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Electronic InstrumentationExperiment 5
Part A: Bridge Circuit BasicsPart B: Analysis using Thevenin
Equivalent Part C: Potentiometers and Strain Gauges
Agenda
Bridge Circuit •
Introduction•
Purpose
•
Structure•
Balance
•
Analysis using the Thevenin Equivalent method
Strain Gauge, Cantilever Beam and Oscillations
What you will know:
What a bridge circuit
is and how it is used
in the experiments.
What a balanced bridge
is and how to
recognize a circuit that is balanced.
How to apply Thevenin
equivalent method
to a voltage divider, a Wheatstone Bridge or other simple configuration.
How a strain gauge
is used in a bridge
circuit
How to determine the damping constant
of
a damped sinusoid given a plot.
Why use a Bridge?Temperature
Light IntensityPressure
Hydrogen Cyanide Gas
Porter, Tim et al., Embedded Piezoresistive Microcantilever Sensors: Materials for Sensing Hydrogen Cyanide
Gas, http://www.mrs.org/s_mrs/bin.asp?CID=6455&DID=176389&DOC=FILE.PDF
.
Strain
Sensors that quantifychanges in physical variables by measuringchanges in resistance ina circuit.
Wheatstone Bridge StructureA bridge is just two voltage dividers in parallel. The output is the difference between the two dividers.
BAoutSBSA VVdVVVRR
RVVRR
RV
41
432
3
A Balanced Bridge Circuit
021
21
111
1111
1
VVVVdV
VKK
KVVKK
KV
rightleft
rightleft
A “zero”
point for normal conditions so changes can be positive or negative in reference to this point.
Thevenin Voltage Equivalents
In order to better understand how bridges work, it is useful to understand how to create Thevenin Equivalents of circuits.
Thevenin invented a model called a Thevenin Source for representing a complex circuit using•
A single “pseudo”
source, Vth
•
A single “pseudo”
resistance, Rth
RL
0
Vo
R4R2
R1 R3
Vth
=
0
Rth
Thevenin Voltage Equivalents
This model can be used interchangeably with the original (more complex) circuit when doing analysis.
Vth
=
0
Rth
The Thevenin source, “looks”
to the load on
the circuit like the actual complex combination of resistances and sources.
Thevenin Model
VsRL
0
Load ResistorRth
Vth
0
RL
Any linear circuit connected to a load can be modeled as a Thevenin equivalent voltage source and a Thevenin equivalent impedance.
Note:
We might also see a circuit with no load resistor, like this voltage divider.
R2
0
R1
Vs
Thevenin Method
Find Vth
(open circuit voltage)
•
Remove load if there is one so that load is open•
Find voltage across the open load
Find Rth
(Thevenin resistance)
•
Set voltage sources to zero (current sources to open) – in effect, shut off the sources
•
Find equivalent resistance from A to B
Vth
=
0
Rth
A
B
Example: The Bridge Circuit
We can remodel a bridge as a Thevenin Voltage source
RL
0
Vo
R4R2
R1 R3
Vth
=
0
Rth
Find Vth
by removing the Load
LetVo=12V R1=2kR2=4kR3=3k R4=1k
RL
0
Vo
R4R2
R1 R3
0
Vo
R4R2
R1 R3
A AB B
V BR 4
R 3 R 4
V o
Step 2: Voltage divider
for points A
and B
VAR2
R2 R1
Vo
Step 3: Subtract VB
from VA
to get Vth
VA 8V VB 3V
Step 1: Redraw and remove load
from circuit
VB
-VA
=Vth Vth=5V
To find Rth
First, short out the voltage source
(turn it
off) & redraw the circuit for clarity.
0
R4R2
R1 R3
A B R12k
R24k
R33k
R41k
BA
Find Rth
Find the parallel combinations of R1 & R2 and R3 & R4.
Then find the series combination of the results.
R R RR R
k kk k
k k12 1 21 2
4 24 2
86
133
.
R R RR R
k kk k
k k34 3 43 4
1 31 3
34
0 75
.
Rth R R k k
12 34 43
34
21.
Redraw Circuit as a Thevenin Source
Then add any load and treat it as a voltage divider.
0
Vth
5V
Rth
2.1k
thLth
LL V
RRRV
Thevenin Applet (see webpage)
Test your Thevenin skills using this applet from the links for Exp 3
Does this really work?
To confirm that the Thevenin method works, add a load and check the voltage across and current through the load to see that the answers agree whether the original circuit is used or its Thevenin equivalent.
If you know the Thevenin equivalent, the circuit analysis becomes much simpler.
Thevenin Method Example
Checking the answer with PSpice
Note the identical voltages across the load.•
7.4 -
3.3 = 4.1 (only two significant digits in Rth)
R3
3kVth
5Vdc
0
7.448V
0V
4.132V
3.310VVo
12VdcRload
10k
R1
2k
12.00V
RL
10kR2
4k
5.000V Rth
2.1k
R4
1k
0
Thevenin’s
method is extremely useful and is an important topic.
But back to bridge circuits –
for a balanced
bridge circuit, the Thevenin equivalent voltage is zero.
An unbalanced bridge is of interest. You can also do this using Thevenin’s
method.
Why are we interested in the bridge circuit?
Wheatstone Bridge
•Start with R1=R4=R2=R3
•Vout
=0
•If one R changes, even a small amount, Vout
≠0
•It is easy to measure this change.
•Strain gauges look like resistors
and the resistance changes with the strain
•The change is very small.
BAoutSBSA VVdVVVRR
RVVRR
RV
41
432
3
Using a parameter sweep to look at bridge circuits.
V1
FREQ = 1kVAMPL = 9VOFF = 0
R1350ohms
R2350ohms
R3350ohms
R4Rvar
0
Vleft Vright
PARAMETERS:Rvar = 1k
V-
V+
• PSpice
allows you to run simulations with several values for a component.
• In this case we will “sweep”
the value of R4 over a range of resistances.
This is the “PARAM”
part
Name the variable that will be changed
Parameter Sweep
Set up the values to use.
In this case, simulations will be done for 11 values for Rvar.
Parameter Sweep
All 11 simulations can be displayed
Right click on one trace and select “information”
to know which Rvar
is shown.
Part B
Strain Gauges
The Cantilever Beam
Damped Sinusoids
Strain Gauges
•When the length of the traces changes, the resistance changes.
•It is a small change of resistance so we use bridge circuits to measure the change.
•The change of the length is the strain.
•If attached tightly to a surface, the strain of the gauge is equal to the strain of the surface.
•We use the change of resistance to measure the strain of the beam.
Strain Gauge in a Bridge Circuit
Cantilever Beam
The beam has two strain gauges, one on the top of the beam and one on the bottom. The strain is approximately equal and opposite for the two gauges.
In this experiment, we will hook up the strain gauges in a bridge circuit to observe the oscillations of the beam.
Two resistors
and two strain gauges
on the beam creates a bridge circuit, and then its output goes to a difference amplifier
Strain Gauge-Cantilever Beam Circuit
Modeling Damped Oscillations
v(t) = A sin(ωt)
Time
0s 5ms 10ms 15msV(L1:2)
-400KV
0V
400KV
Modeling Damped Oscillations
v(t) = Be-αt
Modeling Damped Oscillations
v(t) = A sin(ωt) Be-αt
= Ce-αtsin(ωt)
Time
0s 5ms 10ms 15msV(L1:2)
-200V
0V
200V
Finding the Damping Constant
Choose two maxima at extreme ends of the decay. (t0
,v0
)(t1
,v1
)
Finding the Damping Constant
Assume (t0,v0) is the starting point for the decay.
The amplitude at this point,v0, is C.
v(t) = Ce-αtsin(ωt) at (t1,v1): v1 = v0e-α(t1-t0)sin(π/2) = v0e-α(t1-t0)
Substitute and solve for α: v1 = v0e-α(t1-t0)
Finding the Damping ConstantV0 =(7s, 4.6 – 2)=(7, 2.6)
4.6
2
V1 =(74s, 2.7 – 2)=(74, 0.7)
2.72
V1=V0e-α(t1-t0)
0.7=2.6e-α(74-7)
-α(67)=ln
(0.7/2.6)
α=0.0196/s
7 74
Harmonic Oscillators
Analysis of Cantilever Beam Frequency Measurements
Next week
Electronic InstrumentationExperiment 5 (continued)
Part A: Review Bridge Circuit, Cantilever BeamPart B: Oscillating CircuitsPart C: Oscillator ApplicationsProject 2 Overview
Agenda
Review•
Bridge, Strain Gauge, Cantilever Beam, Thevenin
equivalent
Oscillating Circuits•
Comparison to Spring-Mass
•
Conservation of Energy
Oscillator Applications
Project 2 Overview (velocity measurement)
What you will know:
(Review) How the Bridge, Strain, Gauge and Cantilever Beam relate
Why the Thevenin
equivalent is important
Similarities in concept between a spring-
mass and electronics
Dynamics of Oscillating circuits
Technology using these concepts
Detail about Project 2 (due Oct 27th and 29th
)
Applications
Bridge circuits are used to measure ________ resistances.
What other types of sensors that change resistance in an environment are there?
What are some applications for strain gauges? http://www.hitecorp.com/oemts.cfm
unknown
Review
http://www.allaboutcircuits.com/vol_1/chpt_9/7.html
Question: Why use two strain gauges instead of one?
Why is the Thevenin equivalent important?
Macromodel
used to model electronic sources
Reduces the level of complexity
To predict the behavior under a load for a non-ideal source
Thevenin Method
Find Vth
(open circuit voltage)•
Remove load if there is one so that load is open•
Open circuit means infinite resistance•
Find voltage across the open load
Find Rth
(Thevenin resistance)•
Set voltage sources to zero (current sources to open) –
in effect, shut off the sources (goes to zero) and what is left is Rth
•
Short circuit meaning resistance is zero•
Find equivalent resistance from A to B
Vth
=
0
Rth
A
B
In Class Problem
Find the Thevenin
voltage of the circuit
Find the Thevenin
Resistance
Draw Thevenin
Equivalent
If you place a load resistor of 2K between A and B, what would be the voltage at point A?
V1=6 V, R1=50Ω, R2=500Ω, R3=800Ω, R4=3000Ω
Why is Thevenin’s
method important to this experiment?
When a bridge circuit is unbalanced you get a voltage output rather than zero.
Thevenin’s
method allow for easy prediction of what is happening across a load…but we are using VA
-VB
(essentially Vth
) to a difference amplifier
The result of these non-zero voltage fluctuations for a cantilever beam is….
Modeling Damped Oscillations
v(t) = A sin(ωt) Be-αt
= Ce-αtsin(ωt)
Time
0s 5ms 10ms 15msV(L1:2)
-200V
0V
200V
What causes the dampening? Why doesn’t it oscillate forever?
Examples of Harmonic Oscillators
Spring-mass combination
Violin string
Wind instrument
Clock pendulum
Playground swing
LC or RLC circuits
Others?
Spring
Spring ForceF = ma = -kx
Oscillation Frequency
This expression for frequency holds for a massless
spring with a mass at the end, as
shown in the diagram.
mk
Harmonic Oscillator
Equation
Solution x = Asin(ωt)
x is the displacement of the oscillator while A is the amplitude of the displacement
022
2
xdt
xd
Spring Model for the Cantilever Beam
Where l
is the length, t
is the thickness, w
is the width, and mbeam
is the mass of the beam. Where mweight
is the applied mass and a
is the length to the location of the
applied mass.
Finding Young’s Modulus
For a beam loaded with a mass at the end, a
is
equal to l. For this case:
where E
is Young’s Modulus of the beam.
See experiment handout for details on the derivation of the above equation.
If we can determine the spring constant, k, and we know the dimensions of our beam, we can calculate E and find out what the beam is made of.
3
3
4lEwtk
Finding k using the frequency
Now we can apply the expression for the ideal spring mass frequency to the beam.
The frequency, fn ,
will change depending
upon how much mass, mn
, you add to the end of the beam.
2)2( fmk
2)2( nn
fmm
k
Our Experiment
For our beam, we must deal with the beam mass and any extra load we add to the beam to observe how its performance depends on load conditions.
Real beams have finite mass distributed along the length of the beam. We will model this as an equivalent mass at the end that would produce the same frequency response. This is given by m
=
0.23mbeam
.
Our Experiment
•
To obtain a good measure of k
and m, we will make 4 measurements of oscillation, one for just the beam and three others by placing an additional mass at the end of the beam.
20 )2)(( fmk 2
11 )2)(( fmmk 2
22 )2)(( fmmk 233 )2)(( fmmk
Our Experiment
Once we obtain values for k
and m
we can plot
the following function to see how we did.
In order to plot mn
vs. fn
, we need to obtain a guess for m, mguess
, and k, kguess
. Then we can use the guesses as constants, choose values for mn
(our domain) and plot fn
(our range).
nguess
guessn mm
kf
21
Our Experiment
The output plot should look something like this. The blue line is the plot of the function and the points are the results of your four trials.
Our Experiment
How to find final values for k
and m.•
Solve for kguess
and mguess
using only two of your data points and two equations. (The larger loads work best.)
•
Plot f
as a function of load mass to get a plot similar to the one on the previous slide.
•
Change values of k
and m
until your function and data match.
Our Experiment
Can you think of other ways to more systematically determine kguess
and mguess
?
Experimental hint: make sure you keep the center of any mass you add as near to the end of the beam as possible. It can be to the side, but not in front or behind the end.
Spring vs. Circuits
From your basic physics book the conservation of energy
W=Potential energy + Kinetic Energy
If you pull a spring it gains potential energy
When released you gain kinetic energy at the expense of potential energy then it builds potential again as the spring compresses
A circuit with a capacitor and inductor does the same thing!
Oscillating Circuits
Energy Stored in a Capacitor
CE =½CV²(like potential energy)
Energy stored in an Inductor
LE =½LI²(like kinetic energy)
An Oscillating Circuit transfers energy between the capacitor and the inductor.
http://www.walter-fendt.de/ph11e/osccirc.htm
http://www.allaboutcircuits.com/vol_2/chpt_6/1.html
Voltage and Current
Note that the circuit is in series, so the current through the capacitor and the inductor are the same.
Also, there are only two elements in the circuit, so, by Kirchoff’s
Voltage Law, the
voltage across the capacitor and the inductor must be the same.
CL III
CL VVV
http://www.allaboutcircuits.com/vol_2/chpt_6/1.html
Transfer of Energy
When current is passed through the coils of the inductor, a magnetic field is created
This magnetic field can do work
WM
=(1/2) L I2
When voltage is applied to the capacitor charge flows to the plates positive on one side negative on the other
Force is now between the plates, energy is stored in the electric field created
WE
=(1/2) C V2
Transfer of Energy
Capacitor has been charged to some voltage at time t=0
Charge will flow creating a current through the inductor
(potential to
kinetic)
Charge on capacitor plates is gone
and current in inductor
will reach its maximum
(potential=0,
kinetic=max)
Current will then charge capacitor back up and process starts over!
Oscillator Analysis
Spring-Mass
W = KE + PE
KE = kinetic energy=½mv²
PE = potential energy(spring)=½kx²
W = ½mv²
+ ½kx²
Electronics
W = LE + CE
LE = inductor energy=½LI²
CE = capacitor energy=½CV²
W = ½LI²
+ ½CV²
Oscillator Analysis
Take the time derivative
Take the time derivative
dtdxxk
dtdvvm
dtdW 22 2
12
1
dtdxkx
dtdvmv
dtdW
dtdVVC
dtdIIL
dtdW 22 2
12
1
dtdVCV
dtdILI
dtdW
Oscillator Analysis
W is a constant. Therefore,
Also
W is a constant. Therefore,
Also
0dt
dW 0dt
dW
dtdxv
2
2
dtxd
dtdva
dtdVCI
CI
dtdV
2
2
dtVdC
dtdI
Oscillator Analysis
Simplify
Simplify
kxvdt
xdmv 2
2
0
02
2
xmk
dtxd
CICV
dtVdLIC 2
2
0
012
2
VLCdt
Vd
Harmonic equation for oscillatingcircuits
Oscillator Analysis
Solution
Solution
mk
LC1
x = Asin(ωt) V= Asin(ωt)
Using Conservation Laws
Please also see the write up for experiment 5 for how to use energy conservation to derive the equations of motion for the beam and voltage and current relationships for inductors and capacitors.
Almost everything useful we know can be derived from some kind of conservation law.
Large Scale Oscillators
Tall buildings are like cantilever beams, they allhave a natural resonating frequency.
Petronas
Tower (452m) CN Tower (553m)
Deadly OscillationsThe Tacoma Narrows Bridge went into oscillation when exposed to high winds. The movie shows what happened.
http://www.youtube.com/watch ?v=P0Fi1VcbpAI&feature=rela ted
In the 1985 Mexico City earthquake, buildings between 5 and 15 stories tall collapsed because they resonated at the same frequency as the quake. Taller and shorter buildings survived.
Controlling Deadly OscillationsHow do you prevent this?
Active Mass Damper•
Small auxiliary mass on the upper floors of a building
•
Actuator between mass and structure
•
Response and loads are measured at key points and sent to a control computer
Actuator reacts by applyinginertial forces to reducestructural response
http://www.nd.edu/~tkijewsk/Instruction/solution.html
Atomic Force Microscopy -AFM
This is one of the key instruments driving the nanotechnology revolution
Dynamic mode uses frequency to extract force information Note Strain Gage
AFM on Mars
Redundancy is built into the AFM so that the tips can be replaced remotely.
AFM on Mars
Soil is scooped up by robot arm and placed on sample. Sample wheel rotates to scan head. Scan is made and image is stored.
http://www.nasa.gov/mission_pages/phoenix/main/index.html
AFM Image of Human Chromosomes
There are other ways to measure deflection.
AFM Optical Pickup
The movement of the cantilever is measured by bouncing a laser beam off the surface of the cantilever
MEMS Accelerometer
An array of cantilever beams can be constructed at very small scale to act as accelerometers.
Note Scale
Micro-electro Mechanical Systems
Nintendo Wii uses these for measuring movement and tiltOthers?
Hard Drive Cantilever
The read-write head is at the end of a cantilever. This control problem is a remarkable feat of engineering.
More on Hard Drives
A great example of Mechatronics.