Series Capacitors & RC Circuits

Transcription

1 Lab #19 Series Capacitors page 1 Series Capacitors & RC Circuits Reading: Giambatista, Richardson, and Richardson Chapter 17 ( ), Chapter 18 ( ). Summary: Resistors and capacitors are two of the most important components from which electronic circuits are built. In a camera flash, capacitors are used to store charge. When the bulb goes off, a switch is closed and the capacitor discharges through a resistor. In this lab, you will measure the rate at which charge leaves a capacitor, to accurately determine the capacitance. Then using this information, you will construct a series circuit part of the circuit used in the camera flash and determine its equivalent capacitance (similar to equivalent resistance in the last two labs). Note: I have included three appendices to this lab. When I taught the lecture portion of Physics 2, I found most students had trouble understanding where the charges were going in capacitor circuits. These appendices have explanations that I developed over the course of several years and that many students said they found helpful. If you already understand how capacitors work, you may answer the pre-lab questions without reading the appendices. Pre-Lab Analysis As in the last two labs, the values marked on the circuit components (mostly capacitors in this lab) are only accurate to within 10% to 20%. Thus you will need to measure the capacitances for greater accuracy. Unfortunately capacitance cannot be measured as simply as resistance (with a DVM). Instead you will need to build a circuit, measure the voltages in the circuit at several different times, and from these you will calculate each capacitance. 1. This question will help you determine the formulas you will need to analyze the circuit for the capacitance determination. a.) Write the relation between charge and capacitance, defining all terms in your equation. (This equation is in your textbook.) [5 pts] 1 b.) What is the voltage (VR) drop across a resistor R if a current I flows through it? (This is a review from last week s lab.) [2 pts] 2 c.) How is VR related to the voltage across the capacitor (Vc) in Figure 1? [3 pts] 3 d.) From these two equations, show that I = Q /RC. [3 pts] 4 I R C C C Figure 1: Question 1c, d. As described in Appendix B, both the charge and current in this last equation depend on time. This equation can be rewritten as a differential equation (as discussed in Appendix B): The solution to this equation is: dq dt Ê 1 ˆ Ë Á R C Q = 0 quation 1 Q = CV 0 e - t/ RC quation 2

2 Lab #19 Series Capacitors page 2 where V0 is the initial voltage across the capacitor before it begins to discharge. 2. This last equation can be rewritten in terms of the voltage across the capacitor, VC,discharging (you will measure this voltage in the lab). a.) Do this and show that: [2 pts] 5 V C,discharging = V 0 e - t/ RC quation 3 b.) Sketch this equation, with VC on the y-axis and t on the x-axis. [6 pts] 6 c.) Someplace on the x-axis in your sketch, mark and label the time equal to RC (called the RC time constant). On the y-axis, label the voltage value corresponding to this time. Your answer should be some number (that you determine) times V0. [4 pts] 7 3. Thus to determine C, you just have to measure VC,discharging at several different times and plot the data as in Question 2. Table 1 contains discharging data, with a 1kΩ resistor connected to the capacitor, as shown in Figure 1. a.) Plot this data in xcel, as in Question 2. [7 pts] 8 b.) Fit an exponential Trendline curve to your data. Print your data, with the fit and its equation. [4 pts] 9 c.) From the curve fit equation, find the value of the capacitance? xplain your reasoning. Don't forget units on the capacitance. [6 pts] 10 Capacitance can also be measured by charging up the Table 1: Discharging data. capacitor. Then a power supply would need to be included in the circuit of Figure 1. To do the measurement this way, the capacitor starts with no charge or voltage on it. The power supply charges up the capacitor. The rate at which this happens is the same as the rate at which the capacitor discharges, if the circuit resistance doesn t change. As with the discharging case, by measuring how quickly the capacitor charges up, the capacitance can be found. However, experimentally the charging method frequently gives different results than the discharging method. This is due to internal resistance in the power supply, which changes the overall circuit resistance and hence R in quation 3 should change. However determining the power supply resistance accurately is not always easy. Thus the simplest way to experimentally find C is to measure its discharging rate. The second half of this week s lab consists of determining the equivalent capacitance of two capacitors in series. This is one of two basic 1 2 ways capacitors can be used in circuits. st capacitor nd capacitor 4. Figure 2 shows a power supply attached to two capacitors connected in series. VC (V) Time (sec) a.) Write an expression for the voltage drop (V1) across C1 in terms of C1 and the charge Q the on the left plate of C1. [2 pts] 11 b.) How much charge (in terms of Q) is induced on the left plate of C2? xplain where this R V power supply Figure 2: Charging two series capacitors.

3 Lab #19 Series Capacitors page 3 charge comes from. Write an expression for the voltage drop (V2) across C2 in terms of C2 and the charge on C2. [7 pts] 12 c.) What is the total potential difference (Vtotal) between points 1 and 2 in terms of V1 and V2? Write an expression relating Vtotal to Vpower supply and the voltage drop (VR) across the resistor R. [6 pts] 13 d.) If you were to replace the two capacitors with one equivalent capacitor (Cequivalent), write an expression relating the voltage drop across this equivalent capacitance, the power supply voltage, and the voltage drop across R. Then express the voltage drop across Cequivalent in terms of Cequivalent and the charge Q on this capacitor s plates. xplain why the charge on the equivalent capacitor plates has to be ±Q. (Hint: think of how equivalent capacitance is defined see Appendix C.) [8 pts] 14 e.) Lastly show 1/Cequivalent = 1/C1 1/C2, by combining your answers to the previous parts of this question. [6 pts] Outline the lab following the format of Outline Format posted on the lectronic Reserves web page. [20 pts] 16 quipment to be used in this lab: m 1 digital multimeter m 1 computerized voltage sensor m 1 power supply m 2 small breadboards m 3 capacitors: all capacitors should have values between 100µF and 2.2 mf m 5 resistors: all resistors should have values between 470Ω and 10kΩ m Use ONLY the resistors and capacitors in the box at your lab bench. If you mix them up with another lab group s, you will not be able to do the lab in the 2 hour limit. 1. Accurate Resistance Determination q Connect two leads to the yellow digital multimeter: one to the receptacle labeled COMM and the other to the receptacle with an Ω symbol over it. q Turn the big switch in the center of the meter to either the 2kΩ or 200kΩ setting (depending on the value of the resistance you are measuring). Measure and record all the resistance values and color codes in an appropriately labeled data table in your notebook. [13 pts] 17 If you end up with a 0.L reading, the resistance you are trying to measure is too big for the selected scale. Turn the switch to the next higher scale. 2. Determining Capacitance accurately (by discharging) q As mentioned at the beginning of the Pre-lab, the capacitance values marked on the capacitors are not particularly accurate. In this section you will accurately determine the capacitance of each capacitor on your lab bench by measuring the RC time constant for discharging each capacitor.

4 Lab #19 Series Capacitors page 4 q You will connect each capacitor to one resistor and the power supply as shown in Figure 3 and Figure 4. For each capacitance value, calculate the resistance that will produce an RC time constant of 1 second. Show all calculations (appropriately identified and with correct units) in your notebook. [7 pts] 18 q Next identify the values of the Power Supply Switch resistors at your lab bench that most closely match those you just calculated and record these resistance values in your notebook with the corresponding nominal capacitance with which the resistor is to be paired. [9 pts] 19 q With the power supply off, connect in series the power supply, one capacitor, and the corresponding resistor as shown in Figure 3 and Figure 4. q Connect the red and black clips of the computerized voltage sensor to the universal interface (ULI) box. q On the computer, click on the Capacitors icon. The display should show Potential versus time. q Start with the switch in the open position (switch handle up). q Turn the power supply on and increase the voltage to 5 V. q NVR ever turn the power supply to MOR than 6V. You will blow the switch on the ULI box if you do. q Click on Collect ; close the switch. q When the voltage signal on the computer has leveled out (capacitor is fully charged), open the switch. q Take a wire and locate where you would connect the resistor across the capacitor so that the two are in parallel with each other as in Figure 5. q Click on Collect and immediately connect the wire to Resistor Capacitor Figure 3: Capacitance measurement schematic. Power Supply Computerized Voltage Sensor Red clip Computerized Voltage Sensor Black clip Figure 4: Circuit diagram showing the capacitor in Figure 3 charging up. form the parallel circuit. The computer voltage signal should decrease and level out at zero as the capacitor fully discharges. q Make sure you explain (with sketches) how you charged the capacitor and then measured its discharging rate. [6 pts] 20 q On the computer screen (in LoggerPro) highlight the decaying part of the voltage and fit the data with the appropriate equation from the Pre-lab (see below for instructions). Print your appropriately labeled plot with the curve fit. [7 pts] 21 Curve fitting in LoggerPro: Click on the f(x) icon. R R C C V V Figure 5: Measuring a capacitor by discharging it through a resistor.

5 Lab #19 Series Capacitors page 5 In the window that appears, go to the lower left corner where the types of curves are listed. Click the down arrow, select Natural xponent, and click Try Fit. q What is the RC time constant for this discharging circuit? xplain all work. [7 pts] 22 q From this time constant determine the capacitance of this capacitor and compare it to the value stamped on the capacitor body. Record both the measured and stamped values in your lab notebook along with a column comparing these values. [8 pts] 23 q Follow this same procedure to accurately find the capacitance of the remaining 2 capacitors. Use the resistor/capacitor pairs you determined previously that gave predicted time constants of approximately 1 second. Label each new capacitance measurement with the actual resistance and nominal capacitance values used in the measurement. q Don t forget to include circuit diagrams recording how you did each step of the measurement. In these diagrams use the actual resistance value determined from the DVM measurement and the nominal capacitance value written on the capacitor. Also make sure you record the exact charging voltage of the power supply. [12 pts] 24 q Print each plot of your data with the appropriate curve fit and curve fit equation. Calculate the RC time constants for each measurement (show all work in your notebook). [22 pts] 25 q Record all the calculated capacitances in your notebook and show all your calculations. Don t forget to fill in the comparison column with the printed capacitance values. [14 pts] Series Capacitance q In this last part of the lab, you will be connecting your three capacitors in series with each other. Before doing so, calculate the equivalent capacitance of the three capacitors in series, using the result from Pre-Lab Question 4e and each measured capacitance value (from the first two sections of this lab). Show all work in your notebook. [11 pts] 27 q What size resistor would you need for the circuit in Figure 6 to have a time constant of about 1 second, if the three capacitors in the circuit have the same values as your measured capacitances? Show all work. [12 pts] 28 q How should you combine the resistors on your lab bench to obtain this value (to at least within 10% of the calculated resistance value)? Hint: you may need to use a combination of series and parallel resistors. Show all your calculations and draw the final resistance configuration you will use in place of R in Figure 6. [18 pts] 29 q Build the circuit in Figure 6 with all three of your capacitors R C 1 in series and with your just-determined equivalent resistor combination in place of R. Sketch your final circuit diagram in your lab notebook, labeling the actual value of each circuit component. [9 pts] 30 q Using the same charging/discharging method you used in part 2 above, measure the RC time constant for this circuit. Sketch each charging and discharging circuit in your C3 C 2 Figure 6: Resistor in series with series capacitors.

6 Lab #19 Series Capacitors page 6 lab notebook. Print the discharging graph with the curve fit you will use to determine the RC time constant. [12 pts] 31 q From this RC time constant, determine the equivalent circuit capacitance.[22 pts] 32 q How does your experimentally determined equivalent capacitance compare with the theoretical value determined using the Prelab formulas? If your values are off by more than 10% check your circuit and redo the measurements until you are within 10% of your theoretical calculations (check your theoretical calculations also). [5 pts] 33 q When you are finished with your experiments and calculations, lay out neatly on your lab bench all the capacitors, resistors, cables, switch, and circuit board. q Call your TA over to check off the components against a list of what you were originally given. If anything is missing you are to replace the item before leaving the lab (your TA will show you where to find the replacements). q Once your TA has checked everything, place all the items neatly back in the box on your lab bench. [5 pts] 34 Appendix A Capacitors consist of two parallel metal plates separated by an insulator. With no battery connected to the capacitor, the capacitor plates have no excess charge on them, i.e. they are neutral. (Be careful about this: neutral does not mean there are no charges on the plates. It means there are equal numbers of positive and negative charges on the plates.) Now if a battery is connected to the capacitor, excess charges can accumulate on the capacitor plates. As with resistors, we assume the charges move in the circuit. Thus charges will accumulate on the capacitor plate connected to the side of the battery (see Figure 7, 1 st step). Then Coulomb s Law makes the charges in the other capacitor plate redistribute themselves so the charges are closest to the charges on the first plate (see Figure 7, 2 nd Step). Without the insulator between the metal capacitor plates, the charges of opposite sign, because they are attracted to each other, would hop across the space between the plates and Dielectric prevents charge from moving between plates 1 st Step 2 nd Step neutralize each other. However, the insulator prevents this from happening and the charges are Figure 7: Initial stages of charging a capacitor. forced to stay on their respective plates. An electric field then develops between the separated and charges. The magnitude of this electric field depends on how much charge is on each plate and the amount of charge depends on the capacitor dimensions and materials (similar to a resistor). If a wire is now attached to the rightmost capacitor plate, the wire provides a means of escape for the charges and they will flow along the wire trying to get as far away from the charges on the left capacitor plate. After a very short while only one charge for each charge will be left behind on the right plate. Thus if Q accumulates on the left plate, then Q will be attracted to it on the right plate, see Figure 8. Magnified view of right capacitor plate

7 Lab #19 Series Capacitors page 7 The maximum amount of charge (±)Q that can accumulate on the capacitor plates (called the stored charge) depends solely on its capacitance and the voltage applied to the capacitor. Analogously, in Lab 2, the maximum amount of current through a resistor depended on its resistance and the voltage applied to it. Thus, one can think of the capacitor charge like the current through a resistor and the capacitance like the resistance. Appendix B Discharging a Capacitor With the capacitor fully charged, if one were to connect the two sides of the capacitor with metal wire, the charges would instantaneously flow to the other plate and neutralize the charges. One would learn nothing about the magnitude of the capacitor from such an experiment. However, if a resistor were connected to the capacitor as in Figure 9, the charges would still be pulled towards the charges, but the rate at which they would reach the plate is determined by R and the capacitance of the capacitor. Once the charges reach the charges, the and charges will neutralize each other and the capacitor is said to discharge. Thus as the amount of charge on the left capacitor plate decreases, there will be fewer charges to travel through the circuit. The current in the circuit therefore decreases with time, eventually falling to zero. By measuring how fast the current decreases and knowing R, the capacitance can be determined (as you will do in the first part of the lab). The quantitative relation between the capacitor s discharge rate and R and C is found by examining the voltages in the circuit. The sum of the voltage drop across the resistor and that across the capacitor must total zero. Thus since I is related to Q by I = q / t (or dq /dt for those of you in Phys. 150), this voltage equation can be written as: R DQ Dt Ê 1 ˆ Á Q = 0 or Ë C Dielectric Q dq dt Ê 1 ˆ Ë Á R C Q = 0 quation 4 For those of you in Physics 150, you should have or will shortly see the solution to this differential equation in lecture. For those of you in Physics 108, you only need know the solution to this differential equation, which is: Q = CV 0 e - t RC quation 5 where CV0 = Q0, the maximum charge on the capacitor just before it begins to discharge. Rewriting this equation in terms of the voltage across the capacitor (which you will measure in the lab) gives: V C,discharging = V 0 e - t RC quation 6 Thus by measuring Vc at several different times and knowing R, you can find C even though you don t know V0 (see Pre-lab Question Q Figure 8: If Q accumulates on the left plate, an equal but opposite charge (Q) will be attracted to it on the right plate. The dielectric separates the and charges. Resistor Dielectric Figure 9: Connecting a resistor to capacitor to discharge the capacitor.

8 Lab #19 Series Capacitors page 8 Appendix C Series Capacitors In the second part of this lab, you will 1 st capacitor, C 1 2 nd capacitor, C 2 determine the equivalent capacitance of two capacitors in series, as shown in Figure 10. Suppose Q Q Q Q the left capacitor (C1) has been charged by a power supply, which deposited Q on the left plate of C This positive charge repels an equal amount of Q from the right plate of C1, leaving behind (or inducing) Q on the right plate of C1, as shown. The repelled Q flows through the wire until it hits the first barrier, namely the left plate of C2, and the Q piles up on this plate. Thus regardless of the R Vpower supply magnitude of C2, Q will always be induced in it by the first capacitor if the first capacitor is charged to Figure 10: Charging two series capacitors. ±Q. Thus the second capacitor (C2) charges up (as shown in Figure 10), with its Q coming from the rightmost plate of the first capacitor and not from the power supply, as it does for the first capacitor. The first capacitor then is said to induce the charge on the second capacitor. This means that the charge on the plates of the equivalent capacitor will also be Q and Q and the total potential difference from one side of the capacitor to the other is the same as between points 1 and 2 in Figure 10.