I. RC circuit Charge and Discharge : Theoretical Results

Transcription

1 The University of Hong Kong Department of Physics Experimental Physics Laboratory PHYS2255 Introductory Electricity and Magnetism LABORATORY MANUAL Experiment 2: The A.C. Circuitry This experiment illustrates the main properties of a simple resistance-capacitance (RC) circuit, including the charge and discharge curves of capacitor through a resistor. I. RC circuit Charge and Discharge : Theoretical Results A. Charging to final voltage o (Constant voltage o is applied to the capacitor) R I o o C 1/2 o (1-e -1 )o = o t 1/2 2 t At t 0, the charge at the capacitor q(0)=0. After the switch is closed, o R C i t R q t C, where the current i t dq t dt, so we get dq(t) + 1 dt RC q(t) = o (1) R By solving (1) using the above initial condition, the charge and the voltage on the capacitor can be obtained as: q(t) = o C(1 e t/τ ), C (t) = q(t) C = o(1 e t/τ ), (2) where τ = RC is called the time constant of the circuit. When C (t) reaches o /2, t 1/2 = τ ln 2. B. Discharge from initial voltage 1 ( Charge relaxation ) When t 0, the initial charge and voltage on the capacitor are q(0)=q 1, and 1 Q 1 C, respectively. Now closing the switch of the circuit, then R C 0, i.e., dq t 1 q t 0 dt RC By solving (3) using the initial condition, the voltage across the capacitor is t q t C e t / (4) C 1 (3) 1

2 R I 1 C -1 1 e = t 1/2 2 t II. RC Circuit Charge and Discharge : Experimental Details In this experiment, we aim to test both the charging and discharging for short time constant (order of milliseconds) RC circuits using a signal generator and a CRO. First please check the internal resistance R i of your signal generator using the method described in part (H) of Laboratory 1 on both square and sine waves at frequency 800Hz. Set the Output Multiplier to ( 10). Record your checked result at Output Adjust 5 (i.e. the peak-to-peak amplitude of the trace) on the report sheet. (A) Displaying charge/discharge curves Setup the following circuit, initially taking R = 10k, C = 0.1 F. Note the use of a double-pole switch to allow you to disconnect from the RC circuit and connect the signal generator to the CRO. Be sure you understand the switch connections: they can be a little confusing at first. Set the signal generator to square-wave. Switc h S O Signal Generator O 2 CRO fig.1 If we apply a square wave to the RC circuit, during the positive half-cycle the capacitor C will charge up according to equation (2); and during the negative half-cycle it will discharge according to equation (4). Choose a signal generator frequency such that the half-period of the square wave, T 2, equals to time constant of the circuit. With the switch S in position 1, measure T 2 using the CRO time scale. Record the period T and hence the frequency f (=1/T). Meanwhile, also record the nominal frequency read from the signal generator 2

3 Now turn the switch S to position 2 and adjust the ARIABLE setting. Change R by a factor of at least 4 to make T 2, and then change R again to make T 2. You should get displays as shown below: (i) 1 T ~ (ii) 1 T >> (iii) 1 T << e.g. R = 10k e.g. R ~ 2k e.g. R ~ 35k (B) Measurements of time-constant, It is clear that only in case (ii) above is the majority of the charge or discharge curve observed. From such setting one can measure t 1 2 as a fraction of T 2, both from the charge and the discharge cases, and deduce. Note : You can spread out the waveform to show only slightly over T 2 if you wish. The ratio of t 1 2 to T 2 is of course independent of the ARIABLE setting. Now measure the time-constant in cases (i) and (ii) and record the results on the report sheet. You should be able to get similar display as cases (i), (ii) and (iii) with different input frequencies. In fact any combination of f, C, and R giving the same product CR 2 fcr (or the same ratio /T ) should result in the same display. (C) The "Differentiating" Circuit Now return to your original frequency chosen in part (A) and R = 10k. Interchange C and R in the circuit of Fig.1 and observe the shape of the waveform (i.e. the variation in time of the potential difference) across R. Repeat for larger and smaller values of R and sketch the various waveforms on the report sheet. You should find that with a small enough R (and hence ) you get alternately positive and negative sharp spikes corresponding to each sharp rise and fall of the square wave input voltage. Measure the height of the voltage spikes p. You should find that p corresponds to s from the previous measurements. For CR<<1, i.e., for small time-constant, the output waveform is approximately the derivative of the input waveform. Hence this circuit is called a "differentiating" circuit. Note that for the circuit of section (B) with CR>>1, the output waveform is approximately the integral of the input waveform. That circuit is therefore called an "integrating" circuit. 3

4 (D) oltage across the Capacitor (Please study the course Lecture Notes Chapter 10, or Chapter 29 of the textbook Physics for Scientists and Engineers by Tipler & Mosca for theories on phasors and AC circuits) From the phasor diagram of Fig. 2 below, it is readily shown that the voltage across the capacitor co is given by co= ox/z, where X=1/ C and Z 2 R 2 X 2, so that co o ( RC) 2 1 (5) In this equation co and o may be both peak-to-peak voltage (more convenient when using the CRO) or both rms (more convenient if using a voltmeter). The phase difference between and c is given by tan R X CR (6) The negative sign means that c lags behind. Note that the instantaneous values of is given by o sin t, with being negative. R RI R Signal Generator C CRO - J I wc oi Fig. 2 c Take C=0.01 F, R=20k, and choose a value of f such that CR is about 1, and set up the circuit of Figure 1 with the signal generator on sine wave and set to your chosen frequency. Record the values. According to the chosen values of f, R and C, calculate the expected value of phase difference o with equation(6). Repeat this calculation for a lower and a higher frequency (e.g. f/10, 10f). Complete Table 1 in the Report. The phase difference (representing the time difference between corresponding maxima of and c) can be measured directly in at least three different ways. Each of these illustrates a new technique of using to CRO which you will need to learn and try out first. In this experiment two ways are to be tried. In each case it is useful first to be able to see qualitatively the difference between (i) a low frequency (e.g. f/10) for which ~0 (ii) your chosen frequency f for which ~ /4 (one-eighth of a period) (iii) a higher frequency (e.g. 10f) for which ~ /2 Now you will try the first way. In this method, please use CRO to measure o by switching S to position 1 according to the circuit shown in Figure 1. Then measure co by switching S to position 2. The phase difference measured this way is denoted by 1. Complete Table 2 in the Report. 4

5 (E) Measure Phase Difference using External Triggering If we use internal triggering, the waveforms of and c start at the same point in their own cycle and so no phase difference can be detected. If we instead use the external triggering, the phase difference of and o can be measured from the waveform shown on the screen. Connect the signal generator sine-wave output () to the EXT TRIG (EXT HOR) input terminal, set the SOURCE to EXT, the COUPLING to AC, the SLOPE to +, and the Level lock to "Lock position". Then when we switch c using switch S set to 2, the sweep is still triggered by, so the whole waveform is shifted, as shown below in Figure 3(ii), which shows that c lags, in this case by an angle of / d Fig. 3(i) Fig. 3(ii) Fig. 3(iii) Try this : the phase difference is easily seen from the relative shift in the positions of the peak. Change to a much lower frequency such as f/10 and to a much higher one such as 10f in turn, and see whether you can confirm the effects shown in the figure above i.e. ~0 and ~ /2 respectively. If the period corresponds to a distance d, then the phase difference can be determined from 2 d. Now, measure by this method for the three frequencies used in the first method in. section (D) above. The phase difference measured by this way is denoted by 2. Complete Table 3 in the Report. Switch the SOURCE back to CH1 or CH2 and disconnect the EXT TRIG terminal. Note : We can also use the CH1 and CH2 terminals to show both waveforms simultaneously. The method is : connect the signal generator Output () to the CH1(X) terminal, and the c to the CH2(Y) terminal. All the above switches are set to the original. Set the ERT MODE to DUAL and switch S to 2. Adjust the position so that both waveforms overlap. The phase difference can be easily found. 5

6 (F) The X input terminal, and Lissajou s Figures One may wish to apply one s own signal to the X plate instead of the internal sweep waveform normally used to give a time base. To do this switch the Time/Div to X-Y EXT HOR and ARIABLE to clockwise. And set the SOURCE to EXT. Use the Hz supply to the Y-input, and the signal generator O/P with Output Range set at 1 to the EXT TRRIG input terminal at the upper part of the CRO, and adjust olt/div and output multiplier to get the amplitudes of the two signals roughly equal. Caution : Note that the signal applied to EXT TRIG should not exceed 2 peak-to-peak. Another method is : to apply the Hz supply to the Y input CH1(X), and the signal generator O/P to the CH2(Y). Set the Time/Div to X-Y EXT HOR, the SOURCE to CH1 (X-Y), the ERT MODE to CH2 (X-Y). The same results can thus be obtained. Now adjust the signal generator frequency in turn to 25Hz, 50Hz, 75Hz, 100Hz, etc. When the frequencies are in an exact ratio like 2:1, 3:2, etc, the pattern becomes stable. The followings are some examples. Ratio(1:2) Ratio(2:1) Ratio(2:3) Ratio(1:1) The number of antinodes (maxima) on each side indicates the frequency ratio. This provides a method of adjusting a frequency to be in some exact ratio to a standard or given one; and also a method of calibrating the Signal Generator dial at a number of spot frequencies. Tabulate the signal generator nominal and measured frequencies corresponding to 25Hz, 50Hz, 75Hz, 100Hz and any others you wish, assuming the mains supply (and hence the 6.3 transformer output) is exactly 50Hz while the nominal reading on the signal generator is unreliable. Complete Table 4 in the Report. 6

7 Yet another method to try phase shift, (You may also want to give it a try!) Apart from the methods described in parts (D) and (E) to find the phase difference, it can be done in the following ways: (1) switch off the time base. (2) connect c across the CRO input terminals (i.e. to the Y-plates) (3) connect the signal generator output directly to the EXT TRIG input terminal, or another method as part (E). If the amplitude of the two signals are the same, a diagram like the following can be obtained. And the phase can be determined from sin B A. A=2y1 B=2 y1 sin The basic theory of this method is described here. The x and y coordinates are given by x x 1 cos t, y y 1 cos t, where is the frequency of the c and signal from signal generator. The distance B is just twice the y displacement at time when x 0, such as t 2. At this time, y y 1 cos 2 y 1 sin. So B 2 y 1 sin, as shown in the figure. We also have A 2 y 1, and thus sin can be obtained. 7