Part I: Measurements of AC Voltage using the Oscilloscope and the Multimeter

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1 Phy 3311 Electronics Lab #7: The Series RLC Circuit In this lab we will study AC circuits, in which the principles of reactance, impedance, phase differences and resonance will be explored. You will measure frequency, voltage and current using your multimeter and your oscilloscope. Pay attention to the pros and cons of the multimeter and the oscilloscope, since they are complementary tools for the analysis of AC circuits. Part I: Measurements of AC Voltage using the Oscilloscope and the Multimeter Set your function generator for a sinusoidal output: V(t) = V 0 cos(ωt), and connect it to Channel 1 of your oscilloscope. Using the Measure tools, adjust the function generator s amplitude until V pp = 2 V. Note: the oscilloscopes have a Probe setting which will multiply voltages by a factor of 1x, 10x, 100x or 1000x. You will want to be sure this setting is 1x for both channels. The Probe setting is accessible on the right-hand side of the screen after pressing the CH 1 Menu or CH 2 Menu buttons. If V pp = 2 V, what is V 0? V p? V rms? V avg? (for sinusoidal: V p = V pp /2; V rms = V p / 2; V avg = 2V p /π.) Adjust the frequency until you get f = 1 khz. (You can get a rough value by reading the knob on the function generator, and a more accurate value using the Measure tool on the oscilloscope.) At f = 1 khz, what are T (period) and ω? (Use the Measure tools.) Oscilloscope Coupling: Try adjusting the Coupling you can measure DC Coupling, which sends the signal directly to the oscilloscope. You can set it to AC Coupling, which effectively shorts out any DC component, so you only see the time-varying (AC) part. And you can couple to Ground, which flatlines everything out to V=0 (it s a good way to see where V=0 is located!) Next, set one of the multimeters (either Extech or Fluke) to the Freq or Hz setting. Using the meter like a voltmeter, (i.e. across the source) measure f. Are your results consistent with each other? Then, set your multimeter to the AC Voltage (V ~ or VAC) setting. Using the meter, measure the voltage output of the function generator. (Note that these multimeters will read out rms voltage!) Is the result consistent with what you got from the oscilloscope? (Note that under the CH1 Menu and CH2 Menu on the oscilloscope, there is a Probe setting which will multiply the input voltage by a factor of 1, 10, 100 or 1000, for use with special probes that attenuate the signal voltage. So if there is a factor of 10 n between what your meter and what your oscilloscope say, check the Probe setting!) Finally, set your multimeter to DC Voltage (V - - ). What value do you get? Is this what you expect? Discuss some of the advantages and disadvantages of each measuring tool (oscilloscope and multimeter). What can you get, and what can t you get, with each? What sort of issues would be involved in choosing between a multimeter and an oscilloscope for a given measurement?

2 Part II: The Series RLC Circuit Impedance and Phase Angles Note: For the rest of the experiment, you will be using both channels of the oscilloscope to make simultaneous measurements. The oscilloscope requires that both traces have a common ground, that is, that both of the black terminals are connected to the same place in the circuit. This means that you may have to re-order the components (i.e. switch around the R, C and L). If the two traces do not have a common ground, you will get garbage traces! Set up the circuit shown at right, using f = 1 khz, V 0 = 1 V, R = 1 kω, C = 0.1 µf and L = 35 mh. Measure the capacitance of your capacitor, the resistance of your resistor, and the internal resistance of the inductor using your multimeter. (Unfortunately, there is no straightforward way to measure the L of the inductor, so we will just use the nominal values (which actually are pretty close). As long as the internal resistance of the inductor is small compared to that of the resistor, we can just add the two together: R circuit = R inductor + R resistor and proceed.) At f = 1 khz (ω = 2π krad/s), what are the values of X C (= 1 / ωc ) and X L (= ωl)? For this series RLC circuit, compute the impedance Z, and its magnitude Z and phase angle φ, in both rectangular and polar/phasor form: Z = R + j(x L -X C ) = Z φ Z Z = Z = sqrt( R 2 + (X L -X C ) 2 ) φ Z = tan -1 ( (X L -X C ) / R ) (complex impedance, rectangular and phasor form) (magnitude of impedance) (phase angle of impedance) Now that you know the impedance, compute the current that should be drawn by the circuit: I 0 φ I = (V 0 0) / (Z φ Z ) = V 0 /Z -φ Z (current, expressed in phasor form) I 0 = V 0 /Z φ I = -φ Z (magnitude and phase angle of current) For this circuit, is X L > X C, or is X C > X L? That is, is the circuit capacitive or inductive? What sign would we expect φ Z to have? Would we expect the current to lead the voltage, or vice versa? One point to be aware of: recall that the function generator has an output impedance of Z = 600 Ω. In most cases such an output impedance is resistive, so we can treat it as an extra resistor R int = 600 Ω in series with the circuit. Therefore, the total resistance of the circuit is technically R circuit +R int, not R circuit. However, the output voltage V 0 we are measuring is located after R int, so we will calculate Z using only the R = 1 kω resistor (plus the R of the inductor), and for now pretend that R int doesn t exist. (It becomes important when the value of Z changes, since this circuit acts effectively as a voltage divider between Z internal and Z external ; the effect is that the output voltage of the function generator may change as the load impedance changes. But the circuit impedance is computed using only what s in the circuit.)

3 Connect Channel 1 of your oscilloscope across the output of the function generator. This channel will read out the voltage V(t) being delivered to the circuit by the function generator. Connect Channel 2 of your oscilloscope to measure the voltage V R (t) across the resistor. Because the oscilloscope is in parallel with the resistor, V osc (t) = V R (t). Since the oscilloscope cannot directly measure current, you will use V R (t) as a proxy, since I(t) = V R (t)/r, and the magnitudes relate by, e.g. I pp = V Rpp /R. The magnitude of I (i.e. I pp ) can thus be determined from a measurement of V osc_pp = V Rpp, and the phase angle φ I between V(t) and I(t) is the same as the phase angle φ R between V(t) and V R (t), since for a resistor V R and I are always in phase. (This is a trick we ll use in the future, so remember it!) Remember that the black terminals of both channels must be connected to the same point in your circuit! Using the Measure and/or Cursor tools, measure the values of V Rpp (the resistor s peak-to-peak voltage) and φ R (the phase angle between V(t) and V R (t)), and from them determine the values of I pp and φ I. How does this current compare with what you calculated from AC Ohm s Law (I pp = V pp /Z)? To determine φ R, use the relation φ R = 360 t/t, where t is the time difference ( Delta ) between when the two traces reach a certain point (i.e. minimum or maximum) as determined by your cursor tools, and T = 1/f is the period of your signal. To confirm your measurement of the magnitude of the current I 0, connect a multimeter to directly measure I rms. Is it consistent with what you got from the oscilloscope? In this circuit, does the current lead or lag the voltage? Does this match with what you expect (i.e. ELI the ICE man? ) How does the phase angle φ I compare to what you expect from theory? If not, do you think the amount of error you have is reasonable? Part III: Checking Kirchhoff s Voltage Law in the Series RLC Circuit Just as we measured the magnitude and phase angle (relative to the source voltage) of V R (t): V R0 and φ R in the last part, do the same for V C (t) and V L (t). That is, four measurements: V L0, V C0, φ L and φ C. Note that to do this you will need to re-arrange the order of the components in the circuit so that the black terminals of the two oscilloscope inputs are coincident: the oscilloscope only functions when both inputs share a common ground! Otherwise you re going to get garbage traces try it and see what happens. (Specifically, the circuit should be re-arranged so that the component you re measuring is on the bottom of the circuit diagram on page 2, in place of where the resistor was drawn.) Also note that, just as the voltmeter acts like a large R vm ~ 10 MΩ, our oscilloscope is effectively a big R = 1 MΩ in parallel with a very small C = 20 pf ( very large X C at our frequencies!), connected across the component being measured. This means that, just like in the case of the voltmeter when a resistor R effectively becomes an R eff = R R vm, when an oscilloscope is hooked across an X C or X L that element is no longer a pure X C or a pure X L, but an impedance whose phase angle φ Z is slightly less than 90

4 because it has a large impedance connected in parallel. As long as the parallel impedance is large the effect is hopefully negligible, just as it is for voltmeters in DC circuits. We can write down the four voltages in phasor form as V = V 0 0 V R = V R0 φ R V C = V C0 φ C V L = V L0 φ L From theory, we expect the magnitudes will be (using our AC Ohm s Law relations) V R0 = I 0 R = V 0 R/Z V C0 = I 0 X C = V 0 X C /Z V L0 = I 0 X L = V 0 X L /Z and the phase angles are (with φ I = -φ Z ) φ R = φ I φ L = φ I + 90 φ C = φ I 90 (since resistor voltage is in phase with the current) (since inductor voltage leads the current by 90 : ELI ) (since capacitor voltage lags the current by 90 : ICE ) How do the values you measured compare with what you expect from your calculations? According to our friend Mr. Kirchhoff, the sum of the voltage drops around any loop should be zero. For DC circuits this works beautifully, as long as we re careful to keep track of plus and minus signs. For AC circuits this works beautifully as well, as long as we re careful to keep track of phase angles, or, equivalently, the real and imaginary parts. (DC circuits are sort of a special case of this, where the phase angles are limited to 0 and 180 degrees plus and minus signs!) In time-dependent form, KVL states for our series RLC circuit that V(t) V R (t) V C (t) V L (t) = 0 where the time dependences can be accounted for using phasors: (V 0 0) (V R0 φ R ) (V C0 φ C ) (V L0 φ L ) = 0 To check this, we need to convert these four voltages into rectangular form, using V = V real + jv imag with V real = V 0 cos(0) and V imag = V 0 sin(0) V R = V Rreal + jv Rimag with V Rreal = V R0 cos(φ R ) and V Rimag = V R0 sin(φ R ) V C = V Creal + jv Cimag with V Creal = V C0 cos(φ C ) and V Cimag = V C0 sin(φ C ) V L = V Lreal + jv Limag with V Lreal = V L0 cos(φ L ) and V Limag = V L0 sin(φ L ) and see if the sum of the real parts, and the imaginary parts, separately add up to zero: (V real V Rreal V Creal V Lreal ) = 0 and j (V imag V Rimag V Cimag V Limag ) = 0 If they do not add up to zero, what do you get, and what errors do you suppose could contribute to this?

5 Part IV: The Series RLC Circuit: Resonance and Q If time permits (extra credit) Use the same circuit diagram as in the last section. But now try R ~ 50 Ω, L ~ 35 mh, and C ~ 1 µf. The resonant frequency of this circuit is given by ω res = 2πf res = 1/sqrt(LC) Calculate the ω res of your circuit. At resonance, X L = X C = X res, and the circuit has a Q-factor given by Q = X res /R tot = = ω res L/R tot = (1/R tot ) sqrt(l/c) Again, we have to be a bit careful, as there are two components contributing to R tot : The R ~ 50 Ω resistor, and the R L ~ 25 Ω internal resistance of the inductor. Therefore, R tot = R + R L from which it follows that Z and Z = Z as a function of frequency are given by Z = R tot + j(x L -X C ) Z = Z = sqrt(r tot 2 + (X L -X C ) 2 ) Since there is no way to directly measure Z (no Z-meter! ), we will measure it using our oscilloscope: Z = V/I where V is the terminal voltage (we ll use V pp, which is the easiest to measure with the oscilloscope) delivered by the function generator to the circuit, and I is the current (we ll use I pp ) through it. We will determine I by measuring the voltage V R across the resistor, and using the relation I = V R /R to find Z = R V/V R (where R is only the resistance of the resistor!) Note that the Z int = 600 Ω impedance of the power supply does not contribute to the circuit s impedance. As discussed in Part II, this is because the voltage V is measured at the output terminals; V represents only the voltage left over after passing the Z int = 600 Ω internal impedance, so our calculation for Z is unaffected by Z int. The only effect of Z int is to make V variable, since there is an effective voltage divider in place between Z int and Z, and Z varies with f. Specifically, as the impedance of the load decreases, the terminal voltage V will decrease as well, since the current drawn will be larger and hence a greater fraction of the total voltage will be dropped across the supply s internal impedance. Therefore, at resonance, where Z is minimized with Z = R tot, V will have a minimum value. Tune the frequency up and down (starting around the resonant frequency you calculated), until you find the resonant frequency (it will be a fairly broad range). This is the frequency at which (a) V is minimized, and (b) the ratio of V/V R (which is proportional to I) is maximized. (Note that the value of V R is not necessarily maximized at f res, since our terminal voltage V drops as we approach the resonant frequency as well it s a question of which effect the dropping of V, or the increasing of V/V R is larger.) How does this measured resonant frequency compare with your calculated value?

6 How does the impedance at resonance compare with your expected value? Based on the current drawn, what fraction of the total voltage is being dropped across resistors? Is that what you expect? (Remember that there are two resistors in your circuit: the resistance of your resistor R, and the resistance of your inductor R L!) What is the phase angle between voltage and current at the resonant frequency? (Measure it using your Cursor tools!) How does that compare with your expectations? At frequencies well above and well below resonance, qualitatively note the phase angles φ I between voltage and current. Are they what you expect? (ELI the ICE man, Z is minimized at f res!) Don t bother measuring them, though, just make a note of which one lags the other as you change frequency. Now, tune up and down in frequency, covering from ~200 Hz up to ~ 5 khz over about 10 steps. At each frequency, measure V and V R and determine the circuit s impedance Z = RV/V R. Compare these measured impedances to your theoretically calculated values from Z = sqrt(r tot 2 + (X L -X C ) 2 ). At the lowest and highest frequencies, measure the phase angle between V (total) and I using the Cursor tools. How do these compare with theoretical expectations based on φ I = tan -1 ((X L -X C )/R tot )? Make a table of Z versus frequency. If you see any gaps (i.e. large changes in Z between adjacent points), fill in some extra points, until you have a nice table of data showing Z s continuous variation with frequency. I would suggest doing this in Excel, which will make your calculations easier too! When you get home, plot Z versus ω (or f). Fit this to a curve in order to determine the resonant frequency ω res, and Z at resonance. How do these values compare to your theoretical expectations? (If you have a hard time fitting try making the x-axis ln(ω) rather than ω, and see if it looks better!) When an AC voltage V rms is applied to an RLC circuit, the apparent power S= S and real power P are S = V rms I rms = V rms 2 /Z P = S cos(φ I ) = (V rms 2 /Z) cos(φ I ) To find P, in addition to using the measured value of Z at each frequency you will need to calculate the phase angle of the impedance φ Z. (Yes, calculate the phase angles at each frequency measuring them isn t terribly accurate anyway.) Since φ I = -φ Z = -tan -1 ((X L -X C )/R tot ), the power factor, and real power, are cos(φ I ) = R tot /Z P = (V rms 2 R tot /Z 2 ) The power delivered to the circuit at ω = ω res (since Z = R tot ) is P res = V 2 rms /R tot, and the points at which P = ½P res are known as the 3 db points, and are separated from each other by the bandwidth ω (also known as the full width at half maximum, or FWHM). Determine the 3 db points using your graph of P versus ω (try fitting it to some theoretical function), and determine the bandwidth ω. Finally, determine the Q-factor of this circuit Q = ω res / ω, and compare it to the expected value Q = X Lres /R tot.