Chapter 42: Filter networks

Example 1, page no. 799

In [1]:
from __future__ import division
import math
#initializing  the  variables:
L1  =  2*100E-3;#  in  Henry
C1  =  0.2E-6;#  in  Fareads
L2  =  0.4;#  in  Henry
C2  =  2*200E-12;#  in  Fareads

#calculation:
 #cut-off  frequency
fc1  =  1/(math.pi*(L1*C1)**0.5)
 #nominal  impedance
R01  =  (L1/C1)**0.5
 #cut-off  frequency
fc2  =  1/(math.pi*(L2*C2)**0.5)
 #nominal  impedance
R02  =  (L2/C2)**0.5


#Results
print  "\n\n  Result  \n\n"
print  "\n  cut-off  frequency  ",round(fc1,2),"  Hz  and  the  nominal  impedance  is  ",round(  R01,2),"  ohm  "
print  "\n  cut-off  frequency  ",round(fc2,2),"  Hz  and  the  nominal  impedance  is  ",round(  R02,2),"  ohm  "


  Result  



  cut-off  frequency   1591.55   Hz  and  the  nominal  impedance  is   1000.0   ohm  

  cut-off  frequency   25164.61   Hz  and  the  nominal  impedance  is   31622.78   ohm

Example 2, page no. 801

In [1]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R0  =  600;#  in  ohm
fc  =  5E6;#  in  Hz

 #calculation:
 #capacitance
C  =  1/(math.pi*R0*fc)
 #inductance
L  =  R0/(math.pi*fc)


#Results
print  "\n\n  Result  \n\n"
print  "A  low-pass  T  section  filter  capcitance  is  ",round(C*1E12,2),"pfarad  and  inductance  is",round(  L/2*1E6,2),"uHenry"
print  "A  low-pass  pi  section  filter  capcitance  is  ",round(C/2*1E12,2),"pfarad  and  inductance  is",round(  L*1E6,2),"uHenry"


  Result  


A  low-pass  T  section  filter  capcitance  is   106.1 pfarad  and  inductance  is 19.1 uHenry
A  low-pass  pi  section  filter  capcitance  is   53.05 pfarad  and  inductance  is 38.2 uHenry

Example 3, page no. 805

In [3]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R0  =  500;#  in  ohm
fc  =  100000;#  in  Hz
f  =  90000;#  in  Hz

#calculation:
 #characteristic  impedance  of  the  pi  section
Zpi  =  R0/(1  -  (f/fc)**2)**0.5
 #characteristic  impedance  of  the  T  section
Zt  =    R0*(1  -  (f/fc)**2)**0.5


#Results
print  "\n\n  Result  \n\n"
print  "\ncharacteristic  impedance  of  the  pi  section  is  ",round(Zpi,2),"  ohm"
print  "\ncharacteristic  impedance  of  the  T  section  is  ",round(Zt,2),"  ohm"


  Result  



characteristic  impedance  of  the  pi  section  is   1147.08   ohm

characteristic  impedance  of  the  T  section  is   217.94   ohm

Example 4, page no. 806

In [4]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R0  =  600;#  in  ohm
fc  =  2E6;#  in  Hz
Z1  =  600;#  in  ohm
Z2  =  1000;#  in  ohm
Z3  =  10000;#  in  ohm

#calculation:
 #frequency
f1  =  fc*(1  -  (R0/Z1)**2)**0.5
f2  =  fc*(1  -  (R0/Z2)**2)**0.5
f3  =  fc*(1  -  (R0/Z3)**2)**0.5


#Results
print  "\n\n  Result  \n\n"
print  "frequency  at  which  the  characteristic  impedance  of  the  section  is  600  ohm  is  ",f1,"  Hz "
print  "and  1000  Ohm  is  ",f2*1E-3,"kHz  and  10000  ohm  is  ",round(f3*1E-3,2),"kHz  "


  Result  


frequency  at  which  the  characteristic  impedance  of  the  section  is  600  ohm  is   0.0   Hz 
and  1000  Ohm  is   1600.0 kHz  and  10000  ohm  is   1996.4 kHz

Example 5, page no. 809

In [1]:
from __future__ import division
import math
#initializing  the  variables:
L1  =  100*1E-3;#  in  Henry
C1  =  0.2*1E-6;#  in  Fareads
L2  =  200*1E-6;#  in  Henry
C2  =  4000*1E-12;#  in  Fareads

#calculation:
 #cut-off  frequency
fc1  =  1/(4*math.pi*(L1*C1/2)**0.5)
 #nominal  impedance
R01  =  (L1*2/C1)**0.5
 #cut-off  frequency
fc2  =  1/(4*math.pi*(L2*C2/2)**0.5)
 #nominal  impedance
R02  =  (L2/(C2*2))**0.5


#Results
print  "\n\n  Result  \n\n"
print  "\n  cut-off  frequency  ",round(fc1,0),"  Hz  and  the  nominal  impedance  is  ",round(  R01,0),"  ohm"
print  "\n  cut-off  frequency  ",round(fc2/1000,0),"KHz  and  the  nominal  impedance  is  ",round(  R02,0),"  ohm  "


  Result  



  cut-off  frequency   796.0   Hz  and  the  nominal  impedance  is   1000.0   ohm

  cut-off  frequency   126.0 KHz  and  the  nominal  impedance  is   158.0   ohm

Example 6, page no. 811

In [2]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R0  =  600;#  in  ohm
fc  =  25000;#  in  Hz

#calculation:
 #capacitance
C1  =  2/(4*math.pi*R0*fc)
 #inductance
L1  =  R0/(4*math.pi*fc)
 #capacitance
C2  =  1/(4*math.pi*R0*fc)
 #inductance
L2  =  2*R0/(4*math.pi*fc)


#Results
print  "\n\n  Result  \n\n"
print  "\n  A  low-pass  T  section  filter  capcitance  is  ",round(C1*1E9,2),"nfarad  and  inductance  is",round(L1*1E3,2),"mHenry"
print  "\n  A  high-pass  pi  section  filter  capcitance  is  ",round(C2*1E9,3),"nfarad  and  inductance  is",round(L2*1E3,2),"mHenry"


  Result  



  A  low-pass  T  section  filter  capcitance  is   10.61 nfarad  and  inductance  is 1.91 mHenry

  A  high-pass  pi  section  filter  capcitance  is   5.305 nfarad  and  inductance  is 3.82 mHenry

Example 8, page no. 814

In [5]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R0  =  600;#  in  ohm
fc  =  500;#  in  Hz
Z1  =  0;#  in  ohm
Z2  =  300;#  in  ohm
Z3  =  590;#  in  ohm

 #calculation:
 #frequency
f1  =  fc
f2  =  fc/(1  -  (Z2/R0)**2)**0.5
f3  =  fc/(1  -  (Z3/R0)**2)**0.5


#Results
print  "\n\n  Result  \n\n"
print  "requency  at  which  the  characteristic  impedance  of  the  section  is  0  ohm  is  ",f1,"  Hz "
print  "and  300  Ohm  is  ",round(f2,2),"  Hz  and  590  ohm  is  ",round(f3,2),"  Hz  "


  Result  


requency  at  which  the  characteristic  impedance  of  the  section  is  0  ohm  is   500   Hz 
and  300  Ohm  is   577.35   Hz  and  590  ohm  is   2750.1   Hz

Example 9, page no. 817

In [8]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
r1  =  1.25  + 0.52j;#  propagation  coefficients  
rr  =  1.794;#  propagation  coefficients  
thetar  =  -39.4;#  in  ddegrees

#calculation:
 #r
r2  =  rr*math.cos(thetar*math.pi/180)  +  1j*rr*math.sin(thetar*math.pi/180)
 #attenuation  coefficient
a1  =  r1.real
a2  =  r2.real
 #phase  shift  coefficient
b1  =  r1.imag
b2  =  r2.imag


#Results
print  "\n\n  Result  \n\n"
print  "\nattenuation  coefficient  are  for  (a)  is  ",a1,"  N  and  for  (b)  is  ",round(a2,2),"  N  "
print  "\nphase  shift  coefficient  are  for  (a)  is  ",b1,"  rad  and  for  (b)  is  ",round(b2,2),"  rad  "


  Result  



attenuation  coefficient  are  for  (a)  is   1.25   N  and  for  (b)  is   1.39   N  

phase  shift  coefficient  are  for  (a)  is   0.52   rad  and  for  (b)  is   -1.14   rad

Example 10, page no. 818

In [2]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
ri1  =  0.024;#  in  amperes
ri2  =  0.008;#  in  amperes
thetai1  =  10;#  in  ddegrees
thetai2  =  -45;#  in  ddegrees

#calculation:
 #currents
I1  =  ri1*math.cos(thetai1*math.pi/180)  +  1j*ri1*math.sin(thetai1*math.pi/180)
I2  =  ri2*math.cos(thetai2*math.pi/180)  +  1j*ri2*math.sin(thetai2*math.pi/180)
 #ir
ir  =  I1/I2
irmag  =  ri1/ri2
thetai  =  thetai1-thetai2
 #attenuation  coefficient
a  =  math.log(irmag)
 #phase  shift  coefficient
b  =  thetai*math.pi/180
 #propagation  coefficient  
r  =  a  +  1j*b
 #output  current  of  the  fifth  stage
I6  =  I1/(ir**5)
x  =  ir**5
xmg  =  abs(x)
 #overall  attenuation  coefficient
ad  =  math.log(xmg)
 #overall  phase  shift  coefficient
bd  =  cmath.phase(complex(x.real,x.imag))


#Results
print  "\n\n  Result  \n\n"
print  "\nattenuation  coefficient  is  ",round(a,3),"  N  "
print  "\nphase  shift  coefficient  is  ",round(b,3),"  rad  "
print  "\npropagation  coefficient  is  ",round(a,3),"  +  (",round(b,3),")i  "
print  "\nthe  output  current  of  the  fifth  stage  is  ",round(abs(I6*1E6),1),"/_",round(cmath.phase(complex(I6.real,I6.imag))*180/math.pi,2),"deg   mA "
print  "and  the  overall  propagation  coefficient  is  ",round(ad,2),"  +  (",round(bd+(2*math.pi),2),")i"


  Result  



attenuation  coefficient  is   1.099   N  

phase  shift  coefficient  is   0.96   rad  

propagation  coefficient  is   1.099   +  ( 0.96 )i  

the  output  current  of  the  fifth  stage  is   98.8 /_ 95.0 deg   mA 
and  the  overall  propagation  coefficient  is   5.49   +  ( 4.8 )i

Example 11, page no. 819

In [10]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
XL  =  5j;#  in  ohms
Xc  =  -1j*10;#  in  ohms
RL  =  12;#  in  ohms
I1  =  1;#  in  amperes  (lets  say)

 #calculation:
 #current  I2
I2  =  (Xc/(Xc  +  XL  +  RL))*I1
 #current  ratio
Ir  =  I1/I2
Irmg  =  abs(Ir)
 #attenuation  coefficient
a  =  math.log(Irmg)
 #phase  shift  coefficient
b  =  cmath.phase(complex(Ir.real, Ir.imag))
 #propagation  coefficient  
r  =  a  +  1j*b


#Results
print  "\n\n  Result  \n\n"
print  "\nattenuation  coefficient  is  ",round(a,2),"  N  "
print  "\nphase  shift  coefficient  is  ",round(b,2),"  rad  "
print  "\npropagation  coefficient  is  ",round(a,2),"  +  (",round(b,2),")i  "


  Result  



attenuation  coefficient  is   0.26   N  

phase  shift  coefficient  is   1.18   rad  

propagation  coefficient  is   0.26   +  ( 1.18 )i

Example 12, page no. 823

In [11]:
from __future__ import division
import math
#initializing  the  variables:
L  =  2*0.5;#  in  Henry
C  =  2E-9;#  in  Farad

#calculation:
 #time  delay
t  =  (L*C)**0.5
 #time  delay  at  the  cut-off  frequency
tfc  =  t*math.pi/2


#Results
print  "\n\n  Result  \n\n"
print  "\n  time  delay  is  ",round(t*1E6,2),"usec  "
print  "\ntime  delay  at  the  cut-off  frequency  is  ",round(tfc*1E6,2),"usec"


  Result  



  time  delay  is   44.72 usec  

time  delay  at  the  cut-off  frequency  is   70.25 usec

Example 13, page no. 824

In [12]:
from __future__ import division
import math
#initializing  the  variables:
fc  =  500000;#  in  Hz
t1  =  9.55E-6;#  in  secs
R0  =  1000;#  in  ohm

#calculation:
 #for  a  low-pass  filter  section,  capacitance
C  =  1/(math.pi*R0*fc)
 #inductance
L  =  R0/(math.pi*fc)
 #time  delay
t2  =  (L*C)**0.5
 #number  of  cascaded  sections  required
n  =  t1/t2


#Results
print  "\n\n  Result  \n\n"
print  "\n  for  low-pass  T  section  inductance  is  ",round(L/2*1E6,2),"uH  and  capacitance  is  ",round(C*1E12,2),"pF"
print  "\n  for  low-pass  pi  section  inductance  is  ",round(L*1E6,2),"uH  and  capacitance  is  ",round(C/2*1E12,2),"pF"
print  "\nnumber  of  cascaded  sections  required  is  ",round(n,2)


  Result  



  for  low-pass  T  section  inductance  is   318.31 uH  and  capacitance  is   636.62 pF

  for  low-pass  pi  section  inductance  is   636.62 uH  and  capacitance  is   318.31 pF

number  of  cascaded  sections  required  is   15.0

Example 14, page no. 824

In [13]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
n  =  8;#  sections  in  cascade
R0  =  1000;#  in  ohm
t1  =  4E-6;#  in  secs


#calculation:
 #time  delay
t2  =  t1/n
 #capacitance
C  =  t2/R0
 #inductance
L  =  t2*R0


#Results
print  "\n\n  Result  \n\n"
print  "\n  for  low-pass  T  section  inductance  is  ",L/2*1E6,"uH  and  capacitance  is  ",C*1E12,"pF"
print  "\n  for  high-pass  pi  section  inductance  is  ",2*L*1E6,"uH  and  capacitance  is  ",C*1E12,"pF"


  Result  



  for  low-pass  T  section  inductance  is   250.0 uH  and  capacitance  is   500.0 pF

  for  high-pass  pi  section  inductance  is   1000.0 uH  and  capacitance  is   500.0 pF

Example 15, page no. 829

In [5]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R0  =  600;#  in  ohm
fc = 5000; # in Hz
finf = 5500; #in Hz

#calculation:
m = (1 - (fc/finf)**2)**0.5
C = 1/(math.pi*R0*fc)
L = R0/(math.pi*fc)

LT = m*L/2
CT = m*C
Ls = (1- m**2)*L/(4*m)

Cpi = m*C/2
Lpi = m*L
Cp = (1- (m**2))*C/(4*m)

#Results
print  "\n\n  Result  \n\n"
print  "\n  for mderived  T  section  inductance  is  ",round(Ls*1000,2),"mH  and  capacitance  is  ",round(CT*1E9,2),"nF"
print  "\n  for mderived  pi  section  inductance  is  ",round(Lpi*1000,2),"mH  and  capacitance  is  ",round(Cp*1E9,2),"nF"


  Result  



  for mderived  T  section  inductance  is   18.94 mH  and  capacitance  is   44.2 nF

  for mderived  pi  section  inductance  is   15.91 mH  and  capacitance  is   52.62 nF

Example 16, page no. 832

In [3]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R0  =  500;#  in  ohm
fc = 20000; # in Hz
finf = 16000; #in Hz

#calculation:
m = (1 - (finf/fc)**2)**0.5
C = 1/(4*math.pi*R0*fc)
L = R0/(4*math.pi*fc)

LT = L/m
CT = 4*m*C/(1- m**2)
Csa = 2*C/m

Cpi = C/m
Lpi = 4*m*L/(1- m**2)
Lsa = 2*L/m

#Results
print  "\n\n  Result  \n\n"
print  "\n  For an 'm-derived' high-pass T section: series arm contains a capacitance of  ",round(Csa*1E9,2),"nF"
print  "the shunt arm contains an inductance of",round(LT*1000,3)," mH  in series with a capacitance of",round(CT*1E9,2),"nF"
print  "\n  For an 'm-derived' high pass pi section: shunt arms each contain inductance of  ",round(Lsa*1000,2),"mH"
print  "series arm contains a capacitance of  ",round(Cpi*1E9,2),"nF in parallel with an inductance of",round(Lpi*1E3,3),"mH"


  Result  



  For an 'm-derived' high-pass T section: series arm contains a capacitance of   26.53 nF
the shunt arm contains an inductance of 3.316  mH  in series with a capacitance of 29.84 nF

  For an 'm-derived' high pass pi section: shunt arms each contain inductance of   6.63 mH
series arm contains a capacitance of   13.26 nF in parallel with an inductance of 7.46 mH

Example 17, page no. 835

In [5]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
R0  =  600;#  in  ohm
fc = 10000; # in Hz
finf = 11800; #in Hz

#calculation:
m = (1 - (fc/finf)**2)**0.5
C = 1/(math.pi*R0*fc)
L = R0/(math.pi*fc)

LmT = (1- m**2)*L/(4*m)

mH = 0.6
LmH = (1- mH**2)*L/(2*mH)

#Results
print  "\n\n  Result  \n\n"
print  "\n  For an Prototype T section: series arm contains a Inductance of  ",round(L*1000/2,1),"mH"
print   "the shunt arm contains an Capacitance of",round(C*1E6,4)," uF"
print  "\n  For an 'm-derived' T section: Series arms each contain inductance of  ",round(m*L*1000/2,2),"mH "
print   "Shunt arm contains a capacitance of  ",round(m*C*1E6,4),"uF in Series with an inductance of",round(LmT*1E3,2),"mH"
print  "\n  For an 'm-derived' Half section: Series arms each contain inductance of  ",round(mH*L*1000/2,1),"mH"
print   "Shunt arm contains a capacitance of  ",round(mH*C*1E6/2,4),"uF in Series with an inductance of",round(LmH*1E3,2),"mH"


  Result  



  For an Prototype T section: series arm contains a Inductance of   9.5 mH
the shunt arm contains an Capacitance of 0.0531  uF

  For an 'm-derived' T section: Series arms each contain inductance of   5.07 mH 
Shunt arm contains a capacitance of   0.0282 uF in Series with an inductance of 6.46 mH

  For an 'm-derived' Half section: Series arms each contain inductance of   5.7 mH
Shunt arm contains a capacitance of   0.0159 uF in Series with an inductance of 10.19 mH