THE EXPERIMENT

Purpose

To study the natural oscillations in an electrical circuit; to study the effect of damping on these oscillations; and to gain familiarity with using the oscilloscope to make quantitative measurements.

Equipment

1. Oscilloscope;
2. signal generator (SWG) with sine wave and square wave outputs;
3. multimeter;
4. decade resistor boxes, accuracy about 2%;
5. assortment of components as listed below:
   • Resistor values of 10, 51, 100, 200, 510, 1000 ohms. The value tolerances are +/-10%.
   • Capacitor values of .001, .01, .1, and 1.0 F (this unit is denoted as MFD on the capacitors). The value tolerances are +/-10%.
   • Inductor values of .55, 2.5, 5.0, and 10.0 mH are available at each setup and a limited number of 25 and 50 mH inductors will also be available in each lab section. The inductance tolerances are +/-5%. Inductors also have series resistance, to be measured with the ohmmeter.
   • Numbered unknown inductors.
6. Miscellaneous connecting leads for constructing various circuits.

Preparation and lab work

You will be working with the LRC circuit driven by a signal generator and measured with an oscilloscope. Preparation consists of selecting component values and signal generator frequency to have a hope of being able to do the measurements. You also need to practice using the many formulas relating to what is being measured. A warning: the value of R in the formulas is the total resistance in the circuit, that is R_tot = R+ R_SG + R_L, where R_SG = 50 is the output impedance of the function generator, and R_L is the series resistance of the inductor (R_L 25 for preliminary calculations). The frequency setting of the SWG should be low enough that the natural circuit oscillations will have time to decay before they are restarted by another square wave jump from the signal generator.

Before coming to the lab, select a resistor, an inductor, and a capacitor from those given in the equipment list. Plan to study the oscillations in a series RLC circuit constructed from these components. You should calculate the expected natural oscillation frequency and the Q of the circuit (don't forget the SWG output resistance). The equations to use are all given at the end of the previous section. There are several important constraints on your component choices. The frequency should be within the bandpass limits of the scope (f < 1 Mhz or < 10⁷ ) to be measurable. Also, the Q value should be within the range 4 < Q < 15 so that there will be enough oscillations to permit some measure of the frequency. Select new component values from the list if your first choice is unsatisfactory for any reason. When you arrive in the lab, you should measure the actual values of R and R_L (with uncertainties) and check that the tolerances on the other circuit components are as listed in the manual. Your predicted values for and Q should include uncertainties (found in the usual manner by propagation).

Referring to Eqn(15), be sure you understand how the number of oscillations you might see will depend on Q. Note that the oscillation amplitude will decay to 1/10 of its starting value in the time equal to 2.3 decay times ( = 2.3 : you should show this result for yourself), and Eqn(15) says there will be Q/[[onesuperior]] oscillations in one decay time. Thus it is a simple matter to predict how many oscillations you will observe before the amplitude has decayed by 90%. Remember, you can check all your calculations in the lab by observing the actual circuit you have constructed here only on paper.

Construct the LRC circuit as diagrammed in Fig. 4. To measure the natural oscillation frequency for your circuit in the lab, you must first get the oscillations going and obtain a suitable display on your scope screen. It is recommended that you use an "external trigger" to provide a stable scope trace. Then using the calibrated sweep you can measure the time required for one or for a number of complete oscillations and thus determine their period which is the inverse of the frequency. Consider how you can determine empirically the circuit Q from measurements of the scope trace. One simple method is to make use of Eqn(15). Think through in advance each measurement you will want to perform in the lab. Try to anticipate and resolve any possible problems you might have to face later. Compare your measured values of and Q with the predicted values found earlier. Remember that in this case both the predicted and experimental values have uncertainties.

The decay time t_D can be found accurately by measuring the amplitude of the "peaks" (which define the envelope of the decay) at various times during the oscillations (refer back to Fig. 2). Measure the amplitude (with uncertainty) and time of the "peaks" you observe, trying to get at least ten points. Plot the amplitude as a function of time with error bars and "fit" the data to an exponential decay (i.e. y = A e^-x/B ). From the coefficients of the fit, determine a value (with uncertainty) of the decay time t_D = 2, and from this, a value of the Q (recall Eqn (15) ). Compare this value of Q to the predicted value as above.

Calculate for your circuit the extra resistance you will have to add to obtain critical damping. You can determine the critical damping resistance empirically in the lab by inserting the variable resistor into your circuit and slowly increasing its value while watching the scope trace to see when the oscillations just cease. By adding to this the other circuit resistances, you can compare the actual measured result with the result of your calculation.

Figure 4

Finally, in each laboratory room there are several different "unknown" inductors identified only by a number. Select one of these and determine its inductance as well as you can using only the equipment listed above. Devise your own method and describe it in your report. Be sure to give the unknown inductor number, its value as determined by your measurements, and the error limits you feel are appropriate.

Additional experiments (optional)

Look at the voltage waveforms across each of the components in the circuit and try to understand them in terms of the current, I(t), and the derivative or integral associated with that component. Change the components in your circuit. See how large a Q or how high a frequency you can achieve. What turns out to be the limiting factor in attaining larger Q's? Try to see the expected change in natural oscillation frequency which results from a change in circuit damping. Try other arrangements of the circuit components, such as connecting them in parallel. Can you understand the behavior you observe?