Laboratory 6: Wheastone Bridge and Measurement Uncertainty

1MAE 3340 INSTRUMENTATION SYSTEMS

Laboratory 6: Wheastone Bridge and Measurement Uncertainty

• Lab Objectives:--Understand Wheatstone Bridge Properties--Power Dissipation in Resistor--Reading the Resistor Labeling Code--Balancing a Resistance Bridge--Using a resistance bridge to measure and unknown resistance--Understand how to Calculate statistics

levels for a random population

Chapter 7, pp. 241-249, B.M.L.Chapter 3, pp. 34-63, B.M.L.


You will build this bridge …

KnownPrecision100 resistors

Variable Decade Resistor

Unknown Resistor

• You willBalance this bridge … Using R3

To determine R4


Excitation Voltage …

The Precision 100 resistors (+1%) Have a 1/4 watt capability

… but to be safe … we wantTo limit power dissipatedTo 1/40 watt

… need to calculate Vex

So that power dissipated inR1, R2 ~ 1/40 watt

Vex

Do the analysis before lab, and show it to the lab instructor before lab, staple it to this report.

Pdissipated =I 2R


Resistance Bridge Summary (see Appendix)• .. Infinite Meter Impedance

IM =0

I1 =Vex

R1 + R2( )

I 2 =Vex

R1 + R2( )

I 3 =Vex

R3 + R4( )

I 4 =Vex

R3 + R4( )

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

→

VM =Vex

R1R4 −R2R3

R1 + R2( ) R3 + Rx( )

V1 =Vex

R1

R1 + R2( )

V2 =Vex

R2

R1 + R2( )

V3 =Vex

R3

R3 + R4( )

V4 =Vex

R4

R3 + R4( )

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

→

PM =0

P1 =R1

Vex

R1 + R2( )

⎡

⎣⎢

⎤

⎦⎥

2

P2 =R2

Vex

R1 + R2( )

⎡

⎣⎢

⎤

⎦⎥

2

P3 =R3

Vex

R3 + R4( )

⎡

⎣⎢

⎤

⎦⎥

2

P4 =R4

Vex

R3 + R4( )

⎡

⎣⎢

⎤

⎦⎥

2

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

Current thruresistor

Voltage dropcross resistor

Power dissipated

by resistor


Power Supply


Decade Resistor Box

There are 3 decade resistors that measure higher resistances, and 2 that measure lower resistances. They will be marked “High Resistance” or “Low Resistance”. These can create a resistance anywhere between 0.01 and 900 ohms in 0.01-ohm increments (Low Resistance), or 1 to 99,000 Ohms in 1-Ohm increments (High Resistance).


Wheatstone Bridge on Breadboard


Your Circuit

100

Rx

100

Power supply

Meter

Decade resistance

Unknown resistance

• You will adjust decade resistance until Meter reads zero voltage (balance)


Balancing Technique

• Solve for unknown resistance

• Adjust R3 until IM = 0

I3R3 =I1R1

I xRx =I 2R2

→ Rx =R2 I 2( ) R3I 3( )R1I1( ) I x

=R2R3

R1

⎛

⎝⎜⎞

⎠⎟I 2 I 3I1I x

⎛

⎝⎜⎞

⎠⎟↓

I 3 =I xI1 =I 2

→ Rx =R2R3

R1

⎛

⎝⎜⎞

⎠⎟

Vex

• Good way to senseAn unknown resistance

R3 variable resistance … decade box


Resistance Table

20 resistors selected at random …

You will fill out this table

Table I: Resistance Values Resistor Number

Nominal resistance (Ohms) (color code)

Spec Accuracy (%) (color code)

Actual resistance (Ohms)

Ri/Rnom


Lab Report Summary (1)• When you are finished, get together with the other 4 groups in your section and make a text file containing all 80 samples. . …

• Using all the data, compute an estimate of the normalized mean and the standard deviation for the pool of 80 resistors, your 20 resistors.

x =

Ri

Rnom

⎛

⎝⎜⎞

⎠⎟i

ni=1

n

∑ SX =

Rxi

Rnom

−x⎛

⎝⎜⎞

⎠⎟

2

n−1i=1

n

∑“sample mean” “sample Std. Dev.”


Lab Report Summary (2)• Down Load and Open Histogram Code … http://www.neng.usu.edu/classes/mae/3340/section6/resistor_data_histogram.llb

• Make a histogram of the data with all 80 samples and attach it to this lab (Labview makes this easy). … A VI has been built for this purpose… there is a link on Web page … http://www.neng.usu.edu/classes/mae/3340/section6/resistor_data_histogram.llb


Lab Report Summary (3)• A histogram divides the range of possible values upInto “bins” and plots the % of occurrences of values within that bin

Histogram data


Lab Report Summary (4)• What is the precision uncertainty of the mean based on all 80 resistors? (95% confidence)?

• Using only your 20 resistors, estimate the normalized mean, and the precision uncertainty of the mean to 95% confidence.

• Which one would you expect to have the best precision uncertainty?

80 samples 20 samples


Precision Uncertainty Analysis (1)

• Sample Mean, Standard Deviation Estimates

xn{ } =

RRnom

⎛

⎝⎜⎞

⎠⎟1

RRnom

⎛

⎝⎜⎞

⎠⎟2

RRnom

⎛

⎝⎜⎞

⎠⎟3

.

.

.

RRnom

⎛

⎝⎜⎞

⎠⎟n−1

RRnom

⎛

⎝⎜⎞

⎠⎟n

⎧

⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

• Sample Standard Deviation Estimates

x =

Ri

Rnom

⎛

⎝⎜⎞

⎠⎟i

ni=1

n

∑

SX =

Rxi

Rnom

−x⎛

⎝⎜⎞

⎠⎟

2

n−1i=1

n

∑



• Confidence Interval for Sample Mean Estimate

x −zc/2Sx

n< μ < x+ zc/2

Sx

nSx =Sx

n

Standard Error of

the Sample Mean

95% Confidence Interval --->

0.95 =Px( ) =1

2π ⋅σ x

e−

x−μ( )2

2σ 2⎛

⎝⎜⎜

⎞

⎠⎟⎟

z=−x−μx( )

2

σx2

z=x−μx( )

2

σx2

∫ dx

±zc /2 = ±1.958



→ let....z =x_

− μx_

σx_

→

0.95 = P x( ) =1

2π ⋅σx_

⋅ e−z /2( )z.c/2

z.c/2

∫ dz

Area under curve = 0.95

±zc /2 = ±1.958


Precision Uncertainty Analysis (4)±zc /2 = ±1.958

x − 1.958 ⋅Sx

n

⎛⎝⎜

⎞⎠⎟

< μ x < x + 1.958 ⋅Sx

n

⎛⎝⎜

⎞⎠⎟

....

or... μ x = x ± 1.958 ⋅Sx

n

⎛⎝⎜

⎞⎠⎟

Area under curve = 0.95

• precision uncertainty….in mean estimate


Error Analysis (1)

• Given the relationship between the known resistors and the unknown resistors, find the uncertainty in your measurement that is due to the uncertainty in the values of R1 and R2 (100, +1%). Assume that the decade resistor is exact. (no error)… Hint …

• How do you account for the fact that the meter only has two digits of precision in voltage?… see next page

• Compare this calculated value to the standard deviation you computed

based on your 20 resistorsbased on all 80 resistors

• Comment on the manufacturer’s claim as to the accuracy of these resistors.

Rx = f R1,R2( ) =R2R3

R1

⎛

⎝⎜⎞

⎠⎟→ R3 ~constant


Uncertainty Analysis (6)• How do you account for the fact that the meter only has two digits of precision

VM =Vex

R1Rx −R2R3

R1 + R2( ) R3 + Rx( )→

∂VM

∂Rx

=Vex

R1 R3 + Rx( )R1 + R2( ) R3 + Rx( )

−R1Rx −R2R3

R1 + R2( ) R3 + Rx( )2

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

Vex

R1 R3 + Rx( )−R1Rx + R2R3

R1 + R2( ) R3 + Rx( )2

⎡

⎣⎢⎢

⎤

⎦⎥⎥=Vex

R3

R3 + Rx( )2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

δVM ≈∂VM

∂Rx

⋅δ Rx → δ Rx ≈δVM

∂VM ∂Rx


Uncertainty Analysis (7)• How do you account for the fact that the meter only has two digits of precision

δRx ≈δVM

∂VM ∂Rx

=R3 + Rx( )

2

R3

δVM

Vex

⎡

⎣⎢

⎤

⎦⎥= R3 ⋅ 1 +

Rx

R3

⎛

⎝⎜⎞

⎠⎟

2δVM

Vex

⎡

⎣⎢

⎤

⎦⎥

δ Rx

Rx

⎡

⎣⎢

⎤

⎦⎥

Meter

=δVM

Vex

⎡

⎣⎢

⎤

⎦⎥⋅

R3

Rx

⎛

⎝⎜⎞

⎠⎟⋅ 1 +

Rx

R3

⎛

⎝⎜⎞

⎠⎟

2

• But …. Rx =R2R3

R1

⎛

⎝⎜⎞

⎠⎟→

Rx

R3

=R2

R1

≈100100

=1

δRx

Rx

⎡

⎣⎢

⎤

⎦⎥

Meter

≈ 4δVM

Vex


Uncertainty Analysis (8)

• Meter resolution … 0.01 Volts

Rx = f(R1,R2 ,R3) =R2R3

R1

δRx

Rx

=∂f∂R1

⎛

⎝⎜⎞

⎠⎟

2δR1

R1

⎛

⎝⎜⎞

⎠⎟

2

+∂f∂R2

⎛

⎝⎜⎞

⎠⎟

2δR2

R2

⎛

⎝⎜⎞

⎠⎟

2

+∂f∂R3

⎛

⎝⎜⎞

⎠⎟

2δR3

R3

⎛

⎝⎜⎞

⎠⎟

2

+ δMeter( )2

δRx2 =

R2R3

R12

⎛

⎝⎜⎞

⎠⎟

2

δR1( )2+

R3

R1

⎛

⎝⎜⎞

⎠⎟

2

δR2( )2+

R2

R1

⎛

⎝⎜⎞

⎠⎟

2

δR3( )2+ δRx[ ]Meter

2=

R2R3

R1

⎛

⎝⎜⎞

⎠⎟

2δR1

R1

⎛

⎝⎜⎞

⎠⎟

2

+δR2

R2

⎛

⎝⎜⎞

⎠⎟

2

+δR3

R3

⎛

⎝⎜⎞

⎠⎟

2⎡

⎣⎢⎢

⎤

⎦⎥⎥+ Rx

2 δRx

Rx

⎡

⎣⎢

⎤

⎦⎥Meter

2


Uncertainty Analysis (9)

• Meter resolution … 0.01 Volts• …. δR1/R1 = δR2/R2 =1%

δRx2 =

R2R3

R1

⎛

⎝⎜⎞

⎠⎟

2δ R1

R1

⎛

⎝⎜⎞

⎠⎟

2

+δ R2

R2

⎛

⎝⎜⎞

⎠⎟

2

+δ R3

R3

⎛

⎝⎜⎞

⎠⎟

2⎡

⎣⎢⎢

⎤

⎦⎥⎥

+ Rx2 δ Rx

Rx

⎡

⎣⎢

⎤

⎦⎥

Meter

2

→R2R3

R1

= Rx →δ Rx

Rx

⎛

⎝⎜⎞

⎠⎟

2

=δ R1

R1

⎛

⎝⎜⎞

⎠⎟

2

+δ R2

R2

⎛

⎝⎜⎞

⎠⎟

2

+δ R3

R3

⎛

⎝⎜⎞

⎠⎟

2⎡

⎣⎢⎢

⎤

⎦⎥⎥

+δ Rx

Rx

⎡

⎣⎢

⎤

⎦⎥

Meter

2

→δ Rx

Rx

=δ R1

R1

⎛

⎝⎜⎞

⎠⎟

2

+δ R2

R2

⎛

⎝⎜⎞

⎠⎟

2

+δ R3

R3

⎛

⎝⎜⎞

⎠⎟

2

+ 16δVM

Vex

⎡

⎣⎢

⎤

⎦⎥

2

δRx

Rx

⎡

⎣⎢

⎤

⎦⎥

Meter

≈ 4δVM

Vex

• Vex ---> voltage for 1/4 watt, {R1, R2}

~ 0


Types of Resistors (1)

Fixed resistors

Surface MountResistors


Types of Resistors (2)• Carbon composition resistors consist of a solid cylindrical resistive element with embedded wire leadouts or metal end caps to which the leadout wires are attached.

• Resistive element is made from a mixture of finely ground (powdered) carbon and an insulating material (usually ceramic). The mixture is held together by a resin.

• Resistance is determined by the ratio of the fill material (the powdered ceramic) and the carbon.



• Resistive coating is peeled away in an automatic machine until resistance between ends rod is within tolerance

• Metal leads and end caps are added and resistor is covered insulating coating and painted with colored bands to indicate the resistor value.

•Carbon film resistors are cheap and easily available, with values within 10% or 5% of their marked, or 'nominal' value.

• During manufacture, a thin film of carbon is deposited onto a small ceramic rod.



Metal Film resistors

• Metal-film resistor is a common type of high precision axial resistor today is referred to as a

• Metal Film resistors are usually coated with nickel chromium (NiCr)

• Resistance value is determined by cutting a helix through the coating rather than by etching.

• The result is a reasonable tolerance (0.5, 1, or 2%).


Types of Resistors (5)• Wirewound resistors are commonly made by winding a metal wire around a ceramic, plastic, or fiberglass core.

• The ends of the wire are soldered or welded to two caps, attached to the ends of the core.

• The assembly is protected with a layer of paint, molded plastic, or an enamel coating baked at high temperature.

• For higher power wirewound resistors, either a ceramic outer case or an aluminium outer case on top of an insulating layer is

used.

Wirewound resistors


Types of Resistors (6)• Metal Oxide resistors are non-inductive and substitute for carbon comp in most cases.

• They have a resistance element formed by the oxidation reaction of a vapor or spray of tin chloride solution on the heated surface of a glass or ceramic rod.

• The resulting tin-oxide film is adjusted to value by cutting a helix path through the

film.

Metal Oxide resistors



Fixed resistors

• Resistors are manufactured in values from a few milliohms to about a Gigaohm

• Only a limited range of values from the IEC 60063 preferred number series are commonly available.

• These series are called E6, E12, E24, E96 and E192. The number tells how many standarized values exist in each decade (e.g. between 10 and 100, or between 100 and 1000).

• So resistors confirming to the E12 series, can have 12 distinct values between 10 and 100, whereas those confirming to the E24 series would have 24 distinct values.

Surface MountResistors



• E12 preferred values: 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82 Ohms

• Multiples of 10 of these values are used, … 0.47 , 4.7 , 47 , 470 , 4.7 k , 47 k , 470 k , and so on …. .

• E24 preferred values, includes E12 values and: 11, 13, 16, 20, 24, 30, 36, 43, 51, 62, 75, 91 Ohms

• In practice, the discrete component sold as a "resistor" is not a perfect resistance, as defined above. Resistors are often marked with their tolerance (maximum expected variation from the marked resistance).



▪ Rheostat: variable resistor with two terminals, one fixed and one sliding. It is used with high currents.

▪ Potentiometer: a common type of variable resistor. One common use is as volume controls on audio amplifiers and other forms of amplifiers.

Variable resistors

Construction of a wire-wound variable resistor. The effective length of the resistive element (1) varies as the wiper turns, adjusting resistance.


Resistor Color Code (1)• A resistor is device whose only purpose it is to offer resistance to current flow.

• Nominal resistance of a resistor is printed onto the resistor in code.

• Pattern of colored rings is used.

• Most resistors have three rings to encode the value of the resistance, and one ring to encode the tolerance (uncertainty) in percent.


Resistor Color Code (2)• Ring Colors are internationally defined to as integers 0 -- 9.

• First band is the band closest to one end of the resistor.

• First and second band together make a two-digit integer number. • Multiply number represented by the color of the first band by 10 and add the number represented by the color of the second band.

• Result is a two-digit integer number.



• Number represented by the color of the third band is the number of zeroes that must be appended to the number obtained from the first two bands to get the resistance in Ohms.

• (If this number is 1, you add one zero, or multiply by 101, if the number is 2, you add two zeroes, or multiply by 102, etc.)



• The next band, (i.e. the fourth band), is the tolerance band --typically either gold or silver.

• A gold tolerance band indicates actual value will be within 5% of the nominal value.

• A silver band indicates 10% tolerance.



• If the resistor has one more band past the tolerance band it is a quality band.

• Read the number as the % failure rate per 1000 hours, assuming maximum rated power is being dissipated by the resistor.


Resistor Color Code (6)

(2⋅10 + 3)⋅106 ±5%= 23±1.15( )M

Look for gap


Resistor Color Code (7)

• Another Color Coding Example


Precision Resistor Coding

• 5-band axial resistors5-band identification is used for higher tolerance resistors (1%, 0.5%, 0.25%, 0.1%), to notate the extra digit. The first three bands represent the significant digits, the fourth is the multiplier, and the fifth is the tolerance.


Color Code Quiz

106

5%91

3300 5%

5%


Using Labview for Histogram Plot (1)


Using Labview

for Histogram

Plot (2)


Using Labview for Histogram Plot (3)

“double click”

Set bin intervalsAnd min/maxOn front panel

Documents

Laboratory 6: Wheastone Bridge and Measurement Uncertainty