#8A RLC Circuits: Free Oscillations

Transcription

1 #8A RL ircuits: Free Oscillations Goals In this lab we investigate the properties of a series RL circuit. Such circuits are interesting, not only for there widespread application in electrical devices, but also for their mathematical solutions which simulate other physical systems, including damped mechanical oscillators. Reading A discussion of Inductors can be found in Sec RL circuits are discussed in Young and Freedman, Sec in the th ed. The preceding two chapters discuss RL and L circuits. Theory In Lab. 3 we studied Ohm s law, which states that current i flowing through resistor R results in a potential difference V R = ir. () In Lab. 4 we learned that a capacitor stores charge Q=V. Because it takes time to charge or discharge a capacitor, the current through a capacitor circuit changes exponentially with time. As a result, the charge stored on the capacitor can be expressed as a time integral of the current: Q = idt, so that V = idt/. () In Labs. 5 and 6 we studied and utilized a consequence of Faraday s law, that a changing current in one coil induces an emf in a second coil according to di E M = dt (3) where M is the mutual inductance. However, the same effect is also realized in a single isolated coil: A current in the coil establishes a magnetic field and hence a magnetic flux through the coil. If the current changes, the flux changes, and hence an emf will be induced in the same coil. In analogy with Eq. (3), this emf can be expressed as di E= L (4) dt where L is the self-inductance or simply the inductance of the coil. The unit for inductance is the same as for mutual inductance, i.e., the Henry. The negative sign in Eq. (4) implies that the direction of the induced emf is always such as to oppose the change in the current. For example, if the current through the coil were to increase, the coil would produce an emf or voltage that would reduce the magnitude of the current increase. Note that the induced emf does not oppose the current itself; rather, it opposes, and tends to reduce, the current change. Think of an inductor simply as a coil that slows down the R response of a circuit to changes by generating a changing magnetic flux. The voltage decrease V L across an inductor will always be V L = L di. (5) dt L onsider the series circuit shown in Fig., with the capacitor uncharged when V the switch is closed at t = 0. Kirchoff s law states that at all times the total voltage summed around the circuit is zero, i.e.: V VR VL V di V Ri L dt idt = 0 = 0. (6) Fig.. Series RL circuit.

2 Differentiating with respect to time t: di d i i R + L + = 0 dt dt d i R di + + i= 0 dt L dt L This second-order differential equation has the same form as the equation that describes the oscillation of a pendulum in air, where the mechanical energy transforms back and forth between kinetic and gravitational potential energy, but is gradually reduced or damped by air resistance. In the RL circuit, the electromagnetic energy oscillates between the electric field of the capacitor and the magnetic field of the inductor, but is slowly dissipated by the resistor. Thus the RL series circuit is also an example of a damped oscillator. The current and its derivatives are analogous to the position, velocity, and acceleration of the mechanical oscillator. The solution to Eq. (7) has different forms depending on the relative values of R, L, and. In each case the presence of the inductor demands that i = 0 at t = 0; the capacitor requires i 0 at t. The three most important solutions are: Underdamped Oscillator: R << 4L/ For small R the solution is: i(t) = A exp( Rt/L) sin(ω 0 t) (8) where ω 0 / (L) (9) and A is a constant, approximately proportional to (/L) /. The form of this solution shows that the current oscillates at angular frequency ω 0 but with an exponentially-decaying amplitude characterized by time constant τ =L/R, as illustrated in Fig. (a). ritically damped Oscillator: R = 4L/ As R increases more energy is dissipated in the resistor, and the oscillation ceases. When R = 4L/ i(t) = A t exp( Rt/L). (0) For t <<L/R the current i(t) is proportional to t, but, as indicated in Fig. (b), it ends up by approaching zero exponentially with time constant τ =L/R, as for the underdamped oscillator. (7) Overdamped Oscillator: R >> 4L/ i(t) = A 3 [exp( t/r) exp( Rt/L) ]. () Initially the current increases from zero, but at large times the second exponential goes to zero much faster than the first, so that current ends up dying away with long time constant τ = R, as shown in Fig. (b). A mechanical analogy to the damped oscillator is a swinging two-way door, such as to a restaurant kitchen. An underdamped door swings back and forth with decreasing amplitude. An overdamped door takes for ever to close. A critically damped door is just right, and closes relatively quickly, without oscillating.

3 (a) Relative current R = 50 Ω L = 5 mh = 0.μF (b) Relative current L = 5 mh; = 0. μf ritical damping R = R 0 = 4L/ =.00 kω Overdamping R /R 0 = Time (ms) Time (ms).3.4 Fig.. (a) Underdamped oscillations of series RL circuit. (b) Overdamped and critically damped oscillations. 3

4 Measurements All measurements in this lab will be conducted using the circuit shown in Fig. 3, in which the dc voltage source and switch of the Fig. circuit are replaced with a low frequency square wave derived from a function generator. This automates the procedure of continually opening and closing the switch, and allows us to easily display the voltages on an oscilloscope. The voltage across R 0 monitors the current in the circuit. (i) Underdamped oscillations. Drive the circuit with a 0 Hz square wave and start with = 0. μf and R 0 = 00 Ω, values that are low enough to ensure that the circuit is underdamped. Your first task is to observe how the resonant frequency increases with increasing. The angular frequency ω 0 of the decaying oscillations can be deduced from measurements of the period of the scope trace. ompare the result with the value predicted by Eq. (9). Repeat for two other values of, one below and the other above 0. μf.. Now examine the exponential decay of the oscillations. Using = 0. μf measure the amplitude of the voltage across resistor R 0 proportional to the current i at successive oscillation maxima. On semi-log graph paper plot the voltage versus time t. A straight line should result if the V 0 cos(ωt) Fig. 3. ircuit used throughout this lab. amplitude reduces exponentially, as suggested by Eq. (8). The time constant τ characterizing the decay is equal to the time required for the voltage to drop to /e = 0.37 = 37% of the amplitude of the first maxima. Repeat the measurement of the time constant τ using another value of R 0 between0 and 00 Ω. Note: Because we expect τ = L/R, the measured time constant τ should be inversely dependent on R. A graph of R 0 as a function of /τ will form a straight line, but it will not pass through the origin, because we have neglected the resistance of the inductor and the internal resistance of the function generator. (ii) ritically damped oscillations. With = 0. μf, increase R 0 until the circuit becomes critically damped, i.e., the oscillations cease and the voltage across the resistor decreases rapidly to zero. It is not easy to do this precisely. Perhaps the best way is to gradually increase R 0 until the voltage does not discernibly dip below zero.. For critical damping R =4L/. Use the values of L and to predict R c, the critical value of R. 3. Another way of getting information on R c is from the time constant of the decay at large time. Adjust the oscilloscope trace to examine the large t section of the oscillation well beyond the voltage peak, and from this determine the time constant τ using the 37% rule. As in the underdamped case, we expect τ = L/R. Now, R can be calculated from the observed time constant and the inductance L. 4. omment on the above three values of R. (ii) Overdamped oscillations. Increase R further by about a factor of 0, sufficient to ensure overdamping. Use the oscilloscope to determine the time constant τ at large time t. For under- and critically damped oscillations τ depends on R and L, but for overdamped oscillations we expect τ = R. L Scope R 0 Scope ground 4

5 #8A Laboratory Report Sheet RL ircuits: Free Oscillations Name: Partner: Lab Section: A. Preliminaries: Inductor = mh Date: Inductor resistance = Ω B. Underdamped oscillations:. Oscillation frequencies Note: ωt = π R 0 (Ω) 00 (μf) Osc. period T (ms) Oscillation ω (s ) / L (s ) Difference (%). Oscillation decay time constants Take t = 0 to be at first maximum in the oscillating signal. The other maxima will be at t = T, t = T, etc., which are t/t =, t/t =, etc. Record voltage maxima for each of two resistance values, R 0. Use = 0. μf. R 0 (Ω) T (ms) t/t τ (ms) 00 Ampl. (mv) Ampl. (mv) (a) Using one sheet of semi-log graph paper, plot the oscillation amplitude as a function of time for each R 0, using the ratio t/t as the time axis, and draw the best fit line through the data points. A straight line indicates an exponential decrease. (b) For each of the 3 lines determine the value of τ as the time at which the amplitude graph has decreased to /e = of the initial value. (c) Express this time, τ, in seconds, or in ms. (Multiply by the period T.) Enter τ and /τ in the table.

6 . ritically damped oscillations: (Please show any calculations). Adjust R 0 to obtain critical damping of the oscillations. Value of resistor required for critical damping R 0 = Ω Estimated uncertainty = ± Ω.. L = = Predicted value of R c from nominal L and values: R c = Ω 3. ritical damping time constant Measured time constant at critical damping: τ = ms Predicted value of R from τ and nominal L value: R = Ω 4. omment on the above values. D. Overdamped oscillations: R 0 = Ω (see page 4 of instructions) Ratio R /(4L/) = >> Measured time constant for overdamping: τ = ms E. Observation: Note that the time constant for critical damping is the smallest observed in the experiment.