#19 RLC Circuits and Resonance - Part II Objectives 1. To further investigate the behavior of RLC circuits. 2. To become more familiar with the experimental aspects of resonance. Introduction and Theory Just as with the RC circuit which was studied in Experiment #17, we can ex- tend our study of the RLC circuit to its response to a sinusoidal driving voltage. In fact, many of the most important practical applications of the RLC circuit make use of its frequency response characteristics. Consider the circuit of Fig. 19.1, noting that it is identical to that of Fig. 18.2 (see page 32) except for the addition of a sinusoidal voltage source whose potential is given by V = Vo cos ot (19.1) 020 R L M m V=Vo cosot C puit on Fig. 19.1. Sinusoidally driven RLC circuit. + ce cQ = Vo cosot where the angular frequency o may be set to any value we wish. In analyzing this circuit, the only change from Eq. 18.10 (see page 32) is that the right side is no long- er zero (corresponding to no applied voltage) but rather Vo cos ot. Thus, we have d dQ L +R V. . (19.2) dt? dt The variation with time of the capacitor charge Q is now described by the func- tion which is the solution of Eq. 19.3. In finding the solution, we proceed just as with the RC circuit; i.e., we assume that the solution is sinusoidal with the same frequency as that of the driving voltage, but with a phase difference. That is, Q = Qo cos(ot + o). (19.3) We then substitute this expression and the appropriate derivatives into Eq. 19.2 to find the values which Qo and Ộ must have for it to be a solution. As in Experi- ment #17, we expand the functions sin(ot + ) and cos(ot + Ø) using the sum formu- 36 las, and then collect terms in sinot and cosot, insisting that the coefficients must separately vanish for the same reasons as in Experiment #17 (see page 26). Carrying out this program we find the following results: R tanº= (19.4) OL -1/0C and Qo = Vo sino OR (19.5) Eq. 19.4 shows that at very low frequencies tan ø is very small and negative, so that ø is a small negative angle and sin 0 = 0. Therefore, from Eq. 19.5 we see that Qo~0. At very high frequencies Eq. 19.4 shows that tan ois very small and posi- tive, which means that ø is approximately -180° and sin o is once again = 0 and, from Eq. 19.5, Q. ~ 0. Eq. 19.4 shows that tan ø is a maximum (+00) when the de- * nominator vanishes, i.e., when oL = 1/0C or when a = 1/ Note from V/LC 18.19a (see page 35) that this is simply the natural frequency 0, at which the system will oscillate on its own; that is: Lc. Note from Eq. - 0. 1/ V/LC: (19.6) The frequency 0, is called the resonance frequency. Whenever a system (mechanical, electrical, etc.) is driven at a frequency equal to its natural frequency, resonance occurs. Under resonance conditions, a system will oscillate with its max- imum amplitude. VES of ston o novo ou ( SU 37 Fig. 19.2 shows graphs of (a) Q. vs. o and (b) º vs. o with the resonance fre- quency 0, indicated in each case. Note that resonance occurs when 0= -90°. Qo (a) 00 0 cal 0 001 0 bove 00 -90 + 0 of the gas -180 (b) an R Fig. 19.2 Amplitude and Phase of the response of an RLC Circuit as a function HOT of frequency Ols od Local Bu il os olhos trata sono Apparatus You should have the following items on the lab table. -Oscilloscope -Signal generator -Circuit board -Assorted wires roton WE bogst hos ovo 38 Experimental Procedure 1. Set up the circuit shown in Fig. 19.3. Connect the oscilloscope so that CH1 measures the input signal and CH2 measures the capacitor voltage, and set the os- cilloscope so that both channels are displayed. Set the GND level to the midline of the grid. In order to study the frequency response of the RLC circuit to a sinus- oidal driving voltage, select sine wave on the function generator, choose the 100 kHz range, and turn the FREQUENCY control fully counterclockwise. Set the CH2 vertical scale to 5V per division. Increase the input frequency and you will see the amplitude of the trace rise to a maximum and then begin to decrease; if in this process the trace exceeds the maximum screen height, decrease the AMPLI- TUDE control on the generator so that the trace is fully visible at its maximum value. Carefully adjust the frequency to obtain the maximum amplitude capaci- tor voltage. Record this value as the resonance frequency, fo, along with its un- certainty. R= 1k92 L= 27 mH mm V=Vo cosot +C = 100 pF 5 Fig. 19.3. Experimental RLC circuit. 2. With the function generator set at the resonance frequency of the RLC circuit, ad- just the AMPLITUDE control on the generator so that the CH2 trace completely fills the screen in the vertical direction. By doing this you will have set the peak- to-peak amplitude of the signal to 40 V (8 DIV x 5 VOLTS/DIV). Adjust the CH1 vertical scale so that its trace fills about one-half the screen. Record both values of Vpp along with uncertainties. Do NOT change the AMPLITUDE or the VOLTS/DIV controls for the remainder of this experiment! 3. With the function generator set at the resonance frequency of the RLC circuit, measure and record in the appropriate line of the data table: (1) the capacitor peak- to-peak voltage, V., (2) t, the smaller time interval between adjacent peaks of the two curves using the vertical cursor lines, and (3) T, the period of the input signal. Repeat for frequencies of 60 kHz to 110 kHz in steps of 5 kHz
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