Chapter - 02 : Measurement Errors

Example : 2.1 - Page No : 24

In [2]:
 #Given data
Am= 10.25 # in ohm
A= 10.22 # in ohm
del_A= Am-A # in ohm
print "Absolute error = %0.2f ohm" %del_A

Absolute error = 0.03 ohm

Example : 2.2 - Page No : 24

In [4]:
 #Given data
Am= 6.7 # in A
A= 6.54 # in A
del_A= Am-A # in A
print "Absolute error = %0.2f A" %del_A
print "Correction = %0.2f A" %(-del_A)

Absolute error = 0.16 A
Correction = -0.16 A

Example : 2.3 - Page No : 24

In [5]:
 #Given data
Am= 25.34 # in watt
del_A= -0.11 # in watt
A= Am-del_A 
print "True value = %0.2f watt" %A

True value = 25.45 watt

Example : 2.4 - Page No : 24

In [6]:
#Given data
Am= 205.3*10**-6 # in F
A=  201.4*10**-6 # in F
epsilon_o= Am-A 
epsilon_r= epsilon_o/A*100 # in %
print "Percentage relative error = %0.2f %%" %epsilon_r

Percentage relative error = 1.94 %

Example : 2.5 - Page No : 24

In [13]:
from __future__ import division 
#Given data
PerError= 5 # in %
epsilon_r= PerError/100 
Am=20 # in H
del_A= Am*epsilon_r 
# A= Am+del_A and A= Am-del_A
print "Limiting value of inductance =",int(Am),"±",int(del_A),"in Henry"

Limiting value of inductance = 20 ± 1 in Henry

Example : 2.6 - Page No : 24

In [14]:
 #Given data
V=600 # in volt
A= 400 #in volt
epsilon_r= 2.5/100 
del_V= epsilon_r*V 
PerLimitError= del_V/A*100 # in %
print "The percentage limiting error at 400 volt = ± %0.2f %%" %PerLimitError  

The percentage limiting error at 400 volt = ± 3.75 %

Example : 2.7 - Page No : 25

In [15]:
 #Given data
Am= 500 # in watt
epsilon_r= 1.5/100 # in neg and pos
# for positive value of epsilon_r
A1= Am*(1+epsilon_r) # in watt
# for positive value of epsilon_r
A2= Am*(1-epsilon_r) # in watt
print "Range of reading of wattmeter is ",round(A2,1)," watt to ",round(A1,1)," watt"

Range of reading of wattmeter is  492.5  watt to  507.5  watt

Example : 2.8 - Page No : 25

In [18]:
 #Given data
epsilon_r= 1.5/100 # in neg and pos
A= 10 # in amp
del_A= epsilon_r*A #in amp
# The magnitude of current being measured is 2.5 A. The relative error at this current is 
A= 2.5 # in amp
epsilon_r= del_A/A 
# Hence, the current under measurement is between the limits of
Am= 2.5 #in amp
# for positive value of epsilon_r
A1= Am*(1+epsilon_r) # in amp
# for positive value of epsilon_r
A2= Am*(1-epsilon_r) # in amp
print "Limiting values of current under measurement are ",round(A2,2)," amp to ",round(A1,2)," amp"
LimitingError= del_A/A*100 # in %
print "Limiting Error = %0.f %%" %LimitingError

Limiting values of current under measurement are  2.35  amp to  2.65  amp
Limiting Error = 6 %

Example : 2.9 - Page No : 25

In [22]:
 #Given data
epsilon_r= 1/100 
P=1000 # in watt
del_P= epsilon_r*P # in watt
# The magnitude of the power being measured is 100 watts.
PerLimitError= del_P/100*100 # in %
print "The percentage limiting error at 1000 =",int(PerLimitError),"%"

The percentage limiting error at 1000 = 10 %

Example : 2.10 - Page No : 25

In [25]:
 #Given data
# For positive value of error
R1= 100+100*2/100 #in ohm
R2= 200+200*2.5/100 # in ohm
AddR1R2_pos= R1+R2 # in ohm
# For negative value of error
R1= 100-100*2/100 #in ohm
R2= 200-200*2.5/100 # in ohm
AddR1R2_neg= R1+R2 # in ohm
print "Values of R1+R2 =",int(AddR1R2_neg),"ohm to",int(AddR1R2_pos),"ohm"

Values of R1+R2 = 293 ohm to 307 ohm

Example : 2.12 - Page No : 26

In [31]:
 #Given data
AV= 110.2 # true value of voltage in volt
AI= 5.3 # true value of current in amp
v= 0.2 # uncertainties in voltage in volt
i= 0.6 # uncertainties in current in amp
PLV= v/AV*100 # percentage limiting error to voltage drop
PLC= i/AI*100 # percentage limiting error to current
P= AV*AI # in watt
print "The power dissipated in the resistor = %0.2f watt" %P
LE_P= (PLV+PLC) # limiting error in the power dissipation in pos and neg
print "The limiting error in the power dissipation = ± %0.2f" %LE_P
print "Power dissipation =",round(P-P*LE_P/100,2),"W to ",round(P+P*LE_P/100,2),"W"

The power dissipated in the resistor = 584.06 watt
The limiting error in the power dissipation = ± 11.50
Power dissipation = 516.88 W to  651.24 W

Example : 2.13 - Page No : 26

In [33]:
 #Given data
AR= 100 # true value of resistance in ohm
AI= 2 # true value of current in amp
R= 0.2 # uncertainties in resistance in ohm
I= 0.01 # uncertainties in current in amp
PLR= R/AR*100 # percentage limiting error to resistance
PLC= I/AI*100 # percentage limiting error to current
P=AI**2*AR # in watt
LE_P= 2*PLC+PLR # limiting error in the power dissipation 
print "Power dissipation =",round(P-P*LE_P/100,1),"W to",round(P+P*LE_P/100,1),"W"

Power dissipation = 395.2 W to 404.8 W

Example : 2.14 - Page No : 27

In [36]:
 #Given data
FullScaleReading= 200 # in V
N= 100 # Number of division of scale
SD= FullScaleReading/N # 1 scale division
Resolution = 1/5*SD # in v
print "The value of resolution = %0.1f volts" %Resolution

The value of resolution = 0.4 volts

Example : 2.15 - Page No : 34

In [38]:
 #Given data
# u= 150+2.4 miu F and 150-2.4 miu F
# v= 120+1.5 miu F and 120-1.5 miu F
y=150+120 
del_y = 2.4+1.5 # Pos and neg
print "Limiting error with pos and neg = ± %0.1f miu F" %del_y
RelLimError= del_y/y*100 # in %
print "Relative limiting error with pos and neg = %0.2f %%" %RelLimError  

Limiting error with pos and neg = ± 3.9 miu F
Relative limiting error with pos and neg = 1.44 %

Example : 2.16 - Page No : 35

In [41]:
 #Given data
R1= 1 #in kohm
R1=R1*10**3 #in ohm
del_R1ByR1= 1 
del_R2ByR2= 1 
R2= 500 #in kohm
R= R1*R2/(R1+R2) #in ohm
# Let R= X/Y
X= R1*R2 
Y=R1+R2 
ErrorX= del_R1ByR1+del_R2ByR2 # with pos and neg
# ErrorY= del_R1/Y + del_R2/Y = R1/Y*del_R1ByR1 + R2/Y*del_R2ByR2
ErrorY= R1/Y*del_R1ByR1 + R2/Y*del_R2ByR2 # with pos and neg
PerError= ErrorX+ErrorY # in % with pos and neg
print "Percentage error (maximum posible) in equivalent parallel resistance = ± %0.f %%" %PerError
Error= 333.33*PerError/100 
Error=round(Error) 
print "Error (maximum possible) in equivalent parallel resistance = %0.f ohm" %Error

Percentage error (maximum posible) in equivalent parallel resistance = ± 3 %
Error (maximum possible) in equivalent parallel resistance = 10 ohm

Example : 2.17 - Page No : 35

In [45]:
 #Given data
R1= 200 #in ohm
R2= 100 #in ohm
R3= 50 #in ohm
del_R1ByR1= 5 
del_R2ByR2= 5 
del_R3ByR3= 5 
# Part (i) when the resistance are connected in series
Rse= R1+R2+R3 # in ohm
print "Equivalent resistance when connected in seried = %0.f ohm" %Rse 
LimError= R1/Rse*del_R1ByR1 + R2/Rse*del_R2ByR2 + R3/Rse*del_R3ByR3 
print "Relative limiting error of series resistances = ± %0.f %%" %LimError
LimError= Rse*LimError/100 #relative limiting error of series equivalent resistance in ohm
print "Relative limiting error of series equivalent resistance = ± %0.1f ohm" %LimError

# Part(ii) when the resistance are connected in parallel
Rp= R1*R2*R3/(R1*R2+R2*R3+R3*R1) 
print "Equivalent resistance when connected in parallel = %0.2f ohm" %Rp
# Let Rp= X/Y
X= R1*R2*R3 
Y=R1*R2+R2*R3+R3*R1 
y1= R1*R2 
y2= R2*R3 
y3= R3*R1 
ErrorX= del_R1ByR1 + del_R2ByR2 + del_R3ByR3 
Errory1= del_R1ByR1 + del_R2ByR2  
Errory2= del_R2ByR2 + del_R3ByR3  
Errory3= del_R3ByR3 + del_R1ByR1  
ErrorY= ( y1/Y*Errory1 + y2/Y*Errory2 + y3/Y*Errory3)
LimError= ErrorX + ErrorY 
print "Percentage error (maximum possible) in equivalent parallel resistance = ± %0.f %%" %LimError
LimError= Rp*LimError/100 
print "Error (maximum possible) in equivalent parallel resistance = %0.4f ohm" %LimError

Equivalent resistance when connected in seried = 350 ohm
Relative limiting error of series resistances = ± 5 %
Relative limiting error of series equivalent resistance = ± 17.5 ohm
Equivalent resistance when connected in parallel = 28.57 ohm
Percentage error (maximum possible) in equivalent parallel resistance = ± 25 %
Error (maximum possible) in equivalent parallel resistance = 7.1429 ohm

Example : 2.18 - Page No : 36

In [46]:
 #Given data
epsilon_r= 1.5/100 
V=100 # in volt
I=150 # in mA
del_V= epsilon_r*V # in volt
Vm= 70 # magnitude of voltage being measured in volt
PerLimError_V= del_V/Vm*100 # in %
del_I= epsilon_r*I # in mA
Im= 80 #in mA
PerLimError_C= del_I/Im*100 # in %
P= Vm*Im/1000 # in watt
RelLImError_P= (PerLimError_V+PerLimError_C) # in %
print "Relative limiting error in power measurement = %0.3f %%" %RelLImError_P 

Relative limiting error in power measurement = 4.955 %

Example : 2.19 - Page No : 37

In [47]:
 #Given data
E= 200 # in V
del_E_by_E= 1 
R=1000 # in ohm
del_R_by_R= 5 
P=E**2/R # in watt
print "Normal power consumed = %0.f watt" %P
LimError= 2*del_E_by_E+del_R_by_R # in %
print "Relative limiting error in measurement of power = ± %0.f %%" %LimError
LimError= LimError*P/100 #in watt
print "Limiting error of power = ± %0.1f watt" %LimError

Normal power consumed = 40 watt
Relative limiting error in measurement of power = ± 7 %
Limiting error of power = ± 2.8 watt

Example : 2.20 - Page No : 37

In [48]:
 #Given data
R1= 500 # in ohm
R2= 615 # in ohm
R3= 100 # in ohm
delR1ByR1= 1 
delR2ByR2= 1 
delR3ByR3= 0.5 
# Part(i)
R4=R1*R2/R3 # in ohm
print "Unknown resistance = %0.f ohm" %R4
delR4ByR4= delR1ByR1+delR2ByR2+delR3ByR3 
print "Relative limiting error of unknown resistance = ± %0.1f %%" %delR4ByR4
LimError= R4*delR4ByR4/100 
print "Limiting error = %0.3f ohms" %LimError 

Unknown resistance = 3075 ohm
Relative limiting error of unknown resistance = ± 2.5 %
Limiting error = 76.875 ohms

Example : 2.21 - Page No : 37

In [49]:
 #Given data
del_PbyP=0.5 
del_CbyC=1 
del_VbyV=1 
del_PFbyPF=del_PbyP + del_CbyC + del_VbyV 
print "Relative limiting error = ± %0.1f %%" %del_PFbyPF

Relative limiting error = ± 2.5 %

Example : 2.22 - Page No : 37

In [51]:
 #Given data
C=1 # in miu F
C=C*10**-6 # in F
P=1000 # in ohm
Q=2000 # in ohm
r=200 # in ohm
S=2000 # in ohm
del_C_by_C= 1 
del_P_by_P= 0.4 
del_Q_by_Q= 1 
del_r_by_r= 0.5 
del_S_by_S= 0.5 
Lx= C*P/S*(r*(Q+S)+Q*S) # in Henry
print "Unknown inductance = %0.1f Henry" %Lx
# Let
u=Q+S # in ohm
Error_u= Q/u*del_Q_by_Q + S/u*del_S_by_S # in %
# Let v= r*(Q+S) = r*u
v= r*(Q+S) 
Error_v= del_r_by_r + Error_u # in %
# Let 
x=Q*S 
Error_x= del_Q_by_Q + del_S_by_S # in %
# Let y= r*(Q+S)+Q*S = v+x
y=v+x 
Error_y= v/y*Error_v + x/y*Error_x # in %
del_Lx_by_Lx= del_C_by_C + del_P_by_P + del_S_by_S + Error_y # in %
print "Percentage error in inductance = %0.3f %%" %del_Lx_by_Lx

Unknown inductance = 2.4 Henry
Percentage error in inductance = 3.358 %

Example : 2.23 - Page No : 38

In [53]:
from numpy import pi 
#Given data
R=100 # in ohm
del_R_by_R= 5 
L=2 # in Henry
del_L_by_L= 10 
omega= 2*pi*50 
# Let
u=R**2 
Error_u= 2*del_R_by_R 
# Let
v= omega**2*L**2 
Error_v= 2*del_L_by_L 
# Let 
x= u+v 
Error_x= u/x*Error_u + v/x*Error_v # in %
# Now
Z= x**(1/2) 
Error_Z= 1/2*Error_x 
print "The uncertainly in the measurement of Z = %0.3f %%" %Error_Z

The uncertainly in the measurement of Z = 9.876 %

Example : 2.24 - Page No : 40

In [55]:
from math import sqrt
#Given data
x1= 49.7 
x2= 50.1 
x3= 50.2 
x4= 49.6 
x5= 49.7 
n=5 
x_bar= (x1+x2+x3+x4+x5)/5 
d1= x1-x_bar 
d2= x2-x_bar 
d3= x3-x_bar 
d4= x4-x_bar 
d5= x5-x_bar 
s= sqrt((d1**2+d2**2+d3**2+d4**2+d5**2)/(n-1)) 
print "The value of standard deviation = %0.2f" %s

The value of standard deviation = 0.27

Example : 2.25 - Page No : 44

In [56]:
 #Given data
x1= 41.7 
x2= 42 
x3= 41.8 
x4= 42 
x5= 42.1 
x6= 41.9 
x7= 42.5 
x8= 42 
x9= 41.9 
x10=41.8 
n=10 
# (i)
x_bar= (x1+x2+x3+x4+x5+x6+x7+x8+x9+x10)/10 
print "Arithmetic mean = %0.2f" %x_bar
d1= x1-x_bar 
d2= x2-x_bar 
d3= x3-x_bar 
d4= x4-x_bar 
d5= x5-x_bar 
d6= x6-x_bar 
d7= x7-x_bar 
d8= x8-x_bar 
d9= x9-x_bar 
d10= x10-x_bar 
# (ii)
sigma= sqrt((d1**2+d2**2+d3**2+d4**2+d5**2+d6**2+d7**2+d8**2+d9**2+d10**2)/(n-1)) 
print "The value of standard deviation = %0.3f" %sigma 

# (iii)
r= 0.6745*sigma 
print "Probable error of one reading = %0.3f" %r

Arithmetic mean = 41.97
The value of standard deviation = 0.221
Probable error of one reading = 0.149

Example : 2.26 - Page No : 45

In [63]:
 #Given data
x1= 1.570 
x2= 1.597 
x3= 1.591 
x4= 1.562 
x5= 1.577 
x6= 1.580 
x7= 1.564 
x8= 1.586 
x9= 1.550 
x10=1.575 
n=10 
# (i)
x_bar= (x1+x2+x3+x4+x5+x6+x7+x8+x9+x10)/10 
print "Arithmetic mean        = %0.4f gramme" %x_bar
d1= x1-x_bar 
d2= x2-x_bar 
d3= x3-x_bar 
d4= x4-x_bar 
d5= x5-x_bar 
d6= x6-x_bar 
d7= x7-x_bar 
d8= x8-x_bar 
d9= x9-x_bar 
d10= x10-x_bar 

# (ii)
D= (abs(d1)+abs(d2)+abs(d3)+abs(d4)+abs(d5)+abs(d6)+abs(d7)+abs(d8)+abs(d9)+abs(d10))/n # in gramme
print "Average deviation      = %0.3f gramme" %D

# (iii)
sigma= sqrt((d1**2+d2**2+d3**2+d4**2+d5**2+d6**2+d7**2+d8**2+d9**2+d10**2)/(n-1)) # in gramme
print "Standard deviation     = %0.5f gramme" %sigma 

# (iv)
V= sigma**2 # variance in gramme**2
print "Variance               = %0.3e gramme**2" %V 

# (v)
r= 0.6745*sigma # in gramme
print "Probable error         = %0.4f gramme" %r

# (vi)
rm= r/sqrt(n-1) # in gramme
print "Probable error of mean = %0.4f gramme" %rm

Arithmetic mean        = 1.5752 gramme
Average deviation      = 0.011 gramme
Standard deviation     = 0.01426 gramme
Variance               = 2.033e-04 gramme**2
Probable error         = 0.0096 gramme
Probable error of mean = 0.0032 gramme