# CHAPTER 2: BASICS OF MAGNETIC CIRCUITS¶

## Example 2.1, Page number 53¶

In [1]:
#Variable declaration
l = 4.0      #Length(m)
w = 2.0      #Width(m)
B = 0.12     #Magnetic flux density(Tesla)

#Calculation
A = l*w      #Area(m^2)
flux = B*A   #Magnetic flux(Wb)

#Result
print('Magnetic flux , Φ = %.2f Wb' %flux)
Magnetic flux , Φ = 0.96 Wb

## Example 2.2, Page number 54¶

In [1]:
import math

#Variable declaration
d_in = 3.0      #Inside diameter(cm)
d_out = 6.0     #Outside diameter(cm)
N = 200.0       #Number of turns
I = 3.0         #Current(A)
flux = 0.015    #Flux(Wb)

#Calculation
d = d_in+(d_out-d_in)/2      #Distance(cm)
l = math.pi*d                #Mean length of core(cm)
A = math.pi*d**2/4*10**-4    #Area(m^2)
B = flux/A                   #Flux density(Wb/m^2)
MMF = N*I                    #Magnetomotive force(At)
H = N*I/(l*10**-2)           #Magnetic field intensity(At/m)

#Result
print('Flux density , B = %.2f Wb/m^2' %B)
print('Magnetomotive force , MMF = %.1f At' %MMF)
print('Magnetic field intensity , H = %.2f At/m' %H)
print('\nNOTE : ERROR : Calculation & unit mistakes in textbook')
Flux density , B = 9.43 Wb/m^2
Magnetomotive force , MMF = 600.0 At
Magnetic field intensity , H = 4244.13 At/m

NOTE : ERROR : Calculation & unit mistakes in textbook

## Example 2.3, Page number 55¶

In [1]:
import math

#Variable declaration
u_r = 625.0  #Relative permeability of rectangular core
N = 25.0     #Number of turns
I = 2.0      #Current(A)
a = 5.5      #Lenght of rectangular core(cm)
b = 1.5      #Width of rectangular core(cm)

#Calculation
l = 2*(a+b)                 #Mean length of core(cm)
H = N*I/(l*10**-2)          #Magnetic field intensity(At/m)
u_0 = 4*math.pi*10**-7      #Permeability of free space(H/m)
u = u_0*u_r                 #Permeabilty
B = u*H                     #Magnetic flux density(Wb/m^2)

#Result
print('Magnetic field intensity , H = %.f At/m ' %H)
print('Permeabilty , µ = %.2e ' %u)
print('Magnetic flux density , B = %.2f Wb/m^2 ' %B)
Magnetic field intensity , H = 357 At/m
Permeabilty , µ = 7.85e-04
Magnetic flux density , B = 0.28 Wb/m^2

## Example 2.4, Page number 57¶

In [1]:
#Variable declaration
N = 6.0          #Number of turns
I = 3.0          #Current(A)
flux = 0.056     #Flux(Wb)

#Calculation
MMF = N*I        #Magnetomotive force(At)
R_m = MMF/flux   #Reluctance(At/Wb)

#Result
print('Magnetomotive force , MMF = %.f At' %MMF)
print('Reluctance , R_m = %.1f At/Wb' %R_m)
Magnetomotive force , MMF = 18 At
Reluctance , R_m = 321.4 At/Wb

## Example 2.5, Page number 59¶

In [1]:
import math

#Variable declaration
I = 15.0                      #Current through conductor(A)
N = 10.0                      #Number of turns
u_0 = 4.0*math.pi*10**-7      #Permeability of free space(H/m)
u_r = 1.0                     #Relative permeability of air medium
r = 0.015                     #Distance(m)

#Calculation
B = u_0*u_r*N*I/(2*math.pi*r)   #Magnetic flux density(T)

#Result
print('Magnetic flux density , B = %.1e T' %B)
print('\nNOTE : ERROR : Distance is 1.5 cm & not 2.5 cm as given in textbook')
Magnetic flux density , B = 2.0e-03 T

NOTE : ERROR : Distance is 1.5 cm & not 2.5 cm as given in textbook

## Example 2.6, Page number 60-61¶

In [1]:
import math

#Variable declaration
N = 200.0        #Number of turns
d_in = 7.0       #Inner diameter(cm)
d_out = 10.0     #Outer diameter(cm)
A = 0.005        #Cross sectional area(m^2)
I = 5.0          #Current through coil(A)

#Calculation
u_0 = 4.0*math.pi*10**-7       #Permeability of free space(H/m)
R = d_out-d_in
l = round(2*math.pi*R/100,2)   #Mean circumference length(m)
#For case(i)
H = N*I/l                      #Magnetic field intensity(At/m)
#For case(ii)
B = u_0*H*1000                 #Flux density(mWb/m^2)
#For case(iii)
flux = B*A*1000                #Flux(µWb)

#Result
print('Magnetic field intensity , H = %.1f At/m' %H)
print('Flux density , B = %.1f mWb/m^2' %B)
print('Flux , Φ = %.f µWb' %flux)
Magnetic field intensity , H = 5263.2 At/m
Flux density , B = 6.6 mWb/m^2
Flux , Φ = 33 µWb

## Example 2.7, Page number 62-63¶

In [1]:
import math

#Variable declaration
l = 0.1          #Length(m)
w = 0.01         #Width(m)
h = 0.1          #Height(m)
N = 450.0        #Number of turns
I = 0.2          #Current(A)
u_r = 850.0      #Relative permeability

#Calculation
MMF = N*I                          #Magnetomotive force(At)
l_c = (h-w)*4                      #Mean length of the path(m)
A = w*w                            #Cross sectional area(m^2)
u_0 = 4.0*math.pi*10**-7           #Permeability of free space(H/m)
R_m = l_c/(u_0*u_r*A)              #Reluctance(At/Wb)
flux = MMF/R_m                     #Flux(Wb)
B = flux/A                         #Magnetic flux density(Wb/m^2)
H = B/(u_0*u_r)                    #Magnetic field intensity(At/m)

#Result
print('Flux , Φ = %.2e Wb' %flux)
print('Flux density , B = %.2f Wb/m^2' %B)
print('Field intensity , H = %.1f At/m' %H)
print('\nNOTE : Changes in obtained answer from that of textbook is due to more precision')
Flux , Φ = 2.67e-05 Wb
Flux density , B = 0.27 Wb/m^2
Field intensity , H = 250.0 At/m

NOTE : Changes in obtained answer from that of textbook is due to more precision

## Example 2.8, Page number 64-65¶

In [1]:
import math

#Variable declaration
N = 450.0       #Number of turns wound on left side of limb
A = 4.0         #Cross sectional area(m^2)
phi_2 = 3.0     #Flux(Wb) in the right limb
u_r = 500.0     #Relative permeability
l_1 = 0.12      #Length of middle limb(m)
l_2 = 0.24      #Length of right limb(m)

#Calculation
u_0 = 4.0*math.pi*10**-7        #Permeability of free space(H/m)
phi_1 = phi_2*l_2/l_1           #Flux in middle limb(Wb)
phi = phi_1+phi_2               #Total flux(Wb)
B = phi/A                       #Flux density in the left limb(Wb/m^2)
H = B/(u_0*u_r)                 #Magnetic field intensity(At/m)
MMF = H*l_2                     #Magnetomotive force(At)
B_2 = phi_2/A                   #Flux density in the right limb(Wb/m^2)
H_2 = B_2/(u_0*u_r)             #Magnetic field(At/m)
MMF_2 = H_2*l_2                 #Magnetomotive force(At)
MMF_t = MMF+MMF_2               #Total magnetomotive force(At)
I = MMF_t/N                     #Current(A)

#Result
print('Current , I = %.2f A' %I)
Current , I = 2.55 A

## Example 2.9, Page number 67-68¶

In [1]:
import math

#Variable declaration
l = 0.45               #Mean length(m)
A = 25.0*10**-4        #Cross sectional area(m^2)
l_g = 0.8*10**-3       #Length of air gap(m)
N = 500.0              #Number of turns
I = 1.25               #Current(A)
flux = 1.5*10**-3      #Flux(Wb)

#Calculation
u_0 = 4.0*math.pi*10**-7        #Permeability of free space(H/m)
B = flux/A                      #Magnetic flux density(Wb/m^2)
MMF = N*I                       #Magnetomotive force(At)
R_m = MMF/flux                  #Reluctance(At/Wb)
H = B/u_0                       #Magnetizing force(At/m)
MMF_ag = H*l_g                  #Magnetomotive force(At)
MMF_i = MMF-MMF_ag              #Magnetomotive force for iron ring(At)
H_i = MMF_i/l                   #Magnetic field intensity for iron part(At/m)
u_r = B/(u_0*H_i)               #Relative permeability for iron

#Result
print('Reluctance , R_m = %.2e At/Wb' %R_m)
print('Relative permeability of the iron ring iron , µ_r = %.f ' %u_r)
print('\nNOTE : Reluctance part is not solved in textbook')
print('ERROR : Current is 1.25A not 2.25A & flux is 1.5 mWb not 2.5 mWb as given in textbook')
Reluctance , R_m = 4.17e+05 At/Wb
Relative permeability of the iron ring iron , µ_r = 884

NOTE : Reluctance part is not solved in textbook
ERROR : Current is 1.25A not 2.25A & flux is 1.5 mWb not 2.5 mWb as given in textbook

## Example 2.10, Page number 68¶

In [1]:
import math

#Variable declaration
A = 2.0*10**-4        #Cross sectional area(m^2)
N = 200.0             #Number of turns
flux = 1.5*10**-3     #Flux(Wb)
u_r = 4000.0          #Relative permeability of core
l_g = 0.01            #Length of air gap(m)
a = 9.0               #Length(cm)
w = 3.0               #Width(cm)

#Calculation
u_0 = 4.0*math.pi*10**-7        #Permeability of free space(H/m)
R_mg = l_g/(u_0*A)              #Reluctance of air gap(At/Wb)
l = 4*(a-w-w+(1.5+1.5))-1       #Mean length of iron(cm)
u = u_0*u_r                     #Permeability
R_mi = l*10**-2/(u*A)           #Reluctance of iron(At/Wb)
R_mt = R_mg+R_mi                #Total reluctance(At/Wb)
I = R_mt*flux/N                 #Current(A)

#Result
print('Total reluctance , R_mt = %.3e AT/Wb' %R_mt)
print('Current flowing through the coil , I = %.1f A' %I)
print('\nNOTE : ERROR : Total flux is 1.5 mWB & not 2.5 mWB as given in textbook question')
Total reluctance , R_mt = 4.002e+07 AT/Wb
Current flowing through the coil , I = 300.1 A

NOTE : ERROR : Total flux is 1.5 mWB & not 2.5 mWB as given in textbook question

## Example 2.11, Page number 70¶

In [1]:
#Variable declaration
I = 150.0    #Current through conductor(A)
l = 2.0      #Conductor length(m)
B = 0.35     #Magnetic flux density(T)

#Calculation
F = B*l*I    #Force(N)

#Result
print('Force , F = %.f N' %F)
Force , F = 105 N

## Example 2.12, Page number 76¶

In [1]:
import math

#Variable declaration
l = 25.0*10**-2      #Length of air-core coil(m)
A = 4.0*10**-4       #Cross sectional area(m^2)
N = 65.0             #Number of turns

#Calculation
u_0 = 4.0*math.pi*10**-7        #Permeability of free space(H/m)
u_r = 1.0
u = u_0*u_r                     #Permeability
L = N**2*u*A/l*10**6            #Inductance(µH)

#Result
print('Inductance of the coil , L = %.1f µH' %L)
Inductance of the coil , L = 8.5 µH

## Example 2.13, Page number 80¶

In [1]:
#Variable declaration
k_h = 110.0       #Hysteresis co-efficient(J/m^3)
V_cvol = 0.005    #Volume of core(m^3)
B_m = 1.12        #Maximum flux density(T)
f = 60.0          #Frequency(Hz)
n = 1.6

#Calculation
P_h = k_h*V_cvol*B_m**n*f   #Hysteresis loss(W)

#Result
print('Hysteresis loss , P_h = %.2f W' %P_h)
Hysteresis loss , P_h = 39.56 W