# Chapter 2: Review of Basic Laws of Electromagnetism¶

## Example 2.1, Page 66¶

In [2]:
#Variable declaration
N=1000;#Number of turns
phy_1=100*10**-3;#initial magnetic flux (in webers)
phy_2=20*10**-3;#final magnetic flux (in webers)

#Calculations
phy=phy_2-phy_1;#change in magnetic flux
t=5;#(in seconds)
e=(-1)*N*(phy/t);#induced emf (in volts)

#Result
print 'Induced emf (in volts)=%.f'%e

Induced emf (in volts)=16


## Example 2.6, Page 90¶

In [8]:
import math

#Variable declaration
u_o=4*math.pi*10**-7;#permeablity of air
u_r=1200;#permeablity of magnetic material
N=1500;#No. of turns
I=4;#current in the coil (in Amperes)
r_i=10*10**-2;#inner radii of magnetic core (in meters)
r_o=12*10**-2;#outer radii of magnetic core (in meters)

#Calculations
r_m=(r_i+r_o)/2;#mean radii of magnetic core (in meters)
l_g=1*10**-2;#length of air gap (in meters)
l_m=2*math.pi*(r_m-l_g);#in meters
#Refer to fig:-2.14
A_m=(r_o-r_i)**2;#cross-sectional  area of magnetic path (in meter**2)
R_m=l_m/(u_o*u_r*A_m);#reluctance of magnetic material
R_g=l_g/(u_o*A_m);#reluctance of air gap
#R_m and R_g in sereis
R=R_m+R_g;
B_m=N*I/(R*A_m);#magnetic flux density (in Tesla)

#Result
print 'magnetic flux density (in Tesla)=%.3f T'%B_m

magnetic flux density (in Tesla)=0.716 T


## Example 2.10, Page 103¶

In [13]:
#Variable declaration
#Refer to eqn 2.26
e_21=20.;#voltage induced in coil-2 (in volts)
I1=2000;#rate of change of current in coil-1 (in Amperes/second)

#Calculations
M=e_21/I1;# in henry
L1=25*10**-3;#in henry
L2=25*10**-3;#in henry
#Refer to eqn 2.32
k=(M/L1)*100;#coefficient of coupling

#Result
print 'percentage (%%)=%.f'%k

percentage (%)=40


## Example 2.11, Page 106¶

In [16]:
import math

#Variable declaration
#L1,L2=inductances of coil 1&2
#M=mutual inductance b/w coil 1&2
L_aid=2.38;#effective inductance when connected in sereis aiding
L_opp=1.02;#effective inductance when connected in sereis opposing

#Calculations&Results
#L1+L2+2M=L_aid
#L1+L2-2M=L_opp
M=(L_aid-L_opp)/4;#in henry
print 'mutual inductance (in henry)= %.2f'%M
#L1=16*L2
L1=(L_aid-2*M)/17;#in henry
print 'inductance of coil-1 (in henry)= %.1f'%L1
L2=L_aid-(2*M)-L1;#in henry
print 'inductance of coil-2 (in henry)=%.1f'%L2
k=M/(math.sqrt(L1*L2));
print 'coefficient of coupling=%.2f'%k

mutual inductance (in henry)= 0.34
inductance of coil-1 (in henry)= 0.1
inductance of coil-2 (in henry)=1.6
coefficient of coupling=0.85


## Example 2.12, Page 108¶

In [18]:
#Variable declaration
L1=1.6;#self inductance of coil 1 (in Henry)
L2=0.1;#self inductance of coil 2 (in Henry)
M=0.34;#mutual inductance (in Henry)

#Calculations&Results
#Refer to eqn-2.45
L_aid=((L1*L2)-M**2)*10**3/(L1+L2-(2*M));#in mili-Henry
print 'effective inductance in parallel aiding  (in mili-Henry)=%.1f'%L_aid
#Refer to eqn-2.46
L_opp=((L1*L2)-M**2)*10**3/(L1+L2+(2*M));#in mili-henry
print 'effective inductance in parallel opposing  (in mini-Henry)=%.1f'%L_opp

effective inductance in parallel aiding  (in mili-Henry)=43.5
effective inductance in parallel opposing  (in mini-Henry)=18.7


## Example 2.13, Page 113¶

In [62]:
import math
import numpy

#Variable declaration
#refer to eqn-2.50
#eqn:-2.51,2.52 & 2.53 are obtained
f=numpy.array([25, 25, 60]);#in hertz
T = numpy.array([1.1,1.5,1.1])

#Calculations&Results
B_m=numpy.array([1.1, 1.5, 1.1])
P_m=numpy.array([0.4, 0.8, 1.2])
#On solving eqn:-2.51 & eqn:-2.53
k_e=(0.016-0.02)/(30.25-72.6);
#on solving eqn:-2.51 & eqn:-2.52
n=(math.log((0.016-(30.25*k_e))/(0.032-(56.25*k_e))))/(math.log(1.1/1.5));
k_h=(0.016-(30.25*k_e))/1.1**n;
P_h=k_h*f*B_m**n#hysteresis loss
P_eddy=k_e*(f**2)*B_m**2#eddy current loss

#Results
for n in range(3,):
print 'Frequency(Hz)\t\tFlux Density(T)\t\tHysteresis loss(W/kg)\t\tEddy-current loss(W/kg)\n',(f[n]),"\t\t\t",round(T[n],1),"\t\t\t",round(P_h[n],3),"\t\t\t\t",round(P_eddy[n],3)

Frequency(Hz)		Flux Density(T)		Hysteresis loss(W/kg)		Eddy-current loss(W/kg)
25 			1.1 			0.329 				0.071
Frequency(Hz)		Flux Density(T)		Hysteresis loss(W/kg)		Eddy-current loss(W/kg)
25 			1.5 			0.667 				0.133
Frequency(Hz)		Flux Density(T)		Hysteresis loss(W/kg)		Eddy-current loss(W/kg)
60 			1.1 			0.789 				0.411


## Example 2.14, Page 118¶

In [64]:
import math

#Variable declaration
u_o=4*math.pi*10**-7;#permeablity of air
u_r=500;#permeablity of steel
l_g=1*10**-2;#length of air gap section (in meter)
A_g=10*10**-4;#cross-sectional area of air gap section (in meter**2)
A_m=10*10**-4;#cross-sectional area of magnet section (in meter**2)
A_s=10*10**-4;#cross-sectional area of steel sections (in meter**2)
l_s=50*10**-2;#length of steel section (in meter)
#Refer to fig:-2.29 (Demagnetization and energy-product curves of a magnet)
H_m=-144*10**3;#(in Ampere/meter)
B_m=0.23;#Magnetic flux density (in Tesla)

#Calculations
#refer to eqn:-2.55
l_m=(-1*100)*(((l_g*A_m)/(u_o*A_g))+((2*l_s*A_m)/(u_o*u_r*A_s)))*(B_m/H_m);# (in centimeter)

#Result
print 'minimum length of magnet (in centimeter)=%.2f'%l_m

minimum length of magnet (in centimeter)=1.53


## Example 2.15, Page 119¶

In [1]:
import math
import sympy

#Variable declaration
#From figure 2.32(a)
lm = 52-42   #mean length of magnets,mm
ls = 2.5+2.5+(2*math.pi*54.5/4)  #mean length of yoke,mm
lg = 42-40   #air gap,mm
la = 17.5+17.5+(2*math.pi*22.5/4) #mean length of rotor,mm

#Calculations
#From figure 2.32(b)
Am = 50*(52+42)*math.pi/4   #cross-sectional area of magnet,mm^2
As = 5*50                   #cross-sectional area of yoke,mm^2
Ag = 50*(42+40)*math.pi/4   #cross-sectional area of air-gap,mm^2
Aa = 35*50                  #cross-sectional area of rotor,mm^2

Bm = 0.337   #T
phi = Bm*Am  #Wb
phi_t = round(2*phi*10**-3,3)  #Wb
#We know that, phi_c = 2.488cos100t mWb
from sympy import Symbol,diff,cos
t = Symbol('t')
d_phi_by_dt = diff(cos(100*t),t)
e = -phi_t*d_phi_by_dt
#Result
print "The induced emf is",e,"V"
#Incorrect result in textbook

The induced emf is 248.8*sin(100*t) V