Chapter 14: Alternating voltages and currents

Example 1, page no. 195

In [1]:
from __future__ import division
import math
#initializing  the  variables:
f1  =  50;#  in  Hz
f2  =  20000;#  in  Hz

#calculation:
T1  =  1/f1
T2  =  1/f2

#Results
print  "\n\n  Result  \n\n"
print  "\n  (a)  Periodic  time  T  =  ",T1,"  secs\n"
print  "\n  (b)  Periodic  time  T  =  ",(T2/1E-6),"  usecs\n"


  Result  



  (a)  Periodic  time  T  =   0.02   secs


  (b)  Periodic  time  T  =   50.0   usecs

Example 2, page no. 195

In [2]:
from __future__ import division
import math
#initializing  the  variables:
T1  =  0.004;#  in  secs
T2  =  4E-6;#  in  secs

#calculation:
f1  =  1/T1
f2  =  1/T2


#Results
print  "\n\n  Result  \n\n"
print  "\n  (a)  Frequency  f  =  ",f1,"  Hz\n"
print  "\n  (b)  Frequency  f  =  ",(f2/1E6),"  MHz\n"


  Result  



  (a)  Frequency  f  =   250.0   Hz


  (b)  Frequency  f  =   0.25   MHz

Example 3, page no. 195

In [4]:
from __future__ import division
import math
#initializing  the  variables:
T  =  (8E-3)/5;#  in  secs

#calculation:
f  =  1/T

#Results
print  "\n\n  Result  \n\n"
print  "\n  Frequency  f  =  ",f,"  Hz\n"


  Result  



  Frequency  f  =   625.0   Hz

Example 4, page no. 196

In [4]:
from __future__ import division
import math
#initializing  the  variables:
Ta  =  0.02;#  Time  for  1  complete  cycle  in  secs
Vamax  =  200;#  in  volts
Va1  =  25;#  in  volts
Va2  =  75;#  in  volts
Va3  =  125;#  in  volts
Va4  =  175;#  in  volts
Tb  =  0.016;#  Time  for  1  complete  cycle  in  secs
Ibmax  =  10;#  in  Amperes

#calculation:
#for  Triangular  waveform  (Figure  14.5(a))
fa  =  1/Ta
Aaw  =  Ta*Vamax/4
Vaavg  =  Aaw*2/Ta
Varms  =  (((Va1**2)  +  (Va2**2)  +  (Va3**2)  +  (Va4**2))/4)**0.5
#Note  that  the  greater  the  number  of  intervals  chosen,  the  greater  the  accuracy  of  the  result
Ffa  =  Varms/Vaavg
Pfa  =  Vamax/Varms

#for  Rectangular  waveform  (Figure  14.5(b))
fb  =  1/Tb
Abw  =  Tb*Ibmax/2
Ibavg  =  Abw*2/Tb
Ibrms  =  10
Ffb  =  Ibrms/Ibavg
Pfb  =  Ibmax/Ibrms


#Results
print  "\n\n  Result  \n\n"
print  "\n  (a1)Frequency  f  =  ",fa,"  Hz\n"
print  "\n  (a2)average  value  over  half  a  cycle  =  ",Vaavg,"  V\n"
print  "\n  (a3)rms  value  =  ",round(Varms,2),"  V\n"
print  "\n  (a4)Form  factor  =  ",round(Ffa,2),"\n"
print  "\n  (a5)Peak  factor  =  ",round(Pfa,2),"\n"
print  "\n  (b1)Frequency  f  =  ",fb,"  Hz\n"
print  "\n  (b2)average  value  over  half  a  cycle  =  ",Ibavg,"  A\n"
print  "\n  (b3)rms  value  =  ",Ibrms,"  A\n"
print  "\n  (b4)Form  factor  =  ",Ffb,"\n"
print  "\n  (b5)Peak  factor  =  ",Pfb,"\n"


  Result  



  (a1)Frequency  f  =   50.0   Hz


  (a2)average  value  over  half  a  cycle  =   100.0   V


  (a3)rms  value  =   114.56   V


  (a4)Form  factor  =   1.15 


  (a5)Peak  factor  =   1.75 


  (b1)Frequency  f  =   62.5   Hz


  (b2)average  value  over  half  a  cycle  =   10.0   A


  (b3)rms  value  =   10   A


  (b4)Form  factor  =   1.0 


  (b5)Peak  factor  =   1.0

Example 5, page no. 198

In [3]:
from __future__ import division
import math
import numpy 
from numpy import mean, sqrt, arange
#initializing  the  variables:
Thalf = 5; #in ms
Ta  =  0.02;#  Time  for  1  complete  cycle  in  secs

#calculation:
Tfull = 2*Thalf/1000 # in sec
f = 1/Tfull
A=[3, 10, 19, 30, 49, 63, 73, 72, 30, 2]
Iinst125 = 19
Iinst38 = 70
sq = 0
Ipeak = 76
#B=arange(A)
Imean = (0.5*1E-3)*numpy.mean(A)*10/(5*1E-3)
for h in range(10):
    sq = sq + A[h]**2

Irms = sqrt(sq/10)

#Results
print  "\n\n  Result  \n\n"
print  "\n  (a)Frequency  f  =  ",f,"  Hz\n"
print  "\n  (b)Instantaneous value of current after 1.25 ms  =",Iinst125,"A "
print   "and Instantaneous value of current after 3.8 ms", Iinst38,"A\n"
print  "\n  (c)Peak or maximum value  =  ",Ipeak,"  A\n"
print  "\n  (d)Mean or average value  =  ",round(Imean,2),"A\n"
print  "\n  (e)rms value =  ",round(Irms,1),"A\n"


  Result  



  (a)Frequency  f  =   100.0   Hz


  (b)Instantaneous value of current after 1.25 ms  = 19 A and Instantaneous value of current after 3.8 ms 70 A


  (c)Peak or maximum value  =   76   A


  (d)Mean or average value  =   35.1 A


  (e)rms value =   43.8 A

Example 6, page no. 200

In [5]:
from __future__ import division
import math
#initializing  the  variables:
Imax  =  20;#  in  Amperes

#calculation:
#for  a  sine  wave
Irms  =  Imax/(2**0.5)


#Results
print  "\n\n  Result  \n\n"
print  "\n  Rms  value  =  ",round(Irms,2),"  A\n"


  Result  



  Rms  value  =   14.14   A

Example 7, page no. 200

In [6]:
from __future__ import division
import math
#initializing  the  variables:
Vrms  =  240;#  in  Volts

#calculation:
#for  a  sine  wave
Vmax  =  Vrms*(2**0.5)
Vmean  =  0.637*Vmax


#Results
print  "\n\n  Result  \n\n"
print  "\n  peak  value  =  ",round(Vmax,2),"  V\n"
print  "\n  mean  value  =  ",round(Vmean,2),"  V\n"


  Result  



  peak  value  =   339.41   V


  mean  value  =   216.2   V

Example 8, page no. 200

In [7]:
from __future__ import division
import math
#initializing  the  variables:
Vmean  =  150;#  in  Volts

#calculation:
#for  a  sine  wave
Vmax  =  Vmean/0.637
Vrms  =  0.707*Vmax

#Results
print  "\n\n  Result  \n\n"
print  "\n  peak  value  =  ",round(Vmax,2),"  V\n"
print  "\n  rms  value  =  ",round(Vrms,2),"  V\n"


  Result  



  peak  value  =   235.48   V


  rms  value  =   166.48   V

Example 9, page no. 201

In [8]:
from __future__ import division
import math
#initializing  the  variables:
Vmax  =  282.8;#  in  Volts
w  =  314;#  in  rad/sec
t  =  0.004;#  in  sec

#calculation:
#for  a  sine  wave
Vrms  =  0.707*Vmax
f  =  w/(2*math.pi)
v  =  Vmax*math.sin(w*t)

#Results
print  "\n\n  Result  \n\n"
print  "\n  (a)rms  value  =  ",round(Vrms,2),"  V\n"
print  "\n  (b)frequency  f  =  ",round(f,2),"  Hz\n"
print  "\n  (c)instantaneous  value  of  voltage  at  4  ms  =  ",round(v,2),"  V\n"


  Result  



  (a)rms  value  =   199.94   V


  (b)frequency  f  =   49.97   Hz


  (c)instantaneous  value  of  voltage  at  4  ms  =   268.9   V

Example 10, page no. 202

In [6]:
from __future__ import division
import math
#initializing  the  variables:
Vmax  =  75;#  in  Volts
w  =  200*math.pi;#  in  rad/sec
t  =  0.004;#  in  sec
phi  =  0.25;#  in  radians

#calculation:
#for  a  sine  wave
Vptp  =  2*Vmax
Vrms  =  0.707*Vmax
f  =  w/(2*math.pi)
T  =  1/f
v  =  Vmax*math.sin(w*t)
phid  =  phi*180/math.pi


#Results
print  "\n\n  Result  \n\n"
print  "\n  (a)  Amplitude,  or  peak  value  =  ",Vmax,"  V\n"
print  "\n  (b)  Peak-to-peak  value  =  ",Vptp,"  V\n"
print  "\n  (c)rms  value  =  ",Vrms,"  V\n"
print  "\n  (d)periodic  time,  T  =  ",T,"  sec\n"
print  "\n  (e)frequency  f  =  ",f,"  Hz\n"
print  "\n  (f)phase  angle  =  ",round(phid,2),"deg lagging\n"


  Result  



  (a)  Amplitude,  or  peak  value  =   75   V


  (b)  Peak-to-peak  value  =   150   V


  (c)rms  value  =   53.025   V


  (d)periodic  time,  T  =   0.01   sec


  (e)frequency  f  =   100.0   Hz


  (f)phase  angle  =   14.32 deg lagging

Example 11, page no. 202

In [1]:
from __future__ import division
import math
#initializing  the  variables:
Vmax  =  40;#  in  Volts
T  =  0.01;#  in  sec
v  =  -20;#  when  t  =  0sec,  in  volts
t  =  0;#  in  secs

#calculation:
#for  a  sine  wave
w  =  2*math.pi/T
phir  =  math.asin(v/Vmax)


#Results
print  "\n\n  Result  \n\n"
print  "\n    instantaneous  voltage  v  =  ",  Vmax,"  sin(",round(w,2),"t",round(phir,2),")  V\n"


  Result  



    instantaneous  voltage  v  =   40   sin( 628.32 t -0.52 )  V

Example 12, page no. 203

In [3]:
from __future__ import division
import math
#initializing  the  variables:
Imax  =  120;#  in  Amperes
w  =  100*math.pi;#  in  rad/sec
phi  =  0.36;#  in  rad
t1  =  0;#  in  secs
t2  =  0.008;#  in  secs
i  =  60;#  in  amperes

#calculation:
#for  a  sine  wave
f  =  w/(2*math.pi)
T  =  1/f
phid  =  phi*180/math.pi
i0  =  Imax*math.sin((w*t1)+phi)
i8  =  Imax*math.sin((w*t2)+phi)  
ti  =  (math.asin(i/Imax)  -  phi)/w
tm1  =  (math.asin(Imax/Imax)  -  phi)/w


#Results
print  "\n\n  Result  \n\n"
print  "\n  (a)Peak  value  =  ",  Imax,"  A,  Periodic  time  T  =  ",  T,"  sec, "
print   " Frequency,  f  =  ",  f,"  Hz  Phase  angle  =  ",round(phid,2),"deg leading\n"
print  "\n  (b)  When  t  =  0,  i  =  ",round(i0,2),"  A\n"
print  "\n  (c)When  t  =  8  ms  =  ",  round(i8,2),"  A\n"
print  "\n  (d)When  i  is  60  A,  then  time  t  =  ",round((ti/1E-3),2),"  ms\n"
print  "\n  (e)When  the  current  is  a  maximum,  time,  t  =  ",  round((tm1/1E-3),2),"  ms\n"


  Result  



  (a)Peak  value  =   120   A,  Periodic  time  T  =   0.02   sec, 
 Frequency,  f  =   50.0   Hz  Phase  angle  =   20.63 deg leading


  (b)  When  t  =  0,  i  =   42.27   A


  (c)When  t  =  8  ms  =   31.81   A


  (d)When  i  is  60  A,  then  time  t  =   0.52   ms


  (e)When  the  current  is  a  maximum,  time,  t  =   3.85   ms

Example 13, page no. 204

In [2]:
from __future__ import division
import math
#initializing  the  variables:
i1max  =  20;#  in  Amperes
i2max  =  10;#  in  Amperes
phi1  =  0;#  in  rad
phi2  =  math.pi/3;#  in  rad

#calculation:
#Ig = i1 + i2
Igmax  = 26.5
phiIg = 19*math.pi/180

#Results
print  "\n\n  Result  \n\n"
print  "\n  Current  Ig = i1 + i2 =", Igmax,"sin(wt + ",round(phiIg,3),") Amps\n"


  Result  



  Current  Ig = i1 + i2 = 26.5 sin(wt +  0.332 ) Amps

Example 14, page no. 205

In [1]:
from __future__ import division
import math
#initializing  the  variables:
V1max  =  50;#  in  volts
V2max  =  100;#  in  volts
phi2  =  -1*math.pi/6;#  in  rad

#calculation:
#vR2  =  v1**2  +  v2**2  -  2*v1*v2  cos  150
phidiff  =  math.pi  +  phi2
Vrmax  =  (V1max**2  +  V2max**2  -  2*V1max*V2max*math.cos(phidiff))**0.5
#Using  the  sine  rule
phi  =  math.asin(V2max*math.sin(phidiff)/Vrmax)

#Results
print  "\n\n  Result  \n\n"
print  "\n  VR  =  ",round(Vrmax,2),"sin(wt  -  ",round(phi,2),")  V\n"


  Result  



  VR  =   145.47 sin(wt  -   0.35 )  V

Example 15, page no. 206

In [16]:
from __future__ import division
import math
#initializing  the  variables:
I1max  =  20;#  in  volts
I2max  =  10;#  in  volts
phi2  =  1*math.pi/3;#  in  rad

#calculation:
#iR2  =  i1**2  +  i2**2  -  2*i1*i2cos150
phidiff  =  math.pi  -  phi2
Irmax  =  (I1max**2  +  I2max**2  -  2*I1max*I2max*math.cos(phidiff))**0.5
#Using  the  sine  rule
phi  =  math.asin(I2max*math.sin(phidiff)/Irmax)

#Results
print  "\n\n  Result  \n\n"
print  "\n  IR  =  ",  round(Irmax,2),"sin(wt  +  ",round(phi,2),")  V\n"


  Result  



  IR  =   26.46 sin(wt  +   0.33 )  V