# Chapter 14 : Transducers And The Measurement System¶

## Example14_1,pg 421¶

In [28]:
# find percentage change in resistance

import math
#Variable declaration
delVo=120*10**-3             #output voltage
Vs=12.0                      #supply voltage
R=120.0                      #initial resistance

#Calculations
delR=(delVo*2*R)/Vs          #change in resistance
per=(delR/R)*100             #percent change in resistance

#Result
print("percent change in resistance:")
print("per = %.f"%per)

percent change in resistance:
per = 2


## Example14_2,pg 423¶

In [32]:
# find bridgemann coefficient

import math
#Variable declaaration
lam=175.0                     #gauge factor
mu=0.18                       #poisson's ratio
E=18.7*10**10                 #young's modulus

#Calculations
si=((lam-1-(2*mu))/E)         #bridgemann coefficient

#Result
print("bridgemann coefficient:")
print("si = %.2f * 10^-10 m^2/N"%(math.floor(si*10**12)/100))

bridgemann coefficient:
si = 9.28 * 10^-10 m^2/N


## Example14_3,pg 428¶

In [35]:
# pt100 RTD

import math
#Variable declaration
R4=10*10**3
Ro=-2.2*10**3              #output resistance
R2=R4-0.09*R4

#Calculations
R1=(Ro*((R2**2)-(R4**2)))/(R2*(R2+R4))

#Result
print("resistance R1 and R3:")
print("R1 = R3 = %.1f ohm"%(math.floor(R1*10)/10))

resistance R1 and R3:
R1 = R3 = 217.5 ohm


## Example14_4,pg 435¶

In [37]:
# senstivity in measurement of capacitance

import math
#Variable declaration
#assuming eps1=9.85*10^12
x=4.0                   #separation between plates
x3=1.0                  #thickness of dielectric
eps1=9.85*10**12        #dielectric const. of free space
eps2=120.0*10**12       #dielectric const. of material

#Calculations
Sx=(1/(1+((x/x3)/((eps1/eps2)-1))))

#Result
print("sensitivity of measurement of capacitance:")
print("Sx = %.4f"%Sx)

sensitivity of measurement of capacitance:
Sx = -0.2978


## Example14_5,pg 510¶

In [1]:
# find max gauge factor

import math
#Variable declaration
#if (delp/p)=0, the gauge factor is lam=1+2u
u=0.5                    #max. value of poisson's ratio

#Calculations
lam=1+(2*u)

#Result
print("max. gauge factor:")
print("lam = %.f"%lam)

max. gauge factor:
lam = 2


## Example14_6,pg 510¶

In [3]:
# find Young modulus

import math
#Variable declaration
lam=-150.0          #max. gauge factor
si=-9.25*10**-10    #resistivity change
mu=0.5              #max poisson's ratio

#Calculations
E=((lam-1-(2*mu))/si)

#Result
print("young modulus:")
print("E = %.1f N/m^2"%(E/10**10))

young modulus:
E = 16.4 N/m^2


## Example14_7,pg 510¶

In [6]:
# find capacitance of sensor

import math
#Variable declaration
d1=4*10**-2                #diameter of inner cylinder
d2=4.4*10**-2              #diameter of outer cylinder
h=2.2                      #level of water
H=4.0                      #height of tank
epsv=0.013*10**-5          #dielectric const. of medium(SI)

#Calculations
eps1=((80.37*10**11)/((4*math.pi*10**8)**2))
C=(((H*epsv)+(h*(eps1-epsv)))/(2*math.log(d2/d1)))

#Result
print("capacitance of sensor:")
print("C = %.f micro-F"%(C*10**6))

capacitance of sensor:
C = 60 micro-F


## Example14_8,pg 511¶

In [9]:
# find ratio of collector currents

import math
#Variable declaration
VobyT=0.04                     #extrapolated bandgap voltage
RE1byRE2=(1/2.2)               #ratio of emitter resistances of Q1,Q2
kBbyq=0.86*10**3               #kB->boltzman const., q->charge

#Calcualtions
#(1+a)log(a)=(VobyT/RE1byRE2)*kBbyq, a->ratio of collector currents

#Result
print("ratio of collector currents:")
print("a = 23.094")

ratio of collector currents:
a = 23.094


## Example14_9,pg 511¶

In [22]:
# find normalized output

import math
#Variable declaration
#LVDT parameters
Rp=1.3
Rs=4
Lp=2.2*10**-3
Ls=13.1*10**-3
#M1-M2 varies linearly with displacement x, being maximum 0.4 cm
#when M1-M2=4mH so that k=(4/0.4)=10mH/cm
k=10#*10**-3
f=50.0        #frequency

#Calculations
w=2*math.pi*f
tp=(Rp/Lp)
N=((w*k/Rp)/(math.sqrt(1+(w**2)*(tp**2))))
phi=(math.pi/2)-math.atan(w*Lp/Rp)
phi=phi*(180/math.pi)
phi = 90 -phi
#Result
print("normalized output:")
print("N = %.4f V/V/cm\n"%N)
print("phase angle:")
print("phi = %.2f"%phi)
#Answer do not match with the book

normalized output:
N = 0.0130 V/V/cm

phase angle:
phi = 28.00


## Example14_10,pg 511¶

In [26]:
# find load voltage

import math
#Variable declaration
#for barium titanate, g cost. is taken as 0.04Vm/N. (it varies depending in composition and processing)
t=1.3*10**-3              #thickness
g=0.04                    #const.
f=2.2*9.8                 #force
w=0.4                     #width
l=0.4                     #length
p=13.75                  #pressure

#Calculations

#Result
print("Vo = %.2f V"%Vo)
#Answer in the book is wrong

voltage along load application:
Vo = 70.12 V


## Example14_11,pg 512¶

In [12]:
# find error and senstivity parameters

import math
#Variable declaration
N11=130.0
N22=229.0
N12=220.0
N21=139.0
#variable values
v1=4
v2=6.7
#temperatures
theta1=20
theta2=25

#Calculations
#parameters
B2=((N22+N11-N12-N21)/(v2-v1)*(theta2-theta1))        #temperature coefficient of resistivity
a2=((N22-N21)/(v2-v1))                                #zero error sensitivity
B1=(N22-N12)/(theta2-theta1)                          #temperature coefficient of zero point
a1=N22-(B1*theta2)-(a2*v2)                            #zero error

#Result
print("zero error:")
print("a1 = %.2f\n"%a1)
print("zero error sensitivity:")
print("a2 = %.2f\n"%a2)
print("temperature coefficient of zero point:")
print("B1 = %.2f\n"%B1)
print("temperature coefficient of resistivity:")
print("B2 = %.2f"%B2)

zero error:
a1 = -39.33

zero error sensitivity:
a2 = 33.33

temperature coefficient of zero point:
B1 = 1.80

temperature coefficient of resistivity:
B2 = 0.00