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1 Laboratory Section: Last Revised on December 15, 2014 Partners Names Grade EXPERIMENT 10 Electronic Circuits 0. Pre-Laboratory Work [2 pts] 1. How are you going to determine the capacitance of the unknown capacitor using the oscilloscope in the first part of the experiment? Explain. (1 pt) 2. In theory, what should be the slope of the graph you will make of your data when you plot 1/Q versus Resistance in the second part of the experiment? What value should the y-intercept have? (1 pt) TA or TI Signature 1 of 18

2 Last Revised on December 15, Purpose EXPERIMENT 10 Electronic Circuits To learn about the concept of capacitance, resistance and inductance; to learn about the phenomenon of electrical resonance in a real circuit; to learn how to use the oscilloscope. Please go to the following video link to get familiar with the oscilloscope before : (or just type Scope Primer on the youtube.) 2. Introduction You will be first studying RC circuits and then resonant RLC circuits. To do this, you will first review the use of an oscilloscope, the most versatile electronic measuring instrument. Then you will use this tool to investigate the characteristics of capacitors and resonant circuits. Review of the Oscilloscope An oscilloscope is an instrument principally used to display signals as a function of time. With an oscilloscope it is possible to see and measure the details of wave shape, as well as qualities like frequency, period and amplitude. While these signals are primarily voltages, all manner of signals can be converted into voltages for observation. The heart of an oscilloscope is a cathode ray tube (CRT), similar to that in a TV set, in which an electron beam excites a spot on a phosphor screen (Figure 10.1). The resulting visible spot of light is usually made to draw a graph where the y-axis is the measured signal and the x-axis is time. An "electrostatic deflection tube", in which the electron beam is Figure TA or TI Signature 2 of 18

4 Figure One way to accomplish synchronization is to adjust either the frequency of the input signal or of the sweep so that they match. This is often hard to do satisfactorily because things tend to drift; besides it may be experimentally inconvenient. Connect a function generator to an input of the oscilloscope (check that the input channel is turned on). Set the function generator for sine wave output with a frequency of 1000Hz and set the amplitude control at mid-range. Find the appropriate setting for the time base and the vertical gain. Make sure the oscilloscope is in free-running mode and attempt synchronization both by adjusting the frequency of the signal generator. You should observe the situation described above. A more satisfactory solution to the synchronization problem is provided by "triggered" operation of the oscilloscope. In this mode, rather than having the ramp start over as soon as the previous sweep is finished (called "free-running"), the time base waits to restart the ramp until the input crosses a certain voltage level ("triggers"). As long as this is the same point of the input wave form each time, the sweep across the screen will be synchronized with the repeating signal. This is illustrated in the Figure To use triggered operation, stop the sweep from free-running (set to "Norm" or turn the stability control counter-clockwise to near mid-range). Set the trigger selector to internal and activate the trigger button for the channel from which you want to trigger. Adjust the trigger level to stabilize the display. Observe what happens to the display as you vary the trigger level and change the sign of the trigger selection. (The auto position of the trigger level causes triggering to occur at the mean level of the input wave form.) Change the amplitude of the generator and note how the display changes. Measure the amplitude and frequency of the signal. To do this accurately, the variable controls on the time base and the vertical gain must be in the "cal"(ibrated) position. Also, observe the square and triangle outputs of the signal generator. TA or TI Signature 4 of 18

5 Figure Cautions: Don't let the trace exist as a dot on the screen; it can damage the phosphor. Don't look at big (>50V) voltages (i.e., the AC line). All signals, both input and output, are with respect to a common ground. This is a real earth ground through the chassis and the third prong of the AC plug. The shields of all the BNC connectors are connected together and to this ground and the input signals are applied between it and the center terminal of their BNC connectors. 3. Laboratory Work 3.1 RC Circuits Introduction A capacitor stores electric charge. Typically, a capacitor is made of two parallel plates. Each plate will have a charge of the same magnitude, Q, but of opposite sign between which there will be an electric field. The work (per unit charge) this field would do on a test charge moved between the plates is the potential difference or voltage, V, across the plates of the capacitor. The charge and voltage are linearly related by a constant C, the capacitance. Consider the situation in which a capacitor is charged by a battery through a resistance as shown in Figure 10.4 (with the switch in the Charge position). The charge stored on the capacitor (and therefore the voltage across it) increases with time because of the current, which is a flow of charge through the resistance. This current, by Ohm's law, is proportional to the voltage across the resistor, TA or TI Signature 5 of 18

6 I V B VC Equation 10.1 R This current will decrease as V C rises toward V B so the rate at which the capacitor is charging up will fall. When there is no further flow of charge the capacitor will be fully charged to its final value, Figure 10.4 Q final CV B Equation 10.2 The charge on the capacitor and the voltage across the capacitor do not grow linearly with time. Rather, they follow an exponential law. If the capacitor is initially (at t = 0) discharged, this takes the form, t V RC C VB 1 e Equation 10.3 After a time t = RC, the capacitor is charged to within 1/e (0.37) of its final value. The value RC is known as the time constant of the circuit. If the resistance R is in Ohms and the capacitance C is in Farads then the time constant RC is in seconds. Figure 10.5 displays voltage as a function of time for a charging capacitor. VC VB 0.63 VB t t=rc time Figure Similarly, if a fully charged capacitor is discharged through a resistance (e.g. by moving the switch in Figure 10.4 to the discharge position) the voltage across the capacitor (and the charge on it) will fall to zero exponentially with time, as shown in Figure 10.6, V C B t RC V e Equation 10.4 Again, after a time t = RC the capacitor will discharge to 0.37 of its initial value. TA or TI Signature 6 of 18

7 V C V B V B t=rc time Figure In the first part of this experiment you will investigate the charging and discharging of a capacitor for different values of the time constant. By observing the voltage across the capacitor with an oscilloscope you can measure the voltage as a function of time. Although for purposes of illustration it is convenient to discuss charging and discharging the capacitor with a switch, it is experimentally more practical to apply a square wave which duplicates the switching action using a signal generator, as shown in Figure This is a way of regularly switching the applied voltage between two values so that the capacitor can charge and discharge between them. The charging and discharging traces will be accurately reproducible from cycle to cycle. Then if the oscilloscope is triggered off the periodic input these successive traces will overlay each other on the screen. The "switching" can occur at a fast enough rate that the result will be a bright, stable display from which you can make measurements. If the period of the square wave is greater than several times the time constant, then the capacitor will very closely approach its final value before the square wave switches. R Signal Generator (Square Wave) C To Oscilloscope Figure Hook up the function generator, set it to produce square waves, and build the circuit shown in Figure 10.7 using (variable) resistor and capacitor decade boxes. Find a convenient amplitude and frequency and then look at and trigger on the output of the signal generator. Connect the other channel to observe the capacitor voltage at the same time. You should obtain a display similar to that shown in Figure TA or TI Signature 7 of 18

8 Figure Procedure 1. Set the decade boxes to the values R = 3,000 ; C = 0.01 F and the signal generator to a square wave with a frequency of 3000Hz and an amplitude of 3 Volts. Adjust the oscilloscope so the whole display is used (see figure 10.8) and trace it on a data sheet. Determine both the charging and discharging time constant by measuring the time taken for the voltage to change from its initial value to within 1/e of its final value, as discussed previously. 2. Average the charging time constant and the discharging time constant and compare with the theoretical value of RC. 3. Change the value of R or C by 30% and repeat Steps Get an unknown capacitor from the TA and substitute it for the capacitance decade box. Adjust the oscilloscope for a convenient display and measure the time constant as above. Use the result to obtain an experimental value for the capacitor. 5. Get an unknown resistor from the TA and determine its value in a manner similar to Step 4 6. Turn off the oscilloscope and disconnect all cabling. 3.2 Forced Damped Oscillator RLC Circuits Introduction In this section you will study the electrical version of a mechanical system that you have studied in your mechanics course. There is much similarity between the two cases as Figure 10.9 indicates. TA or TI Signature 8 of 18

9 On the right we have a mass, m, on a spring of constant, k and with damping b. It is being driven by an oscillating force of constant amplitude at an angular frequency (which can be varied). This system has a resonant frequency, Figure k o 2 f o Equation 10.5 m for which the response is a maximum when the driving frequency is equal to o. In the vicinity of o the response curve looks like Figure 10.10, where the width of the curve, f, (measured from the point where A A max 2 ) is given by = o /b. amplitude Amax f 0 Amax/1.4 Q f 0 f frequency width = f Figure TA or TI Signature 9 of 18

10 In this experiment you will investigate the general phenomenon of resonance in the form of the particular example of an RLC (resistor, inductor and capacitor) circuit. You will be able to determine steady state behavior as well as the Q (quality) factor. In the process you will gain some experience with electronic circuits and components. You will set up the circuit shown in Figure Be sure that the ground to the signal generator is the same as the voltmeter. Figure The basic equation that describes the phenomenon of resonance is that of a driven, damped, harmonic oscillator. In the case of the above RLC circuit, this takes the form, 2 d q dq q L R V cos t Equation dt dt c where q is the charge on the capacitor and V is the peak amplitude of the signal generator and 2f is the angular frequency. It is important to understand that the phenomena are characteristic of the mathematics and not peculiar to its electrical realization. For instance, if it were a mechanical oscillator the phenomena of resonance would be the same. The general solution of Equation 10.6 (written in terms of I dq, which is more practical dt to measure) has two parts, I = I s + I t, a steady state and a transient term. Theory on Steady State RLC The steady state term is, V I s cos t Equation 10.7 Z where the quantity Z, impedance is, and the phase offset is, Z L R Equation 10.8 C TA or TI Signature 10 of 18

11 tan L 1 C R Equation 10.9 You are going to measure the RMS voltage across the resistor, so V S I S R. The steady state current, Is is the term caused by the driving voltage; it is all that is left after initial transients have died away. The physical significance of these quantities can be made more apparent by expressing them in terms of the more universal quantities: resonant angular frequency, o, and the quality factor, Q. and Equation o 2f o Q 1 R L C LC Equation The quality factor Q is roughly the number of oscillations it takes for the transient to die down. Thus in terms of and Q we have, and o tan Q Z R Q R cos Equation Equation In the steady state it is apparent that at the resonant frequency the phase angle will be zero and the circuit will act just like a resistance R and the current will be maximum. Away from the resonant frequency, the amplitude of the current will decrease. The "full width at half maximum (half power =.707 of maximum current)" of the resonance curve will be about, 0 Q Equation that is it will be narrower, in terms of a fraction of the resonant frequency, as Q increases. Procedure The object in this part of the experiment is to identify the resonant frequency of an RLC circuit and measure the response curve similar to the one in Figure You will not be using the oscilloscope in this section. 1. Assemble the experimental circuit shown in Figure using the values: R 400 : C 0.01 F : L 25 mh TA or TI Signature 11 of 18

12 2. Calculate the resonant frequency (not the angular frequency), f o, in Hz. Set the signal generator frequency near the value calculated with an voltage amplitude of about 3 Volts. You can set the signal generator output precisely by unhooking the RLC circuit from the generator and directing the output only to the multimeter. Set the multimeter to the AC Volts setting, with the sensitivity set to the 20V setting. The AC Volts setting is the one with a symbol like so: V~. 3. With the RLC circuit set up, measure the voltage across the resistor. Now vary the frequency f to find the maximum voltage (it will likely not be the full 3V). This frequency will probably be a little different from your calculation. 4. Take many (at least 4 on each side of f o ) measurements of V in the vicinity of the resonant frequency ensuring that you tune the signal generator through a broad range of frequencies so you see the voltage drop off by at least one-third on each side of the peak voltage. Record your data in the tables provide in Section Now change R to 200 and repeat the measurements. 6. Now change R to 100 and repeat the measurements. TA or TI Signature 12 of 18

13 Last Revised on December 15, 2014 EXPERIMENT 10 Electronic Circuits 4. Post-Laboratory Work [18 pts] 4.1 RC Circuit [8 pts] Data [4pts] (2pts for raw data) A. Known Values Data Sketch the pattern you see on your oscilloscope, as in Figure You will want to note what value one division on the x-axis is equal to in seconds. R = : C = : RC = Time of Charging (to within 1/e of final value) Time of Discharging (to within 1/e of final value) Average TA or TI Signature 13 of 18

14 B. 30% Values Data Sketch the pattern you see on your oscilloscope, as in Figure 10.8 R = : C = : RC = Time of Charging (to within 1/e of final value) Time of Discharging (to within 1/e of final value) Average C. Unknown Capacitance Data Sketch the pattern you see on your oscilloscope, as in Figure R = : C = Unknown : RC = Unknown Time of Charging (to within 1/e of final value) Time of Discharging (to within 1/e of final value) Average TA or TI Signature 14 of 18

15 1. Calculate the unknown capacitance value from your measured average for RC (1pt): D. Unknown Resistance Data Sketch the pattern you see on your oscilloscope, as in Figure 10.8 R = Unknown : C = : RC = Unknown Time of Charging (to within 1/e of final value) Time of Discharging (to within 1/e of final value) Average 2. Calculate your unknown Resistance from your measured average of RC (1pt): TA or TI Signature 15 of 18

16 Analysis [4 pts] 3. What systematic or random errors can you think of which might account for differences between the measured and calculated values of the time constant? List at least two.(2 pts). 4. How accurately do you think you can measure a time constant from your traces of the oscilloscope pattern? Justify your answer. (2 pts) 4.2 RLC Circuit [10 pts] Data [3 pts] L C f o Frequency (Hz) R = 400 Voltage (V) Frequency (Hz) R = 200 Voltage (V) Frequency (Hz) R = 100 Voltage (V) TA or TI Signature 16 of 18

17 Analysis [7 pts] 5. Plot Voltage vs. Frequency ( y vs. x!) for all 3 data sets on a single grid, plotting the data for R = 400 first as it should be broadest. Use 3 different markers: solid circles for 400, hollow squares for 200, and small x s for 100. Connect each different data set with a smooth line forming a bell-shape curve. (2 pts) 6. How do the resonant frequencies compare in the 3 plots? Is this expected? Explain. (1 pt) 7. From your plots, measure the Quality factor, Q, of each of the 3 curves. Show your work for at least one value. Use this variant of Equation (Hint: See Figure 10.10) (1 pt) f o Q f TA or TI Signature 17 of 18

18 8. Calculate Q for each of the three resistances: 100, 200, and 400 ohms. Show work for at least one of the calculations. Use Equation (1 pt) 9. Plot 1/Q (measured from your plots) versus R on the small grid below. Is this a straight line, as expected? Draw a best-fit line and measure its slope. How close is your measured slope to what it should be? Hint : Remember Equation (2 pts) TA or TI Signature 18 of 18

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