQ sample rate until one to two periods of the function fit within the display, and then saved an ensemble of data. A graph of those results is shown below:

Graph 4: A graph of the voltage across the inductor over time

In this graph, the function jumps to high and low voltage values, and then slowly approaches a voltage of 0. This makes sense because when the voltage changes, the inductor starts with a high impedance, much higher than the R1 so that the voltage drop is mainly across the inductor. However, as the current through the inductor increases, the impedance of the inductor gets lower, until the impedance approaches 0, and the voltage drop becomes mainly across the resistor. This is also shown by the equation for the voltage across the inductor:

$V_L (t) = V_b e^{-tR/L}$ (Eq. 4)

The equation for the inductor is a downward sloping exponential relationship, which is clearly shown in the graph each time the voltage jumps to a value and then slowly approaches zero.

For the last part of the experiment, we built an RLC circuit with $R = 7.5 \Omega$, $C =1.0 uF$, and $L = 20 mH$, using the setup shown in Fig. 4.

Fig. 4: RLC circuit diagram

We then connected the myDAQ so that channel 0 measured the voltage across the capacitor and channel 1 measured the voltage across the inductor. We also used Ch. A01 of the myDAQ as the positive terminal of the voltage source, and the ground of the myDAQ as the negative terminal of the voltage source. Then, using the BODE Analyzer software, we adjusted the start/stop frequency, the number of steps, and the peak amplitude until we got a decently smooth curve with a distinct peak. The graph is shown below:

Graph 4: A graph of the RLC voltage amplitude based on varying frequencies of a AC voltage source

The shape of the curve matches the equation for the response function of a RLC circuit:

$G(\omega) = \frac{V_R}{V_{in}} = \frac{IR}{IZ} = \frac{R}{R + i(\omega L - \frac{1}{\omega C})}$ (Eq. 5)

This equation reaches a maximum value as the impact of the imaginary part of the denominator decreases, when $\omega^2 = 1/LC$, showing that the resonant frequency of the circuit corresponds to the maximum value of the graph. The graph illustrates this pattern exactly, verifying the relationship.

Data Analysis

Part 1

For the first part, we wanted to find the relationship between the current and voltage of resistor $R$. We first calculated the current through the circuit using the voltage and resistance across resistor $r$, and the equation:

$I = V/R$ (Eq. 6)

We then performed a linear regression between the voltage values and current values, giving us a slope of $0.001013 \pm 0.0000004$ Amps/Volt. According to Ohm’s law,

$R = V/I = 1/slope$ (Eq. 7)

which, when combined with the formula for error of multiplicative inverse (Eq. iv), gives us $R = 986.85 \pm 0.0004 \Omega$. Our measured value of $998 \pm 0.5 \Omega$ is very close, but does not fall within the error bounds of our theoretical value. This could be due to the resistance of the wires between the myDAQ and the resistor, lowering the actual voltage across the resistor R and thus giving a lower calculated resistance.

For the part involving the diode, we used a similar method of linear regression, but first we needed to convert the exponential function into a linear function to perform the linear regression on. To do so, we first cropped the data, taking only the portion that seemed to fit an exponential curve the closest. Then, by taking the natural log of the equation for the current-voltage relationship of the diode (given in Eq. 2), we were able to extract a linear relationship:

$ln(I) = ln(I_0) + (\frac{e}{nk_B T})V$ (Eq. 8)

Thus, we first took the natural log of the current values of our cropped data. We then performed a linear regression to get a slope of $\frac{e}{nk_B T} = 20.176 \pm 0.0141$. Multiplying this value by $n =2$ and $T=293 K$ and using the formula for error of multiplying by a constant (Eq. i) gives us $\frac{e}{k_B} = 11823 \pm 8.3$, which is very close to the actual value of $\frac{e}{k_B} = 11694$. However, the actual value does not fall within the error bounds of our calculated value. This is probably because of the data cropping we did to better fit an exponential relation, a decision which could have shifted our result either direction.

Part 2

For the RC circuit, in order to calculate the time constant from our voltage-time relationship we first have to convert the exponential function into a linear function, just as we did with the diode’s current-voltage relationship. Using the same approach, we simply take the natural log of both sides of voltage-time equation, from Eq. 3, giving us

$ln(V_b - V) = ln V_b - \frac{1}{RC} t$ (Eq. 9)

This equation shows that taking the natural log of $V_b - V$, or the base voltage minus the voltage over time, gives a linear relationship. We crop the data so that the rising edge of the square-wave voltage source marks the start of the data, and the next falling edge marks the end of the data. We then use the maximum voltage of the square-wave voltage source as the base voltage, and create a column of $V_b - V$ values. Taking the natural log of those values gives the linear relationship that we want, and a linear regression gives us a slope of $- \frac{1}{RC} = -1181.42 \pm 24.32$.

Calculating the time constant is then trivial:

$\tau = RC = - 1/slope$

which, with the formula for error of multiplicative inverse (Eq. iv), we calculate to be $\tau = 0.000846 \pm 0.0206 s$, of which our measured time constant of $\tau = RC = 0.000982 \pm 0.0491$ (with uncertainty calculated using error of products, Eq. iii) falls within the error bounds, confirming the equation for time constant $\tau = RC$.

For the RLC circuit, we start by comparing the measured resonance frequency $f_{res}$ to the calculated $f_{res}$, for which the equation is:

$f_{res} = \frac{1}{2\pi \sqrt{LC}}$ (Eq. 10)

We get the measured value from the frequency at which the amplitude of the voltage reaches its peak, with uncertainty measured from the distance to adjacent frequency data points. Thus, the measured resonance frequency is $1258.925 \pm 258.925 Hz$. On the other hand, for the calculated resonance frequency, we first need to derive the expression for the uncertainty. Using the propagation of errors equation, we get:

$\sigma_{f_{res}} = \frac{f_{res}}{4\pi} \sqrt{(\frac{\sigma_L}{L})^2 + (\frac{\sigma_C}{C})^2}$ (Eq. 11)

Using both Eq. 10 and 11 gives us $f_{res} = 1125.40 \pm 44.83 Hz$, which falls within the error bounds of the measured resonance frequency.

To find the circuits Q-factor, we first calculated $V_{max}/\sqrt{2}$ and found which two data points (one on each side of the curve) were the closest to this value, again using the distance to adjacent data points as the uncertainty. The Q-factor is then calculated using

$Q = f_{res}/(f_2 - f_1)$ (Eq. 12)

With the uncertainty calculated through chaining together Eq. ii, iii, and iv, giving us a result of $Q = 0.9228 \pm 0.3665$.

Conclusion

For part 1, we were able to verify Ohm’s law using a linear regression of the voltage-current relationship of a resistor. In addition, we were able to use linear regression to find the voltage multiplier for the diode, and the calculated and actual values for $e/k_B$ were very close. However, they were not within the error bounds, probably due to the data cropping that was necessary to make the voltage-current curve fit the exponential curve better.

For part 2, we were able to use linear regression to verify the voltage-time relationship of a capacitor, our theoretical and measured time constants falling within the error bounds. We were also able to verify the formula for the resonance frequency of an RLC circuit, and were able to calculate the Q-Factor of our circuit.