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Physics 2306 Experiment 10: Time-Dependent Circuits, Part 2

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Physics 2306 Experiment 10: Time-Dependent Circuits, Part 2 Name___________________ ID number_________________________ Date____________________ Lab partner_________________________ Lab CRN________________ Lab instructor_______________________ Physics 2306 Experiment 10: Time-dependent Circuits, part 2 Objectives • To study the frequency dependent properties of capacitors in RC circuits • To study the frequency dependent properties of inductors in RL circuits • To study the time dependent behavior of a series RLC circuit Required background reading • Young and Freedman, sections 30.1 – 30.6 Introduction In this lab, we will revisit capacitors and study more of their properties in a circuit where the driving signal varies with time. Please review the introduction section to Lab 7, Time-Dependent Circuits part 1. It will refresh your memory about the properties of RC circuits and the general behavior of exponential growth and decay curves. We will then study inductors as circuit elements and their behavior in circuits where the driving signal varies with time. Inductors are circuit elements that make use of the phenomenon known as electromagnetic induction. As you learned in last week’s lab on magnetism, the principle of electromagnetic induction is described by Faraday’s law. It states that a changing magnetic flux through a loop causes an induced emf around the loop. A typical inductor consists of a coil of many loops of wire. An example of one is the coil on the upper right hand side of your circuit board. When a time-varying current is passed through the inductor, there is time-varying magnetic field generated. This causes a changing magnetic flux between the coils and a resulting induced emf (or voltage) across the inductor. This can be stated mathematically with the relation: dI V = −L dt 1 where V is the voltage across the inductor, and I is the current passing through it. L is the inductance which is measured in a unit called henrys (1 henry = 1 H = 1 V*s/A). Notice that the voltage gets larger the more rapidly the current is changing. We can build simple circuits with inductors and resistors just like you have done with resistors and capacitors. A simple RL circuit is shown in Figure 1. L 1 2 + DC power R supply _ Figure 1: Simple RL circuit with a single-pole, double-throw switch that allows a DC power supply to be hooked up at position 1 (for building up current in the circuit) or a short circuit hooked up at position 2 (for dissipating the current). In the case of the capacitor, we were concerned with the charge that was stored on it. The capacitor stored energy in its electric field when charge was stored on it. In contrast, for the inductor we are interested in the current flowing through it. The current flowing through it generates a magnetic field in the inductor, which stores energy. Initially, consider the inductor in Figure 1 to have no current flowing through it. Inductors tend to oppose rapid changes in current, so when the switch is flipped to position 1 to a DC power supply with voltage V0, the current through the circuit slowly builds up to its final value of V0/R. As shown in Young and Freedman section 30.4, the current build-up is described by an exponential build-up: ⎛ R ⎞ V ⎛ −⎜ ⎟ t ⎞ I (t) = 0 ⎜1 − e ⎝ L ⎠ ⎟ (build - up) ⎜ ⎟ R ⎝ ⎠ Just like in the RC circuits we studied earlier, there is a “time constant” that characterizes this build-up. The time constant is: τ=L/R. After one time constant has elapsed, the −1 current through the inductor has built up to1− e = .63 , or 63% of its final value of V0/R. As you know from your study of exponential functions the current approaches the final value of V0/R asymptotically. Theoretically, after an infinite time the current is equal to V0/R. From a practical point of view the current reaches V0/R after a few time constants have elapsed. This is clear if you look at Table 1 and Figure 2 (same table and figure you 2 used in lab 7). These indicate the percent of the maximum current, V0/R, that is achieved after a certain number of time constants have elapsed. Note that the curve shown in Figure 2 is a universal curve. Once you know the time constant of an RL circuit, this curve gives you everything you need to know about its time behavior during current build-up. Exponential Growth 100% 90% 80% e 70% 60% 50% 40% 30% % of valu maximum 20% 10% 0% 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Time constants Figure 2: Current exponential build-up (growth) curve in an RL circuit. Table 1: Behavior of growth and decay curves as a function of the number of time constants. Number % % Decay of time Growth constants 0.0 100.00% 0.00% 0.5 60.65% 39.35% 1.0 36.79% 63.21% 1.5 22.31% 77.69% 2.0 13.53% 86.47% 2.5 8.21% 91.79% 3.0 4.98% 95.02% 3.5 3.02% 96.98% 4.0 1.83% 98.17% 4.5 1.11% 98.89% 5.0 0.67% 99.33% 5.5 0.41% 99.59% 6.0 0.25% 99.75% 6.5 0.15% 99.85% 7.0 0.09% 99.91% 7.5 0.06% 99.94% 8.0 0.03% 99.97% 3 After the current through the inductor has settled to its asymptotic value, the switch is flipped to position 2. This creates a circuit with just an inductor and resistor. There is magnetic energy stored in the inductor, so this energy is gradually transferred to the resistor. The current flowing in the circuit gradually decays to zero with the same characteristic time constant (τ=L/R): ⎛ R ⎞ −⎜ ⎟ t ⎝ L ⎠ I(t) = I 0 e (decay) where I0 is the original current in the circuit. After one time constant has elapsed, the −1 current through the inductor has decayed to e = .37 , or 37% of its initial value of I0. Once again, theoretically it would take an infinite time for the current to reach zero, but for practical purposes the voltage reaches zero after a few time constants have elapsed. This is clear if you look at Figure 3 and Table 1. Once you know the time constant of an RL circuit, this curve gives you everything you need to know about its time behavior during current decay. Exponential Decay 100% 90% 80% e 70% 60% 50% 40% 30% % of maximum valu % of maximum 20% 10% 0% 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Time constants Figure 3: Current exponential decay curve in an RL circuit. 4 Name___________________ ID number_________________________ Date____________________ Lab partner_________________________ Lab CRN________________ Lab instructor_______________________ Physics 2306 Experiment 10: Pre-lab assignment (complete and turn in at the beginning of your lab session) 1. In this lab, you will be revisiting the series RC circuit that you studied in Lab 7: Time Dependent Circuits, Part 1. In that lab, you constructed a series RC circuit that contained a 560 Ω resistor and a 1 µF capacitor. a) What is the RC time constant for this circuit? b) If the function generator driving the RC circuit is set to output a square wave at a frequency of 200 Hz, what is the period of the signal? What is the period of a signal at a frequency of 2000 Hz? c) How many RC time constants elapse during the time that the capacitor is charging when a 200 Hz square wave is applied to the circuit? (Remember that the capacitor is only charging during half of the square wave period; the charge is decaying away during the other half of the period.) 5 2. Now consider a series RL circuit like that from Figure 1 of the introduction to this lab. a) What is the RL time constant for a series RL circuit with a 500 Ω resistor and a 10 mH inductor? b) Assume that you are driving the RL circuit with a square wave voltage source. You would like five time constants to elapse during the portion of the square wave (half of the period) when the current is building up. What frequency do you need to set the function generator at to obtain this condition? 6 Equipment You will use the following equipment: • Tektronix TDS 1002 Oscilloscope • BK Precision 4017A Function Generator • BK Precision 1670A Adjustable Power Supply • LCR meter (BK Precision 875B) • Manual range digital multimeter (DMM) • Pasco EM-8656 AC/DC Electronics Laboratory • Components: (in Ziploc bag) o Resistor: 560 Ω o Capacitors: one 1 μF, one 10 μF • One BNC tee (attached to function generator output) • One BNC – BNC cable • Two BNC – banana plug cables • Several wires of varying lengths (5, 10, and 25 cm) (in Ziploc bag) • Three red and three black banana plug – banana plug cables • Four red and four black banana plug to alligator clip adapters • Knife switch (single pole, double throw) Other general notes about this lab: 1. In your Class Notes folder for this lab, there is a file (ph2306_lab10_sample_data.doc) that contains sample data showing what you should expect to see for each of the circuits you construct in this lab. If you are confused about what you are seeing, refer to this document for help. 2. If the measured value (using your DMM) of any of the components (resistors or capacitors) varies by more than 10% from the value on the label, then don’t use it.
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