# Chapter 3 Generalized Performance Characteristics of Instruments¶

## Example 3_1 pgno:60¶

In :
#Chapter_3 Generalized Performance Characteristics Of Instruments
#Caption:Gaussian Distribution
# Example 1

me=7. ;
stddev=0.5;
x = 6.  #('enter the lower limit of the range=:')
y= 7.5  #('enter the upper limit of the range=:')
n= 200.  #('enter the number of samples=:')
print"using    k =abs((x-me)/((2**0.5)*stddev));"
k =abs((x-me)/((2**0.5)*stddev));
print'Value of eta1 is  \n',k

p=abs((y-me)/((2**0.5)*stddev));
print'Value of eta2 is  \n',p
#Using the gaussian probability error function table, find the error function corresponding to the value of k and p
#LET IT BE s
s= 0.95  # ('enter the error function corresponding to k value=:')
Fx=(1./2.)+(1./2.*s);# Probability of having lengths less than x
l= 0.68   # ('enter the error function corresponding to p value=:')
Fy=(1./2.)+(1./2.*l);# Probability of having lengths less than y

print'probability of having length less than 6 cm is  ',Fx
print'probability of having length less than 67.5cm is  ',Fy

Px=abs(Fy-Fx);
print"Number of samples in the given length range="
m=(n*Px);
print(m);

using    k =abs((x-me)/((2**0.5)*stddev));
Value of eta1 is
1.41421356237
Value of eta2 is
0.707106781187
probability of having length less than 6 cm is   0.975
probability of having length less than 67.5cm is   0.84
Number of samples in the given length range=
27.0


## Example 3_2 pgno:62¶

In :
#Caption:Combination of component errors in overall system-accuracy calculations
#example2
#page 62

#Consider an experiment for measuring, by means of a dynamometer, the average power transmitted by a rotating sheft

from math import pi
R=1202.  #('Enter the revolutions of shaft during time t=:')
F=45.  #('Enter the force at end oftorque arm=:')
L=0.397 #('Enter the length of torque arm=:')
t=60.  #('Enter the time length of run=:')
W=(2*pi*R*F*L)/t;
#Computing various partial dervatives
dWF=(2*pi*R*L)/t;
print(dWF)   #dWF represents dW/dF
dWR=(2*pi*F*L)/t;
dWL=(2*pi*F*R)/t;
dWt=-(2*pi*R*F*L)/(t**2);
#Let f, r, l and t represent the uncertainties

f=0.18   #('Enter the uncertainty in force=:')
r=1  #('Enter the uncertainty in the no of revolutions=:')
l=0.00127   #('Enter the uncertainty in the length=:')
t=0.5  #('Enter the uncertainty in the time length of run=:')
Ea=(dWF*f)+(dWR*r)+(dWL*l)+abs(dWt*t);      #absolute error
print"The absolute error is  "
print(Ea);
#To find total uncertainty
U=(((dWF*f)**2)+(dWR*r)**2+(dWL*l)**2+abs(dWt*t)**2)**0.5
print"Total uncertainty is "
print(U)

49.9716388246
The absolute error is
36.798728247
Total uncertainty is
22.075336595


## Example 3_3 pgno:64¶

In :
#Caption:Combination of component errors in overall system-accuracy calculations
#example3
#page 64

#Consider an experiment for measuring, by means of a dynamometer, the average power transmitted by a rotating sheft

from math import pi
R=1305.  #('Enter the revolutions of shaft during time t=:')
F=85.  #('Enter the force at end oftorque arm=:')
L=0.467 #('Enter the length of torque arm=:')
t=10.  #('Enter the time length of run=:')
W=(2*pi*R*F*L)/t;
#Computing various partial dervatives
dWF=(2*pi*R*L)/t;
print(dWF)   #dWF represents dW/dF
dWR=(2*pi*F*L)/t;
dWL=(2*pi*F*R)/t;
dWt=-(2*pi*R*F*L)/(t**2);
#Let f, r, l and t represent the uncertainties

f=0.18   #('Enter the uncertainty in force=:')
r=1  #('Enter the uncertainty in the no of revolutions=:')
l=0.00127   #('Enter the uncertainty in the length=:')
t=0.5  #('Enter the uncertainty in the time length of run=:')
Ea=(dWF*f)+(dWR*r)+(dWL*l)+abs(dWt*t);      #absolute error
print"The absolute error is  "
print(Ea);
#To find total uncertainty
U=(((dWF*f)**2)+(dWR*r)**2+(dWL*l)**2+abs(dWt*t)**2)**0.5
print"Total uncertainty is "
print(U)

382.919303768
The absolute error is
1809.7878357
Total uncertainty is
1631.45984422


## Example 3_4 pgno:94¶

In :
# Chapter_3 Generalized Performance Characteristics Of Instruments
#Caption:First order instrument
#Example 4
#Page no. 94
d=.006   #('Enter the diameter of the diameter of the sphere in meters=:')
p=14500.  #('Enter the density of the liquid in glass bulb=:')
c=180.   #('Enter the specific heat of liquid(in j/kg degree centigrade)=:')
U=20.  #('Enter the heat transfer coefficient in W/m^2-degree centigrade=:')
from math import pi
Vb=(pi*d*d*d)/6;    #Volume of sphere
Ab=pi*d*d;    #Surface area of sphere
timconstant=(p*c*Vb*1000)/(U*Ab);    #time constant
print "the value of the time constatn is ",(timconstant)

the value of the time constatn is  130500.0


## Example 3_5 pgno:96¶

In :
# Chapter_3 Generalized Performance Characteristics Of Instruments
#Caption:First order instrument
#Example 5
#Page no. 96
d=.004   #('Enter the diameter of the diameter of the sphere in meters=:')
p=13600.  #('Enter the density of the liquid in glass bulb=:')
c=150.   #('Enter the specific heat of liquid(in j/kg degree centigrade)=:')
U=40.  #('Enter the heat transfer coefficient in W/m^2-degree centigrade=:')
from math import pi
Vb=(pi*d*d*d)/6;    #Volume of sphere
Ab=pi*d*d;    #Surface area of sphere
timconstant=(p*c*Vb*1000)/(U*Ab);    #time constant
print "the value of the time constatn is ",(timconstant)

the value of the time constatn is  34000.0


## Example 3_6 pgno:¶

In :
#Caption:Step response of first order systems
#Example 6
# page 100

# Given:In air, probe dry          timeconstant(tc)=30s
#       In water                                tc=5s
#       In air, probe wet                       tc=20s
# for t<0,T=25 degree C(initial temperature)
#     0<t<7, T=35 degree C(dry probe in air)
#     7<t<15, T=70 degree C(probe in water)
#     15<t<30, T=35 degree C(wet probe in air)
from math import e
#case i T(a)=25
T7=35+(25-35)*e**(-(7/30))
print"Temperature at the end of first interval"
print(T7);
#case ii T(a)=T7
T15=70+(T7-70)*e**(-((15-7)/5))
print"Temperature at the end of second interval"
print(T15);
#case iii T(a)=T15
T30=35+(T15-35)*e**(-((30-15)/20))
print"Temperature at the end of third interval"
print(round(T30));

Temperature at the end of first interval
25.0
Temperature at the end of second interval
53.4454251473
Temperature at the end of third interval
53.0


## Example 3_7 pgno:103¶

In :
#Caption:Adequate frequency response conditions for first order instruments
#Example 7
#Page 103
# To measure qi given by
# qi=sin2t+0.3sin20t
# timeconstant=0.2s
from math import pi,atan
H=1/((0.16+1)**0.5);       #H(jw)=qo/qiK
phi=((atan(-2*0.2))*180)/pi ;
H2=1/((16+1)**0.5);
phi2=((atan(-20*0.2))*180)/pi;
print"sinusoidal transfer function at 2 rad/sec is"
print(H);
print(phi)
print"sinusoidal transfer function at 20rad/sec is"
print(H2)
print(phi2)

print"qo/K can be written as"

print"      qo=0.93K sin(2t-21.8)+(0.24K) 0.3sin(20t-76)"
#Suppose we consider use of an instrument with timeconstant=0.002s
H=1/((1.6*(10)**(-5)+1)**0.5);
phi=((atan(-2*.002))*180)/pi ;
H2=1/((1.6*(10**-3)+1)**0.5);
phi2=((atan(-20*0.002))*180)/pi;
print"sinusoidal transfer function at 2 rad/sec is"
print(H);
print(phi)
print"sinusoidal transfer function at 20rad/sec is"
print(H2)
print(phi2)
print"qo/K can be written as"
print"qo=K sin(2t-0.23)+K 0.3sin(20t-2.3)"
print"Clearly, this instrument measures the given qi faithfully"

sinusoidal transfer function at 2 rad/sec is
0.928476690885
-21.8014094864
sinusoidal transfer function at 20rad/sec is
0.242535625036
-75.9637565321
qo/K can be written as
qo=0.93K sin(2t-21.8)+(0.24K) 0.3sin(20t-76)
sinusoidal transfer function at 2 rad/sec is
0.999992000096
-0.229181895754
sinusoidal transfer function at 20rad/sec is
0.999200958722
-2.29061004264
qo/K can be written as
qo=K sin(2t-0.23)+K 0.3sin(20t-2.3)
Clearly, this instrument measures the given qi faithfully