Chapter 43: Magnetically coupled circuits

Example 1, page no. 842

In [1]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
Na  =  1200;  
Nb  =  1000;
Ia  =  0.8;#  in  amperes
Phia  =  100E-6;#  in  Wb
xb  =  0.75;

 #calculation:
 #self  inductance  of  coil  A
La  =  Na*Phia/Ia
 #mutual  inductance,  M
Phib  =  xb*Phia
M  =  Nb*Phib/Ia


#Results
print  "\n\n  Result  \n\n"
print  "\n  self  inductance  of  coil  A  is  ",La,"  H"
print  "\n  mutual  inductance,  M  is  ",M,"H"


  Result  



  self  inductance  of  coil  A  is   0.15   H

  mutual  inductance,  M  is   0.09375 H

Example 2, page no. 843

In [2]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
M  =  600E-3;#  in  Henry
Ia  =  5;#  in  amperes
dt  =  0.2;#  in  secs

 #calculation:
 #change  of  current
dIa  =  2*Ia  
dIadt  =  dIa/dt
 #secondary  induced  e.m.f.,  E2
E2  =  -1*M*dIadt


#Results
print  "\n\n  Result  \n\n"
print  "\n  secondary  induced  e.m.f.,  E2  is  ",E2,"  V"


  Result  



  secondary  induced  e.m.f.,  E2  is   -30.0   V

Example 3, page no. 844

In [3]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
La  =  250E-3;#  in  Henry
Lb  =  400E-3;#  in  Henry
M  =  80E-3;#  in  Henry

 #calculation:
 #coupling  coefficient,
k  =  M/(La*Lb)**0.5


#Results
print  "\n\n  Result  \n\n"
print  "\n  coupling  coefficient,  is  ",round(k,3)


  Result  



  coupling  coefficient,  is   0.253

Example 4, page no. 845

In [4]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
Lx  =  80E-3;#  in  Henry
Ly  =  60E-3;#  in  Henry
Nx  =  200;#  turns
Ny  =  100;#  turns
Ix  =  4;#  in  Amperes
Phiy  =  0.005;#  in  Wb

#calculation:
 #mutual  inductance,  M
M  =  Ny*Phiy/(2*Ix)
 #coupling  coefficient,
k  =  M/(Lx*Ly)**0.5


#Results
print  "\n\n  Result  \n\n"
print  "\n  mutual  inductance,  M  is  ",M*1E3,"mH"
print  "\n  coupling  coefficient,  is  ",round(k,3)


  Result  



  mutual  inductance,  M  is   62.5 mH

  coupling  coefficient,  is   0.902

Example 5, page no. 846

In [5]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
La  =  40E-3;#  in  Henry
Lb  =  10E-3;#  in  Henry
L  =  60E-3;#  in  Henry

 #calculation:
 #mutual  inductance,  M
M  =  (L  -  La  -  Lb)/2
 #coupling  coefficient,
k  =  M/(La*Lb)**0.5


#Results
print  "\n\n  Result  \n\n"
print  "\n  mutual  inductance,  M  is  ",M*1E3,"mH"
print  "\n  coupling  coefficient,  is  ",k


  Result  



  mutual  inductance,  M  is   5.0 mH

  coupling  coefficient,  is   0.25

Example 6, page no. 847

In [6]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
V  =  240;#  in  Volts
Ra  =  5;#  in  Ohm
La  =  1;#  in  Henry
Rb  =  10;#  in  Ohm
Lb  =  5;#  in  Henry
I  =  8;#  in  amperes
dIdt  =  15;#  in  A/sec

 #calculation:
 #Kirchhoff’s  voltage  law
L  =  (V  -  I*(Ra  +  Rb))/dIdt
 #mutual  inductance,  M
M  =  (L  -  La  -  Lb)/2
 #coupling  coefficient,
k  =  M/(La*Lb)**0.5


#Results
print  "\n\n  Result  \n\n"
print  "\n  mutual  inductance,  M  is  ",M,"H"
print  "\n  coupling  coefficient,  is  ",round(k,3)


  Result  



  mutual  inductance,  M  is   1.0 H

  coupling  coefficient,  is   0.447

Example 7, page no. 848

In [1]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
k  =  0.7;#  coefficient  of  coupling
L1  =  15E-3;#  in  Henry
L2  =  10E-3;#  in  Henry

#calculation:
 #L1  =  La  +  Lb  +  2*k*(La*Lb)**0.5
 #L2  =  La  +  Lb  -  2*k*(La*Lb)**0.5
 #self  inductance  of  coils
a  =  ((L1  -  (L1    +  L2)/2)/(2*k))**2
La1  =((L1    +  L2)/2  +  (((L1    +  L2)/2)**2  -  4*a)**0.5)/2
La2  =((L1    +  L2)/2  -  (((L1    +  L2)/2)**2  -  4*a)**0.5)/2
Lb1  =  (L1  +  L2)/2  -  La1
Lb2  =  (L1  +  L2)/2  -  La2
 #mutual  inductance,  M
M  =  (L1  -  L2)/4


#Results
print  "\n\n  Result  \n\n"
print  "\nself  inductance  of  coils  are  ",round(La1*1E3,2),"mH  and  ",round(  Lb1*1E3,2),"mH"
print  "\n  mutual  inductance,  M  is  ",round(M*1E3,2),"mH"


  Result  



self  inductance  of  coils  are   12.24 mH  and   0.26 mH

  mutual  inductance,  M  is   1.25 mH

Example 8, page no. 850

In [8]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
E1  =  8;#  in  Volts
thetae1  =  0;#  in  degrees
w  =  2500;#  in  rad/sec
R  =  15;#  in  ohm
L  =  5E-3;#  in  Henry
M  =  0.1E-3;#  in  Henry

 #calculation:
 #voltage
E1  =  E1*math.cos(thetae1*math.pi/180)  +  1j*E1*math.sin(thetae1*math.pi/180)
 #Impedance  of  primary
Z1  =  R  + 1j*w*L
 #Primary  current  I1
I1  =  E1/Z1
 #E2
E2  =  1j*w*M*I1


#Results
print  "\n\n  Result  \n\n"
print  "\nE2  is  ",round(abs(E2),3),"/_",round(cmath.phase(complex(E2.real,E2.imag))*180/math.pi,2),"deg  V"


  Result  



E2  is   0.102 /_ 50.19 deg  V

Example 9, page no. 850

In [9]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
Lx  =  20E-3;#  in  Henry
Ly  =  80E-3;#  in  Henry
k  =  0.75;#  coupling  coeff.
Ex  =  5;#  in  Volts

 #calculation:
 #mutual  inductance
M  =  k*(Lx*Ly)**0.5
 #magnitude  of  the  open  circuit  e.m.f.  induced
Ey  =  M*Ex/Lx


#Results
print  "\n\n  Result  \n\n"
print  "\n  mutual  inductance  is  ",M,"  H"
print  "\n  magnitude  of  the  open  circuit  e.m.f.  induced  is  ",Ey,"  V"


  Result  



  mutual  inductance  is   0.03   H

  magnitude  of  the  open  circuit  e.m.f.  induced  is   7.5   V

Example 10, page no. 852

In [1]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
E1  =  2;#  in  Volts
thetae1  =  0;#  in  degrees
f  =  1000/math.pi;#  in  Hz
R1  =  4;#  in  ohm
R2  =  16;#  in  ohm
R3  =  16;#  in  ohm
R4  =  50;#  in  ohm
L  =  10E-3;#  in  Henry
M  =  2E-3;#  in  Henry

#calculation:
w  =  2*math.pi*f
 #R1e  is  the  real  part  of  Z1e
R1e  =  R1  +  R2  +  ((R3  +  R4)*(M**2)*(w**2))/((R3  +  R4)**2  +  (w*L)**2)
 #X1e  is  the  imaginary  part  of  Z1e
X1e  =  w*L  -  (L*(M**2)*(w**3))/((R3  +  R4)**2  +  (w*L)**2)
Z1e  =  R1e  +  1j*X1e
Z2e  =  R3  +  R4  +  1j*w*L
 #primary  current,  I1
I1  =  E1/Z1e
 #E2
E2  =  1j*w*M*I1
 #secondary  current  I2
I2  =  E2/Z2e


#Results
print  "\n\n  Result  \n\n"
print  "secondary  current  I2  is  ",round(abs(I2)*1E3,3),"/_", round(cmath.phase(complex(I2.real,I2.imag))*180/math.pi,2),"deg  mA"


  Result  


secondary  current  I2  is   4.085 /_ 28.55 deg  mA

Example 11, page no. 853

In [11]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
E1  =  50;#  in  Volts
thetae1  =  0;#  in  degrees
w  =  500;#  in  rad/sec
R1  =  300;#  in  ohm
L1  =  0.2;#  in  Henry
L2  =  0.5;#  in  Henry
L3  =  0.3;#  in  Henry
R2  =  500;#  in  ohm
C  =  5E-6;#  in  farad
M  =  0.2;#  in  Henry

#calculation:
 #  Self  impedance  of  primary  circuit
Z1  =  R1  +  1j*w*(L1  +  L2)
 #Self  impedance  of  secondary  circuit,
Z2  =  R2  +  1j*(w*L3  -  1/(w*C))
 #reflected  impedance,  Zr
Zr  =  (w*M)**2/Z2
 #Effective  primary  impedance,
Z1e  =  Z1  +  Zr
 #Primary  current  I1  
I1  =  E1/Z1e
 #Secondary  current  I2
E2  =  1j*w*M*I1
I2  =  E2/Z2


#Results
print  "\n\n  Result  \n\n"
print  "\n  Self  impedance  of  primary  circuit,  Z1  is  ",Z1.real,"  +  (",  Z1.imag,")i  ohm"
print  "\n  Self  impedance  of  secondary  circuit,  Z2  is  ",Z2.real,"  +  (",  Z2.imag,")i  ohm"
print  "\n  reflected  impedance,  Zr  is  ",Zr.real,"  +(",  Zr.imag,")i  ohm"
print  "\n  Effective  primary  impedance  Z1(eff)  is  ",Z1e.real,"  +(",Z1e.imag,")i  ohm"
print  "\n  primary  current  I1  is  ",round(abs(I1),2),"/_",round(cmath.phase(complex(I1.real,I1.imag))*180/math.pi,2),"deg  A"
print  "\n  secondary  current  I2  is  ",round(abs(I2),2),"/_",round(cmath.phase(complex(I2.real,I2.imag))*180/math.pi,2),"deg  A"


  Result  



  Self  impedance  of  primary  circuit,  Z1  is   300.0   +  ( 350.0 )i  ohm

  Self  impedance  of  secondary  circuit,  Z2  is   500.0   +  ( -250.0 )i  ohm

  reflected  impedance,  Zr  is   16.0   +( 8.0 )i  ohm

  Effective  primary  impedance  Z1(eff)  is   316.0   +( 358.0 )i  ohm

  primary  current  I1  is   0.1 /_ -48.57 deg  A

  secondary  current  I2  is   0.02 /_ 68.0 deg  A

Example 12, page no. 855

In [2]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
E1  =  20;#  in  Volts
thetae1  =  0;#  in  degrees
R1  =  15;#  in  ohm
C1  =  400E-12;#  in  farad
R2  =  30;#  in  ohm
L1  =  0.001;#  in  Henry
L2  =  0.0002;#  in  Henry
R3  =  50;#  in  ohm
M  =  10E-6;#  in  Henry

#calculation:
 #voltage
E1  =  E1*math.cos(thetae1*math.pi/180)  +  1j*E1*math.sin(thetae1*math.pi/180)
 #the  resonant  frequency,  fr  
fr  =  1/(2*math.pi*(L1*C1)**0.5)
 #The  secondary  is  also  tuned  to  a  resonant  frequency
 #capacitance,C2
C2  =  1/(L2*(2*math.pi*fr)**2)
 #the  effective  primary  impedance  Z1eff
w = 2*math.pi*fr
Z1e  =  R1 + R2 +  ((w*M)**2)/R3
 #Primary  current  I1  
I1  =  E1/Z1e
 #Secondary  current  I2
E2  =  1j*w*M*I1
I2  =  E2/Z1e
 #voltage  across  capacitor  C2
Vc2  =  I2*(-1*1j/(w*C2))
 #coefficient  of  coupling,  k  
k  =  M/(L1*L2)**0.5


#Results
print  "\n\n  Result  \n\n"
print  "\n  the  resonant  frequency,fr  is  ",round(fr/1000,2),"KHz"
print  "\n  capacitance,C2  is  ",round(C2*1E9,2),"nF"
print  "\n  Effective  primary  impedance  Z1(eff)  is  ",round(abs(Z1e),2)," ohm"
print  "\n  primary  current  I1  is  ",round(abs(I1),2),"/_",round(cmath.phase(complex(I1.real,I1.imag)),0),"deg  A"
print  "\n  voltage  across  capacitor  C2  is ",round(abs(Vc2),2),"/_",round(abs(cmath.phase(complex(Vc2.real,Vc2.imag))),0),"deg  V"
print  "\n  coefficient  of  coupling,  k    is  ",round(k,4),""


  Result  



  the  resonant  frequency,fr  is   251.65 KHz

  capacitance,C2  is   2.0 nF

  Effective  primary  impedance  Z1(eff)  is   50.0  ohm

  primary  current  I1  is   0.4 /_ 0.0 deg  A

  voltage  across  capacitor  C2  is  40.0 /_ 0.0 deg  V

  coefficient  of  coupling,  k    is   0.0224

Example 13, page no. 858

In [13]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
E1  =  250;#  in  Volts
thetae1  =  0;#  in  degrees
R1  =  50j;#  in  ohm
R2  =  10;#  in  ohm
R3  =  10;#  in  ohm
R4  =  50j;#  in  ohm
R5  =  50;#  in  ohm
M  =  10j;#  in  ohm

 #calculation:
 #voltage
E1  =  E1*math.cos(thetae1*math.pi/180)  +  1j*E1*math.sin(thetae1*math.pi/180)
 #Applying  Kirchhoff’s  voltage  law  to  the  primary  circuit  gives
 #(R1  +  R2)*I1  -  M*I2  =  E1
 #Applying  Kirchhoff’s  voltage  law  to  the  secondary  circuit  gives
 #-1*M*I1  +  (  R3  +  R4  +  R5)*I2  =  0
 #solving  these  two
I2  =  E1/((R1  +  R2)*(R3  +  R4  +  R5)/(-1*M)  +  (-1*M))
I1  =  I2*(R3  +  R4  +  R5)/(-1*M)


#Results
print  "\n\n  Result  \n\n"
print  "\n  primary  current  I1  is  ",round(I1.real,2),"  +(",round(  I1.imag,2),")i  A"
print  "\n  secondary  current  I2  is  ",round(I2.real,2),"  +(",round(  I2.imag,2),")i  A"


  Result  



  primary  current  I1  is   0.85   +( -4.77 )i  A

  secondary  current  I2  is   -0.54   +( 0.31 )i  A

Example 14, page no. 859

In [2]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
E1  =  40;#  in  Volts
thetae1  =  0;#  in  degrees
R1  =  5;#  in  ohm
L1  =  0.001;#  in  Henry
L2  =  0.006;#  in  Henry
R2  =  40;#  in  ohm
rzl  =  200;#  in  ohm
thetazl  =  -60;#  in  degrees
k  =  0.70
f  =  20000;#  in  Hz

 #calculation:
w  =  2*math.pi*f
 #voltage
#E1  =  E1*math.cos(thetae1*math.pi/180)  +  1j*E1*math.sin(thetae1*math.pi/180)
 #impedance
ZL  =  rzl*math.cos(thetazl*math.pi/180)  +  1j*rzl*math.sin(thetazl*math.pi/180)
 #mutual  inductance,  M
M  =  k*(L1*L2)**0.5
 #Applying  Kirchhoff’s  voltage  law  to  the  primary  circuit  gives
 #(R1  +  1j*w*L1)*I1  -  1j*w*M*I2  =  E1
 #Applying  Kirchhoff’s  voltage  law  to  the  secondary  circuit  gives
 #-1j*w*M*I1  +  (  R2  +  ZL  +  1j*w*L2)*I2  =  0
 #solving  these  two

a = R1  +  1j*w*L1
b = 1j*w*M
c = R2  +  ZL  +  1j*w*L2
I1  =  E1/(1*a - (b**2)/c)
d = -1*cmath.phase(complex(I1.real,I1.imag))
e = abs(I1)
I2  =  (b/c)*(e*math.cos(d)  +  1j*e*math.sin(d))
pd2 = I2*ZL


#Results
print  "\n\n  Result  \n\n"
print  "\n  mutual  induction  M  is  ",round(M*1E3,3),"mH"
print  "\n  primary  current  I1  is  ",round(abs(I1),3),"/_",round(-1*cmath.phase(complex(I1.real,I1.imag))*180/math.pi,2),"deg  A"
print  "\n  secondary  current  I2  is  ",round(abs(pd2),1),"/_",round(cmath.phase(complex(pd2.real,pd2.imag))*180/math.pi,2),"deg  V"


  Result  



  mutual  induction  M  is   1.715 mH

  primary  current  I1  is   0.724 /_ 65.15 deg  A

  secondary  current  I2  is   52.2 /_ 18.7 deg  V

Example 15, page no. 860

In [15]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
E1  =  50;#  in  Volts
thetae1  =  0;#  in  degrees
r  =  5;#  in  ohm
R1  =  20;#  in  ohm
L1  =  0.2;#  in  Henry
L2  =  0.4;#  in  Henry
R2  =  25;#  in  ohm
RL  =  20;#  in  ohm
M  =  0.1;#  in  Henry
f  =  75/math.pi;#  in  Hz

#calculation:
w  =  2*math.pi*f
 #Applying  Kirchhoff’s  voltage  law  to  the  primary  circuit  gives
 #(r  +  R1  +  1j*w*L1)*I1  -  1j*w*M*I2  =  E1
 #Applying  Kirchhoff’s  voltage  law  to  the  secondary  circuit  gives
 #-1*1j*w*M*I1  +  (  R2  +  RL  +  1j*w*L2)*I2  =  0
 #solving  these  two
I2  =  E1/((r  +  R1  +  1j*w*L1)*(R2  +  RL  +  1j*w*L2)/(1j*w*M)  +  (-1*1j*w*M))
I1  =  I2*(R2  +  RL  +  1j*w*L2)/(1j*w*M)


#Results
print  "\n\n  Result  \n\n"
print  "\n  primary  current  I1  is  ",round(abs(I1),2),"/_",round(cmath.phase(complex(I1.real,I1.imag))*180/math.pi,2),"deg  A"
print  "\n  load  current  I2  is  ",round(abs(I2),2),"/_",round(cmath.phase(complex(I2.real,I2.imag))*180/math.pi,2),"deg  A"


  Result  



  primary  current  I1  is   1.3 /_ -45.84 deg  A

  load  current  I2  is   0.26 /_ -8.97 deg  A

Example 16, page no. 862

In [3]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
E1  =  50;#  in  Volts
thetae1  =  0;#  in  degrees
r  =  5;#  in  ohm
R1  =  20;#  in  ohm
L1  =  0.2;#  in  Henry
R  =  8;#  in  ohm
L  =  0.1;#  in  Henry
L2  =  0.4;#  in  Henry
R2  =  25;#  in  ohm
RL  =  20;#  in  ohm
M  =  0.1;#  in  Henry
f  =  75/math.pi;#  in  Hz

#calculation:
w  =  2*math.pi*f
 #Applying  Kirchhoff’s  voltage  law  to  the  primary  circuit  gives
 #(r  +  R1  +  1j*w*L1  +  R  +  1j*w*L)*I1  -  (1j*w*M  +  R  +  1j*w*L)*I2  =  E1
 #Applying  Kirchhoff’s  voltage  law  to  the  secondary  circuit  gives
 #-1*(1j*w*M  +  R  +  1j*w*L)*I1  +  (R2  +  RL  +  1j*w*L2  +  R  +  1j*w*L)*I2  =  0
 #solving  these  two
I2  =  E1/((r  +  R1  +  1j*w*L1  +  R  +  1j*w*L)*(R2  +  RL  +  1j*w*L2  +  R  +  1j*w*L)/(1j*w*M  +  R  +  1j*w*L)  +  (-1*(1j*w*M  +  R  +  1j*w*L)))
I1  =  I2*(R2  +  RL  +  1j*w*L2  +  R  +  1j*w*L)/(1j*w*M  +  R  +  1j*w*L)
 #reversing
 #Applying  Kirchhoff’s  voltage  law  to  the  primary  circuit  gives
 #(r  +  R1  +  1j*w*L1  +  R  +  1j*w*L)*I1r  -  (-1*1j*w*M  +  R  +  1j*w*L)*I2r  =  E1
 #Applying  Kirchhoff’s  voltage  law  to  the  secondary  circuit  gives
 #-1*(-1*1j*w*M  +  R  +  1j*w*L)*I1r  +  (R2  +  RL  +  1j*w*L2  +  R  +  1j*w*L)*I2r  =  0
 #solving  these  two
I2r  =  E1/((r  +  R1  +  1j*w*L1  +  R  +  1j*w*L)*(R2  +  RL  +  1j*w*L2  +  R  +  1j*w*L)/(-1*1j*w*M  +  R  +  1j*w*L)  +  (-1*(-1*1j*w*M  +  R  +  1j*w*L)))
I1r  =  I2r*(R2  +  RL  +  1j*w*L2  +  R  +  1j*w*L)/(-1*1j*w*M  +  R  +  1j*w*L)


#Results
print  "\n\n  Result  \n\n"
print  "primary  current  I1  is  ",round(abs(I1),2),"/_",round(cmath.phase(complex(I1.real,I1.imag))*180/math.pi,2),"deg  A"
print  "load  current  I2  is  ",round(abs(I2),2),"/_",round(cmath.phase(complex(I2.real,I2.imag))*180/math.pi,2),"deg   A"
print  "reversed  primary  current  I1r  is  ",round(abs(I1r),2),"/_",round(cmath.phase(complex(I1r.real,I1r.imag))*180/math.pi,2),"deg   A"
print  "reversed  load  current  I2r  is  ",round(abs(I2r),2),"/_",round(cmath.phase(complex(I2r.real,I2r.imag))*180/math.pi,2),"deg   A"


  Result  


primary  current  I1  is   1.03 /_ -45.47 deg  A
load  current  I2  is   0.35 /_ -25.16 deg   A
reversed  primary  current  I1r  is   0.89 /_ -54.42 deg   A
reversed  load  current  I2r  is   0.08 /_ -109.17 deg   A