sin( D t), where D is called the driving frequency of the amplitude 2

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1 Physics 241 Lab: R ircuits A Source Name: Section 1: 1.1. Last week you studied R circuits, examining the exponential time dependence of the capacitor voltage as you charged and discharged the capacitor with a constant voltage. To do this you used a square wave with a D offset. Today you will examine the behavior of a capacitor when a sinusoidal voltage is applied: V (T) V circuit (note that it is an angular frequency). t), where D is called the driving frequency of the The capacitor voltage will no longer exhibit exponential time behavior. Instead the capacitor voltage will oscillate sinusoidally with the same frequency as the driving frequency. This can be proven by writing the differential equation for the circuit, finding its solution, and checking the solution. However, this requires knowledge of solving inhomogenous differential equations. Instead, the most useful results of that calculation are provided: the time dependent voltages across each component. Thus, you are not required to be able to derive the solutions to the A-driven R circuit, but you must understand and be able to use these results. Each component of the sinusoidally driven R circuit has a sinusoidally varying voltage across it, but each peaks at a different time determined by a phase shift. The solutions for the time dependent voltages of each component are given by the equations: V (t) V t ) V R (t) V Z t) V (t) V Z sin D t 2 There are several new parameters to discuss. First notice that the voltage is now written with a phase shift, the capacitor voltage has a phase shift of /2, and the resistor voltage has no phase shift. What this means in practice is that we will use the resistor voltage as a reference for all other components in the circuit: i.e. we will measure the phases of each component in relation to what is happening inside the resistor. This is because the resistor is Ohmic and can always provide the time dependent current via Ohm s law, which can often be useful to know.

2 1.2. The Source Voltage Equation: V (t) V t ) The voltage equation is straightforward. It oscillates sinusoidally, i.e. it is a sine function of time. The maximum voltage applied across the whole circuit is V. The oscillates with an angular driving frequency D 2 f D (which you will set later with your function generator). The voltage is phase shifted from the resistor voltage by an amount arctan where X is the reactive R capacitance of the capacitor given by the equation 1 (more on this later). Note that this x-like D variables are really the capital Greek letter hi (pronounced kai). If you look at the equation for resistor voltage, you will see no phase shift. Again, what this means is that we measure all phases in relation to the resistor not the. The resistor will have its maximum voltage at a different time than when the voltage is maximum. Imagine a sinusoidally driven R circuit. If the voltage has an of V =1.8 volts, a linear driving frequency f D =555 Hz, a resistance R=150, and a capacitance =1.5x10-5 F, find the phase shift of the voltage compared to the resistor. Your work and answer: 1.3. The Resistor Voltage Equation: V R (t) V Z t) The resistor voltage oscillates sinusoidally without a phase shift while R is simply the resistance. Z is the impedance of the whole circuit. Z acts like the total resistance of the circuit. Z is measured in SI units of Ohms and is given by the equation Z 2 L 2. This definition has new stuff, too. X L and X are like the resistances of the inductor and capacitor, respectively. We won t study inductors until later in the semester, but it is easier to memorize the complete equation. Since we don t have an inductor (coil) in the circuit, you can set this to zero. So we have Z 2 2. is called the reactive capacitance and is measured in Ohms. Now examine the resistor equation as V R (t) V resistor t). The maximum and minimum voltage would oscillate across the resistor is V resistor Z V. Imagine a sinusoidally driven R circuit. If the capacitance is increased, explain what happens to the of the resistor voltage? Your explanation: If the frequency is increased what happens to the of the resistor voltage? Your answer:

3 Explain what happens to the current through the circuit if the resistor voltage decreases? Your explanation: Explain what happens to the power lost through heating the resistor if the resistor voltage decreases? (Remember that P R =I R V R.) Your explanation: Imagine a sinusoidally driven R circuit with voltage V S, resistance R, and capacitance. Explain whether the resistor will become hotter if you increase the driving frequency? Use the concept that Z = total impedance of the circuit. Your answer and explanation: 1.4. The apacitor Voltage Equation: V (t) V Z sin D t 2 The capacitor voltage oscillates sinusoidally and lags behind the resistor voltage by 90 o. The reactive capacitance is like the resistance of the capacitor and is measured in SI units of Ohms. The resistance of the capacitor is related to the capacitance of the capacitor and the driving frequency. This relationship 1 can be derived from the differential equation modeling the D circuit, but you must memorize it. The larger the capacitance, the less resistance in the capacitor. But just as importantly if the driving frequency is increased, the resistance of the capacitor decreases. This is why a capacitor is often used as a high pass filter in electronics: the capacitor has less resistance to more quickly oscillating currents. BE SURE TO REMEMBER THIS DURING TODAY S LAB! If we rewrite the capacitor equation as V (t) V capacitor sin D t, the capacitor voltage 2 is given by V capacitor Z V. That means that the ratio of the capacitive reactance and the total circuit impedance times the gives the of the voltage across the capacitor. In a previous equation, you found that the resistor voltage increases when the frequency is increased. Since the voltage across the resistor and capacitor must add to the voltage across the, if the resistor voltage increases, then the capacitor voltage must decrease. Therefore, as you increase the driving frequency, the resistor voltage increases while the capacitor voltage decreases. (Not a question)

4 Section 2: 2.1. Work though an example before beginning. Remember the equations below as you work. V (t) V t ) V R (t) V Z t) V (t) V Z sin D t 2 If your circuit has V 2 Volts, R 10,000, 1x10-7 Farads, and D 1,500 radians/sec find the following values with correct units. Your answers: X = Z = = V R,. = V,. = Now examine V R, + V, = Your answer to this previous question adds to more than V!!! No, you didn t make a mistake. Since the voltages are out of phase, their maximums do not add together at the same time. Now let s try and visualize this circuit s behavior: Write the functions for V S (t), V R (t) and V (t) using the numerical solutions to the previous questions. Quickly sketch V R (t) and V (t) on the oscilloscope screen below using a graphing calculator. Don t worry about providing the scale of the time axis. Then sketch V R (t) + V (t) onto the screen using a dotted line. This should equal the function V S (t) so check it using your graphing calculator.

5 Section 3: 3.1. Now you will set up the sinusoidally driven R circuit with R 10,000, and 1x10-7 Farads. Set your function generator to create a sin wave with a voltage of a nice round number like 3 Volts. You may want to adjust your frequency later, but start at about 400 Hz. Set up a middle ground to view the voltage across both the resistor and the capacitor simultaneously making sure to invert the correct channel (a necessary step when using a middle ground). Make a sketch on the oscilloscope screen below. Label the signals V R (t) and V (t) on your sketch Explain which signal is phase shifted to lag by 90 o. Your explanation: Find the s of each signal by measuring the peak-to-peak voltage of each signal. Your observation: Use the labeled values to determine the impedance of your circuit for this driving frequency. Remember Z Your work and answer:

6 Use your previous answer to determine what the signal s should and then compare these predicted (calculated) s to your measured s in the other previous question (they should be close). Your work and answers: Find the frequencies f of each signal using oscilloscope measurements. Your observation: Use your answers to the previous questions to write equations for V R (t), V (t) and V S (t) entirely with numerical values (no free parameters). (Don t forget the phase shift.) Your solutions: 3.2. Set your oscilloscope to plot V R (t) on the x-axis and V (t) on the y-axis (an XY plot). Sketch the result on the oscilloscope screen below. Your sketch:

7 In an XY plot, if the signal on the y-axis oscillates twice as fast as the signal on the x-axis and the signals are 90 o out of phase, then sketch what will appear on the oscilloscope screen below. Your sketch: Section 4: 4.1. Next you will test the relationship 1 by observing a sinusoidally driven R circuit using D many different driving frequencies. Use the same circuit set up as in the previous part of the lab. As you increase the driving frequency, the of the resistor voltage will increase because the total circuit impedance is decreasing, i.e. V resistor Z V (work through this logic!). Meanwhile, as the driving frequency increases, the capacitor decreases. This makes sense because the resistor and the capacitor are the only two components in the circuit other than the. Since the voltages across both must add up to the voltage, if the voltage of one increases, then the other must decrease. Therefore, there must be some specific driving frequency when the of the resistor voltage is the same as the capacitor voltage: V resistor V capacitor for a specific D. Substitute V resistor Z V and V capacitor Z V and you get Z V Z V for a specific D. The first method for finding the capacitance of an unknown capacitor makes use of the previous equation. Adjust the driving frequency of your circuit until the capacito voltage and the resistor voltage are equal. Then use 1 D and Z V Z V for the specific D to find the capacitance. Obtain an accurate measurement for R using a DMM. Your observations, work and answer for determined experimentally:

8 4.2. The second method for finding an unknown capacitance is more involved. The voltage s of the sinusoidally driven R are: V resistor Z V and V capacitor Z V. V V capacitor Z Dividing these two equations gives V capacitor V resistor R V R. Therefore,. V resistor Z 1 In order to test the relationship and experimentally determine for your solenoid, simply drive combine the last two equations and rearrange: V 1 resistor drive. R V capacitor Therefore if you graph 1 R V resistor V capacitor vs. drive, you should obtain a linear graph with a slope equal to. Find by collecting data for multiple driving frequencies, making a graph and finding the slope. Make your observations and graph now. Then write your work and result for : Section 5: Test Yourselves - Each lab partner should take a turn doing this section. Quickly set up a working circuit that simultaneously uses a random capacitor and a 1000 resistor in series powered by a sinusoidal voltage on your function generator. Then make the necessary measurements to determine the capacitance of the capacitor. Be sure your experimentally determined measurements give the correct capacitance. Your lab partners can give you verbal feedback, but only you are allowed to touch the equipment. Record your results below:

9 Section 6: (Open-ended question / creative lab design) Make a capacitor from the square cardboard pieces covered in conductive aluminum foil. Sandwich a non-foil square of cardboard between the foiled boards, and be sure your makeshift capacitor is not shorted out by accident. Measure the capacitance of your homemade capacitor. The equation for the capacitance of two parallel plates is given by A o. Use this equation to report the dielectric d constant of the sandwiched cardboard between the plates with correct units. Note: 2 o 8.85x10 12 N m. 2 At the following prompts, design an experiment to determine the capacitance of your cardboard capacitor and the dielectric constant of the cardboard. Then implement your experiment and record your observations. You may cheat by talking to other groups for ideas, but not cheat by already knowing the answer or looking it up. Your planned experiment, sketch of actual implementation and any theoretical calculations: Your observations: Your explanations & conclusion:

10 Report Guidelines: Write a separate section using the labels and instructions provided below. You may add diagrams and equations by hand to your final printout. However, images, text or equations plagiarized from the internet are not allowed! Title A catchy title worth zero points so make it fun. Goals Write a 3-4 sentence paragraph stating the experimental goals of the lab (the big picture). Do NOT state the learning goals (keep it scientific). [~1-point] oncepts & Equations [~5-points] Be sure to write a separate paragraph to explain each of the following concepts. o ompare and contrast how to find the capacitance of a capacitor using a D (square wave) versus a sinusoidal. o Discuss at length the three time dependent voltage equations that describe the A-driven R circuit. Be sure to explain: o impedance o reactive capacitance o phase shifts o Discuss how to find the of the current through the resistor and what combination of parameters gives this value. Procedure & Results Write a 2-4 sentence paragraph for each section of the lab describing what you did and what you found. Save any interpretation of your results for the conclusion. [~4- points] onclusion Write at least three paragraphs where you analyze and interpret the results you observed or measured based upon your previous discussion of concepts and equations. It is all right to sound repetitive since it is important to get your scientific points across to your reader. Write a separate paragraph analyzing and interpreting your results from your open-ended experiment. Do NOT write personal statements or feeling about the learning process (keep it scientific). [~5-points] Graphs All graphs must be neatly hand-drawn during class, fill an entire sheet of graph paper, include a title, labeled axes, units on the axes, and the calculated line of best fit if applicable. [~5- points] o The graph from section 4.2. Worksheet thoroughly completed in class and signed by your TA. [~5-points.]