Experiment # 2
The Wheatstone Bridge—Practical Applications
ENGN/PHYS 207—Fall 2015
Due: Wed, Sep 30, 5 pm
The goals of this lab are to:
• Understand limitations of measurement equipment (multi-meter and oscilloscope)
• Become (re)acquainted with the oscilloscope and its various features
• 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.
Every measurement device has limitations. In the case of making a voltage measurement with your multi-meter or oscillscope, you are adding another element to the circuit. How does making making a measurement affect your circuit? Hopefully, not at all. But this is never the case. The order of the universe and the laws of physics dictate that we always change the system we are actually
trying to measure: Heisenberg bites us again! (See Fig. 1).
Figure 1: Say hello to Werner and Heisenberg. They aren’t my pet dogs, but I wish they were.
While they look cute, beware: Werner likes to lick, Heisenberg likes to bite.
Every time you make a voltage measurement, you must “bleed off” a little bit of current to go through the meter. This is very similar to how you have to actually release a little bit of air from a tire in order to find its pressure. In circuits parlance, the voltmeter loads the circuit you are measuring. Now, let’s dive in to figure out just how good or bad the measurement situation might be.
The left side of Fig 2 shows a voltmeter measuring the voltage across R2. The right side, shows
your voltmeter modeled as a resistor R input
—this is commonly referred to as the meter’s input resistance . Manufacturer’s will always specify the input resistance of a meter. Sometimes you have to dig into their tech specs, but it is there, if you look. The manufacturer of our voltmeters
(Fluke, Inc.) specifies an input resistance of R input
= 10 MΩ. This is a number that you should remember, carry around in your back pocket (surely this will impress your friends!).
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Vs
R1
R2
Vs
R1
R2 V R input
Figure 2: Left: Series circuit with ideal voltmeter.
Right: Same series circuit, but this time accounting for the non-ideal behavior of the voltmeter. The voltmeter is modeled as a resistance
R input
.
1. What is the resistance of an ideal voltmeter? How much current will flow through an ideal voltmeter? Briefly explain your rationale.
2. Build the circuit shown in Figure 2. Use values of
R
1
= R
2
= 10 kΩ. What voltage do you expect to measure across R
2
, assuming the volt meter (multimeter or oscilloscope) is ideal?
3. What is the voltage measured with the multimeter?
4. Time for a quick sanity check! Compare the predicted and measured values: Are the predicted and measured values in reasonable accord (say, a few % difference)?
5. Now change the value of the resistors R
1 and R
2
. Use R
1
= R
2
= 6.8 MΩ.
6. Calculate the voltage you would expect to measure across R
2 if the voltmeter were ideal (or the voltmeter were not connected in the circuit).
7. What is the measured voltage across R
2
?
8. Time for the quick sanity check again! Compare the predicted value (part 6 above) and
measured values: Are the predicted and measured values in reasonable accord (say, a few % difference)? Hint: they should not be.
9. Knowing that the meter has an input resistance R input
= 10 MΩ, go back and calculate the
voltage you expect to measure (i.e., see Fig. 2, right). How do these newly calculated values
compare to your measured values? Sanity check: Are they in good agreement?
10. What’s the take-home message? Suggest a reasonable “rule of thumb” for how large R input must be relative to the resistance of the element whose voltage you want to accurately measure. Briefly justify (1–2 sentences is sufficient.)
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The Wheatstone Bridge (WB) 1 , is the circuit shown in Figure 3. This typical configuration 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 supply at nodes a and d. Note that node d is marked as ground , the point in the circuit designated as 0V. The output is obtained by reading the potential difference between nodes b and c:
V out
= V b
− V c
= ( V b
− V d
) − ( V c
− V d
) .
Figure 3: 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 R
4 will make V out
= 0 .
In practice, R
4 might be a strain gage, thermocouple, flexible resistor—anything that transduces one physical property into a change in resistance.
The WB is basically a very sensitive resistance measurement device. Thought it was invented circa
1833 (people have been clever for a very long time...), it still finds widespread use today in many engineering applications—mechanical, aerospace, and civil engineering, to name a few. The basic idea is that the one (or more) of the bridge legs has a variable resistance. For example, the variable leg of the bridge could be a thermocouple. If the temperature changes, then the resistance changes,
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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.
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so the output changes accordingly. Or it could be a strain gage
2
. If the structure to which it is
attached deflects or vibrates, the bridge will report this motion as a change in its output voltage.
Clearly, then, it would be fun (and instructive!) to build and analyze one.
1. The bridge output is defined as the voltage difference between nodes b and c
: V out
= V b
− V c
.
On which respective nodes would you place the (+) and (-) probes of the DMM?
2. Show that:
V out
=
R
1
R
4
− R
2
R
3
( R
1
+ R
2
)( R
3
+ R
4
)
V s
(1)
Hint: Think voltage dividers.
3. The bridge is said to be “balanced” when V out resistances R
1
, R
2
, R
3
, R
4
= 0. What is the relationship between the when the bridge is balanced?
4. Assume all resistors in your WB are equal R
1
= R
2
= R
3
= R
4
= R , so your bridge is balanced
( V out
= 0). Then let R
4
For the case that
∆ R
R increase its resistance by a relatively small amount:
<< 1, show that:
R
4
→ R
4
+ ∆ R .
V out
≈
∆ R
V s
4 R
(2)
5. Make a quick sketch of V out this line?
vs. ∆ R . Explain how you interpret the meaning of the slope of
6. What is the range over which the linear apprxoximation made in Eqn 2 remains valid? Addi-
tionally, what are some inherent trade offs with the setting the proper ratio of
∆ R
R
? 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)?
7. 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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A strain gage is basically a variable resistor. It essentially consists of long, thin piece of metal patterned in a shape designed to detect specific types of deformations (stretching, compressing, twisting). Changing it’s shape basically changes the values for A and L in R =
ρ
A L
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“Is the section title really involve The WB?,” you ask. Good news: No Dawson’s Creek, Buffy the
Vampire Slayer, or other half-baked, crappy reruns here. Just the good ole Wheatstone Bridge is the subject of this investigation.
1. Now build the bridge (see Fig. 3). Set
V s
= +5 V. Use 1 kΩ resistors for R
1
, R
2
, and R
3
.
(Remember to carefully measure and record the actual resistance of each of these.) For R
4
, 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 R
4 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
Make a plot of V out for each setting.
vs. ∆ R . Remember: ∆ R is the change in the resistance relative to the balanced condition.
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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These section describes an experiment with a vibrating beam—great for future MEs and CEs. If that doesn’t float your boat, check out section 5 below.
Now you will use actual strain gage elements (see Fig. 4(a)) configured as a Wheatstone Bridge to measure deflections in a beam (e.g., see Fig. 4(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.
(a) (b)
Figure 4: (a) Strain gage. The resistance changes when thin metal film “comb” shape deforms. So the electrical resistance of the strain gage changes when the gage contracts or expands. In a nonstressed 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
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 INA126
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. 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
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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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of the amplifier is given by (refer to Fig. 5):
V out
= 50 V
W B
= 50( V
+
− V
−
) .
(3) where the difference on the right hand side 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
W B
= 50 mV
(possible to read with a scope, but just barely), 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: Use an oscilloscope to do it! The positive scope probe should connect INA126 pin 6 (the amplifier’s output). Additionally, connect the ground probe to INA126 pin5 (the amplifier’s ground).
Figure 5: 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 measure the frequency of oscillation.
• 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.
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• A quick bit of vibrations theory. A beam can be modeled as a mass, spring, damper system, which obeys the ODE:
¨ + 2 ζω o
˙ + ω
2 o x = 0 where x denotes the vertical position of the beam. The damping constant is given by ζ , and has to do with the material properties of the beam. The natural frequency, from simple harmonic oscillations, given by p k/m.
ω o is your old pal
In this case, m is the total mass oscillating in the system, the applied load plus the “effective
mass” of the beam
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. When the damping is not too severe–which is the case here–the equation
of motion for the beam can be written as: x ( t ) ≈ Ae
− ζω o t cos( ω o t + φ )
The upshot of all this madness is that the damped frequency of oscillation should be inversely proportional to the square root of the total mass in vibration: f ∝ √
1 m
.
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.
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 photoresistors (resistance of which increases as the intensity of incident light increases); flex
sensors (kind of like a flexible potentiometer, see Fig. 6(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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The effective mass in this case is
330) if you want to know more!
1
3 m beam
. We won’t derive this here. Shameless plug: take Vibrations (ENGN
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(a) (b)
Figure 6: (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.
Your report will consist of 3 main parts. Do only section 4 OR section 5.
1. Section 1 (Input resistance of a volt meter): Be sure to complete questions 1–10.
2. Section 2.2 (WB Theoretical Considerations ): Be sure to complete questions 1–7.
3. Section 3 (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).
4. Section 4 (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.
5. Section 5 (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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