Resistor-Capacitor (RC) Circuits Introduction In this second exercise dealing with electrical circuitry, you will work mainly with capacitors, which are devices that are used to store charge for later use. Some examples of capacitors in everyday life: In computer-based circuits, large capacitors continue to provide power to the memory circuits even when the power is off. Here, capacitors function like batteries. Capacitors (along with rectifiers) are used to convert alternating current (AC) into smooth direct current (DC) for use in typical household appliances. Here, the capacitors supply energy to the circuitry in between "refills" from the rectifier. Combined with inductors, capacitors form frequency-selective circuits. These are used extensively in radio and TV to allow the device to select the desired station or channel. Groups of large, specially constructed, low-inductance high-voltage capacitors (capacitor banks) are used to supply huge pulses of current for many pulsed power applications, such as pulsed lasers and particle accelerators. In combination, capacitors and resistors are used to make time-delay circuits that control devices like the blinkers on your car. Equipment DC power supply (do not exceed 7 volts: can damage voltage probes) Lab Pro + 2 Differential Voltage Probes Two 1000-µF capacitors (black cylindrical) One 80-µF capacitor (oblong silver) Two resistors (unknown value) Fluke Digital Multimeter Various wire leads Background Capacitors are charge-storage devices consisting of two electrodes (metallic plates or sheets, usually aluminum) separated by an insulating material called a dielectric such as vacuum, air, oil, etc. The device is packaged with a separate wire connected to each electrode (see Figure 1). Work is required to put the charge onto the plates. The energy invested in charging a capacitor is stored in the electric field created between the plates. The capacitance (C) of a capacitor is an inherent property of the device that depends on its geometry and the type of dielectric used. It can be related to the voltage across its plates, V, and the charge on the plates, Q: V = Q C " C = Q V The units of capacitance are Farads (F): 1 Farad = 1 Coulomb/1 Volt. Capacitors typically store charge on the order of micro- (µf) to pico-farads (pf). The potential energy stored in a capacitor can be expressed as U = 1 2 CV 2 1
Figure 1: Components of a Parallel-Plate Capacitor Procedure Charging a Capacitor Verify that the power supply is indeed turned off. You will need to assemble your circuits first before energizing the system. Begin building your first circuit by connecting one of the 1000-µF capacitors directly to the power supply using a set of banana plug wires (see Figure 2), connecting with proper polarity (red-to-red and blackto-black). Here, red & black indicate the positive & negative sides of the power supply, respectively. Attach the leads of the Differential Voltage probe to the back inserts of the banana plug wires, again, keeping polarity in mind. The probe will record the voltage across the capacitor (V C ) as it is charged by the power supply. Open Logger Pro and verify that the Differential Voltage Probe is the only sensor plugged into the LabPro interface. Set the data collection time to 15 seconds, with a sampling rate of 10 samples/second. Set the dial on the power supply setting to 20, which corresponds to approximately 5 volts. Begin your data collection by clicking on the Collect button, and then turn on the power supply. After 5 seconds, turn off the power supply, but continue collecting voltage data to observe how the capacitor retains voltage. Figure 2: Circuit Diagram for Charging a Capacitor Print a copy of this graph for your notebook. What type of behavior does this capacitor exhibit as its energy dissipates? If capacitors are supposed to act as charge storage devices, why does it dissipate energy at all? 2
Discharging a Capacitor Now, unplug the wire leads from the power supply, and attach an alligator clip and nail to the end of each wire - this will enhance the effect in the next observation. Begin collecting data, again for a 15-second window of observation. Recharge the capacitor by touching the nails to the posts on the power supply, as opposed to plugging the wires in directly. Once the capacitor is fully charged, remove the tips of the nails from the power supply, and touch them to each other. You can repeat this several times per data set, as it only takes a few seconds to complete this procedure. The voltage (and hence, the stored energy) should drop to nearly zero. Explain why that happens. Capacitors Arranged in Series & Anti-Series Series: Connect a set of wires and a single Differential Voltage Probe to each of the 1000-µF capacitors, but do not attach them to the power supply. Make sure that both voltage probes are detected by Logger Pro; you will only see one graph, as before, but along the y-axis, you should see headers that say "Potential 1" & "Potential 2". Set the data collection time to 60 seconds at a rate of 10 samples/second, and begin collecting data. In this exercise, it may be helpful to have a digital reading for the values of the potentials. You can bring up such a display by clicking: Insert Meter Digital Meter ; the meter display will appear on top of the graphical display in Logger Pro. Repeat this step so that you have two digital meters available, one for each potential measurement. Turn on the power supply (which should still be set to a dial reading of 20), and keeping polarity in mind, touch the unconnected end of the wires from each capacitor to the plugs of the power supply. A mere fraction of a second is needed to build a full charge on each capacitor, and you can view the voltage on each capacitor on your graph. Assemble the circuit in Figure 3a, using the Fluke multimeter to measure the voltage across both capacitors as a set (be sure that the multimeter is set to measure DC voltage (V), not AC voltage (V~)). These capacitors are now connected in series. Record your reading from the multimeter for the total voltage of this combination. In principle, if you connected 100 of these charged capacitors in series, what would the resultant output voltage be? Anti-Series: Disconnect the circuit in Figure 3a, and discharge each of the capacitors by touching the positive and negative wires to each other. Re-charge each capacitor as you did previously, and configure the capacitors and the multimeter as shown in Figure 3b, which is similar to the previous exercise, but in this case, the polarities of the capacitors are opposite. The capacitors are now connected in anti-series. Record your reading from the multimeter for the total voltage of this combination. If both capacitors are charged to the same voltage, what would the potential difference across the pair be when they are connected in anti-series? Do your experimental results agree with this? Explain. 3
Figure 3a: Series Figure 3b: Anti-Series Figure 3c: Parallel Capacitors Arranged in Parallel: Recharge both 1000-µF capacitors, unplug the wires from the power supply, and plug them into the voltmeter as shown in Figure 3c. The capacitors are now connected in parallel. Record your reading from the multimeter for the total voltage of this combination. What should the voltage be in this case? Explain your data. If you are unsure, ask your TA whether your answer is sensible. Resistor & Capacitor Circuits In this section, you will study the nature of the capacitor as part of a resistor-capacitor (RC) circuit. Elementary circuit theory describes the charging equation for an RC circuit as: V R (t) + V C (t) = " R dq dt + q C = " where: V R & V C are the voltages across the resistor and capacitor, respectively, as a function of time R is the resistance of the circuit C is the capacitance of the circuit q is the total charge on the capacitor as a function of time dq/dt is the rate at which charge builds up on or leaves the capacitor. This is also the current flowing through the resistor for this particular circuit. ε is the electromotive force (or EMF), which provides an energy source to the circuit (ex, voltage supplied by the power supply) For a charging circuit, the charge on the capacitor as a function of time can be described by: $ q t & when there is no charge on the capacitor at time t = 0. $ ( ) = C" 1 # exp # t & '' ) RC ( ) ( 4
Recalling the relationship between charge, voltage, and capacitance [i.e. q(t) = C V(t)], we can study the voltage across the capacitor as a function of time: Using the definition of current, we find that: V (t) = q(t) C I = dq dt = # " & $ R' ( e ) t RC = # " & $ R ( e ) t * ' The time constant of the circuit, defined as τ = RC, is a measure of how long it takes for a capacitor to charge or discharge. Charging Capacitors In the following exercises, you will investigate time-varying charging and discharging circuits created by placing a resistor and capacitor in series with a power supply. Figure 4: Diagram for Charging RC Circuit Construct a series RC circuit using one of the resistors, the 80-µF capacitor, and the power supply (see Figure 4 for a circuit diagram). Be sure that the power supply is OFF while doing this. Make a physical sketch (NOT a circuit diagram) in your notebook including all of the components that you used, and the flow of current once the circuit is energized. Connect one of the voltage probes to record the voltage across the capacitor, and the other to measure the voltage across the resistor. Verify that the dial on the power supply is still set to approximately 20 (ε 5V). Reset the data collection time to 60 seconds, with a sampling rate of 10 samples/second. Make sure that you have a clear graph with the voltage across the capacitor versus time; it may be Potential 1" or "Potential 2," depending on which probe you are using to measure V C. If you wish to change the value being shown on either axis, click on the header, scroll down to "More", and select by hand the values you wish to be displayed. Turn on the power supply, and begin collecting data. Turn off the power supply after the data collection has elapsed, and discharge the capacitor. 5
Review the graph of the potential difference across the capacitor as a function of time as the capacitor charges - it increases quickly within the first few seconds that the power supply is turned on, but then levels off after 30-40 seconds. Does this agree with the theory discussed above? Explain. To determine the time constant, you will need to know the curvature of this data set. Draw a highlighted box around as much of the non-zero portion of your curve, then click on the "Curve Fit" feature. There are several curve fits to choose from, but only a few exponential-type fits. Select the fit you think to be the best fit for your curve. Given the equation that describes the voltage across the capacitor versus time for the charging circuit, determine the time constant (τ) from the variables given on your graph; this can be done in just a few steps. Print a copy of this graph for your notebook, and explain how you determined the time constant. Lastly, determine the total amount of charge (Q) on the capacitor once it is fully charged, based on your values for voltage and capacitance. Kirchoff s Loop Rule In the Simple Circuits lab, you studied Kirchoff s Loop Rule for circuits, which stated that the sum of all voltages across the circuit elements in a closed loop is zero. Recall that this is a statement of energy conservation in the electrical circuit. In the previous section, you observed the voltage across a capacitor as it was charged, but we did not analyze the voltage across the resistor in this circuit. Here you will look at the voltage across both the capacitor and the resistor during the charging portion of the experiment. Using your previous graph of voltage as a function of time, click on the y-axis variable and make a selection so that both voltages (Potential 1 & Potential 2) are displayed on the graph. During the charging portion, describe qualitatively the voltage across the capacitor and the resistor? At t=0, which circuit element has no (or very small) potential difference across it? As t (meaning times when the the capacitor is fully charged), which element has no (or very small) potential difference across it? Explain why this is so. Calculate the sum of the potential differences across the capacitor and resistor as the capacitor charges and graph this new function over time. Does your result support the idea that energy should be conserved in the circuit? Explain. Discharging Capacitors Similar to charging a capacitor in an RC circuit, we can describe the discharging of a capacitor as: R dq dt + q C = 0 where the discharging circuit consists of an initially charged capacitor that begins to discharge through a resistor at t=0. In this circuit the electrical potential energy initially stored in the capacitor is dissipated as heat through the resistor as time passes. The general solution to the above equation yields the charge on the capactior as a function of time: # q = q 0 exp " t & ( $ RC ' when there is a charge of q 0 on the capacitor at time t = 0. 6
In this exercise, you will study how the charge on a capacitor decays over time when the capacitor is behaving as a source of energy (i.e., capacitor is like a battery whose voltage decays over time). Change the data collection settings in Logger Pro to: 120 second duration, 10 samples/second. Charge up the capacitor in the RC circuit in the same manner that you did in the previous section. Once the capacitor is fully charged, unplug the wires going into power supply and connect them to each other - in essence, completing the circuit. Now, the capacitor is acting as the power supply for the circuit. Begin your LoggerPro data collection. Allow the voltage in the capacitor to approach zero (which should happen within the data collection time being used.) In your notebook, draw a circuit diagram of this discharging circuit. As you did previously, select a curve fitting function that appropriately describes the voltage across the capacitor as the capacitor discharges. From your curve fit to the discharging data, determine the time constant for the circuit. How does this time constant compare to the one found in the charging circuit? Should the two values be the same or different? Print out a copy of this graph for your notebook. Concluding Questions When responding to the questions/exercises below, your responses need to be complete and coherent. Full credit will only be awarded for correct answers that are accompanied by an explanation and/or justification. Include enough of the question/exercise in your response that it is clear to your teaching assistant to which problem you are responding. 1. Figure 2 shows a power supply hooked up to a capacitor. Explain in words how a voltage builds up across the capacitor plates. Consider the flow of electrons in the circuit. 2. Explain in words why the voltage across a capacitor goes to zero when wires connected to each terminal of a charged capacitor are brought into contact. 3. A charged capacitor stores electrical potential energy. Keeping in mind conservation of energy, describe what happens to the stored energy in the capacitor when the capacitor is discharged and where it goes. 4. If we doubled the capacitance of the capacitor, would you expect it to take more or less time to fully charge the capacitor in a charging circuit? Explain. * Turn off power supply and multimeter before leaving class. 7