Formula Sheets: Error Analysis | Electrical and Electronic Measurements - Electrical Engineering (EE) PDF Download

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Electrical and Electronic Measuremen ts: Error
Analysis F orm ula Sheet for Electrical GA TE
T y p es of Errors
• Absolute Error : Difference b et w een measured and true v alue.
E =X
m
-X
t
where E is absolute error, X
m
is measured v alue, X
t
is true v alue (units dep end on
quan tit y , e.g., V , ? ).
• Relativ e Error : Ratio of absolute error to true v alue.
Relativ e Error=
E
X
t
• P ercen tage Error : Relativ e error expressed as a p ercen tage.
P ercen tage Error=
E
X
t
×100
• Limiting Error : Maxim um p ossible error based on instrumen t accuracy .
Limiting Error=± F ull Scale V alue× A ccuracy
where A ccuracy is sp ecified as a fraction or p ercen tage.
Statistical Error Analysis
• Mean V alue : A v erage of m ultiple measuremen ts.
¯
X =
?
n
i=1
X
i
n
where
¯
X is mean, X
i
are individual measuremen ts, n is n um b er of measuremen ts.
• Deviation : Difference from mean for eac h measuremen t.
d
i
=X
i
-
¯
X
where d
i
is deviation of the i -th measuremen t.
• Standard Deviation : Measure of disp ersion for a set of measuremen ts.
s=
v
?
n
i=1
(X
i
-
¯
X)
2
n
where s is standard deviation.
• V ariance : Square of standard deviation.
V ariance=s
2
=
?
n
i=1
(X
i
-
¯
X)
2
n
1
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Electrical and Electronic Measuremen ts: Error
Analysis F orm ula Sheet for Electrical GA TE
T y p es of Errors
• Absolute Error : Difference b et w een measured and true v alue.
E =X
m
-X
t
where E is absolute error, X
m
is measured v alue, X
t
is true v alue (units dep end on
quan tit y , e.g., V , ? ).
• Relativ e Error : Ratio of absolute error to true v alue.
Relativ e Error=
E
X
t
• P ercen tage Error : Relativ e error expressed as a p ercen tage.
P ercen tage Error=
E
X
t
×100
• Limiting Error : Maxim um p ossible error based on instrumen t accuracy .
Limiting Error=± F ull Scale V alue× A ccuracy
where A ccuracy is sp ecified as a fraction or p ercen tage.
Statistical Error Analysis
• Mean V alue : A v erage of m ultiple measuremen ts.
¯
X =
?
n
i=1
X
i
n
where
¯
X is mean, X
i
are individual measuremen ts, n is n um b er of measuremen ts.
• Deviation : Difference from mean for eac h measuremen t.
d
i
=X
i
-
¯
X
where d
i
is deviation of the i -th measuremen t.
• Standard Deviation : Measure of disp ersion for a set of measuremen ts.
s=
v
?
n
i=1
(X
i
-
¯
X)
2
n
where s is standard deviation.
• V ariance : Square of standard deviation.
V ariance=s
2
=
?
n
i=1
(X
i
-
¯
X)
2
n
1
Error Propagation
• Error in Sum or Difference : F or Z =X±Y .
?Z =
v
(?X)
2
+(?Y)
2
where ?X,?Y are absolute errors of X and Y , ?Z is error in Z .
• Error in Pro duct or Quotien t : F or Z =X·Y or Z =
X
Y
.
?Z
Z
=
v
(
?X
X
)
2
+
(
?Y
Y
)
2
where
?X
X
,
?Y
Y
are relativ e errors.
• Error in P o w er : F or Z =X
n
.
?Z
Z
=n·
?X
X
where n is the p o w er,
?X
X
is relativ e error of X .
Instrumen t Sp ecifications
• A ccuracy : Closeness to true v alue, often giv en as p ercen tage of full scale.
A ccuracy=±
Max Error
F ull Scale V alue
×100
• Precision : Rep eatabilit y of measuremen ts, related to standard deviation.
• Sensitivit y : Output c hange p er unit input c hange.
S =
? Output
? Input
Units dep end on instrumen t (e.g., mVA
-1
for ammeters).
• Resolution : Smallest detectable c hange in input.
Resolution= Smallest Division on Scale
Key Notes
• Units dep end on measured quan tit y: V oltage (V ), Curren t (A ), Resistance (? ), etc.
• Systematic errors (e.g., calibration) are consisten t; random errors v ary and are ana-
lyzed statistically .
• Limiting error is critical for GA TE n umerical problems in v olving instrumen t sp ecifi-
cations.
• Error propagation form ulas assume indep enden t errors.
• Use SI units for GA TE calculations.
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