Chapter 44: Transmission lines

Example 1, page no. 873

In [1]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
f  =  1910;#  in  Hz
b  =  0.05;#  in  rad/km

#calculation:
w  =  2*math.pi*f
 #wavelength  
Y  =  2*math.pi/b
 #speed  of  transmission
u  =  f*Y


#Results
print  "\n\n  Result  \n\n"
print  "\n  wavelength  Y  is  ",round(Y,1),"  km"
print  "\n  speed  of  transmission  ",round(u,1),"km/sec"


  Result  



  wavelength  Y  is   125.7   km

  speed  of  transmission   240017.7 km/sec

Example 2, page no. 873

In [2]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
L  =  0.004;#  in  Henry/loop
C  =  0.004E-6;#  in  F/loop
f  =  1000;#  in  Hz

 #calculation:
w  =  2*math.pi*f
 #phase  delay
b  =  w*(L*C)**0.5
 #wavelength  
Y  =  2*math.pi/b
 #speed  of  transmission
u  =  f*Y


#Results
print  "\n\n  Result  \n\n"
print  "\n  phase  delay  is  ",round(b,3),"  rad/km"
print  "\n  wavelength  Y  is  ",Y,"  km"
print  "\n  speed  of  transmission  ",u,"km/sec"


  Result  



  phase  delay  is   0.025   rad/km

  wavelength  Y  is   250.0   km

  speed  of  transmission   250000.0 km/sec

Example 3, page no. 874

In [1]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
a  =  0.25;#  in  Np/km
b  =  0.20;#  in  rad/km
Vs  =  5;#  in  Volts
n  =  10;#  in  km
f  =  2000;#  in  Hz

 #calculation:
w  =  2*math.pi*f
 #the  voltage  10  km  down  the  line
r  =  a  +  1j*b
VR  =  Vs*cmath.e**(-1*n*r)


#Results
print  "\n Result  \n\n"
print  "voltage  10  km  down  the  line is ",round(abs(VR),2),"/_",round(cmath.phase(complex(VR.real,VR.imag))*180/math.pi,2),"deg  V"

 Result  


voltage  10  km  down  the  line is  0.41 /_ -114.59 deg  V

Example 4, page no. 875

In [2]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
a  =  0.5;#  in  Np/km
b  =  0.25;#  in  rad/km
rvs  =  2;#  in  Volts
thetavs  =  0;#  in  degrees
rzo  =  800;#  in  ohm
thetazo  =  -25;#  in  degrees
n  =  5;#  in  km

#calculation:
 #voltage
Vs  =  rvs*math.cos(thetavs*math.pi/180)  +  1j*rvs*math.sin(thetavs*math.pi/180)
 #characteristic  impedance
Zo  =  rzo*math.cos(thetazo*math.pi/180)  +  1j*rzo*math.sin(thetazo*math.pi/180)
 #  receiving  end  voltage
r  =  a  +  1j*b
VR  =  Vs*cmath.e**(-1*n*r)
 #Receiving  end  current,
IR  =  VR/Zo


#Results
print  "\n\n  Result  \n\n"
print  "Receiving end current, IR is  ",round(abs(IR)*1E3,3),"/_",round(cmath.phase(complex(IR.real,IR.imag))*180/math.pi,2),"deg  mA"


  Result  


Receiving end current, IR is   0.205 /_ -46.62 deg  mA

Example 5, page no. 875

In [5]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
Vs  =  8;#  in  Volts
VR  =  2;#  in  Volts
x  =  2;  

#calculation:
 #  receiving  end  voltage  VR  =  Vs*e**(-nr)
 #e**-nr  =  p
p  =  VR/Vs
 #If  the  line  is  doubled  in  length,  then
VR  =  Vs*(p)**2


#Results
print  "\n\n  Result  \n\n"
print  "\n  Receiving  end  voltage  If  the  line  is  doubled  in  length,  VR  is  ",abs(VR)," V"


  Result  



  Receiving  end  voltage  If  the  line  is  doubled  in  length,  VR  is   0.5  V

Example 6, page no. 876

In [3]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
rzoc  =  800;#  in  ohm
thetazoc  =  -50;#  in  degrees
rzsc  =  413;#  in  ohm
thetazsc  =  -20;#  in  degrees
f  =  1500;#  in  Hz

 #calculation:
 #open  circuit  impedance
Zoc  =  rzoc*math.cos(thetazoc*math.pi/180)  +  1j*rzoc*math.sin(thetazoc*math.pi/180)
 #short  circuit  impedance
Zsc  =  rzsc*math.cos(thetazsc*math.pi/180)  +  1j*rzsc*math.sin(thetazsc*math.pi/180)
 #characteristic  impedance  Zo
Zo  =  (Zoc*Zsc)**0.5


#Results
print  "\n\n  Result  \n\n"
print  "characteristic impedance Zo is",round(abs(Zo)),"/_",round(cmath.phase(complex(Zo.real,Zo.imag))*180/math.pi,2),"deg  ohm"


  Result  


characteristic impedance Zo is 575.0 /_ -35.0 deg  ohm

Example 7, page no. 877

In [4]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R  =  15;#  in  ohm/loop  km
L  =  0.0034;#  in  H/loop  km
C  =  10E-9;#  in  F/km
G  =  3E-6;#  in  S/km
f  =  2000;#  in  Hz

 #calculation:
w  =  2*math.pi*f
 #characteristic  impedance  Zo
Zo  =  ((R  +  1j*w*L)/(G  +  1j*w*C))**0.5


#Results
print  "\n\n  Result  \n\n"
print  "characteristic impedance Zo  is  ",round(abs(Zo),0),"/_",round(cmath.phase(complex(Zo.real,Zo.imag))*180/math.pi,2),"deg  ohm"


  Result  


characteristic impedance Zo  is   600.0 /_ -8.99 deg  ohm

Example 8, page no. 879

In [8]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
L  =  0.0005;#  in  H/loop  km
C  =  0.12E-6;#  in  F/km
f  =  400000;#  in  Hz

#calculation:
w  =  2*math.pi*f
 #characteristic  impedance  Zo
Zo  =  (L/C)**0.5
 #the  propagation  coefficient
r  =  1j*w*(L*C)**0.5
 #the  attenuation  coefficient  
a  =  r.real
 #the  phaseshift  coefficient
b  =  r.imag
 #wavelength
Y  =  2*math.pi/b
 #velocity  of  propagation  
u  =  f*Y


#Results
print  "\n\n  Result  \n\n"
print  "\n  characteristic  impedance  Zo  is  ",abs(Zo),"ohm"
print  "\n  propagation  coefficient  is  ",a,"  +(",round(b,2),")i"
print  "\n  wavelength  Y  is  ",round(Y*1E3,0),"m"
print  "\n  speed  of  transmission  ",round(u,2),"km/sec"


  Result  



  characteristic  impedance  Zo  is   64.5497224368 ohm

  propagation  coefficient  is   0.0   +( 19.47 )i

  wavelength  Y  is   323.0 m

  speed  of  transmission   129099.44 km/sec

Example 9, page no. 880

In [5]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R  =  25;#  in  ohm/loop  km
L  =  0.005;#  in  H/loop  km
C  =  0.04E-6;#  in  F/km
G  =  80E-6;#  in  S/km
f  =  1000;#  in  Hz

 #calculation:
w  =  2*math.pi*f
 #characteristic  impedance  Zo
Zo  =  ((R  +  1j*w*L)/(G  +  1j*w*C))**0.5
 #the  propagation  coefficient
r  =  ((R  +  1j*w*L)*(G  +  1j*w*C))**0.5
 #the  attenuation  coefficient  
a  =  r.real
 #the  phaseshift  coefficient
b  =  r.imag


#Results
print  "\n\n  Result  \n\n"
print  "characteristic impedance Zo  is",round(abs(Zo),2),"/_",round(cmath.phase(complex(Zo.real,Zo.imag))*180/math.pi,2),"deg  ohm"
print  "\n  propagation  coefficient  is  ",round(abs(r),4),"/_",round(cmath.phase(complex(a,b))*180/math.pi,2),"deg"
print  "\n  attenuation  coefficient  is  ",round(a,4),"  Np/km"
print  "\n  the  phase-shift  coefficient  ",round(b,4),"  rad/km"


  Result  


characteristic impedance Zo  is 390.16 /_ -10.43 deg  ohm

  propagation  coefficient  is   0.1029 /_ 61.92 deg

  attenuation  coefficient  is   0.0484   Np/km

  the  phase-shift  coefficient   0.0908   rad/km

Example 10, page no. 881

In [6]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R  =  8;#  in  ohm/loop  km
L  =  0.003;#  in  H/loop  km
C  =  7500E-12;#  in  F/km
G  =  0.25E-6;#  in  S/km
f  =  1000;#  in  Hz
n  =  300;#  in  km
Zg  =  400  +  1j*0;#  in  ohm
Vg  =  10;#  in  Volts

 #calculation:
w  =  2*math.pi*f
 #characteristic  impedance  Zo
Zo  =  ((R  +  1j*w*L)/(G  +  1j*w*C))**0.5
 #the  propagation  coefficient
r  =  ((R  +  1j*w*L)*(G  +  1j*w*C))**0.5
 #the  attenuation  coefficient  
a  =  r.real
 #the  phaseshift  coefficient
b  =  r.imag
 #the  sending-end  current,
Is  =  Vg/(Zg  +  Zo)
 #the  receiving-end  current,
IR  =  Is*cmath.e**(-1*n*r)
 #wavelength
Y  =  2*math.pi/b
 #velocity  of  propagation  
u  =  f*Y


#Results
print  "\n\n  Result  \n\n"
print  "characteristic impedance Zo is",round(abs(Zo),1),"/_",round(cmath.phase(complex(Zo.real,Zo.imag))*180/math.pi,2),"deg  ohm"
print  "propagation  coefficient  is  ",round(abs(r),5),"/_",round(cmath.phase(complex(r.real,r.imag))*180/math.pi,2),"deg"
print  "attenuation  coefficient  is  ",round(a,5),"  Np/km  and  the  phaseshift  coefficient  ",round(b,5),"  rad/km"
print  "sending-end  current  Is  is  ",round(abs(Is)*1E3,3),"/_",round(cmath.phase(complex(Is.real,Is.imag))*180/math.pi,2),"deg  mA"
print  "receiving-end  current  IR is",round(abs(IR)*1E3,3),"/_",round(cmath.phase(complex(IR.real,IR.imag))*180/math.pi,2),"deg  mA"
print  "wavelength  Y  is  ",round(Y,1),"  km"
print  "speed  of  transmission  ",round(u,1),"km/sec"


  Result  


characteristic impedance Zo is 659.2 /_ -11.35 deg  ohm
propagation  coefficient  is   0.03106 /_ 78.35 deg
attenuation  coefficient  is   0.00627   Np/km  and  the  phaseshift  coefficient   0.03042   rad/km
sending-end  current  Is  is   9.485 /_ 7.07 deg  mA
receiving-end  current  IR is 1.445 /_ -155.88 deg  mA
wavelength  Y  is   206.5   km
speed  of  transmission   206521.1 km/sec

Example 11, page no. 884

In [11]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R  =  10;#  in  ohm/loop  km
L  =  0.0015;#  in  H/loop  km
C  =  0.06E-6;#  in  F/km
G  =  1.2E-6;#  in  S/km

 #calculation:
 #the  condition  for  minimum  distortion  is  given  by  LG  =  CR,  from  which,
Lm  =  C*R/G
dL  =  Lm  -  L


#Results
print  "\n\n  Result  \n\n"
print  "\n  inductance  should  be  increased  by  ",round(dL*1E3,1),"mH/loop  km  for  minimum  distortion"


  Result  



  inductance  should  be  increased  by   498.5 mH/loop  km  for  minimum  distortion

Example 12, page no. 884

In [7]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R  =  80;#  in  ohm/loop  km
C  =  5E-9;#  in  F/km
G  =  2E-6;#  in  S/km
f  =  1500;#  in  Hz

 #calculation:
w  =  2*math.pi*f
 #the  condition  for  minimum  distortion  is  given  by  LG  =  CR,  from  which,  inductance
L  =  C*R/G
 #attenuation  coefficient,
a  =  (R*G)**0.5
 #phase  shift  coefficient,
b  =  w*(L*C)**0.5
 #propagation  coefficient,
r  =  a  +  1j*b
 #velocity  of  propagation,
u  =  1/(L*C)**0.5
 #wavelength
Y  =  u/f


#Results
print  "\n\n  Result  \n\n"
print  "\n  inductance  is  ",round(L,2),"  H"
print  "\n  propagation  coefficient  is  ",round(a,2),"  +(",round(b,2),")i"
print  "\n  speed  of  transmission  ",round(u,2),"km/sec"
print  "\n  wavelength  Y  is  ",round(Y,2),"  km"


  Result  



  inductance  is   0.2   H

  propagation  coefficient  is   0.01   +( 0.3 )i

  speed  of  transmission   31622.78 km/sec

  wavelength  Y  is   21.08   km

Example 13, page no. 888

In [13]:
 
from __future__ import division
import math
import cmath
#initializing  the  variables:
Zo  =  75;#  in  ohm
ZR  =  250;#  in  ohm
VR  =  10;#  in  Volts

#calculation:
 #reflection  coefficient
p  =  (Zo  -  ZR)/(Zo  +  ZR)
 #Current  flowing  in  the  terminating  load
IR  =  VR/ZR
 #incident  current,  Ii
Ii  =  IR/(1  +  p)
 #incident  voltage,  Vi    
Vi  =  Ii*Zo
 #reflected  current,  Ir
Ir  =  IR  -  Ii
 #reflected  voltage,  Vr
Vr  =  -1*Ir*Zo


#Results
print  "\n\n  Result  \n\n"
print  "\n  reflection  coefficient  is  ",round(p,3),""
print  "\n  incident  current,  Ii  is  ",round(Ii,4),"  A"
print  "\n  incident  voltage,  Vi  is  ",round(Vi,2),"  V"
print  "\n  reflected  current,  Ir  is  ",round(Ir,4),"  A"
print  "\n  reflected  voltage,  Vr  is  ",round(Vr,2),"  V"


  Result  



  reflection  coefficient  is   -0.538 

  incident  current,  Ii  is   0.0867   A

  incident  voltage,  Vi  is   6.5   V

  reflected  current,  Ir  is   -0.0467   A

  reflected  voltage,  Vr  is   3.5   V

Example 14, page no. 889

In [14]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
Zo  =  500  -  1j*40;#  in  ohm
ZR1  =  500  +  1j*40;#  in  ohm
ZR2  =  600  +  1j*0;#  in  ohm

 #calculation:
 #reflection  coefficient
p1  =  (Zo  -  ZR1)/(Zo  +  ZR1)
p2  =  (Zo  -  ZR2)/(Zo  +  ZR2)
p1mag  =  abs(p1)
p2mag  =  abs(p2)


#Results
print  "\n\n  Result  \n\n"
print  "\n  reflection  coefficient  (a)",p1mag,"  and  (b)", round(p2mag,2),""


  Result  



  reflection  coefficient  (a) 0.08   and  (b) 0.1

Example 15, page no. 890

In [8]:
#(b) the incident voltage
from __future__ import division
import math
import cmath
#initializing  the  variables:
rzo  =  500;#  in  ohm
thetazo  =  0;#  in  degrees
ZR  =  320  +  1j*240;#  in  ohm
rvr  =  20;#  in  volts
thetavr  =  35;#  in  degrees

 #calculation:
 #voltage
VR  =  rvr*math.cos(thetavr*math.pi/180)  +  1j*rvr*math.sin(thetavr*math.pi/180)
 #characteristic  impedance
Zo  =  rzo*math.cos(thetazo*math.pi/180)  +  1j*rzo*math.sin(thetazo*math.pi/180)
 #the  ratio  of  the  reflected  to  the  incident  voltage  
 #vr  =  VR/Vi
vr  =  (ZR  -  Zo)/(Zo  +  ZR)
vrmag  =  abs(vr)
 #incident  voltage,  Vi
Vi  =  VR/vr


#Results
print  "\n\n  Result  \n\n"
print  "\n  the  magnitude  of  the  ratio  Vr  :  Vi  is  ",round(vrmag,3),""
print  "\n  incident  voltage,  Vi  is  ",round(abs(Vi),1),"/_",round(cmath.phase(complex(Vi.real,Vi.imag))*180/math.pi,2),"deg  V"


  Result  



  the  magnitude  of  the  ratio  Vr  :  Vi  is   0.351 

  incident  voltage,  Vi  is   57.0 /_ -75.56 deg  V

Example 16, page no. 895

In [16]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
rzo  =  600;#  in  ohm
thetazo  =  0;#  in  degrees
ZR  =  400  +  250j;#  in  ohm

 #calculation:
 #characteristic  impedance
Zo  =  rzo*math.cos(thetazo*math.pi/180)  +  1j*rzo*math.sin(thetazo*math.pi/180)
 #reflection  coefficient
p  =  (Zo  -  ZR)/(Zo  +  ZR)
pmag  =  abs(p)
 #standing-wave  ratio,
s  =  (1  +  pmag)/(1  -  pmag)


#Results
print  "\n\n  Result  \n\n"
print  "\n  reflection  coefficient,  is  ",round(abs(p),4),"/_",round(cmath.phase(complex(p.real,p.imag))*180/math.pi,2),"deg"
print  "\n  standing-wave  ratio,  s  is  ",round(s,3),""


  Result  



  reflection  coefficient,  is   0.3106 /_ -65.38 deg

  standing-wave  ratio,  s  is   1.901

Example 17, page no. 896

In [9]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
rp  =  0.2;  
thetap  =  -120;#  in  degrees
Zo  =  80;#  in  ohm
Ir  =  0.01;#  in  Amperes

#calculation:
 #reflection  coefficient
p  =  rp*math.cos(thetap*math.pi/180)  +  1j*rp*math.sin(thetap*math.pi/180)
 #standing-wave  ratio,
s  =  (1  +  rp)/(1  -  rp)
 #load  impedance  ZR  
ZR  =  Zo*(1  -  p)/(1  +  p)
 #incident  current
Ii  =  Ir*(s  +  1)/(s  -  1)


#Results
print  "\n\n  Result  \n\n"
print  "\n  standing-wave  ratio,  s  is  ",s,""
print  "\n  load  impedance  ZR  is  ",round(ZR.real,2),"  +(",round(ZR.imag,1),")i  ohm"
print  "\n  incident  current  is  ",Ii,"  A"


  Result  



  standing-wave  ratio,  s  is   1.5 

  load  impedance  ZR  is   91.43   +( 33.0 )i  ohm

  incident  current  is   0.05   A

Example 18, page no. 897

In [18]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
s  =  1.6;
Pi  =  0.2;#  in  Watts

#calculation:
 #reflected  power,  Pr
Pr  =  Pi*((s  -  1)/(s  +  1))**2


#Results
print  "\n\n  Result  \n\n"
print  "\n  reflected  power,  Pr  is  ",round(Pr*1E3,2)," mW"


  Result  



  reflected  power,  Pr  is   10.65  mW