PHYSICS 122 Lab EXPERIMENT NO. 6 AC CIRCUITS


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1 PHYSICS 122 Lab EXPERIMENT NO. 6 AC CIRCUITS The first purpose of this laboratory is to observe voltages as a function of time in an RC circuit and compare it to its expected time behavior. In the second part the resonance frequency of a series RLC circuit is measured and compared with the expected value. Equipment: 1 AC Signal Generator 1 Oscilloscope 1 Board with resistors, a capacitor and an inductance 7 Wires (4 red, 3 black) 6 Clamps (crocodile) AC Voltage Generator Oscilloscope Clamps Wires Board with Components. Note: The 1000 ohm resistor is not always in the same place! Fig 1 1
2 2 Inductor L= H 7 Capacitor C= 10 F Resistor R=100 Ω Resistor R=1 kω Fig 2. Two of the component boards used, note that you need to be careful as not all boards have the different components in the same place! In this lab, you use the oscilloscope to study some properties of alternating current (AC) circuits which involve capacitors and inductors. In the previous lab you worked with simpler direct current (DC) components, specifically, resistors. The important difference between the two types of components is that the behaviors of the AC components components depend on the rate of change of the input voltage/current, i.e., the frequency f of the driving signal (see Ch20 sheet 24 and 26.). In this experiment the AC voltage is supplied by the AC signal generator. Part I: RC Circuits Your goal is to measure the capacitive time constant τ C in an RC circuit and compare it to the predicted value of τ C = RC. One way to charge and discharge a series RC circuit (see the summary in Ch20 sheet 39, 7,8 for charging the capacitor, sheet 9 for discharging) to use a DC source of electrical potential V 0 such as a battery, and a switch for connecting and bypassing the battery. This placed in series with a resistor and capacitor which for a series RC circuit are also in series with each other (see Fig 3a below). In this experiment, we will use a different approach; the voltage will be generated using the square wave output from the AC signal generator (see Fig 3b below). As you saw in Lab 2 the signal you would produce manually if you used the battery and switch would be the same as a square wave switching between 0 V and V B. One advantage of using the signal generator is that it can switch the voltage on and off much faster and more reproducibly than you could with your hand. 2
3 V B V R +V B 0 V R V C V C (a) (b) Fig 3. The two figures above show two circuits that are essentially equivalent in behavior If you think about Fig 3 (a) at time t = 0, we can assume that we begin with no charge difference on the capacitor C (C is measured in Farads) and that the switch is then set to connect the battery to the circuit. The voltage V C across the capacitor is then given by V C = Q/C, so that at t = 0 we know V C = 0. As the charge Q builds up on the capacitor in time, V c increases until it equals V B. The voltage across the capacitor when the capacitor is charging is given in Ch20 sheet 7. V 0 (1e 1 )V V 0 V C V 1 e t = RC 0 (1) t=τ C =RC t There are two important things to notice about this formula. First when t = 0 the exponential factor becomes 1. In formula (1) you should notice that this gives V C = 0. Second, notice that if the capacitor starts out with no charge on it then as t approaches infinity the exponential factor goes to 0. This means that if the capacitor is left charging long enough that its voltage will eventually equal to V 0. In the example above V 0 =V B and so t V = RC C VB 1 e (1A) The voltage drop V R across the resistor must satisfy V B = V C + V R. Using this formula and equation (1), you can eliminate the variable V C and derive what V R is equal to in terms of V B and e t/ τ c. Which gives us Equation (2). 3
4 V 0 (e 1 )V V 0 = V e t RC V R B (2) t=τ C =RC As with V C you can look at V R for the cases with t = 0 and t approaching infinity. In our experiment, unlike in the example above, the AC generator output voltage changes between V0 and + V0 (not between 0 and + V0 ). The magnitude of the voltage change is thus 2V 0 (not V 0 )! You can see that for the capacitor voltage in equation (1) above that the curve approaches V o exponentially (as demonstrated in the formula for capacitor voltage) when it rises from 0 to V o. In your lab, it will do the same thing when it rises from V o to +V o. When charging from +V o to V o, the curve approaches exponentially V o (ie looks like a falling exponential). For this lab you will look at the capacitor charging from V o to +V o. Also, instead of voltmeters we will be using the oscilloscope to look at the voltage coming from the signal generator, the voltage across the capacitor, and the voltage across the resistor. The diagrams that represent the two experimental configurations, the first to measure V C and the second to measure V R are shown below in Fig. 4(a) and 4 (b). +V 0 V CH1 CH V 0 V CH1 CH2   In the lab prep assignment you need to calculate the result predicted for τ C from τ C = RC. Since the resistor and capacitor in your circuit may be slightly inaccurate, use the values for R (with a 10% error) and C (with a 10% error) in your setup to calculate the predicted time constant and its error using equation (7) in Error and Uncertainty ( EU ). Make a note of the values you obtain for RC and it s error as you will need to enter them in to the Excel Analysis Sheet. 4
5 Procedure: Begin your set up by creating a series RC circuit following the diagram in Fig. 4a shown above. In your test setup, C = 107 F. There are several resistors available to be used in series. These values are shown in Fig. 2. First, connect the resistor and capacitor in series. For this part you will use the 1 kω resistor. Make sure that you are using this resistor labeled with this value, which is not in the same place on all the boards. Connect Ch. 2 of the oscilloscope in parallel with the signal generator. Connect the positive terminal of Ch. 1 of the oscilloscope to the point between the resistor and capacitor. Connect the negative terminal of Ch. 1 of the oscilloscope to the negative terminal of the signal generator. Now connect the positive terminal of the signal generator to the end of the resistor that is not connected to the capacitor. After that, connect the negative terminal of the signal generator to the end of the capacitor that is not connected to the resistor. When you are finished your set up should look like Fig. 4a. Make sure the ground connections are made exactly as given in the diagram 4a. It is important that all 3 grounds, ie the function generator ground, the CH1 ground and the CH2 ground are all connected together. Record the values for the resistance and capacitance in Execution Sheet 1. ground of generator both ground terminals of oscilloscope low voltage side of capacitor all connected together with black wires + (red) terminal of generator to + (red) terminal of oscilloscope CH2 high voltage side of capacitor to + terminal (red) of oscilloscope CH1 resistor to high voltage side of capacitor + (red) terminal of generator to resistor Fig 5(a): Wiring for Fig 4(a) 5
6 The Capacitor Voltage V C (t): In this part of the lab you will observe the voltage across the capacitor V C (t) with CH1 of the oscilloscope and the AC generator output voltage V AC (t) with Ch2. Use the AC generator output MAIN OUT LO with AMPLITUDE full clockwise. Dial a frequency of ~ Hz: FREQUENCY ~ 510 MULT 100 Set the oscilloscope to: COUPLING SOURCE AC, CH2 Both CH1 And CH2 inputs DC Both VAR (red buttons) CAL D (full clockwise) VERT MODE CH2 Both VOLT/DIV ~0.5 V ( such that you see the square wave) VAR SWEEP CAL D (full clockwise) TIME/DIV ~ 0.2 ms (such that you see at least one period of the square wave). Adjust the TRIG LEVEL so that you see a stable picture on the oscilloscope. Now set the oscilloscope to VERT MODE DUAL. You should see both voltage signals V AC (t) and V C (t). Using the VERTICAL POSITION buttons center your voltage signals on the vertical center of the screen. Record the settings for FREQUENCY, VOLT/DIV for CH1 and CH2 and TIME/DIV in Execution Sheet 1. Sketch the observed pattern of V AC (t)and V C (t) on Execution Sheet 1. Your sketch should try to be as accurate as possible and must include axes scales and labels. Next you are going to analyze the curve to obtain τ C. First, look back at Eq. 1A describing the capacitor voltage. Notice that when t = τ C the equation becomes V C = V 0 (1e 1 ). When you put in the value of e, this will give you V C = 0.63*V 0. Note however that as you are going from V 0 to V 0 you need to find the point that is 0.63*2V 0 above V 0. Find this voltage on your curve and then find the time at which is occurs. This time will be your experimental value for τ C. Excel Analysis Sheet Questions: Enter the values you calculated earlier for τ C and its error in the lab prep assignment in to the appropriate boxes on the Excel Analysis Sheet. Then, enter you measured value for τ C and estimate the error for your measured value of the time constant. Are the two values consistent within error? 6
7 The easiest way to wire this up is to simply swap the wires that come from the function generator to the series resistor/capacitor combination. Fig 5(b): Wiring for Fig 4(b) To measure V R you need to exchange the positions of the resistor and capacitor and connect Channel 1 of the oscilloscope parallel to the resistor so that you can monitor V R. Your set up should look like as Fig. 4b. Again, make sure the ground connections are made exactly as given in the diagram 4b. It is important that all 3 grounds, ie the function generator ground, the CH1 ground and the CH2 ground are all connected together. Observe the voltages V AC (t) and V R (t) and sketch them into your Execution Sheet. You will notice that V R (t) approaches zero (and NOT + V 0 or V 0 ). This follows from V 0 = V C + V R : since V C approaches V 0, V R approaches zero. You should also notice that the initial V R is (2 V 0 ) above 0, since the voltage change is (2 V 0 ) and the initial current is (2 V 0 )/R. You are now going to obtain a measurement of τ C from the voltage across the resistor. As before, adjust the TIME/DIV and VOLT/DIV settings so that you have one decaying exponential curve on the screen. Sketch this curve on Execution Sheet 2. At t = τ C, V R = e 1 *2V 0, which gives V R = 0.37*2V 0 Find the point on your curve where this value occurs for V R. Then find the time associated with this voltage. This will be your 7
8 measured time constant, τ C from this part of the experiment. Make sure you estimate the error for your value. Excel Analysis Sheet Questions: Enter your measured value of the capacitive time constant τ C from V R (t) and include an estimate of the error. Is this value compatible with the τ C measured from V C (t) within your estimated errors? Part II: Resonant AC Circuits: Your goal is to measure the resonance frequency of an in series RLC circuit and compare it to the predicted frequency you find by setting the reactance of the circuit to zero, i.e. minimizing the impedance and hence maximizing the current for a given voltage. The following exercise is in the MapleTA Lab Prep Assignment and the resulting equation is important for this part of the lab. You want to find the resonance frequency f 0, which is the frequency which minimizes the impedance Z (see Ch20 sheet 31) of the in series RLC circuit by causing the reactance to be zero (see Ch20 sheet 32). From Ch20 sheet 31, enter the equation for reactance (use f 0 to denote the frequency in your equation). Then solve for f 0, the resonance frequency. Calculate the resonance frequency f 0 from the equation and get its value, using the values of the inductance L and the capacitance C in your setup (see Fig 2). As before, since you cannot count on these values being completely accurate, propagate the errors of L and C ( assume 10% for both) into the error for f 0 using expressions (7) and (8) in EU. Make a note of the values you get, you will need to enter them in to the Excel Analysis Sheet. As the resonance frequency f 0 maximizes the current I = V/Z it also maximizes the voltage across the resistor R which is V R = I R. To find f 0 you will vary the frequency of the AC generator until you observe maximum voltage V R. Procedure: Wire the circuit as given in the diagram below in Fig. 6. Begin with a series loop with the signal generator, the resistor, the capacitor, and the inductor Make sure you connect the low voltage side of the resistor to the ground (black) of the generator and the oscilloscope. Make sure you wire the capacitor, inductor and the resistor in series exactly as shown in the diagram. Use the 100 Ω resistor (see Fig 2 above) for this part. You will be observing the voltage across the resistor so attach Channel 1 of the oscilloscope parallel to the resistor. You will not need to look at the voltage from the signal generator, so disconnect it from Channel 2. When you have wired your circuit it should look like Fig. 6. 8
9 High voltage side of the resistor Low voltage side of the resistor Fig 6 Switch the AC generator from square wave to a sine wave by pushing in the button labeled with a sine curve. Set the generator frequency at ~3000 Hz. Set the VOLTS/DIV to ~0.2 V and set the TIME/DIV at ~0.1 ms. Set the scope VERT MODE and the COUPLING SOURCE to CH1. You should see a couple of periods of the sine curve. Enter the set values and the value of the inductance (see Fig.2 ) into Execution Sheet 3. Vary the frequency of the generator smoothly between ~1000 and ~10000 Hz and observe the size of V R. Set the frequency of the signal generator to the setting where you think the maximum V R occurs. Record this frequency as your first guess for the resonance frequency on Execution Sheet 3. Leave the frequency at this setting and measure the period of the wave you see on the oscilloscope, convert this to frequency and record the value on Execution Sheet 3. The ratio r of the frequency determined with the oscilloscope over the dialed FREQUENCY of the generator will be used as a correction factor when extracting f 0 from your graph below. Now dial FREQUENCIES in ~ 5 steps of ~ 500 Hz below the maximum and record the dialed frequencies and your measured values for V R in Execution Sheet 3. Repeat for ~ 5 steps of ~ 1000 Hz above the maximum. (Note that you may have to readjust the TRIGGER LEVEL 9
10 knob closer to 12 o clock for a stable picture as your signal becomes smaller. You also may have to change the TIME/DIV in order to see a full period clearly). Excel Analysis Sheet Questions: Enter the ratio r of the actual frequency over the dialed frequency into the box provided on the Excel Sheet. Transfer your values of V R and dialed frequency into the table on the Excel Sheet. In the third column of the table calculate the dialed frequency times r which gives you the actual frequency. The Sheet will graph V R versus frequency. Estimate your best value f 0 from your graph including an error estimate and enter it in the box on the sheet. Enter the value of the resonance frequency and its error that you calculated in the lab prep assignment in to the appropriate boxes on the Excel Sheet. Are the measured f 0 from your graph and the calculated value of f 0 consistent within error? 10
11 PHY 122/124 EXPERIMENT 6 AC circuits EXECUTION SHEET #1 (to be signed by instructor) Value: 12 points Name: Section: SB#: Date: TA: TA Signature: The capacitor voltage V C (t): Quanities/Settings for the measurement Capacitance Resistance Frequency Volt/Div(CH1) Volt/Div(CH2) Time/Div Unit Value Sketch your observed voltage traces for the RC circuit V AC (t) and V C (t) into the graph paper below. Label the axes including units and indicate which trace is which voltage. Further indicate the range in voltage and range in time you used for the calculation of the time constant τ C. The measured τ C including an estimate of the error is:. 11
12 PHY 122/124 EXPERIMENT 6 AC circuits EXECUTION SHEET #2 (to be signed by instructor) Value: 12 points Name: Section: SB#: Date: TA: TA Signature: The resistor voltage V R (t): Quantities/Settings for the measurement Capacitance Resistance Frequency Volt/Div(CH1) Volt/Div(CH2) Time/Div Unit Value Sketch your observed voltage traces for the RC circuit V AC (t) and V R (t) into the graph paper below. Label the axes including units and indicate which trace is which voltage. Further indicate the range in voltage and range in time you used for the calculation of the time constant τ C. The measured τ C including an estimate of the error is:. 12
13 PHY 122/124 EXPERIMENT 6 AC circuits EXECUTION SHEET #3 (to be signed by instructor) Value: 6 points Name: Section: SB#: Date: TA: TA Signature: Part II: Resonant AC Circuits: Quantities/Settings for the measurement Capacitance Resistance Inductance Frequency Volt/Div(CH1) Time/Div Unit Value First guess for resonance frequency as shown on frequency generator dial: [ ] First guess for resonance frequency (1/period) as measured on oscilloscope screen: [ ] The ratio r of the more accurate frequency measured with the oscilloscope over the nominal FREQUENCY setting of the generator is (no error). Enter the values for the voltage V R across the resistor in Fig 6 and the FREQUENCY settings of the generator for the 10 steps into the table below: step FREQUENCY Voltage V R [ ] [ ]
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