ENTC 4350

Theories of Measurement

Basics of Measurements

Measurement = assignment of numerals to represent physical properties

• Two Types of Measurements for Data• Qualitative

• Quantitative

Qualitative Measurements

Qualitative = Non-numerical or verbally descriptive also have 2 types

• Nominal = no order or rank eg. list

• Ordinal = allows for ranking but differences between data is meaningless eg. alphabetical list

Quantitative Measurements

Quantitative = Numerical Ranking also have 2 types

• Interval = meaningless comparison eg. calendar

• Ratio = based on fixed or natural zero point eg. weight, pressure, Kelvin

Definition Decibels

dB = 20 log (Gain) where Gain = Voutput/ Vinput can also be in current or power• Why bother?

• Easier math because you can add and subtract db instead of multiplying and dividing

V1 V2 V3A1 A2

Definition Decibels

A1 = V2/V1 A2 = V3/V2 Total Gain = A1*A2 = V2/V1 * V3/V2

Definition Decibels

Now if everything was in dB

• Total Gain = A1 (dB) + A2 (dB)

Calculation of Gain given dB

dB = 20 Log (output/ input)• Output = input 10dB/20

Decibel example

Question An amplifier has 3 amplifier states and a

1 db attenuator in cascade. Assuming all impedances are matched, what is the overall gain if the amplifiers are 5, 10, 6 dB? Express your answer in dB and nondB form.

Decibel example

Solution:

• Gain = 5 dB + 10 dB + 6 dB -1 dB = 20 dB

or

• 20 dB = 20 log (Gain)

• Gain = 1020/20 =10

Variation and Error

Variation caused by small errors in measurement process

Error caused by limitation of machine Data will exhibit variation where you will

see a distribution in data. You can quantify distribution by calculating mean, variance, and standard deviation

Variation and Error

Data will exhibit variation where you will see a distribution in data.

You can quantify distribution by calculating the• mean,

• variance, and

• standard deviation

MEAN

Mean

• where Xi = data point and N = Total number of points

Example data points = 2,3,3,4,3 Mean • Xbar = (2 + 3 + 3 + 4 + 3 ) / 5 = 3

N

i

i

N

XX

1

Variance

Variance

• Example Variance =[(2-3)2 + ( 3-3) 2 + (3-3)2 + (4 – 3)2 + (3 – 3)2] /5 = 2 / 5 = 0.4

N

XXiN

i

1

2

2

Standard Deviation

Standard Deviation

• Example Standard Deviation = (0.4)1/2

• Note with small populations use N-1 instead of N

N

XXiN

i

1

2

Root Mean Square (RMS)

RMS used in electrical circuits

dttVT

Vt

tRMS

2

1

21

Root Mean Square (RMS)

VRMS= RMS value in voltage T = time interval from t1 to t2

V(t) = time varying voltage signal With a sine wave

pP

RMS VV

V 70702

.

Voltage Indicators

Vrms = Vp · .707 (Sine wave)

VppVrms Vp

f = 1/T = 2f

Frequency and PeriodPeriod, T

f1( )t

RMS: Root-Mean-Square RMS is a measure of a signal's

average power. • Instantaneous power delivered to a

resistor is: P= [v(t)]2/R.

• To get average power, integrate and divide by the period:

Solving for Vrms:

R

Vdttv

TRP rms

Tot

otavg

22211

)(

221

Tot

otrms dttv

TV )(

RMS: Root-Mean-Square

An AC voltage with a given RMS value has the same heating (power) effect as a DC voltage with that same value.

RMS: Root-Mean-Square

All the following voltage waveforms have the same RMS value, and should indicate 1.000 VAC on an rms meter:

1.414 v

WaveformVpeakVrms

Sine1.4141

1.733 v

Triangle1.7331

1 v

DC11

1 v

Square11

1

All = 1 WATT

Three Categories of Measurement

Direct Measurement: Indirect Measurement: Null Measurement:

Direct Measurement

Direct Measurement: holding a measurand up to a calibrated standard and comparing them eg. meter stick

Indirect Measurement

Indirect Measurement: Measuring something other than an actual measurement • This is typically done when direct measurement is

difficult to obtain or is dangerous.

• Example blood pressure can be obtained using a catheter with pressure transducer or can be obtained using Korotkoff Sounds

• Neural activity of brain, direct measurement would be implanting of electrodes or use of indirect measurement of MRI

Null Measurement

Null Measurement: Compared calibrated source to an unknown measurand and adjust till one or other until difference is zero

• Electrical Potentiometer used in Wheatstone Bridge

Definitions of Factors that Affect Measurements

• Error normal random variation not a mistake, • If you have a nonchanging parameter and

you measure this repeatedly, the measurement will not always be precisely the same but will cluster around a mean Xo. • The deviation around Xo = error term where

you can assume your measurement is Xo as long is deviation is small.

Definitions of Factors that Affect Measurements

Validity = Statement of how well instrument actually measures what it is supposed to measure • Eg. you’re developing a blood pressure

sensor with a diaphragm that has a strain gauge.

• This instrument is only valid if the deflection of the strain gauge is correlated to blood pressure.

Definitions of Factors that Affect Measurements

• Reliability and Repeatability

• Reliability statement of a measurement’s consistency of getting the same values of measurand on different trials

• Repeatibility getting the same value when exposed to the same stimulus

Definitions of Factors that Affect Measurements continued

Accuracy and Precision:•Accuracy Freedom from error, how

close is a measurement to a standard •Precision exactness of successive

measurements, has small standard deviations and variance under repeated trials

Definitions of Factors that Affect Measurements continued

Xo Xo Xo XoXi

Xi = Where the measurement is supposed to beXo = Mean of Data

Xi Xi Xi

Good Precision (Sm. Std)Good Accuracy (Xi ~ Xo)

Good Precision (Sm. Std)Bad Accuracy (Xi << Xo or Xi >> Xo)

Bad Precision (Large. Std)Good Accuracy (Xi ~ Xo)

Bad Precision (Large. Std)Bad Accuracy (Xi << Xo or Xi >> Xo)

Example of Precision and AccuracyGood Precision (Sm. Std)Good Accuracy (Xi ~ Xo)

Good Precision (Sm. Std)Bad Accuracy (Xi << Xo or Xi >> Xo)

Bad Precision (Large. Std)Good Accuracy (Xi ~ Xo)

Bad Precision (Large. Std)Bad Accuracy (Xi << Xo or Xi >> Xo)

Tactics to Decrease Error on Practical Measurements:

1. Make Measurements several Times

2. Make Measurements on Several Instruments

3. Make successive Measurements on different parts of instruments (different parts of ruler)

Definitions of Factors that Affect Measurements cont.

Resolution – Degree to which a measurand can be broken into identifiable adjacent parts ex pictures dpi (dots per square inch)

More Resolution Less Resolution

Definitions of Factors that Affect Measurements cont.

Binary Resolution

• If you have 8 Bits that will represent 10 V what is the resolution of the system? • Resolution = 10 – 0 / 255 = 39 mV per bit

• 8 bits gives you 28 = 256 values or 256 -1 = 255 segments

1

2

3

11.5

2.52

3

Error

Measurement Error Deviation between actual value of measurand and indicated value produced by instrument• Categories of Error

• Theoretical Error:

• Static Error:

• Dynamic

• Instrument Insertion Error

Theoretical Error:

The difference between the theoretical equation and the simplified math equation.

Static Error:

Errors that are always present even in unchanging system and therefore are not a function of time or frequency.• Reading Static Error:

• Environmental Static Error:

• Characteristic Static Errors:

• Quantization Error:

Reading Static Error:

Misreading of Digital display output• Parallax Reading Error error when not

measuring straight on (water in measuring cup).

• Interpolation Error Error in estimating correct value

• Last Digit Bobble Error Digital display variations when the LSB varies between 2 values .

Environmental Static Error:

Temperature, pressure, electromagnetic fields, and radiation can change output • Eg. electrical components are rated as

industrial temperature, temp = -50 to 85C.

Characteristic Static Errors:

Residual error that is not reading or environment • Eg. zero offset, gain error, processing error,

linearity error, hysteresis, repeatibility or resolution, or manufacturing deficiencies.

Quantization Error:

Error due to digitization of data and is the value between 2 levels.

Dynamic Error:

When a measurand is changing or is in motion during measurement process

• Eg. inertia of mechanical indicating devices during measurement of rapidly changing parameters

• Eg. analog meters or frequency, slew rate limitation of instrumentation

Instrument Insertion Error:

Measurement process should not significantly alter phenomenon being measured • Eg. If you are measuring body temp and

performing laser surgery the laser will heat the surrounding area and not give an accurate body temperature

Error Contribution Analysis

Error Budget = Analysis to determine allowable error to each individual component to ensure overall error not too high.

• Error Calculation =

N

i

i

1

2

Error Contribution Analysis

Why not take just summation of the average?

• Because noise error can be positive and negative thus canceling and showing less error that what truly exists.

• Also need to depict standard deviation because need to denote spread in your data

Operation Definitions To keep procedure constant so that the

results are repeatable.• Example of Standards

•ANSI—American National Standard Institute

• ITU—International Telecommunication Union

•AAMI—Association for the Advancement of Medical Instrumentation

• IEEE—Institute for Electrical and Electronic Engineers

Summary

Define and understand how to depict system gain in dB and non dB format

Define 2 Types of Measurement Calculate Mean, Variance and Standard

Deviation Define 3 categories of Measurement Explain 5 factors that Affect

Measurement

Summary

Define Accuracy and Precision Define 4 types of Error Describe one way to avoid Error What is an Error Budget and how do you

calculate Error What are Standards and why are they

important