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.