Download Foreword 1 Input Resistance of your DC voltmeter

Experiment # 2
The Wheatstone Bridge—Practical Applications
ENGN/PHYS 207—Fall 2014
Foreword
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.
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Input Resistance of your DC voltmeter
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 Rinput —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 Rinput = 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
R1
R2
V
Vs +
−
R2
Rinput
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
Rinput .
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 R1 = R2 = 10 kΩ. What voltage do you
expect to measure across R2 , 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 R1 and R2 . Use R1 = R2 = 6.8 MΩ.
6. Calculate the voltage you would expect to measure across R2 if the voltmeter were ideal (or
the voltmeter were not connected in the circuit).
7. What is the measured voltage across R2 ?
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 Rinput = 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 Rinput
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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2.1
The Wheatstone Bridge
Introduction
The Wheatstone Bridge 1 , (WB) 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:
Vout = Vb − Vc = (Vb − Vd ) − (Vc − Vd ).
Figure 3: Basic Wheatstone Bridge Circuit. The voltage source Vs provides excitation at nodes
a and d. The output (gage reading) Vout is measured across as the difference in voltages between
nodes b and c. The bridge is said to be “balanced” when Vout = 0. Starting in a balance, changing
the value of R4 will make Vout 6= 0. In practice, R4 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 gage2 . 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.
2.2
Theoretical Considerations
1. The bridge output is defined as the voltage difference between nodes b and c: Vout = Vb − Vc .
On which respective nodes would you place the (+) and (-) probes of the DMM?
2. Show that:
Vout =
R1 R4 − R2 R3
Vs
(R1 + R2 )(R3 + R4 )
(1)
Hint: Think voltage dividers.
3. The bridge is said to be “balanced” when Vout = 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
(Vout = 0). Then let R4 increase its resistance by a relatively small amount: R4 → R4 + ∆R.
For the case that ∆R
R << 1, show that:
Vout ≈
∆R
Vs
4R
(2)
5. Make a quick sketch of Vout vs. ∆R. Explain how you interpret the meaning of the slope of
this line?
6. What is the range over which the linear apprxoximation made in Eqn 2 remains valid? Additionally, 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. 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?
2
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 = AρL
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Experiment I: Getting Acquainted with the WB
“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 Vs = +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 Vout for each setting.
Make a plot of Vout 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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Experiment II: Real-life Application of the WB
Note: Do only section 4 OR section 5. 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. (I did promise
that circuits is actually useful...hopefully this helps convince you).
(a)
(b)
Figure 4: (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
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 INA1263 . 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 (refer to Fig. 5):
Vout = 50VW B = 50(V+ − V− ).
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(3)
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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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 VW B = 50 mV
(possible to read with a scope, but just barely), the amplifier will multiply this by a factor of 50
and output Vamp = 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.
• A quick bit of vibrations theory. A beam can be modeled as a mass, spring, damper system,
which obeys the ODE:
ẍ + 2ζωo ẋ + ωo2 x = 0
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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 thep
beam. The natural frequency, ωo is your old pal
from simple harmonic oscillations, given by k/m.
In this case, m is the total mass oscillating in the system, the applied load plus the “effective
mass” of the beam4 . 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−ζωo t cos(ωo 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:
1
f∝√ .
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.
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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. 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 Write-Up
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 Vout 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).
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The effective mass in this case is
330) if you want to know more!
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mbeam .
3
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.
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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