MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2005 Experiment 11: Driven RLC Circuit OBJECTIVES 1. To measure the resonance frequency and the quality factor of a driven RLC circuit by creating a resonance (frequency response) curve. 2. To see the phase relationships between driving voltage and driven current in such a circuit at, below, and above the resonance frequency. 3. To use DataStudio display capabilities to carry out these objectives. INTRODUCTION The presence of inductance in an electric circuit gives the current an “inertia” (resistance to change), 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 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 RLC Circuit Suppose we have an AC (Alternating-Current) voltage source given by ()=V sin ⎛⎜ 2πt ⎞=V sin (2πf t )=V sin (ωt ) (11.1)Vt 0 0 0⎝ T ⎠⎟ where V0 is the amplitude (maximum value). The voltage varies between V0and −V0 since a sine function varies between +1 and −1. A typical graph of voltage as a function of time is shown in Figure 1. E11-1Figure 1 Sinusoidal voltage source The sine function is periodic in time. This means that the value of the voltage at time t will be texactly the same at a later time t′=+T where T is the period. The frequency, f , is defined to be the reciprocal of the period, f =1 T . The units of frequency are inverse seconds, s-1 , which are called hertz [Hz]. The angular frequency ω is defined to be ω =2πf. Suppose the voltage source is connected in a series circuit consisting of a coil with self-inductance L , a resistor of resistance R and a capacitor with capacitance C , as shown in Figure 2. Figure 2 Driven RLC Circuit The expression for the current in the series circuit as a function of time is derived in the 8.02 Course Notes, Section 12.3. The result is reproduced here as it 0() =I (ω)sin(ωt −φ), I0 ()= V0 , (11.2)ω 2 R2 +⎜⎛ω L −ω 1 C ⎠⎟⎞ ⎝ 1ω L − ωtan φ()=ω C R (((these experiment instructions will use lowercase it) and qt) for time-varying currents and (charges, but Vt) for time-varying voltages). Note that we explicitly indicate the dependence of these quantities on the driving frequency ω . This quantity is something we get to choose in this E11-2experiment, and once we select the frequency we can calculate how the circuit responds using Equations (11.2). The amplitude I0(ω) is at a maximum when the term in parentheses in the denominator vanishes: 1ω L −= 0 . (11.3)ωC We can solve Equation (11.3) for the driving frequency and see that resonance occurs at 1 (11.4)ωmaximum current = ω0 =LC (see the 8.02 Course Notes, Section 11.5 for a further discussion of resonance). The amplitude at resonance is then = (11.5)I0,max VR0 −and φ= 0 at resonance since tan 1 0 = 0 . The corresponding current response is given by () = V0 sin ω0t . (11.6)it R The power delivered to the circuit by the voltage source is Pt () () =V02 sin2 ω0t (11.7)() = V t it R and its time-averaged is 1 T 12 () =∫0 Pt dt =∫0 TV02 sin2 ω0t dt = V0 (11.8)Pt T () T R 2R , which is the same as the Joule heating in the circuit; 1 T 21 2 () =∫0 i R dt =∫0 TV02 sin2 ω0t dt = V0 . (11.9)Pt T T R 2R Thus, all the power introduced into the circuit is dissipated in the resistor. 1 1When the driving angular frequency is such that ωω = , then ω L −< 0, and< 0 LC ωC E11-3⎛ 1 ⎞ φ(ω)tan −1 ⎜ ωL − ωC ⎟< 0 (11.10)= ⎜ ⎟ ⎝ R ⎠A negative phase shift means that circuit behaves in a capacitive fashion: the driven current leads the driving voltage. When the oscillations take a very long time to complete a cycle, the charge builds up on the plate. Hence, the charged capacitor will determine the current the flow of current. > >On the other hand, when ωω0 , the phase shift is positive, φ(ω)0 . This implies that circuit behaves in an inductive fashion: the driven current lags the driving voltage. This shouldn’t be surprising because the oscillations are getting very rapid, which means the self-inductance dominates. EXPERIMENTAL SETUP Components needed • 100-µF capacitor • 750 Interface • AC/DC Electronics Lab Circuit Board • Two Data Studio files: exp11a.ds, exp11b.ds. Computer Setup: If not already done, connect the 750 Interface to the computer using the SCSI cable. Connect the power supply to the 750 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: On your AC/DC circuit board, place the 100-µF capacitor in series with the inductor coil on the electronics board (make sure the iron core is removed from the inductor). Connect the Signal Generator of the 750 Interface into the electronics board using the banana plugs (see Figure 3), and connect the Signal Generator output in series with the capacitor and the inductor coil. Note that in Figure 3, one capacitor lead is connected to the same port as the black lead from the output, but the other is connected to the inductor coil lead, not the red lead from the output. E11-4Figure 3 Setup of the AC/DC Electronics Lab Circuit Board Data studio Files: Download the file exp11a.ds from the web page. This file has a Signal Generator Display and two scope displays (see figure below). In the Signal Generator dialog box make sure that a Sine Wave Function has been selected. If necessary,
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