Chapter 6: Alternating Quantities

Example 6.1, Page 202

In [16]:
import math

# Comparing alternating voltage v = 35*sin(314.2*t) with the standard Eq.
# Part (a)
V_m = 35;  # Maximum value of alternating voltage, volt

# Part (b)
#We know that v= vm8sin(2*pi*f*t)
#Comparing the alternating voltage equation with the above one, we get,
f = 314.2/(2*math.pi)     # Frequency of waveform, Hz

# Part (c)
T = 1/f;    # Time period of waveform, sec

# Part (d)
t = 3.5;    # Time with reference to zero crossing, sec
v = 35*math.sin(2*math.pi*50*3.5*1e-03);    # Volatge value after the waveform passes through zero, going positive

#Results
print "The maximum value of alternating voltage = %2d volt"%V_m
print "The frequency of alternating voltage = %2d Hz"%f
print "The time period of alternating voltage = %3.1f ms"%(T/1e-03);
print "The volatge value after the waveform passes through zero = %5.2f volt"%v

The maximum value of alternating voltage = 35 volt
The frequency of alternating voltage = 50 Hz
The time period of alternating voltage = 20.0 ms
The volatge value after the waveform passes through zero = 31.19 volt

Example 6.2, Page 202

In [17]:
import math

# Part (a)
#I = Im*sin(2*pi*f*t)
# Given i = 75*sin(200*%pi*t) mA which on comparing with the general expression gives
#Comparing the above two equations, we get,
f = (200*math.pi)/(2*math.pi)   # Frequency of alternating current, Hz

# Part(b)
i = 35.;     # Alternating current after passing through zero, mA
t = math.asin(i/75)/(200*math.pi*1e-03);   # Time taken for current to reach 35 mA, ms

#Results
print "The frequency of alternating current = %2d Hz"%f
print "The time taken for current to reach 35 mA = %5.3f mA"%t

The frequency of alternating current = 100 Hz
The time taken for current to reach 35 mA = 0.773 mA

Example 6.3, Page 204

In [3]:
#Variable declaration
V_av = 3.5;     # Average value of sinusoidal alternating voltage, V
T = 6.67e-03;   # Time period of alternating current, s

#Calculations
V_m = V_av/0.637;   # Peak value of alternating current, V
f = 1/T;    # Frequency of alternating volatge, Hz

#Result
print "The standard expression for %3.1f voltage = %3.1f sin(%3d*pi*t) volt"%(V_av, V_m,round(2*f))

The standard expression for 3.5 voltage = 5.5 sin(300*pi*t) volt

Example 6.4, Page 204

In [18]:
import math

#Variable declaration
V_av = 3.5;     # Average value of sinusoidal alternating voltage, V
T = 6.67e-03;   # Time period of alternating voltage, s

#Calculations&Results
V_m = V_av/0.637;   # Peak value of alternating voltage, V
f = 1/T;    # Frequency of alternating volatge, Hz

# Part (a)
t = 0.5e-03;    # Time taken by the waveform after passing through zero, s
v = V_m*math.sin(2*math.pi*f*t);     # Instantaneous value of alternating voltage, s
print "The instantaneous value of alternating voltage after %3.1f ms = %3.1f volt"%(t/1e-03, v)

# Part (b)
t = 4.5e-03;    # Time taken by the waveform after passing through zero, s
v = V_m*math.sin(2*math.pi*f*t);     # Instantaneous value of alternating voltage, s
print "The instantaneous value of alternating voltage after %3.1f ms = %3.1f volt"%(t/1e-03, v);

# Part (c)
v = 3;     # Alternating voltage after passing through zero, mA
t = math.asin(v/V_m)/(2*math.pi*f);   # Time taken for current to reach 3 V, s
print "The time taken for voltage to reach %1d volt = %5.3f ms"%(v, t/1e-03);

The instantaneous value of alternating voltage after 0.5 ms = 2.5 volt
The instantaneous value of alternating voltage after 4.5 ms = -4.9 volt
The time taken for voltage to reach 3 volt = 0.613 ms

Example 6.5, Page 206

In [19]:
import math

#Variable declaration
V = 240;    # Rms vlaue of alternating voltage, volt

#Calculations
V_m = math.sqrt(2)*V;   # Peak value of alternating voltage, volt

#Result
print "The amplitude of household %3d volt supply = %5.1f volt"%(V, V_m);

The amplitude of household 240 volt supply = 339.4 volt

Example 6.6, Page 207

In [6]:
#Variable declaration
pf = 2.5;   # Peak factor of non-sinusoidal alternating voltage
V = 240;    # Rms vlaue of alternating voltage, volt

#Calculations
V_m = pf*V;   # Peak value of alternating voltage, volt

#Result
print "The absolute minimum working voltage = %3d volt"%V_m

The absolute minimum working voltage = 600 volt

Example 6.7, Page 207

In [20]:
import math

#Variable declaration
l = 0.25;   # Length of the rectangular coil, m
d = 0.2;    # Width of rectangular coil, m
N = 80;     # Number of turns of the rectangular coil
B = 0.075;  # Magnetic flux density, tesla
n = 3000/60;   # Frequency of revolution of the coil, rev/s
v = n*math.pi*d;    # Linear speed with which the coil sides move, m/s
t = 2e-03;      # Time after the emf crosses zero, s

#Calculations
# Part (a)
# As e = 2*N*B*l*v*sin(2*pi*f*t) volt, and for maximum value of sin(2*pi*f*t) = 1
E_m = 2*N*B*l*v*(1);  # Amplitude of emf, volt
E = 0.707*E_m;      # rms value of emf, volt
E_av = 0.637*E_m;   # Average value of emf, volt
# For a two pole field system, 
f = n;      # Frequency of generated waveform, Hz

# Part (b)
T = 1./f;    # Time period of generated waveform, Hz

# Part (c)
e = E_m*math.sin(2*math.pi*f*t);     # Instantaneous value at time 2 ms after zero, volt

#Results
print "The amplitude, rms and average value of emf = %5.2f V, %5.2f V and %5.2f V resp."%(E_m, E, E_av);
print "The frequency and time period of generated waveform = %2d Hz and %2d ms resp."%(f, T/1e-03);
print "The instantaneous value of emf at time 2 ms after crossing zero = %4.1f V"%e

The amplitude, rms and average value of emf = 94.25 V, 66.63 V and 60.04 V resp.
The frequency and time period of generated waveform = 50 Hz and 20 ms resp.
The instantaneous value of emf at time 2 ms after crossing zero = 55.4 V

Example 6.8, Page 212

In [21]:
import math

#Variable declaration
R_c = 50;   # Resistance of the coil of meter, ohm
K = 10e+03;     # Figure of merit of the moving coil meter, ohm per volt
V = 10;     # d.c. range of coil meter, volt

#Calculations&Results
# Part (a)
I_fsd = 1/K;    # Full scale deflection for moving coil meter, ampere
R = V/I_fsd;    # Total meter resistance, ohm
# As R = R_m + R_c, solvign for R_m
R_m = R - R_c;      # Multiplier resistance required by the meter, ohm
print "The multiplier resistance required for 10 V d.c. range = %5.2f k-ohm"%(R_m/1e+03)

# Part(b)
I_av = I_fsd;   # Average value of ac current, A
I_rms = math.pi/(2*math.sqrt(2))*I_av;    # rms value of ac current, A
V = 10 ;     # a.c. range of coil meter, volt
R = V/I_rms;    # Total meter resistance, ohm
# As R = R_m + R_c, solvign for R_m
R_m = R - R_c;      # Multiplier resistance required by the meter, ohm
print "The multiplier resistance required for 10 V a.c. range = %5.2f k-ohm"%(R_m/1e+03);

The multiplier resistance required for 10 V d.c. range = 99.95 k-ohm
The multiplier resistance required for 10 V a.c. range = 89.98 k-ohm

Example 6.9, Page 213

In [9]:
# Case_I: Square_wave
ff = 1.11;  #  Form factor of calibrated meter
ff_square = 1;  # Form factor for square wave
V_apparent = 5;     # Meter reading for sqaure wave, volt
V_true = V_apparent*1*(ff_square/ff);    # True rms value of square wave voltage, volt
print "The true rms value of square wave voltage = %5.3f V"%V_true

# Case_II: Triangular_wave
ff_triangle = 1.15;  # Form factor for triangular wave
V_apparent = 5;     # Meter reading for triangular wave, volt
V_true = V_apparent*(ff_triangle/ff);    # True rms value of triangular wave voltage, volt
print "The true rms value of triangular wave voltage = %4.2f V"%V_true

The true rms value of square wave voltage = 4.505 V
The true rms value of triangular wave voltage = 5.18 V

Example 6.10, Page 215

In [22]:
import math

#Variable declaration
# The general expression for alternating current is I = Io*sin(2*pi*f*t + phi)
#Comparing the given equations with the above, we get,
f = (80*math.pi)/(2*math.pi)    # Frequency of alternating current, Hz

#Calculations
# I2 is the reference waveform with zero phase angle, so that
phi2 = 0;   # Phase angle for reference waveform I2, degrees
Im2 = 3;    # Current amplitude of reference waveform I2, A
Im1 = 5;    # Current amplitude of reference waveform I1, A
Im3 = 6;    # Current amplitude of reference waveform I3, A
phi1 = math.pi/6*(180/math.pi); # Phase angle for reference waveform I1, degrees
phi3 = math.pi/4*(180/math.pi); # Phase angle for reference waveform I3, degrees

#Results
print "The frequency of all three waveforms = %2d Hz"%f
print "I1 leads I2 by = %2.0f degrees"%(phi1-phi2);
print "I3 lags I2 by = %2d degrees"%(phi3-phi2);
print "Current amplitude of reference waveform I1 = %1d A"%Im1
print "Current amplitude of reference waveform I2 = %1d A"%Im2
print "Current amplitude of reference waveform I3 = %1d A"%Im3

The frequency of all three waveforms = 40 Hz
I1 leads I2 by = 30 degrees
I3 lags I2 by = 45 degrees
Current amplitude of reference waveform I1 = 5 A
Current amplitude of reference waveform I2 = 3 A
Current amplitude of reference waveform I3 = 6 A

Example 6.12, Page 218

In [23]:
import math

#Variable declaration
Im1 = 7;    # Current amplitude of reference waveform I1, A
Im2 = 6;    # Current amplitude of reference waveform I2, A
Im3 = 5;    # Current amplitude of reference waveform I3, A
Im4 = 4;    # Current amplitude of reference waveform I4, A

#Calculations
phi1 = 70*math.pi/180;   # Phase angle for reference waveform I1, rad
phi2 = 0*math.pi/180;   # Phase angle for reference waveform I2, rad
phi3 = -50*math.pi/180;   # Phase angle for reference waveform I3, rad
phi4 = -90*math.pi/180;   # Phase angle for reference waveform I4, rad

#Results
print "i1 = %dsin(wt + %4.2f) amp"%(Im1, phi1)
print "i2 = %dsin wt amp"%Im2;
print "i3 = %dsin(wt + %4.2f) amp"%(Im3, phi3);
print "i4 = %dsin(wt + %4.2f) amp"%(Im4, phi4);

i1 = 7sin(wt + 1.22) amp
i2 = 6sin wt amp
i3 = 5sin(wt + -0.87) amp
i4 = 4sin(wt + -1.57) amp

Example 6.13, Page 221

In [24]:
import math

#Variable declaration
omega = 314.;    # Angular frequency of voltage, rad per sec
Vm1 = 25.;   # Peak value of first phasor, V
Vm2 = 15.;   # Peak value of second phasor, V

#Calculations
H_C = Vm1*math.cos(math.pi/3)+Vm2*math.cos(-math.pi/6); # Horizontal component of phasor sum, V
V_C = Vm1*math.sin(math.pi/3)+Vm2*math.sin(-math.pi/6); # Vertical component of phasor sum, V
Vm = math.sqrt(H_C**2+V_C**2); # Peak value of phasor sum, V
phi = math.atan(V_C/H_C);   # Phase angle, degrees
print "v = %5.2fsin(%3dt + %5.3f) volt"%(Vm, omega, phi);

v = 29.15sin(314t + 0.507) volt

Example 6.14, Page 222

In [25]:
import math

#Variable declaration
Im1 = 6;   # Peak value of first phasor, A
Im2 = 8;   # Peak value of second phasor, A
Im3 = 4;   # Peak value of third phasor, A

#Calculations
H_C = Im1*math.cos(0)+Im2*math.cos(-math.pi/2)+Im3*math.cos(math.pi/6); # Horizontal component of phasor sum, A
V_C = Im1*math.sin(0)+Im2*math.sin(-math.pi/2)+Im3*math.sin(math.pi/6); # Vertical component of phasor sum, A
Im = math.sqrt(H_C**2+V_C**2); # Peak value of phasor sum, V
phi = math.atan(V_C/H_C);   # Phase angle, rad


print "i = %4.1fsin(wt%5.3f) amp"%(Im, phi);

i = 11.2sin(wt-0.565) amp

Example 6.15, Page 222

In [26]:
import math

# Part (a)
omega = 628;    # Angular frequency of voltage, rad per sec
f = omega/(2*math.pi);  # Frequency of the waveforms, Hz
Vm1 = 10.;   # Peak value of first phasor, V
Vm2 = 8.;   # Peak value of second phasor, V
Vm3 = 12.;   # Peak value of third phasor, V
phi1 = -math.pi/6*180/math.pi;      # Phase angle for first voltage, degrees
phi2 = math.pi/3*180/math.pi;      # Phase angle for second voltage, degrees
phi3 = math.pi/4*180/math.pi;      # Phase angle for third voltage, degrees
print "The frequency of all three waveforms = %3d Hz"%f
print "The phase angle and frequency of first voltage : %2d degrees, %2d V"%(phi1, Vm1);
print "The phase angle and frequency of second voltage : %2d degrees, %2d V"%(phi2, Vm2);
print "The phase angle and frequency of third voltage : %2d degrees, %2d V"%(phi3, Vm3);

# Part (b)
H_C = Vm1*math.cos(phi1)+Vm2*math.cos(phi2)+Vm3*math.cos(phi3); # Horizontal component of phasor sum, V
V_C = Vm1*math.sin(phi1)+Vm2*math.sin(phi2)+Vm3*math.sin(phi3); # Horizontal component of phasor sum, V
Vm = math.sqrt(H_C**2+V_C**2); # Peak value of phasor sum, V
phi = math.atan(V_C/H_C);   # Phase angle, rad
print "v = %5.2fsin(%3dt + %5.3f) volt"%(Vm, omega, phi);

The frequency of all three waveforms =  99 Hz
The phase angle and frequency of first voltage : -29 degrees, 10 V
The phase angle and frequency of second voltage : 59 degrees,  8 V
The phase angle and frequency of third voltage : 45 degrees, 12 V
v = 17.65sin(628t + 1.558) volt

Example 6.16, Page 228

In [27]:
import math

#Variable declaration
tb1 = 0.1e-03;  # Timebase of channel 1, s/cm
tb2 = 10e-06;  # Timebase of channel 2, s/cm
Y_amp1 = 5.; # Y-amp setting for channel 1, V/cm
Y_amp2 = 0.5; # Y-amp setting for channel 2, V/cm

#Calculations&Results
# Channel 1
V_pp = 3*Y_amp1;    # Peak-to-peak value of waveform in channel 1, V
Vm = V_pp/2;    # Amplitude of waveform in channel 1, V
V = Vm/math.sqrt(2); # rms value of sine wave in channel 1, V
T = 4*tb1;  # Time period of sine wave, second
f = 1./(T*1000);    # Frequency of sine wave, kHz
print "The amplitude of sine waveform in channel 1 = %3.1f V"%Vm
print "The rms value of sine wave in channel 1 = %3.1f V"%V
print "The frequency of sine wave in channel 1 = %3.1f kHz"%f

# Channel 2
V_pp = 2*Y_amp2;    # Peak-to-peak value of waveform in channel 2, V
Vm = V_pp/2;    # Amplitude of waveform in channel 2, V
V = Vm; # rms value of square wave in channel 2, V
T = 2./3*tb2;  # Time period of square wave, second
f = 1./(T*1000);    # Frequency of square wave, kHz
print "The amplitude of square waveform in channel 2 = %3.1f V"%Vm
print "The rms value of square wave in channel 2 = %3.1f V"%V
print "The frequency of square wave in channel 2 = %3d kHz"%f

The amplitude of sine waveform in channel 1 = 7.5 V
The rms value of sine wave in channel 1 = 5.3 V
The frequency of sine wave in channel 1 = 2.5 kHz
The amplitude of square waveform in channel 2 = 0.5 V
The rms value of square wave in channel 2 = 0.5 V
The frequency of square wave in channel 2 = 150 kHz