Chapter 7 - Basic Measuring Instruments

Example 1 - pg 7-13

In [1]:
#Chapter-7,Example7_1,pg 7-13
#calculate the deflection angle
#given
N=100.
B=0.15
A=10*8*10**-6
I=5*10**-3
K=0.2*10**-6#spring const.
#calculations
Td=N*B*A*I#deflecting torque
theta=Td/K#deflecting angle
#results
print"deflection theta (deg) = ",theta

deflection theta (deg) =  30.0

Example 2 - pg 7_21

In [3]:
#Chapter-7,Example7_2,pg 7-21
#given
#calculate the deflection
import math,sympy
from sympy import Symbol,horner
x=Symbol('x')
L=(12+6*x-(x**2))#x is deflection in rad from zero
dl=L.diff(x)
K=12.
I=8.
#calculations
x=6./(((2*K)/(I**2))+2)#x=((I**2)dl)/(2*k)
z=x*(180./math.pi)
y=horner(x,L)
#results
print"deflection for given current (degrees) = ",round(z,2)
print"inductance for given deflection (muH) = ",round(y,2)

deflection for given current (degrees) =  144.75
inductance for given deflection (muH) =  2.53

Example 3 - pg 7_23

In [4]:
#Chapter-7,Example7_3,pg 7-23
#calculate the value of shunt resistance
#given
Rm=100.
Im=2.*10**-3
I=150.*10**-3
#calculations
Rsh=(Im*Rm)/(I-Im)
#results
print"value of shunt resistance (ohm) = ",round(Rsh,3)

value of shunt resistance (ohm) =  1.351

Example 4 - pg 7_23

In [5]:
#Chapter-7,Example7_4,pg 7-23
#calculate the current, voltage and meter resistance
#given
Vsh1=400.*10**-3
Rsh=0.01
Ish1=50.
Rm=750.#coil resistance
#calculations
Ish=Vsh1/Rsh
Vsh=Ish1*Rsh
Im=Vsh1/Rm
Rm1=Vsh/Im#meter resistance
#results
print"current through shunt (A) = ",Ish
print"voltage through shunt (V) = ",Vsh
print"meter resistance (ohm) = ",Rm1

current through shunt (A) =  40.0
voltage through shunt (V) =  0.5
meter resistance (ohm) =  937.5

Example 5 - pg 7_25

In [6]:
#Chapter-7,Example7_5,pg 7-25
#calculate the full scale deflection and meter resistance
#given
I1=10*10**-3
Im=2*10**-3
Rm=75
I2=50*10**-3
I3=100*10**-3
#calculations
R1=(Im*Rm)/(I1-Im)
R2=(Im*Rm)/(I2-Im)
R3=(Im*Rm)/(I3-Im)
#results
print"designed multi-range ammeter"
print"full scale deflection Im (mA) = ",Im*1000.
print"meter resistance Rm (ohm) = ",Rm
print"R1(ohm) = ",R1
print"R2(ohm) = ",R2
print"R3(ohm) = ",round(R3,2)

designed multi-range ammeter
full scale deflection Im (mA) =  2.0
meter resistance Rm (ohm) =  75
R1(ohm) =  18.75
R2(ohm) =  3.125
R3(ohm) =  1.53

Example 6 - pg 7_27

In [7]:
#Chapter-7,Example7_6,pg 7-27
#calculate the meter resistance and full scale deflection
#given
I1=10.
Im=1*10**-3
Rm=50.
#in position-1 R1 is in shunt with R2+R3+Rm
#R1=10**-4(R2+R3+50)......(1)
#in position-2 (R1+R2) is in shunt with R3+Rm
#R1+R2=2*10**-4(R3+50).....(2)
#in position-3 R1+R2+R3 is in shunt with Rm
#R1+R2+R3=0.05............(3)
#from.....(3)
#R1+R2=0.05-R3
#substituting in........(2)
#calculations
R3=0.04/1.0002
#R2=0.01-R1........(4)
#substituing in (1)
R1=5.00139*10**-3/1.0001
R2=0.01-R1#from........(4)
#results
print"various sections of aryton shunt are\n"
print"full scale deflection Im (mA) = ",Im*1000.
print"meter resistance Rm (ohm) = ",Rm
print"R1 (ohm) = ",round(R1,3)
print"R2 (ohm) = ",round(R2,3)
print"R3 (ohm) = ",round(R3,5)

various sections of aryton shunt are

full scale deflection Im (mA) =  1.0
meter resistance Rm (ohm) =  50.0
R1 (ohm) =  0.005
R2 (ohm) =  0.005
R3 (ohm) =  0.03999

Example 7 - pg 7_30

In [8]:
#Chapter-7,Example7_7,pg 7-30
#calculate the multiplier resistance
#given
Rm=500.
Im=40*10**-6
V=10
#calculations
Rs=(V/Im)-Rm
#results
print"multiplier resistance (kohm) = ",Rs/1000.

multiplier resistance (kohm) =  249.5

Example 8 - pg 7_30

In [10]:
#Chapter-7,Example7_8,pg 7-30
#calculate the required shunt and multipler resistances
#given
Im=20*10**-3
Vm=200*10**-3
I=200
#calculations
Rm=(Vm/Im)
Rsh=(Im*Rm)/(I-Im)
V=500.
Rs=(V/Im)-Rm
#results
print"required shunt resistance (ohm) = ",round(Rsh,3)
print"required multipler resistance (kohm) = ",Rs/1000.

required shunt resistance (ohm) =  0.001
required multipler resistance (kohm) =  24.99

Example 9 - pg 7_33

In [11]:
#Chapter-7,Example7_9,pg 7-33
#calculate the series string of multiplers
#given
Rm=50
Im=2*10**-3
#calculations
#for position V4 multipler is R4
V4=10.
R4=(V4/Im)-Rm#Rs=(V/Im)-RmV3 m
#for position V3 multipler is R3+R4
V3=50.
R3=(V3/Im)-Rm-R4
#for position V2 multiplier is R2+R3+R4
V2=100.
R2=(V2/Im)-Rm-R3-R4
#for position V1 multiplier is R1+R2+R3+R4
V1=500.
R1=(V1/Im)-Rm-R3-R4-R2
#results
print"series string of multipliers"
print"R1 (kohm) = ",R1/1000.
print"R2 (kohm) = ",R2/1000.
print"R3 (kohm) = ",R3/1000.
print"R4 (kohm) = ",R4/1000.

series string of multipliers
R1 (kohm) =  200.0
R2 (kohm) =  25.0
R3 (kohm) =  20.0
R4 (kohm) =  4.95

Example 10 - pg 7_35

In [12]:
#Chapter-7,Example7_10,pg 7-35
#calculate the series string of multipliers
#given
Rm=50
Im=2*10**-3
V1=500.
V2=100.
V3=50.
V4=10.
#calculations
S=1/Im#senstivity
R4=S*V4-Rm
R3=S*V3-(R4+Rm)
R2=S*V2-(R4+Rm+R3)
R1=S*V1-(R4+Rm+R3+R2)
#results
print"series string of multipliers"
print"R1 (ohm) = ",R1/1000.
print"R2 (ohm) = ",R2/1000.
print"R3 (ohm) = ",R3/1000.
print"R4 (ohm) = ",R4/1000.

series string of multipliers
R1 (ohm) =  200.0
R2 (ohm) =  25.0
R3 (ohm) =  20.0
R4 (ohm) =  4.95

Example 11 - pg 7_36

In [13]:
#Chapter-7,Example7_11,pg 7-36
#calculate the multipler resistance
#given
Im=50*10**-6
Rm=200.
V=500.#V is voltage range
#calculations
S=1/Im
Rs=S*V-Rm
#results
print"multipler resistance (Mohm) = ",round(Rs/10**6,4)

multipler resistance (Mohm) =  9.9998

Example 12 - pg 7_36

In [14]:
#calculate the senstivity of meters A and B
#Chapter-7,Example7_12,pg 7-36
#given
#for meter A
Rs=25.*10**3
Rm=1.*10**3
V=100.
#calculations
S=(Rs+Rm)/V
#for meter B
Rs=150.*10**3
Rm=1.*10**3
V=1000.
S2=(Rs+Rm)/V
#results
print"senstivity of meter A (ohm/volt) = ",S
print"senstivity of meter B (ohm/volt) = ",S2
print 'Meter A is more sensitive than meter B'

senstivity of meter A (ohm/volt) =  260.0
senstivity of meter B (ohm/volt) =  151.0
Meter A is more sensitive than meter B

Example 13 - pg 7-37

In [16]:
#Chapter-7,Example7_13,pg 7-37
#calculate the voltmeter readings
#given
R1=20.*10**3
R2=25.*10**3
V=250#voltage supply
#calculations and results
VR2=R2*V/(R1+R2)#voltage across R2
#case-1
S=500
Vr=150#voltage range of resistor
Rv=S*Vr
Req=R2*Rv/(R2+Rv)
VReq=Req*V/(Req+R1)#voltage across Req
print"first voltmeter reading (V) = ",round(VReq,2)
#case-2
S=10*10**3
Rv=S*Vr
Req=R2*Rv/(R2+Rv)
VReq=Req*V/(Req+R1)
print"second voltmeter reading (V) = ",round(VReq,2)

first voltmeter reading (V) =  120.97
second voltmeter reading (V) =  137.87

Example 14 - pg 7_38

In [18]:
#Chapter-7,Example7_14,pg 7-38
#calculate the voltmeter reading case and percentage error,accuracy
#given
Rb=1.*10**3
Ra=5.*10**3
V=25.
#calculations and results
VRb=Rb*V/(Ra+Rb)#voltage across Rb
Vr=5.
#case-1
S=1.*10**3
Rv=S*Vr
Req=Rb*Rv/(Rb+Rv)
VReq=Req*V/(Req+Ra)
err=(VRb-VReq)*100/VRb
acc=100-err
print "true voltage (V) = ",round(VRb,3)
print"voltmeter reading case-1 (V) = ",round(VReq,3)
print"percentage error (percent) = ",round(err,1)
print"percentage accuracy = ",round(acc,1)
#case-2
S=20*10**3
Rv=S*Vr
Req=Rb*Rv/(Rb+Rv)
VReq=Req*V/(Req+Ra)
err=(VRb-VReq)*100/VRb
acc=100-err
print"voltmeter reading case-2 (V) = ",round(VReq,3)
print"percentage error (percent) = ",round(err,2)
print"percentage accuracy = ",round(acc,2)

true voltage (V) =  4.167
voltmeter reading case-1 (V) =  3.571
percentage error (percent) =  14.3
percentage accuracy =  85.7
voltmeter reading case-2 (V) =  4.132
percentage error (percent) =  0.83
percentage accuracy =  99.17

Example 15 - pg 7_41

In [19]:
#Chapter-7,Example7_15,pg 7-41
#calculate the shunt and series resistances
#given
Rm=50.
Im=20.*10**-3
I=10.
#calculatiosn and results
Rsh=(Im*Rm)/(I-Im)
print"shunt resistance for I=10A (ohm) = ",round(Rsh,4)

I=20
Rsh=(Im*Rm)/(I-Im)
print"shunt resistance for I=20A (ohm) = ",round(Rsh,2)

V=150
Rs=(V/Im)-Rm
print"series resistance for V=150V (kohm) = ",round(Rs/1000.,2)

V=300
Rs=(V/Im)-Rm
print"series resistance for V=300V (kohm) = ",round(Rs/1000.,2)

shunt resistance for I=10A (ohm) =  0.1002
shunt resistance for I=20A (ohm) =  0.05
series resistance for V=150V (kohm) =  7.45
series resistance for V=300V (kohm) =  14.95

Example 16 - pg 7_42

In [21]:
#Chapter-7,Example7_16,pg 7-42
#calculate the shunt current,resistance
#given
Rsh=0.02
R=1000.
Vm=500.*10**-3
#calculations and results
Im=Vm/R
Ish=Vm/Rsh
print"shunt current (A) = ",Ish
Ish1=10.
V=Ish1*Rsh
R=V/Im
print"resistance for Ish=10A (ohm) = ",R
Ish2=75.
V=Ish2*Rsh
R=V/Im
print"resistance for Ish=75A (ohm) = ",R

shunt current (A) =  25.0
resistance for Ish=10A (ohm) =  400.0
resistance for Ish=75A (ohm) =  3000.0

Example 17 - pg 7_50

In [22]:
#Chapter-7,Example7_17,pg 7-50
#calculate the final inductance
#given
import math
K=5.73*10**-6
I=20.
theta=110*(math.pi/180)#full scale deflection
L=4*10**-6
#calculations
dtheta=theta#change in theta
dm=(theta*K/(I**2))*dtheta#change in inductance
Lf=L+dm
#results
print"final inductance (muH) = ",round(Lf*10**6,4)

final inductance (muH) =  4.0528

Example 18 - pg 7_50

In [23]:
#Chapter-7,Example7_18,pg 7-50
#calculate the deflecting torque
import math
#given
I=10*10**-3
x=30#deflection
#calculations
dM=5*math.sin((x+45)*(math.pi/180))*10**-3#diffrentiate M w.r.t x
Td=(I**2)*dM#deflecting torque
#results
print"deflecting torque (muNm) = ",round(Td*10**6,5)

deflecting torque (muNm) =  0.48296

Example 19 - pg 7_51

In [24]:
#Chapter-7,Example7_19,pg 7-51
#calculate the difference in readings
import cmath
import math
#given
I=100*10**-3
Td=0.8*10**-4
dtheta=90*math.pi/180#in radians
theta=90#deflection
#calculations
dM=Td*dtheta/(I**2)
Mo=0.5#original M
M=Mo+dM#total M
#case-1 
Vdc=100
R=Vdc/I
w=2*math.pi*50
Z=R+(1j*w*M)
Z=abs(Z)
Vac=R*Vdc/Z
dif=Vdc-Vac#difference between readings
#case-2
Vdc1=50
I1=Vdc1/R
theta1=theta*((I1/I)**2)#theta=kI**2
theta1=theta1*math.pi/180#in radians
dM1=Td*theta1/(I**2)
M1=dM1+Mo
Z1=R+(1j*w*M1)
Z1=abs(Z1)
Vac1=R*Vdc1/Z1
dif1=Vdc1-Vac1
#results
print"difference in readings Vdc=100V (V) = ",round(dif,4)
print"difference in readings Vdc=50V (V) = ",round(dif1,4)

difference in readings Vdc=100V (V) =  1.2718
difference in readings Vdc=50V (V) =  0.6132

Example 20 - pg 7_65

In [25]:
#Chapter-7,Example7_20,pg 7-65
#calculate the percentage error
#given
I=20.
V=230.
Pf=0.8#power factor
t=3600.
K=100.
#calculations
Et=V*I*Pf*t
Et=Et/(3600*10**3)#in kWh
N=360.
Er=3.6#in kWh
err=(Er-Et)/Et
err=err*100
#results
print"percentage error = ",round(err,3)
print"negative sign shows that meter is slow and Er<Et"

percentage error =  -2.174
negative sign shows that meter is slow and Er<Et

Exampe 21 - pg 7_65

In [26]:
#Chapter-7,Example7_21,pg 7-65
#calculate the percentage error
#given
K=1800.
V=230.
I=10.
Pf=1.#half load
Ihl=I/2.#half load current
t=138.
#calculations
Et=V*Ihl*Pf*t
Et=Et/(3600*10**3)
N=80#no. of revolutions
Er=N/K#in kWh
err=(Er-Et)/Et
err=err*100
#results
print"percentage error = ",round(err,3)
print"positive sign shows that meter is fast and Er>Et"

percentage error =  0.819
positive sign shows that meter is fast and Er>Et

Example 22 - pg 7_66

In [37]:
#Chapter-7,Example7_22,pg 7-66
#calculate the meter constant and power factor
#given
V=230.
I=4.
t=6.
Pf=1.
N=2208.
#calculations
Et=V*I*Pf*t
K=N/Et
V=230
I=5
t=4
N=1472
Et=V*I*Pf*t
Er=N/K
Pf=(Er/Et)
#results
print"meter constant (rev/kWh) = ",K*1000
print"power factor (lagging) = ",Pf

meter constant (rev/kWh) =  400.0
power factor (lagging) =  0.8

Example 23 - pg 7_66

In [29]:
#Chapter-7,Example7_23,pg 7-66
#calculate the speed of disc and percentage error
#given
I=5.
V=220.
Pf=1.
K=3275.
t=1/60.#in hr
#calculations
E=V*I*Pf*t
E=E/10**3#in kWh
Rev=E*K#no. of revolutions
#at half load
I=I/2
t=59.5
Et=V*I*Pf*t
Et=Et/(3600*10**3)#in kWh
N=30
Er=N/K
err=(Er-Et)/Et
err=err*100.
print"speed of disc (rpm) = ",round(Rev,2)
print"percentage error = ",round(err,2)
print"Er>Et meter is fast"

speed of disc (rpm) =  60.04
percentage error =  0.77
Er>Et meter is fast

Example 24 - pg 7_67

In [30]:
#Chapter-7,Example7_24,pg 7-67
#calculate the speed and percentage error
import math
from math import cos,sin
#given
V=240.
I=10.
Pf=0.8
t=1/60.
K=600.
#calculations
E=V*I*Pf*t
E=E/10**3#in kWh
Rev=E*K#no. of revolutions 
dela=90#for correct lag adjustment
dela1=86*math.pi/180#given in radian
phi=0#case-1 unity power factor
err=(sin(dela1-phi)-cos(phi))/cos(phi)
err=err*100
print"in case 1, percentage error in case-1 = ",round(err,4)
Pf=0.5#case-2
phi=60*math.pi/180#in radians
err=(sin(dela1-phi)-cos(phi))/cos(phi)
err=err*100
print"speed of disc (rpm) = ",Rev
print"in case 2, percentage error in case-2 = ",round(err,3)

in case 1, percentage error in case-1 =  -0.2436
speed of disc (rpm) =  19.2
in case 2, percentage error in case-2 =  -12.326

Example 25 - pg 7_67

In [31]:
#Chapter-7,Example7_25,pg 7-67
#calculate the percentage error upperlimt and lowerlimt
#given
V=240.
I=5.
K=1200.
N=40.
t=99.8
Td=500#total divisions
#calculations
Er=N/K
W=V*I
div=K/Td#1 division
We=0.1*div#wattmeter error
Ce=0.05*K/100#construction wattmeter error
Te=We+Ce#total error
Wru=K+Te
Wrl=K-Te#wattmeter reading limits
He=0.05#human error
Se=0.01#stopwatch error
Tte=He+Se#total timing error
Sru=t+Tte#stopwatch reading limits
Srl=t-Tte
Eu=Wru*Sru*1/(3600*10**3)#energy obtained limits
El=Wrl*Srl*1/(3600*10**3)
errl=(Er-El)/El
errl=errl*100
erru=(Er-Eu)/Eu#error limits
erru=erru*100
#results
print"percentage error upperlimt = ",round(erru,3)
print"percentage error lowerlimt = ",round(errl,3)

percentage error upperlimt =  0.07
percentage error lowerlimt =  0.331

Example 26 - pg 7_79

In [33]:
#Chapter-7,Example7_26,pg 7-79
#calculate the line current
#given
I1=250.
I2=5.
#calculations
I=I1/I2
#as ammeter is in secondary I2=2.7
I1=I*2.7#line current
#results
print"line current (A) = ",I1

line current (A) =  135.0

Example 27 - pg 7_82

In [34]:
#Chapter-7,Example7_27,pg 7-82
#calculate the line voltage
#given
V1=11000.
V2=110.
#calculations
V=V1/V2
V2=87.5
V1=87.5*V#line voltage
#results
print"line voltage (V) = ",V1

line voltage (V) =  8750.0

Example 28 - pg 7_88

In [35]:
#Chapter-7,Example7_28,pg 7-88
#calculate the percentage ratio error
#given
Im=120.
Ic=38.
Kn=1000./5 #at full load
Is=5.
Ns=1000.
Np=5.
#calculations
n=Ns/Np#turns ratio
R=n+(Ic/Is)
err=(Kn-R)/R#ratio error
err=err*100.
#results
print"percentage ratio error = ",round(err,2)

percentage ratio error =  -3.66

Example 29 - pg 7_88

In [36]:
#Chapter-7,Example7_29,pg 7-88
#calculate the percentage ratio error
#given
import math
Im=90.
Ic=40.
delta=28*(math.pi/180)#in radians
Is=5.
Ns=400.
Np=1.
#calculations
n=Ns/Np
Kn=n
R=n+((Im*math.sin(delta)+Ic*math.cos(delta))/Is)
Ip=R*Is#actual primary current
err=(Kn-R)/R
err=err*100
#results
print"percentage ratio error = ",round(err,3)

percentage ratio error =  -3.734