MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8. Spring 3 Experiment 8: Driven LC Circuit OBJECTIVES To measure the resonance frequency and the quality factor of a driven LC circuit INTODUCTION The presence of inductance in an electric circuit gives the current an inertia since inductors try to prevent changes in the flow of current. The presence of capacitance in a circuit means that charge can flow into one side of the capacitor to be stored there, and later on this charge can restore the electric current as the capacitor discharges. These two properties of inertia and energy storage are analogous to the inertia and energy storage of a mass-spring combination, which you studied in mechanics. In a mechanical system viscous friction causes damping and in electric circuits resistance causes the damping. If a mechanical system that has a natural frequency of oscillation is driven by a periodic external force whose frequency matches the natural frequency of oscillations, then the system is said to be in resonance with the driving force and the amplitude of oscillations can grow very large. An electric circuit driven by a periodic external voltage exhibits the same behavior. In this experiment you will study the properties of circuits consisting of an inductor, capacitor and resistor in series. You will observe the behavior near resonance and measure the resonant frequency. Driven LC Circuit Suppose we have an ac voltage source given by π t V () t = Vsin = Vsin ( π f t) (8.) T where V is the amplitude (maximum value). The voltage varies between V and V since a sine function varies between + and -. A graph of voltage as a function of time is shown in Figure 8.. Figure 8. Sinusoidal voltage source E8-
The sine function is periodic in time. This means that the value of the voltage at time t will be exactly the same at a later time t = t+ T where T is the period. The frequency, f, is defined to be f = T. The units of frequency are inverse seconds [sec - ] which are called hertz [Hz]. The angular frequency is defined to be ω = π f. Suppose the voltage source is connected in a series circuit consisting of a coil, a resistor, and a capacitor with capacitancec, as shown in Figure 8.. The circuit has a total self-inductance L due primarily to the coil. In the circuit diagram, the resistor symbol stands for the total resistance in the circuit, and the inductor symbol stands for the total self-inductance L in the circuit. Figure 8. Driven LC Circuit Faraday s Law states that electromotive force around a closed loop is equal to the back emf, Q di I+ V() t = L C dt (8.) Since the current in the circuit is given by I = dq dt, the above equation becomes dq dq Q L + + = V() t (8.3) dt dt C An ac current will flow in the circuit as a response to the driving voltage source. The current will oscillate with the same frequency as the voltage source, but will have an amplitude I and phase φ that depends on the driving frequency: It () = Isin( π ft φ ) (8.4) We have already investigated how an undriven LC circuit will undergo free oscillations with frequency f = (8.5) π LC E8-
If the driving frequency is increased from Hz, the amplitude of the current will increase until its reaches a maximum when the driving frequency of the voltage source is the same as the natural frequency, f, associated with the undriven LC circuit. This is called a resonance. When the driving frequency is increased above the resonant frequency the amplitude of the current diminishes. The dependence of the amplitude on the driving angular frequency ω is given by I ( ω) = V + ωl ωc (8.6) The response current is not in phase with the driving frequency. The phase shift is also a function of the driving angular frequency: esonance ωl ( ) tan C φω = ω (8.7) The amplitude I ( ) ω is at a maximum when the term in the denominator vanishes: ωl = ωc (8.8) We can solve this equation for the driving frequency and see that resonance occurs at The amplitude at resonance is then ω = ω = (8.9) LC I,max = V (8.) where φ = since tan =. The corresponding current response is given by I t V = ωt (8.) () sin The power delivered to the circuit by the voltage source is V = = (8.) Pt () VtIt () () sin ωt E8-3
and its time-averaged is T T V V ω Pt () = Ptdt () = sin tdt= (8.3) T T which is the same as the Joule heating in the circuit: T T V V ω Pt () = I dt= sin tdt= (8.4) T T Thus, all the power introduced into the circuit is dissipated in the resistor. When the driving angular frequency is such thatω < ω =, then ωl <, and LC ωc ωl C = ω < φω ( ) tan (8.5) A negative phase shift means that circuit behaves in a capacitive fashion, the voltage lags the current. When the oscillations take a very long time to complete a cycle, the charge will builds up on the plate. Hence, the charged capacitor will determine the current the flow of current On the other hand, when ω > ω, the phase shift is positive, φ( ω ) >. This implies that circuit behaves in an inductive fashion, i.e., the voltage leads the current. This shouldn t be surprising because the oscillations are getting very rapid, which means the self-inductance dominates. EXPEIMENTAL SETUP Components needed µf capacitor 75 Interface AC/DC electronics laboratory Two or Three Data Studio files: 7-LC.ds, 7-LCcurve.ds, 7-LCCurrentVsVoltage.ds Computer Setup: Connect the Science Workshop 75 Interface to the computer using the SCSI cable. Connect the power supply to the 75 Interface and turn on the interface power. Always turn on the interface before powering up the computer. Turn on your computer. AC/DC Electronics Lab Circuit Board: Take your AC/DC electronics laboratory and the µ F capacitor and connect the capacitor in series with the coil on the electronics board. Connect the E8-4
Signal Generator of the Science Workshop 75 Interface into the electronics board using the banana plugs (see Figure 8.3), and connect the Signal Generator output in series with the capacitor and the coil. Figure 8.3 Setup of the AC/DC Electronics Lab circuit board Data studio Files: Download the file 7-LC.ds from the web page. This file has a Signal Generator Display, and a Scope Display (see figure below). In the Signal Generator dialog box choose a Sine Wave Function. If necessary, adjust the Amplitude to. V, the Frequency to Hz and the Sampling ate to 5 Hz. Question : As you determined in a previous lab, the resistance of the coil is 5.4Ω and its inductance is 8.5 mh. Calculate the resonant frequency you should expect for your experiment in Hertz (cycles per second) using the formula f =, with L = 8.5 mh and C = µf π LC MEASUEMENTS Part : esonance Frequency Click Start to begin collecting data. On the Scope display you should see two waveforms for the Output Voltage and the Output Current, as shown in Figure 8.4 (if necessary, click the Trigger control on the Scope display to stabilize the picture). You can adjust the display using the controls. Increase the frequency of the sine wave in Hz steps (you change the frequency in the Signal Generator window). Identify a range of frequencies where the Output Current seems to have a maximum. Then use smaller frequency steps in order to locate the resonant frequency more precisely. E8-5
Figure 8.4 Output Voltage and output current Question : ecord the resonant frequency at which the Output Current reaches a maximum. Does your value agree with the theoretical value you calculated above? Question 3: How does the phase relation between Output Current and Output Voltage change as you increase the frequency from below the resonant frequency to above it? How could you use this information to help pinpoint the resonant frequency? Use the Smart Tool to measure the amplitude of the Output Current at the resonant frequency. NOTE on using the Smart Tool in the Scope Display: When you have more than one signal being displayed on the Scope, you can have a Smart Tool for EACH trace. You need separate Smart Tools because each signal can have a different scale. Look at the right side of the display. Click the name of the signal you want to make active and a box will appear around it. Then click on the Smart Tool button and a Smart Tool cursor will appear in the color of that trace. You can do the same for the other trace. Since the Output Voltage and Output Current are in phase with each other at resonance, the output power P varies as the square of a sinusoidal function. The maximum value of the output power is P max = Imax Vmax ; and the average power supplied is one half of the maximum (because the average value of sine is ½.) Question 4: Compute the average power consumed by your circuit at resonance. Question 5: The power lost in a resistor is I. For a sinusoidal current, average value of I is ½ of I max. Compute the average value of I at the resonant frequency. Compare that with the average power consumed by the circuit. E8-6
Part : esonance curve and the Quality Factor Now examine the behavior of the circuit near resonance in more detail. For this purpose it is more convenient to use a Graph display than the scope, so download and open the file 7-LCcurve.ds, which should simplify data collection. Starting at Hz on the Signal Generator, click Start to obtain the maximum value of the output current for this frequency. As the data is taken, this maximum value is automatically calculated and placed in the box identifying the data run. ecord the maximum value of the current for this Hz frequency in the esonance Data Table on the right, in the first row of the column to the right of the frequency column. Then increase the frequency of the Signal Generator to the next value of the current in the esonance Data Table and again record the maximum current in the data table. As you enter data in the table a curve will be drawn. It should look like Figure 8.5. Figure 8.5 esonance curve Continue doing this until you have taken measurements at all of the listed frequencies in the esonance Data Table. If you want to better define the shape of the curve around the resonance, take extra data points near the resonant frequency so you can see clearly what happens to the current there. What this curve tells you is that if a large number of different sine-wave voltage signals were all added together and connected to this circuit, only those with frequencies near the resonant frequency will be able to produce significant currents, that is, the circuit would be tuned to hear only those signals and ignore the others. Obviously, something of this sort occurs in the tuner of a radio. defined as The quality factor of a circuit is a measure of how sharp the resonance curve is. It is f o Q, where f o is the resonant frequency and f and f are the frequencies for f f The graph has some quirks: the frequencies in the data table need to be entered in order from low to high, so you will need to use the Insert ows and Delete ows buttons at the top of the table to re-arrange the data. E8-7
which the Output Current drops to ½ of its value at f o (see Figure 8.5 above). In a tuner, Q may range from to or so. Question 6: Compute Q for your circuit. Part 3: Phase elationships An interesting way of exploring the phase relationships between current and voltage in a resonant circuit is to graph the Output Current versus the Output Voltage. You can either do this following the instructions below using the file 7-LC.ds (re-open it), or you can download a different file named 7-LCCurrentVsVoltage.ds for which the display is already set up. To amend the re-opened 7-LC.ds file, go to the Scope display click on the box labeled Output Voltage and delete it from the display. Then go to the Data window on the left and click and drag the icon for Output Voltage into the scope window and drop it on the time setting at the bottom of the scope display. Now Start observing data, using either this amended 7-LC.ds file, or the newly downloaded 7-LCCurrentVsVoltage.ds file. Adjust the displayed signal so that it is wide enough and tall enough to be seen clearly. Change the frequency of the Signal Generator. Question 7: Observe what happens at the resonance frequency of the circuit. When you are exactly at resonance, the scope should display a straight line. Can you explain this behavior? E8-8