MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2003 Experiment 17: RLC Circuit (modified 4/15/2003) OBJECTIVES
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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8. Spring 3 Experiment 7: R Circuit (modified 4/5/3) OBJECTIVES. To observe electrical oscillations, measure their frequencies, and verify energy relationships in an R circuit.. To determine the quality factor of the circuit. INTRODUCTION The presence of inductance in an electric circuit gives the current inertia since inductors are reluctant to allow changes in the flow of current. The presence of capacitance in a circuit gives the current springiness since current can flow into one side of the capacitor storing energy there, and later the current can flow out as the capacitor discharges. The charges on the capacitor plates and the current in the electric circuit have a natural oscillation frequency: f = π (7.) In electric circuits, resistance causes the amplitude of the oscillations to decay exponentially. Damping affects the natural oscillation frequency (lowering the frequency) until the damping is so strong that the current will no longer oscillate. 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 fact, the equations for the electric and mechanical systems are identical if you replace x() t by Qt (), m by L, and k (the spring constant) by C. The mechanical system has a natural frequency of oscillation ( fo = ( π ) km). In a mechanical system viscous friction causes damping and the amplitude of the oscillations decays exponentially. Free Oscillations in the and R circuit Consider a series R circuit shown in Figure 7.. E7-
2 Figure 7. R Circuit with external voltage removed Applying the Kirchhoff s voltage rule, the circuit equation for the R circuit without any external voltage is dq dq Q R dt dt C = L + + (7.) Case : Resistance is negligible Let s assume that the resistance in the circuit is negligible, R. Our equation for the charge on the capacitor is then dq Q = L + (7.3) dt C The general solution to the above differential equation is π Qt () = Q cos( t+ φ) = Q cos( t+ φ) (7.4) T which indicates that charge on the capacitor will oscillate between Q and Q. The period of oscillation is T = π (7.5) The frequency and the angular frequency are given by and f = T = ω π = π (7.6) ω = π f = (7.7) E7-
3 respectively. The corresponding current in the circuit is dq Q It () = = sin( t+ φ) = I sin( t+ φ) (7.8) dt So the current will also oscillate between I and I, where the imum current and the imum charge are related by π I = Q = Q (7.9) T Suppose initially the charge on the capacitor at t= isqt ( = ) = Q, we then have Q = Q cosφ (7.) If we assume further that Q = Q, i.e., charge is at a imum at t =, then cosφ = or φ =. Case : Resistance is not negligible Suppose the resistance is no longer negligible but still has only a small effect. When the external voltage is non-zero, energy is introduced into the circuit and is stored in the electric and magnetic fields. When the external voltage is turned off, the current and the charge will still oscillate. However, the resistance acts like a damping force. This shouldn t be surprising because the energy in the system is no longer conserved due to the energy loss from Joule heating in the resistor. For fixed values of self-inductance L and the capacitance C, the behavior of the charge and the current will depend on the amount of resistance present. In our circuit, R 5Ω, L 8mH and C µ F. The angular frequency associated with the circuit (no resistance) is 4 ω = π f =.8 rad s (7.) The resistance is relatively small with 4 = 3Ω > R = 4Ω. The circuit is an example of an underdamped R circuit since R < 4.. Charge on the capacitor: The solution for the charge on the capacitor is where ( ω φ) Qt () = Q cos t+ e γ t (7.) ω R = 4L (7.3) E7-3
4 is the angular frequency of oscillation, γ = R L is parameter that measured the exponential decay of the oscillations, Q is the amplitude and φ is the phase factor.. Voltage across the capacitor: The voltage across the capacitor is Qt () Q γ t VC () t = = cos( ωt+ φ) e (7.4) C C 3. Current in the circuit: The current in the circuit is the derivative of the charge so dq I() t = = Q sin( t+ ) Q cos( t+ ) e dt γ t ω ω φ γ ω φ (7.5) Differences between the Circuit and the R Circuit There are several important differences between the R oscillations and the oscillations. The first is that the angular frequency of oscillations is different. For the R circuit, the current is not zero when the charge is at a imum. The charge and current oscillate but the amplitudes of the oscillations decay exponentially.. Angular frequencies of oscillations are different: For the circuit the angular frequency of oscillation is ω ( ) = ( R 4L ) ω =.. For the R circuit, the angular frequency is slightly less. Phase relation between charge and current: In the circuit the charge and the current are 9 out of phase. This means that when the charge is at a imum, the current is zero, and all the energy is stored in the electric field. For the R circuit, the current is not zero when the charge is at a imum. The energy in the circuit is stored in both the electric and magnetic fields. 3. Amplitudes of the oscillations decay exponentially: The third significant difference is that the charge and current oscillate but the amplitudes of the oscillations decay exponentially. In the figure below, the current is plotted as a function of time for the underdamped R circuit. (A plot of charge vs. time is identical.) E7-4
5 Figure 7. Exponential decay of current in R circuit Suppose we now reset time so that the current is a imum at t = : I( t = ) = I = I (7.6) The current decreases in time exponentially according to γ t I() t = I e cosωt (7.7) Since the coefficient of exponential decayγ = R L is proportional to the resistance, we see that the current will fall off more rapidly as the resistance increases. We can introduce a time constant τ = γ = L R. Thus when t = τ, the current is I Ie cosωτ = (7.8) Although the current is still oscillating, the envelope of exponential decay has now decreases by a factor of e, i.e. the amplitude can be at most I e. In our circuit, we have R 4L rad/s (7.9) Therefore, the angular frequency is approximately equal to the same angular frequency of the circuit: R 4L ω = = ω (7.) E7-5
6 During this time t = τ, the current has undergone a number of oscillations. The total number of ωτ = ω LR. The closest integral number of cycles is then radians is given by ( ) Quality factor of the R circuit n ωτ ω L π (7.) πr = We define the quality Qquality of this circuit to be proportional to the number of integral cycles it takes for the exponential envelope of the current to fall off by a factor of e. (Unfortunately the same letter is used for quality and charge). The constant of proportionality is chosen to be π. (this choice is made to correspond to different definition for the quality of the R circuit when an oscillating sinusoidal voltage drives the circuit). Then Qquality = π n (7.) For the weakly damped case, we have Q quality L R ω (7.3) Energy Relationships in R circuits As the current sloshes back and forth in such circuits, energy may be stored in both the magnetic field of the inductor UB = LI (7.4) and in the electric field of the capacitor U E = CVC (7.5) The energy stored in the electric and magnetic fields is simply the sum: U LI CV C = + (7.6) However, this energy is gradually being lost as heat in the resistor at the rate ( () ( ) ) t I R : U = U t U t = = I Rdt (7.7) E7-6
7 EXPERIMENTAL SETUP A. Computer 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. B. AC/DC Electronics Lab circuit board Connect the black banana plug cord from the OUTPUT ground port of the 75 Interface to the banana jack located in the lower right corner on the AC/DC Electronics Lab circuit board. Connect the red banana jack with alligator clip to the positive OUTPUT port of the 75 Interface. During the course of the exercises below, you will connect the alligator clip to various places in order to close the circuit. MEASUREMENTS Part : Free Oscillations a: R Circuit Circuit Diagram: Put a C = µ F capacitor in series with the coil (without its core) and the R = Ω resistor. Connect the capacitor to a spring on the right side of the coil. Connect, using the alligator clip, the positive OUTPUT port to the other side of your capacitor (see Figure 7.6). Figure 7.3 Circuit diagram for the R circuit Use the multimeter to measure the resistance of your coil. Make sure you zero the multimeter first. Calculate the total resistance of your circuit by adding the coil resistance to ohms. Data Studio File: Download and open the Data Studio file 6-.ds.. This file contains a Graph Display to show the Output Current from the Signal Generator, which has been set to a Positive Square Wave at Hz and Volt. When the output of the Signal Generator switches to Volts, the current in the circuit oscillates back and forth through the inductor and capacitor until its E7-7
8 energy is all turned into heat within the resistance of the coil and the external resistor. Click Start to obtain a graph similar to Figure 7. above. Question : Determine the period T of these oscillations by using the Smart Tool to measure the time interval between when the current is zero. The period is twice the time interval between successive zeroes. Calculate the frequency, f = T, of these oscillations and record your results here. Question : For small values of resistance, the oscillation frequency is approximately f ( π ). For your circuit parameters, compute the expected value of f and compare it to your measured value. Do you expect your result to be greater, equal, or less than the measured valued? Question 3: What is the quality of this circuit? How many cycles do you predict the current oscillates until the exponential envelope of the current falls off by a factor of e? Question 4: Using your plot of current vs. time, measure the number of cycles the current oscillates until the exponential envelope of the current falls off by a factor of e. How does your measurement agree with your prediction? Can you explain any discrepancies? b: Circuit To see what effect the resistor has on the circuit, connect the wire directly to the left side of the coil, bypassing the Ω resistor (see Figure 7.4). Obtain a new graph. Figure 7.4 Circuit diagram for the circuit Question 5: Describe the qualitative difference between the two graphs. Calculate the frequency of the oscillations for the circuit without the external resistance. Now measure the frequency. Does your measured value of the frequency correspond to your theoretical prediction? E7-8
9 Part : Observe the Energy in the R Circuit Circuit Diagram: In order to investigate the energy in an R circuit, you use a circuit consisting of just an inductor and capacitor, with the resistance of the circuit due to the internal resistance of your coil. Insert a Voltage Sensor into Channel B on the 75 Interface and connect it across the terminals of the capacitor (see Figure 7.5). Figure 7.5 Circuit diagram for investigating the energy relationship in an R circuit Data Studio File: Then download and open the Data Studio file 6-energy.ds, which is similar to 6-.ds, but allows for the new sensor and adds one new window in which you calculate the electric energy, magnetic energy and total energy in the circuit as a function of time. [The calculated energies assume the values L = 8mH and C = µf]. We calculate the energies by taking the voltage V from the voltage sensor and calculating and plotting CV. We then calculate and plot L I. We then plot the sum of these two quantities to get the total energy. Click Start and take data. The oscillations you previously observed should appear in the same Graph window. In Graph, the energies are plotted. Because the energies are very small, they have been multiplied by a million, that is, the units are micro-joules. You will want to manually expand the scale and move the data around within the Graph window in order to make it more visible. Figure 7.6 shows an example of what you are looking for. E7-9
10 Figure 7.6 Energy in an R circuit Question 6: The circuit is losing energy most rapidly at times when the graph of total energy is steepest and that these times occur at about the same times that the magnetic energy reaches a local imum. Briefly explain why. E7-
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