In [1]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
V = 240; # in Volts
f = 50; # in Hz
x = 0.2;
phi3 = 3*math.pi/4; # in Rad
#calculation:
Vamp = V*2**0.5
w = 2*math.pi*f
T = 1/f
V3 = Vamp*x
f3 = 3*f
w3 = 3*w
#Results
print "\n\n Result \n\n"
print "\n voltage, V =",round(Vamp,1),"sin(",round(w,1),"t) + ",round(V3,1),"sin(", round(w3,1),"t - ", round(phi3,1),") volts"
```

In [2]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
A1 = 0.100;# in amperes
A3 = 0.020;# in amperes
A5 = 0.010;# in amperes
#calculation:
#the rms value of current is given by
Irms = ((A1**2 + A3**2 + A5**2)/2)**0.5
#Results
print "\n\n Result \n\n"
print "\n the rms value of current is ",round(Irms*1000,2)," mA"
```

In [3]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
A1 = 10;# in volts
A3 = 3;# in volts
A5 = 2;# in volts
#calculation:
#the rms value of voltage is given by
Vrms = ((A1**2 + A3**2 + A5**2)/2)**0.5
#the mean value of voltage is given by
#x = wt
Vav = (1/math.pi)*((10 + 1 + 2/5)-(-10 - 1 - 2/5))
#form factor is given by
ff = Vrms/Vav
#Results
print "\n\n Result \n\n"
print "\n (a)the rms value of voltage is ",round(Vrms,2)," V"
print "\n (b)the mean value of voltage is ",round(Vav,2)," V"
print "\n (c)form factor is ",round(ff,3)," "
```

In [1]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
V = 240;# in volts
x = 0.3;# for third harmonic
y = 0.1;# for fifth harmonic
f = 31.83;# in Hz
#calculation:
#V3 = x*V1
#V5 = y*V1
#the rms value of the fundamental,
V1 = ((V**2)/(1 + x**2 + y**2))**0.5
#Rms value of the third harmonic
V3 = x*V1
#the rms value of the fifth harmonic,
V5 = y*V1
#Maximum value of the fundamental,
V1m = V1*2**0.5
#Maximum value of the third harmonic,
V3m = V3*2**0.5
#Maximum value of the fifth harmonic,
V5m = V5*2**0.5
w = 2*math.pi*f
#Results
print "\n\n Result \n\n"
print "v = ",round(V1m,2),"sin",round(w,2),"t + ",round(V3m,2),"sin",round((3*w),2),"t + ",round(V5m,2),"sin",round((5*w),2),"t Volts"
```

In [5]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
A1 = 12;# in amperes
A3 = 5;# in amperes
A5 = 2;# in amperes
R = 20;# in ohms
#calculation:
#rms current
Irms = ((A1**2 + A3**2 + A5**2)/2)**0.5
#average power
P = R*Irms**2
#Results
print "\n\n Result \n\n"
print "\n average power ",P," W"
```

In [6]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
Ia1 = 2;# in amperes
Ia3 = 0.3;# in amperes
Ia5 = 0.1;# in amperes
Va1 = 60;# in volts
Va3 = 15;# in volts
Va5 = 10;# in volts
Phii1 = -1*math.pi/6;# in radians
Phii3 = -1*math.pi/12;# in radians
Phii5 = -8*math.pi/9;# in radians
Phiv1 = 0;# in radians
Phiv3 = math.pi/4;# in radians
Phiv5 = -1*math.pi/2;# in radians
#calculation:
#rms values;
I1 = Ia1/(2**0.5);# in amperes
I3 = Ia3/(2**0.5);# in amperes
I5 = Ia5/(2**0.5);# in amperes
V1 = Va1/(2**0.5);# in volts
V3 = Va3/(2**0.5);# in volts
V5 = Va5/(2**0.5);# in volts
#total power supplied,
P = V1*I1*math.cos(Phiv1 - Phii1) + V3*I3*math.cos(Phiv3 - Phii3) + V5*I5*math.cos(Phiv5 - Phii5)
#rms current
Irms = ((I1**2 + I3**2 + I5**2))**0.5
#rms voltage
Vrms = ((V1**2 + V3**2 + V5**2))**0.5
#overall power factor
pf = P/(Vrms*Irms)
#Results
print "\n\n Result \n\n"
print "\n(a)the total active power supplied to the circuit ",round(P,2)," W"
print "\n(b)overall power factor ",round(pf,2)
```

In [2]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
R1 = 40;# in ohm
L = 7.96E-3;# in Henry
C = 25E-6; # in Farad
f = 1000; # in Hx
#calculation:
wL = 2*math.pi*1000*L
wC = 2*math.pi*1000*C
#Results
print "\n\n Result \n\n"
print "(a)i = ",round(100/R1,2),"sin(wt) +",round(30/R1,2),"sin(3wt - pi/3) +",round(10/R1,2),"sin(5wt - pi/6) A"
print "(b)i = ",round(100/wL,2),"sin(wt - pi/2) +",round(30/(3*wL),2),"sin(3wt - pi/6) +",round(10/(5*wL),2),"sin(5wt - 2pi/3) A"
print "(c)i = ",round(100*wC,2),"sin(wt + pi/2) +",round(30*3*wC,2),"sin(3wt + 5pi/6) +",round(10*5*wC,2),"sin(5wt + pi/3) A"
```

In [7]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
V1m = 240;# in volts
V3m = 40;# in volts
V5m = 30;# in volts
w1 = 314;# fundamental
R = 12;# in ohm
L = 0.00955;# in Henry
#calculation:
#fundamental or first harmonic
#inductive reactance,
XL1 = w1*L
#impedance at the fundamental frequency,
Z1 = R + 1j*XL1
#Maximum current at fundamental frequency
I1m = V1m/Z1
I1mag = abs(I1m)
phii1 = cmath.phase(complex(I1m.real,I1m.imag))
#Third harmonic
XL3 = 3*XL1
#impedance at the third harmonic frequency,
Z3 = R + 1j*XL3
#Maximum current at third harmonic frequency
I3m = V3m/Z3
I3mag = abs(I3m)
phii3 = cmath.phase(complex(I3m.real,I3m.imag))
#fifth harmonic
XL5 = 5*XL1
#impedance at the third harmonic frequency,
Z5 = R + 1j*XL5
#Maximum current at third harmonic frequency
I5m = V5m/Z5
I5mag = abs(I5m)
phii5 = cmath.phase(complex(I5m.real,I5m.imag))
#rms voltage
Vrms = ((V1m**2 + V3m**2 + V5m**2)/2)**0.5
#rms current
Irms = ((I1mag**2 + I3mag**2 + I5mag**2)/2)**0.5
#power dissipated
P = R*Irms**2
#overall power factor
pf = P/(Vrms*Irms)
#Results
print "\n\n Result \n\n"
print "\n(a)i = ",round(I1mag,2),"sin(",round(w1,2),"t + (",round(phii1,2),")) + ",round(I3mag,2),"sin(",round((w1*3),2),"t + (",round(phii3,2),")) + ",round(I5mag,2),"sin(",round((w1*5),2),"t + (",round(phii5,2),")) A"
print "\n(b)the rms value of current is ",round(Irms,2)," A"
print "\n(c)the rms value of voltage is ",round(Vrms,2)," V"
print "\n(d)the total power dissipated ",round(P,2)," W"
print "\n(e)overall power factor ",round(pf,2)
```

In [3]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
Vom = 50;# in volts
V1m = 200;# in volts
V2m = 40;# in volts
V4m = 5;# in volts
f = 50;# in Hz
R = 50;# in ohm
C = 100E-6;# in farad
phiv1 = 0;# in rad
phiv2 = -1*math.pi/2;# in rad
phiv4 = math.pi/4;# in rad
#calculation:
#voltage
V1 = V1m*math.cos(phiv1) + 1j*V1m*math.sin(phiv1)
V2 = V2m*math.cos(phiv2) + 1j*V2m*math.sin(phiv2)
V4 = V4m*math.cos(phiv4) + 1j*V4m*math.sin(phiv4)
#Inductance has no effect on a steady current. Hence the d.c. component of the current, i0, is given by
Iom = 0
#fundamental or first harmonic
w1 = 2*math.pi*f
#inductive reactance,
Xc1 = 1/(w1*C)
#impedance at the fundamental frequency,
Z1 = R + 1j*Xc1
#Maximum current at fundamental frequency
I1m = V1/Z1
I1mag = abs(I1m)
phii1 = cmath.phase(complex(I1m.real,I1m.imag))
#second harmonic
Xc2 = Xc1/2
#impedance at the third harmonic frequency,
Z2 = R + 1j*Xc2
#Maximum current at third harmonic frequency
I2m = V2/Z2
I2mag = abs(I2m)
phii2 = cmath.phase(complex(I2m.real,I2m.imag))
#fourth harmonic
Xc4 = Xc1/4
#impedance at the third harmonic frequency,
Z4 = R + 1j*Xc4
#Maximum current at third harmonic frequency
I4m = V4/Z4
I4mag = abs(I4m)
phii4 = cmath.phase(complex(I4m.real,I4m.imag))
#rms current
Irms = (Iom**2 + (I1mag**2 + I2mag**2 + I4mag**2)/2)**0.5
#Results
print "\n\n Result \n\n"
print "(a)i = ",round(Iom,2)," + ",round(I1mag,2),"sin(",round(w1,2),"t + (",round(phii1,2),")) + ",round(I2mag,2),"sin(",round((w1*2),2),"t + (",round(phii2,2),")) + ",round(I4mag,2),"sin(",round((w1*4),2),"t + (",round(phii4,2),")) A"
print "(b)the rms value of current is ",round(Irms,2)," A"
```

In [4]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
Vom = 25;# in volts
V1m = 100;# in volts
V3m = 40;# in volts
V5m = 20;# in volts
w1 = 10000;# fundamental
R = 5;# in ohm
L = 500E-6;# in Henry
phiv1 = 0;# in rad
phiv3 = math.pi/6;# in rad
phiv5 = math.pi/12;# in rad
#calculation:
#voltage
V1 = V1m*math.cos(phiv1) + 1j*V1m*math.sin(phiv1)
V3 = V3m*math.cos(phiv3) + 1j*V3m*math.sin(phiv3)
V5 = V5m*math.cos(phiv5) + 1j*V5m*math.sin(phiv5)
#Inductance has no effect on a steady current. Hence the d.c. component of the current, i0, is given by
Iom = Vom/R
#fundamental or first harmonic
#inductive reactance,
XL1 = w1*L
#impedance at the fundamental frequency,
Z1 = R + 1j*XL1
#Maximum current at fundamental frequency
I1m = V1/Z1
I1mag = abs(I1m)
phii1 = cmath.phase(complex(I1m.real,I1m.imag))
#Third harmonic
XL3 = 3*XL1
#impedance at the third harmonic frequency,
Z3 = R + 1j*XL3
#Maximum current at third harmonic frequency
I3m = V3/Z3
I3mag = abs(I3m)
phii3 = cmath.phase(complex(I3m.real,I3m.imag))
#fifth harmonic
XL5 = 5*XL1
#impedance at the third harmonic frequency,
Z5 = R + 1j*XL5
#Maximum current at third harmonic frequency
I5m = V5/Z5
I5mag = abs(I5m)
phii5 = cmath.phase(complex(I5m.real,I5m.imag))
#rms current
Irms = (Iom**2 + (I1mag**2 + I3mag**2 + I5mag**2)/2)**0.5
#power dissipated
P = R*Irms**2
#Results
print "\n\n Result \n\n"
print "\n(a)i = ",round(Iom,2)," + ",round(I1mag,2),"sin(",round(w1,2),"t + (",round(phii1,2),")) + ",round(I3mag,2),"sin(",round((w1*3),2),"t + (",round(phii3,2),")) + ",round(I5mag,2),"sin(",round((w1*5),2),"t + (",round(phii5,2),")) A"
print "\n(b)the rms value of current is ",round(Irms,2)," A"
print "\n(c)the total power dissipated ",round(P,3)," W"
```

In [7]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
Vom = 30;# in volts
V1m = 40;# in volts
V2m = 25;# in volts
V4m = 15;# in volts
Iom = 0;# in amperes
I1m = 0.743;# in Amperes
I2m = 0.781;# in Amperes
I4m = 0.636;# in Amperes
phii1 = 1.190;# in rad
phii2 = 0.896;# in rad
phii4 = 0.559;# in rad
w = 1000;# in rad
#calculation:
#the average power P is given by
P = Vom*Iom+(0.707*V1m)*(0.707*I1m)*math.cos(phii1)+(0.707*V2m)*(0.707*I2m)*math.cos(phii2) + (0.707*V4m)*(0.707*I4m)*math.cos(phii4)
#rms current
Irms = (Iom**2 + (I1m**2 + I2m**2 + I4m**2)/2)**0.5
#resistance R
R = P/(Irms**2)
#impedance
Z1 = V1m/I1m
#Xc1
Xc1 = (Z1**2 - R**2)**0.5
#capacitance
C = 1/(w*Xc1)
#Results
print "\n\n Result \n\n"
print "\n(a)the average power P is ",round(P,2)," W"
print "\n(c)the resistance R ",round(R,2)," ohm and capacitance ",round(C*1E6,2),"uF"
```

In [2]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
V1m = 300;# in volts
V3m = 120;# in volts
phiv1 = 0;# in rad
phiv2 = 0.698;# in rad
w1 = 314;# in rad
C = 2.123E-6;# in farads
R1 = 560;# in ohms
R2 = 2000;# in Ohm
#calculation:
#voltage
V1 = V1m*math.cos(phiv1) + 1j*V1m*math.sin(phiv1)
V3 = V3m*math.cos(phiv2) + 1j*V3m*math.sin(phiv2)
#capacitive reactance,
Xc1 = 1/(w1*C)
#impedance at the fundamental frequency,
Z1 = R1 - 1j*Xc1*R2/(R2 - 1j*Xc1)
#Maximum current at fundamental frequency
I1m = V1/Z1
I1mag = abs(I1m)
phii1 = cmath.phase(complex(I1m.real,I1m.imag))
#Third harmonic
Xc3 = Xc1/3
#impedance at the third harmonic frequency,
Z3 = R1 - 1j*Xc3*R2/(R2 - 1j*Xc3)
I1m = V1m/Z1
I1mag = abs(I1m)
phii1 = cmath.phase(complex(I1m.real,I1m.imag))
#Maximum current at third harmonic frequency
I3m = V3/Z3
I3mag = abs(I3m)
phii3 = cmath.phase(complex(I3m.real,I3m.imag))
#Percentage harmonic content of the supply current is given by
percent = I3mag*100/I1mag
#total active power
P = (0.707*V1m)*(0.707*I1mag)*math.cos(phiv1 - phii1) + (0.707*V3m)*(0.707*I3m)*math.cos(phiv2 - phii3)
I1 = I1m*R2/(R2 - 1j*Xc1)
I3 = I3m*R2/(R2 - 1j*Xc3)
I1nmag = abs(I1)
phini1 = cmath.phase(complex(I1.real,I1.imag))
I3nmag = abs(I3)
phini3 = cmath.phase(complex(I3.real,I3.imag))
#Results
print "\n\n Result \n\n"
print "\n(a)supply current, i=", round(I1mag,3),"sin(", w1,"t +",round(phii1,3),") + ",round(I3mag,3),"sin(", 3*w1,"t +",round(phii3,3),") A"
print "\n(b)Percentage harmonic content of the supply current is ",round(percent,2)," percent"
print "\n(c)total active power is ",round(abs(P),2)," W"
print "\n(d)Voltage, v1 =", round(I1mag*R1,3),"sin(", w1,"t +",round(phii1,3),") + ",round(I3mag*R1,3),"sin(", 3*w1,"t +",round(phii3,3),") A"
print "\n(e)current, ic =", round(I1nmag,3),"sin(", w1,"t +",round(phini1,3),") + ",round(I3nmag,3),"sin(", 3*w1,"t +",round(phini3,3),") A"
```

In [8]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
V1m = 400;# in volts
V3m = 10;# in volts
C = 0.2E-6;# in farads
R = 2;# in ohms
L = 0.5;# in Henry
#calculation:
#Resonance with the third harmonic means that
w = (1/(9*L*C))**0.5
#fundamental frequency, f
f = w/(2*math.pi)
#At the fundamental frequency,
#impedance Z1
Z1 = R + 1j*(w*L - 1/(w*C))
Z1mag = abs(Z1)
phiZ1 = cmath.phase(complex(Z1.real,Z1.imag))
#Maximum value of current at the fundamental frequency,
I1m = V1m/Z1mag
#At the third harmonic frequency,
Z3 = R + 1j*(3*w*L - 1/(3*w*C))
Z3mag = abs(Z3)
phiZ3 = cmath.phase(complex(Z3.real,Z3.imag))
#Maximum value of current at the third harmonic frequency,
I3m = V3m/Z3
#Results
print "\n\n Result \n\n"
print "(a)fundamental frequency for resonance with the third harmonic is ",round(f,2)," Hz"
print "(b)Maximum value of current at fundamental freq. is",round(abs(I1m),3),"A "
print "and at the third harmonic frequency ", abs(I3m)," A"
```

In [14]:

```
from __future__ import division
import math
import cmath
#initializing the variables:
V1m = 800;# in volts
f = 50;# in Hz
x = 0.015;
C = 0.122E-6;# in farads
R = 5;# in ohms
L = 0.369;# in Henry
#calculation:
#voltage at nth harmonic
Vnm = x*V1m
w = 2*math.pi*f
#For resonance at the nth harmonic nwL = 1/nwC
n = 1/(w*(L*C)**0.5)
#At resonance, impedance
Zn = R
#the maximum value of current at the nth harmonic
Inm = Vnm/Zn
#capacitive reactance, at nth harmonic
Xcn = 1/(n*w*C)
#the p.d. across the capacitor at the nth harmonic
Vcn = Inm*Xcn
#At the fundamental frequency, inductive reactance,
XL1 = w*L
#capacitive reactance
Xc1 = 1/(w*C)
#Impedance at the fundamental frequency,
Z1 = R + 1j*(XL1 - Xc1)
Z1mag = abs(Z1)
phiZ1 = cmath.phase(complex(Z1.real,Z1.imag))
#Maximum value of current at the fundamental frequency,
I1m = V1m/Z1mag
#Results
print "\n\n Result \n\n"
print "\n(a)n = ",round(n,2),""
print "\n(b)the maximum value of current at the nth harmonic ",round(Inm,2)," A"
print "\n(c)the p.d. across the capacitor at the nth harmonic is ",round(Vcn,2),""
print "\n(d)the maximum value of the fundamental current. ",round(I1m,2)," A"
```