Transcript
Experiment # 3 The Wheatstone Bridge—Practical Applications. ENGN/PHYS 207—Fall 2012
Foreword The goals of this lab are to:
• Build a Wheatstone Bridge, and understand why it is able to make sensitive measurements. • Utilize the bridge to measure deflection of a beam (a very practical setting!). • Enjoy a first introduction to the wonderful world of amplifiers. 1
The The Whea Wheats tsto tone ne Brid Bridge ge
1.1 1.1
Intr Introdu oduct ctio ion n
The Wheatstone Bridge 1 , (WB) (WB) is the the circ circui uitt show shown n in Figu Figure re 1. This This typica typicall configura configuration tion of the WB consists of four resistors in a series-parallel configuration, a constant voltage source (“excitation” to the WB), and a voltage gage (“output” of the WB). Excitation is supplied by your power power supply at nodes a and d. The output is obtained by reading reading the potential potential difference difference between between nodes b and c: V out V c out = V b
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The WB is basically a very sensitive resistance measurement device, owing its high level of sensitivity to the fact that it is basically a difference difference amplifier. amplifier. Thought Thought it was invent invented ed circa 1833 (people have been clever for a very long time...), it still finds widespread use today in many engineering applications applications—mec —mechanica hanical, l, aerospace, aerospace, and civil engineering, engineering, to name a few. The basic idea is that the resistance resistance of one of the bridge legs can vary vary with time. This leg of the bridge could be a thermothermocouple. If the temperature changes, then the resistance changes, so the output changes accordingly. Or one of the bridge legs could be a strain gage2 . If the structure structure to which which it is attached attached deflects or vibrates, vibrates, the bridge bridge will report this motion as a change change in its output output voltage. Clearly Clearly,, then, it would be fun (and instructive!) to build and analyze one. 1.2
Theore Theoretic tical al Consid Considerat eration ionss
1. The bridge output output is defined defined as the voltage voltage difference difference between between nodes b and c: V out out = V b On which respective nodes would you place the (+) and (-) probes of the DMM?
− V . c
2. Show that: V out out =
R1 R4 R2 R3 V s (R1 + R2 )(R )(R3 + R4 )
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(1)
Interesting historical side note: Wheatstone didn’t actually invent this circuit. Credit for the first description of the circuit goes to S.H. Christie; but Wheatstone is the one who found widespread practical use for this circuit. 2 A strain gage is basically a variabl variablee resistor. resistor. It essentiall essentially y consists consists of long, thin piece of metal metal patterne patterned d in a shape designed to detect detect specific types of deformati deformations ons (stretching, (stretching, compressing, compressing, twisting). twisting). Changing Changing it’s shape basically changes the values for A and L in R = AρL
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Figure 1: Basic Wheatstone Bridge Circuit. The voltage source V s provides excitation at nodes a and d. The output (gage reading) V out is measured across as the difference in voltages between nodes b and c. The bridge is said to be “balanced” when V out = 0. Starting in a balance, changing the value of R4 will make V out = 0. In practice, R4 might be a strain gage, thermocouple, flexible resistor—anything that transduces one physical property into a change in resistance.
Hint: V b and V c nodal voltages are be easy to compute. Once you get them, V out is in the bag. 3. The bridge is said to be “balanced” when V out = 0. What is the relationship between the resistances R1, R2, R3, R4 when the bridge is balanced? 4. Assume all resistors in your WB are equal R1 = R2 = R3 = R4 = R, so your bridge is balanced (V out = 0). Then let R4 increase its resistance by a relatively small amount: R4 R4 + ∆R. For the case that ∆RR << 1, show that:
→
V out
V ≈ ∆R 4R
(2)
s
5. Make a quick sketch of V out vs. ∆R. Explain how you interpret the meaning of the slope of this line? 6. Imagine you build a WB circuit to measure the deflection of a beam in a building during an earthquake. R4 might be a strain gage whose resistance changes as the beam flexes. Thus, V out is indicative of the strain in the beam. What are some inherent trade offs with the setting the proper ratio of ∆RR ? In other words, what is good and bad about having this ratio be really small (<< 1)? What is good and bad about having this ratio be relatively large ( 1)?
≈
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7. As we’ve oft discussed in class, electrical circuits can often be thought of in terms of their analogy to mechanical or fluid system. So, before we finish this exercise, let’s think about what the mechanical/fluid analog of the WB would be. Devise an mechanical/fluid analog to the WB. Fully explain how your proposed fluid/mechanical measurement system is analogous to the WB. Describe this analogy in terms of what it would mean for your mechanical/fluid system to be “balanced” or “imbalanced”. What would you use in place of a voltmeter to measure any imbalances in your fluidic bridge?
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Experiment I: Getting Acquainted with the WB
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1. Now build the bridge (see Fig. 1). Set V s = +5 V. Use 1 kΩ resistors for R1, R2, and R3. (Remember to carefully measure and record the actual resistance of each of these.) For R4, use a 2 kΩ pot. Balance your bridge. Carefully measure and record the resistance of the pot with your circuit balanced. How do your measurements compare to the expected result developed in question 3 above? 2. Imagine R4 to be a strain gage element attached to an airplane wing to measure its vibrations during flight (as you might do in a real engineering project one day!). If the wing flexes upor downward, the resistance will change by an amount ∆R. We’ll simulate this for now by turning the dial on the 2 kΩ pot. Sweep the pot through a range of resistance from about 500 – 1500 Ω, in about 50–100 Ω increments. Carefully measure and record V out for each setting. Make a plot of V out vs. ∆R. 3. Analyze and discuss your result in the context of Eqn 2. Is the graph linear? Everywhere? Or are there seemingly non-linear regions? In your analysis/discussion, carefully consider the validity of the assumptions made when deriving Eqn 2.
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Experiment II: Real-life Application of the WB
Now you will use actual strain gage elements (see Fig. 2(a)) configured as a Wheatstone Bridge to measure deflections in a beam (e.g., see Fig. 2(b)). Where would you do something similar in real life? There are many examples! For instance, you might test the structural integrity of a bridge with a strain gage in a WB bridge configuration; you might examine the response of an airplane wing to an impact; you might test how earthquake isolated a building is; and so on. (I did promise that circuits is actually useful...hopefully this helps convince you). The output of strain gage bridge is quite small (approximately 2–20 mV), just barely large enough to see on your oscilloscope. We will amplify this signal to make it easier to see and measure. To do so, we’ll make use of a very, very useful piece of electronics called an instrumentation amplifier . We’ll use the model known as the INA 1264 Later in the term, we’ll discuss how its inner workings. For now, you just need to know that it amplifies the input signal by a gain factor G. Your amplifier will be wired for a gain G = 50. Therefore, the output of the amplifier is given by: V amp = 50V out . 3
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No Dawson’s Creek, Buffy the Vampire Slayer, or other half-baked, crappy reruns here. Just good ole plain useful circuits. 4 Amplifiers are kind of like cars. By analogy, general category is automobile (amplifier); a specific model is a Honda Civic (INA 126). The datasheet for the INA 126 is available at: http://www.ti.com/lit/ds/symlink/ina126.pdf
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(a)
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Figure 2: (a) Strain gage. The resistance changes when thin metal film “comb” shape deforms. In a non-stressed state, the gage typically has a resistance of a few hundred ohms. (b) Building undergoing vibration. Strain gauges can be used to measure and record deflections vs time. Image credit: http://w-ave.de/resources/vibration-measurement/building-vibration.html where V out is the output of your strain gage—which is also the input to your instrumentation amplifier. For instance, if your bridge has an output V out = 50 mV (possible to read, but difficult with scope), the amplifier will multiply this by a factor of 50 and output V amp = 2500 mV = 2.5 V (much easier to cleanly read on a scope). Detailed instructions how to wire your circuit are as follows:
• Bridge excitation: Connect the Red wire to +10 V. Connect the Black wire to GND. • Route bridge output to amplifier: Connect INA126 pin 2 to the White conductor of the bridge. Connect INA126 pin 3 to the Green conductor of the bridge.
• Route power connections to your amplifier: Connect INA126 pin 4 to -10 V and pin 7 to +10 V.
• Provide a reference for the amplifier: Connect INA126 pin 5 to GND. • Setting the gain of the amplifier: Connect a 1780 Ω resistor between pins 1 and 8 of the INA126. This sets the gain to be G = 50.
• Measure the amplified bridge output: You an oscilloscope to do it! The positive scope scope should connect INA126 pin 6 (the amplifier’s output). Additionally, connect the ground probe to INA126 pin5 (the amplifier’s ground).
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Figure 3: Pinout of INA126. Note that pin 1 is in the upper left corner, with the chip properly oriented.
• Use a C-clamp to fasten your the beam, with attached strain gage sensors to the bridge.
Orient your beam so that the strain gauges are near the table, the eyelet for loading your bridge at the distal end.
• With your beam firmly fastened to the table, simulate an impact on the beam.
View the resulting signal on the oscilloscope to determine the frequency of oscillation. Measure the
• Modify your beam by adding weight in reasonable increments to the end of the beam with
a mass m. This might simulate changing the design for load-bearing beams in buildings, or on a car chassis; or it might simulate changing the overall design of the airplane wing. At any rate, investigate how response of the beam changes depending on the size of the mass attached to it. Repeat for at least 5 different loads.
• A beam can be modeled as a spring, and—as you will recall—a spring-mass simple harmonic oscillator obeys the equation:
k x=0 m where x denotes the vertical position of the beam, m is the mass/load applied to the end of the beam, and k is the effective spring constant of the beam. x ¨+
This leads to a solution of: x(t) = A cos(ωt + φ)
where the natural frequency is ω = k/m. Therefore, the frequency of oscillation should be inversely proportional to the square root of the applied load: f
∝ √ 1m .
How does your data compare with this theory? To investigate this question, make a log-log plot of frequency of oscillation (f ) vs the applied load (m). What is the slope of the best-fit line to your data? What should it be according to simple-harmonic oscillator theory? In this context, analyze and discuss the result you obtained. Wrapping up, note that you could also measure static deflections as a result of loading your bridge. It is easiest to use the digital multi-meter for this task, sine they are best suited to measuring dc voltages. If you have time, you could make a plot of applied load vs. bridge output (voltage). This plot would serve as a reference if you wanted to measure how much, say, a bridge sags when cars and trucks are loading it. Hopefully this portion of the experiment convinced you that knowing a little circuits really does pay off no matter what field/discipline you ultimately work in. 5
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Part II: Alternate—Choose your own Adventure
The beam oscillation experiment was just one example application of a Wheatstone bridge. By no means is it the only one. You should feel free to invent your own quick study/application. We have thermocouples (changes resistance upon changing temperature); flex sensors (kind of like a flexible potentiometer, see Fig. 4(b)). There must be a technical element of experiment with some theory to which you can reference your results. Please be sure to run your idea by the instructor before beginning—we’ll make sure the project is feasible and of appropriate technical scope.
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Figure 4: (a) Airplane wing: come fly the friendly skies. (b) Flexible sensor: the resistance increases as the sensor is bent. Let’s hope none of us ever see an airplane wing doing that! However, note that a bat’s wing might work well in such a configuration.
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The Write-Up
Your report will consist of 3 main parts: 1. Section 1.2 (Theoretical Considerations): Be sure to complete questions 1–7. 2. Section 2 (Experiment I): Be sure to address all all questions in 1–3. The figure of V out vs. ∆R is the center piece, and you should write a brief discussion to explain/interpret what you saw (something like 1 paragraph should be sufficient). 3. Section 3 (Application to Beams): No need to write an Intro or Methods. Write the Results and Discussion/Conclusions section. Be certain to address the following: What did theory predict? What did you measure? How do those compare/contrast? Regarding this last point, make a quantitative comparison. Hard numbers are golden. This also means you must use your data to st back up any claims you make. 4. Section 4 (Application: Choose your Own Adventure): In addition to Results and Conclusion/Discussion, please include a brief Intro and Methods if you opt for your choose your own adventure. Basically, say what system you are studying, why it is important (and/or cool and fun); and describe your setup in sufficient detail that a Circuits-knowledgeable friend could replicate the experiment.
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