23a HW p18 WB in Lab 9

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2023 Winter EECS 314 HW p18 Wheatstone bridge in Lab 9 HW problem 18 (80 points) Wheatstone bridge in Lab 9 For each part of this problem, show you work to get credit. Part One (10 points) Balancing the Wheatstone bridge in theory Wheatstone bridge circuit is a pair of voltage divider powered from the same source V S ; the output voltage of the bridge is measured / calculated between the middle points of these voltage dividers: see HW p18 Figure 1. HW p18 Figure 1. Wheatstone bridge circuit is balanced when its output voltage equals zero. Refer to HW p18 Figure 1: derive the algebraic relation for resistors in a balanced bridge. Do not assume that any resistor equals any other resistor in this circuit. Pay attention to the notation: they may differ from those in the readings, lecture notes, etc. Write your answer: R 1 /R 2 = R 3 /R 4 © 2023 Alexander Ganago Page 1 of 12 Last printed 3/28/23 2:13:00 PM File: 451ddf4ad071cb7adefe42edad1bacf745289d33.docx

2023 Winter EECS 314 HW p18 Wheatstone bridge in Lab 9 Part Two (30 points) Unbalanced Wheatstone bridge with a potentiometer Assume that all resistors in a Wheatstone bridge equal each other, for instance, R 1 = R 2 = R 3 = R 4 = 120 Ω Then, you definitely obtain in calculations that the bridge is balanced (HW p18 Figure 2 panel a). When you build a real circuit of four resistors from a 120 Ω bin, you will most likely obtain an unbalanced Wheatstone bridge, because the actual resistances may differ from their nominal values. For balancing a real Wheatstone bridge circuit, you need to include a small potentiometer such as R P = 10 Ω (HW p18 Figure 2 panel b). At both limit positions of the potentiometer’s tap, the bridge will be unbalanced (HW p18 Figure 2 panels c, d). (a) (b) (c) (d) HW p18 Figure 2. Panel a shows a balanced Wheatstone bridge circuit with 4 identical resistors, which is a theoretical model. Panel b shows a Wheatstone bridge circuit with a balancing potentiometer. Panels c and d explain the conditions, at which the output voltage reaches its limits. © 2023 Alexander Ganago Page 2 of 12 Last printed 3/28/23 2:13:00 PM File: 451ddf4ad071cb7adefe42edad1bacf745289d33.docx

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2023 Winter EECS 314 HW p18 Wheatstone bridge in Lab 9 Part Two, continued (10 points) Use the numerical values of the resistors and potentiometer listed above, and V S = 10 V , calculate the minimal and maximal output voltages. Write your answers: V WB,Min = -200 mV V WB,Max = 200 mV V WB,Min = V S * (R 2 /(R 1 + R 2 + R P ) – R 4 /(R 3 + R 4 )) V WB,Min = 10 * (120/(120 + 120 + 10) – 120/(120 + 120)) = -0.2 V V WB,Max = V S * ((R 2 + R P )/(R 1 + R 2 + R P ) – R 4 /(R 3 + R 4 )) V WB,Max = 10 * ((120 + 10)/(120 + 120 + 10) – 120/(120 + 120)) = 0.2 V (a) © 2023 Alexander Ganago Page 4 of 12 Last printed 3/28/23 2:13:00 PM File: 451ddf4ad071cb7adefe42edad1bacf745289d33.docx

2023 Winter EECS 314 HW p18 Wheatstone bridge in Lab 9 (b) HW p18 Figure 3. The output voltage of a Wheatstone bridge circuit with the nominal values as listed above is measured with AD2 Scope channel 1 (the red trace). Panel a shows the reading of V WB, Min ; panel b shows the reading of V WB, Max . Disregard the readings in channel 2 (blue trace). Part Two, continued (10 points) Calculate the % difference between the measured and the calculated values of the difference V WB, Max − V WB, Min , which you calculated and which you determined from HW p18 Figure 3. V WB, Max − V WB, Min expected from theory V WB, Max − V WB, Min measured in HW p18 Figure 3 % Difference (measured – expected) 400 mV 362.98112 mV 9.25% Theoretical = 200 – (-200) = 400 mV Measured = 190.46 – (-172.52112) = 362.98112 mV % difference = (400 – 362.98112)/400 * 100 = 9.25% (5 points) Assume that the potentiometer is circular, and the resistance between its end connection and the tap is a linear function of the tap’s angular displacement. Calculate the change of the output voltage in mV when the tap is moved by 1 Ω: ∆V WB 1 Ω = 40 mV ∆V WB /1Ω = (200 – (-200))/10 = 40 mV (5 points) Assume that the potentiometer’s tap is moved by the angle of 300˚ between its limit positions. Calculate the displacement in ˚ between the positions of the tap, which corresponds to the change of output voltage by 1 mV. ∆angle 1 mV = 0.75 ˚ ∆angle/1mV = 300/400 = 0.75˚ © 2023 Alexander Ganago Page 5 of 12 Last printed 3/28/23 2:13:00 PM File: 451ddf4ad071cb7adefe42edad1bacf745289d33.docx

2023 Winter EECS 314 HW p18 Wheatstone bridge in Lab 9 Part Three (10 points) Signals from the tuning fork, before amplification In Lab 9, you will build a Wheatstone bridge circuit (HW p18 Figure 4 panel a), in which resistor R SG = 120 Ω belongs to the strain gauge mounted on the tuning fork (HW p18 Figure 4 panel b). When the tuning fork vibrates, the strain gauge is compressed and stretched, which alters its resistance R SG ; therefore, the output voltage of the Wheatstone bridge varies. Typical data are shown in HW p18 Figure 4 panel c. (a) (b) (c) HW p18 Figure 4. Panel a shows Wheatstone bridge circuit, which includes strain gauge R SG = 120 Ω bonded to the tuning fork (panel b). Panel c presents typical output signal of the Wheatstone bridge circuit caused by the vibrations of the tuning fork. © 2023 Alexander Ganago Page 6 of 12 Last printed 3/28/23 2:13:00 PM File: 451ddf4ad071cb7adefe42edad1bacf745289d33.docx

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2023 Winter EECS 314 HW p18 Wheatstone bridge in Lab 9 Part Three, continued (5 points) From HW p18 Figure 4 panel c, assess the peak-to-peak amplitude ∆V TF in mV of the Wheatstone bridge output voltage due to the vibrations of the tuning fork. Write your result: ∆V TF = 3.4 mV (5 points) Use your result from Part Two to calculate the change ∆ R SG of the strain gauge resistance in Ω. ∆ R SG = ¿ 0.085 Ω ∆V TF /1Ω = (200 – (-200))/10 = 40 mV ∆R SG = 3.4/40 © 2023 Alexander Ganago Page 8 of 12 Last printed 3/28/23 2:13:00 PM File: 451ddf4ad071cb7adefe42edad1bacf745289d33.docx

2023 Winter EECS 314 HW p18 Wheatstone bridge in Lab 9 Part Four (10 points) Amplified signals from the tuning fork In Lab 9 you will build a high-gain difference amplifier for the output signals from the Wheatstone bridge circuit with the tuning fork (HW p18 Figure 5 panel a), and measure the signals before and after amplification (HW p18 Figure 5 panel a). HW p18 Figure 5. The Wheatstone bridge output signals are fed to the inputs of difference amplifier (panel a). Typical data in panel b provide information about the amplifier’s gain and the frequency of the tuning fork’s vibrations: the red trace belongs to channel 1, which shows the signal before amplification; the blue trace belongs to channel 2, which shows the amplified signal. (a) (b) (10 points) From HW p18 Figure 5 panel b, assess the amplifier’s gain: V OUT , ppk V ¿ , ppk = 571.94 V OUT, ppk /V In, ppk = 2.1406/0.0037427 = 571.94 © 2023 Alexander Ganago Page 9 of 12 Last printed 3/28/23 2:13:00 PM File: 451ddf4ad071cb7adefe42edad1bacf745289d33.docx

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2023 Winter EECS 314 HW p18 Wheatstone bridge in Lab 9 Part Five (20 points) Avoid clipping of the amplified signals In HW p15 and in Lab 8 you learned about voltage clipping of amplifiers’ output signals. HW p18 Figure 6 shows a clipped output signal from the tuning fork, which was hit too hard but this is not the most important cause of clipping. HW p18 Figure 6. Amplified signals can be clipped if the tuning fork was hit too hard. The most important cause of clipping in Lab 9 is poor balancing of the Wheatstone bridge. For example, consider the following: o The amplifier’s output voltages V OUT may be clipped if V OUT > 3 V or V OUT ← 3 V . o The amplifier’s gain equals 1,000. o Then, to avoid voltage clipping of the output, the safe range of input voltages from the Wheatstone bridge must be from –3 mV to +3 mV. o This safe range is much narrower than the range V WB, Max − V WB, Min , which can be achieved by moving the potentiometer’s tap. o Therefore, to obtain output signals free from clipping, the potentiometer’s tap should be very close to the position, which corresponds to a perfectly balanced bridge. (5 points) From the safe range of amplifier’s output voltages –3 V to +3 V and the amplifier’s gain, which you calculated in Part Four, calculate the safe range of amplifier’s input voltages in mV. Your result: The amplifier’s input voltages should be between -5.2453 mV and 5.2453 mV. -3/571.94 * 1000 ≤ V In ≤ 3/571.94 * 1000 -5.2453 ≤ V In ≤ 5.2453 Part Five, continued © 2023 Alexander Ganago Page 10 of 12 Last printed 3/28/23 2:13:00 PM File: 451ddf4ad071cb7adefe42edad1bacf745289d33.docx

2023 Winter EECS 314 HW p18 Wheatstone bridge in Lab 9 The sketch in HW p18 Figure 7 shows two limits of the amplifier’s input (=Wheatstone bridge) voltages, within which the amplifier’s output voltages would not be clipped. HW p18 Figure 7. The long-dashed lines show the limits of input voltages, which correspond to the range of output voltages free of clipping. The short-dashed lines show the input voltages, which are obtained when the fork is not vibrating. ∆V ¿ + ¿ and ∆V ¿ − ¿ correspond to the limit positions of the potentiometer’s tap, within which the output signals are not clipped. (15 points) Refer to HW p18 Figure 7. Calculate the limits of angular displacement (in degrees) of the potentiometer’s tap, which correspond to ∆V ¿ + ¿ and ∆V ¿ − ¿ In other words, determine by how much can the tap be moved to ensure that the amplifier’s output voltage would be free from clipping. Use the following numerical values: o The amplitude of the Wheatstone bridge signal (= the amplifier’s input ) equals 1 mV peak-to-peak. o Use the amplifier’s gain, which you calculated in Part Four. o Use the ratio ∆angle 1 mV , which you calculated in Part Two. Your answer: The range of safe angular positions is from - 214.4775 ˚ to 214.4775 ˚. Output = Input * Gain = 0.5 * 571.94 = 285.97 mV © 2023 Alexander Ganago Page 11 of 12 Last printed 3/28/23 2:13:00 PM File: 451ddf4ad071cb7adefe42edad1bacf745289d33.docx

2023 Winter EECS 314 HW p18 Wheatstone bridge in Lab 9 -285.97 * 0.75 ≤ Angular position ≤ 285.97 * 0.75 -214.4775 ≤ Angular position ≤ 214.4775 © 2023 Alexander Ganago Page 12 of 12 Last printed 3/28/23 2:13:00 PM File: 451ddf4ad071cb7adefe42edad1bacf745289d33.docx

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