# Chapter 3: Principles of Electromechanical Energy Conversion¶

## Example 3.1, Page 144¶

In :
import math

#Variable declaration
A=20*10**-4;#surface area of each  capacitor's plate
d=5*10**-3;#separation between the plates
e=(10**-9)/(36*math.pi);#permetivity of air
V=10*10**3;#potential diff. between the plates

#Calculations&Results
F_e=(e*A*V**2)/(2*d**2);#electric force
g=9.81;#acceleration due to gravity (in meter/second**2)
#For condt of balancing electric force=weight of object
#F_e=m*g
m=F_e/g;
print 'mass of object (in grams)=%.2f'%(m*1000)
W_f=(e*A*V**2)/(2*d);
print 'energy stored in the feild (in micro-joules)=%.f'%(W_f*1000000)

mass of object (in grams)=3.61
energy stored in the feild (in micro-joules)=177


## Example 3.3, Page 148¶

In :
import scipy

#Variable declaration
#i=current in the ckt (in Amperes)

#Calculations
def f(x):
return x/(6-(2*x))
#Refer to eqn:3.18

#Result
print 'Energy stored in magnetic feild (in Joules)=%.3f'%W_m

Energy stored in magnetic feild (in Joules)=0.648


## Example 3.4, Page 151¶

In :
import math
import scipy
from scipy.misc import derivative

#Variable declaration
N=100;#no. of turns of coil
A=10**-4;#area
x=1*10**-2;#length of air gap
u_o=4*math.pi*10**-7;#permeablity of air
u_r=2000;#permeablity of magnetic material
D=7.85*10**3;#density of material (in kg/m**3)
V=11*10**-6;#volume of material
m=D*V;#mass of material
g=9.81;#acceleration due to gravity

#Calculations&Results
#Refer to fig:3.7
R_o=(15.5*10**-2)/(u_o*u_r*A);#reluctance of outer legs
R_c=(5.5*10**-2)/(u_o*u_r*A);#reluctance of central leg
def L( x ):#inductance
return (N**2)/ R ( x );

def R( x ):#total reluctance
return R_c+R_g(x)+(0.5*(R_o+R_g(x)));

def R_g( x ):#reluctance of air gap
return x/(u_o*A);

x = 0.01 ;     # Points of interest
t = derivative(L,x)
#t=[diag(derivative(L,x))];#t=dL/dx (at x=0.01m)
#since t<o,i.e,F_m is acting in opp direction that of weight
#for equilibrium F_m=m*g
I=math.sqrt((m*g)/(0.5*t*(1)));#Refer to eqn3.23
print 'current in the coil (in Amperes)= %.2f'%(I/100)
L_o=L(.01);
W_f=0.5*L_o*I**2;
print 'energy stored in the magnetic feild  (in mili-Joules)= %.1f'%(W_f*10**3/10000)

current in the coil (in Amperes)= 14.22
energy stored in the magnetic feild  (in mili-Joules)= 8.4


## Example 3.5, Page 153¶

In :
import math
import scipy
from scipy.misc import derivative

#Variable declaration
T=20;#torque exerted by spring (in Newton-meter)

#Calculations
F_s=T/r;#force exerted by spring on magnetic plate
N=1000;#no. of turns in coil
u_o=4*math.pi*10**-7;#permablityof air
A=9*10**-4;#area (in meter**2)
def L( x ):#inductance
return (N**2)/ R ( x );

def R( x ):#reluctance of air gap
return (2*x)/(u_o*A);

x = 0.001;     # Points of interest
t = derivative(L,x)
#t=[diag(derivative(L,x))];#t=dL/dx (at x=0.001m)
#since t<o i.e,F_m is acting in opp direction that of weight
#for equilibrium F_m=F_s
I=math.sqrt((2*F_s)/(t*(1)));#Refer to eqn3.23

#Result
print 'current in the coil (in Amperes)=%.3f'%(I/1000)

current in the coil (in Amperes)=0.595


## Example 3.6, Page 168¶

In :
import math

#Variable declaration
N=100;#no. of turns in coil
P=4;#number of poles
N_m=1800;#rotor speed (in rpm)

#Calculations&Results
flux_p=4.5*10**-3;#flux per pole (in Wb)
f=(P*N_m)/120;#Refer to eqn:3.30a
print '(a) frequency of induced emf (in Hertz)=%.f'%f
#refer to eqn:3.31
E_m=(2*math.pi*P*flux_p*N_m)/120;#max value of induced emf per turn
E_mc=N*E_m;
print '(b) m/ax value of induced emf in coil (in Volts)= %.2f'%E_mc
E_rms=E_mc/math.sqrt(2);
print '(c) rms value of induced emf (in Volts)= %.f'%E_rms
E_avg=(2*E_mc)/math.pi;
print '(d) average value of induced emf (in Volts)= %.f'%E_avg

(a) frequency of induced emf (in Hertz)=60
(b) m/ax value of induced emf in coil (in Volts)= 169.65
(c) rms value of induced emf (in Volts)= 120
(d) average value of induced emf (in Volts)= 108


## Example 3.7, Page 177¶

In :
#Variable declaration
P=4;#no. of pole
f=50;#frequency (in Hz)
N_r=1200.;#speed of rotor(in rpm)

#Calculations&Results
N_s=(120*f)/P;
print 'synchronous speed (in rpm)= %.f'%N_s
s=(N_s-N_r)/N_s;#slip
s_p=s*100;
print 'percent slip of the motor(%%)= %.f'%s_p

synchronous speed (in rpm)= 1500
percent slip of the motor(%)= 20


## Example 3.8, Page 185¶

In :
import math

#Variable declaration
N=2;#no. of poles
f=60;#frequency in Hz
I_rms=10;#current intake
L_q=1;#min inductance (in H)
L_d=2;#max inductance(inH)

#Calculations&Results
w=2*math.pi*f;
#Refer to eqn:3.52
T_avg=(-1)*0.125*(L_d-L_q)*((I_rms*math.sqrt(2))**2)*math.sin(2*45*math.pi/180);
if ( T_avg <0 ):
print "average torque developed by motor (in Newton-meter)= %.f"%(T_avg*(-1))
else:
print "average torque developed by motor (in Newton-meter)= %f"%T_avg

rotor speed(in rad/sec)= 376.991118
average torque developed by motor (in Newton-meter)= 25


## Example 3.9, Page 188¶

In :
import math

#Variable declaration
N=500;#no. of turns
u_o=4*math.pi*10**-7;#Permeablity of air
I=4.2;#main winding current(in A)
A=2.25*10**-4;#area of air gap(in m**2)
x=0.002;#length of air gap(in m)

#Calculations
i=I*1.50;#min current needed for activating relay
F_m=u_o*A*0.5*((N*i)/x)**2;#Refer to eqn 3.53

#Result
print 'restraining force of the spring(in Newton)=%.2f'%F_m

restraining force of the spring(in Newton)=350.69