1. Measurements Measurements and errors :
- Here , the goal is to have some understanding of the operation and behavior of electrical test instruments . Also , to gain practical information on their use .

2. Instruments : is a device for determining the value or magnitude of a quantity or a variable.
Accuracy : is a measure of how close the output reading of an instrument is to the correct value .
Note : In practice , it’s more usual to express it by inaccuracy which is the extent to which a reading might be wrong ( uncertainty ) .

3. Example : a voltmeter with a voltage range of ( 0-10 V ) with uncertainty of ±1 % of the full scale (f.s) reading .
Notice that when measuring 10V (f.s) we obtain a reading of 9.9 and when measuring 1V we obtain a reading of 1.1V which had an inaccuracy of 10% in the other hand for the 10V just 1% .
Hence , it is very important that instruments are chosen such that their range is appropriate to the value being measured .

4. 2) precision : it is a measure of the reproducibility of the reading that describes an instrument’s degree of agreement within a group of readings . This means that if a large numbers of reading are taken of the same quality by a high precision instrument , then the spread of readings will be very small .

5. Note : High precision does not imply anything about measurement accuracy , ( a high precision instrument may have a low accuracy ) .
The reproducibility describes the closeness of the output readings for the same input when there are changes in the method of measurement , observer , measuring instrument , location , conditions of use and time of measurement .
The reproducibility in measurements from an instrument is one way of expressing it’s precision .

6. - Example :

7. 3) Sensitivity : it is the ratio of the output signal ( or response ) of the instrument to a change of input .
4) Resolution : is the lowest limit on the magnitude of the change in the input measured quantity that produces an observable change in the instrument output .

8. Example : A car speedometer , the subdivisions are typically 20 km/h this means that when the needle is between the scale markings , we cannot estimate speed more accurately than to the nearest 5 km/h and this is the meter resolution .

9. 5) Error : is the deviation from the true value of the measured variable .
Note : precision is composed by two things :-
1- conformity ( necessary , but not a sufficient condition for precision (lack of significant figures ) .
2- significant figures .
- Precision is a necessary , but not sufficient for accuracy .

10. Significant figures :
- None zero integers : they always count as significant figures .
Zeros : (a) Leading zeros : They never count as significant figures .
Example :  2 S.F
 3 S.F

11. (b) captive zeros : They always count as S.F .
Example :  4 S.F
(c) Trailing zeros : These zeros are S.F only if the number contains a decimal point .
Example : 100  1 S.F
100.0  4 S.F
100.  3 S.F
1.00 X  3 S.F

12. Note : The expression ( 1.00 X 102 ) has some advantages :
The number of significant figures can be easily indicated .
Fewer zeros are needed to write a very large/small numbers .

13. Applications : Multiplication / Division :
Here , the number of S.F. in the result is the same as the number in the least precise measurement used in the calculation .
Example : 4.56
X 1.4  least S.F
________
 6.4

14. Subtraction / Addition :
Here , the result has the same number of decimal places as the least precise measurement .
Example :
+18.0  1 decimal
1.013
______
 31.1

15. Example : we have obtained the following voltage measurement :
One way in getting close to the true value is taking several measurements and then obtaining their arithmetic mean ( average ) . The range of possible error is known as the largest deviation from the mean .
Example : we have obtained the following voltage measurement :
( V , V , V , V )
Find : 1) Average voltage .
2) Range of error .

16. Error range = Vmax - Vavg = 0.05 = VH
Answer : Vavg = ∑ V =
____________ 4
Error range = Vmax - Vavg = 0.05 = VH
= Vavg – Vmin = 0.04 = VL
Error range = ( VH + VL ) = ±
_____________________ 2

17. Range of doubt : (RD) Addition : X1 = 826 ± 5  X= X1 + X2 = 1454 ± 8
_________________
1454
X1+X2 ± (a1+a2)
Rd = ( (a1+a2) / ( x1+x2) ) X100%

18. (2) Subtraction : same X1 , X2  X = X1-X2 = 198 ± 8 8 X 100% = 4.04 %
_________________
198
- Here , we see that the percentage doubt differs greatly after subtraction compared to addition .
This tells us that you should avoid measurement techniques that depends on subtraction of results .
X1-X2 ± (a1+a2)
Rd = ( (a1+a2) / ( x1+x2) ) X100%

19. (3) Multiplication :
Example : X1= 8.62 ± 0.02
X2= ± 0.04

20. * Division EX : X1=10.1 ± 0.1 X2=5.3 ± 0.2 X=X1/X2=?
Steps : 1) Divide the numerator (added to it the (+Rd) ) by the denominator ( added to it the (-Rd) ) to get the worst answer .
10.2 / 5.1 = (X1+a1) / ( X2 - a2 ) = R
2) Perform the division using the original number .
10.1 /5.3 = X1/X2 = D
3) Obtain the difference between 1 and 2 .
2.0 – 1.9 = R-D
4) X= X1/X2 = 1.9 ± D ± ( R-D) OR X1/X2 ± (R-D)

21. Types of errors :
In measurements , Errors may come from different sources :
Gross error : This error comes from human mistakes in reading or using the instrument .
Example : A voltmeter with sensitivity of 1000 Ω/V reads 100V on its 150 V scale when connected across an unknown resistor in series with a milliameter when it reads 5 mA.
Apparent resistance of the unknown.
The actual resistance of the unknown.
Error due to the loading effect of the voltmeter .

22. RT  V=IR RT = V/I = 100/5 mA = 20 KΩ RV = 1000Ω/V X 150 V = 150 KΩ
Answer :
RT  V=IR RT = V/I = 100/5 mA = 20 KΩ
RV = 1000Ω/V X 150 V = 150 KΩ
RT = ( Rx RV ) / ( Rx+ RV ) = 20 
׃ Rx= RT RV
RT – RV = 23.05
3) Error = (actual – app ) / actual.
= (23.05 – 20 ) / = %

23. 2) Systematic error : Instrumental error . Environmental error .
This error is due to the instruments mechanical structures . ( i.e. calibration error ).
Here , there is a need for setting the instrument at the zero reference before taking any measurement .

24. Hence , in general :
One should select a suitable instrument .
Apply correction factor after determining the amount of instrument error .
One should check for erratic behavior and stability and reproducibility of result .

25. b. Here, external conditions affecting the instrument (i. e
b. Here, external conditions affecting the instrument (i.e. change in temperature , humidity, pressure , magnetic field … etc ).
The Instrumental error is subdivided into 2 categories :
1- static : this error is cause by the limitation of the device.
2- Dynamic : this error is caused by the instrument not responding fast enough to follow changes in the measured value.

26. 3) Random error :
Is an error that is caused by unknown causes.
This error may occur even when systematic errors have been accounted for.
Example : A voltmeter monitoring a voltage for half an hour , readings that vary slightly will be seen over the period of observation , the only way to offset ( overcomes) these errors is by increasing the number of readings and using the statistical mean .

27. Arithmetic mean ( average ) :
It represents the most probable value of a measured variable. The more readings we have, the more accurate results we get.