3-phase VSI (PWM)

A three-phase voltage source inverter is supplied from a $600\,$V DC source as shown in the figure below and is operated using sinusoidal pulse width modulation technique. The modulating signals for inverter legs a, b, c are, repectively, $m_a(t)=m_1\,\sin(100\,\pi \,t) + m_0(t)$, $m_b(t)=m_1\,\sin(100\,\pi \,t-2\,\pi/3) + m_0(t)$, $m_c(t)=m_1\,\sin(100\,\pi \,t-4\,\pi/3) + m_0(t)$, where $m_0(t)=m_2\,\sin(300\,\pi \,t)$. The switching frequency of the inverter is $2\,$kHz. The inverter is connected to a balanced star-connected series $RL$ load with $R=10\,\Omega$, $L=0.01\,$H.
  1. Determine the RMS values of the fundamental components of phase voltage and load current.
  2. Determine the amplitude of the 150-Hz component in phase voltage ($V_{an}$) and pole voltage ($V_{a0}$).
  3. What is the average current drawn from the DC supply?
  4. Which are the dominant harmonics present in the DC source current?
In [1]:
from IPython.display import Image
Image(filename =r'VSI_3ph_4_fig_1.png', width=600)
Out[1]:
No description has been provided for this image
In [2]:
# run this cell to view the circuit file.
%pycat VSI_3ph_4_orig.in

We now replace the strings such as \$Vdcby2, \$L, with the values of our choice by running the python script given below. It takes an existing circuit file VSI_3ph_4_orig.in and produces a new circuit file VSI_3ph_4.in, after replacing \$Vdcby2, \$L, etc. with values of our choice.

In [3]:
import gseim_calc as calc
Vdc = 600.0
s_Vdcby2 = ("%11.4E"%(Vdc/2)).strip()

s_L = "0.01"
s_R = "10"
s_f_carrier = "2e3"
s_f2 = "150"
s_m1 = "1.12"
s_m2 = "0.186"
s_dt_min = "0.1e-6"
s_dt_nrml = "5e-6"

f1 = 50.0
s_f1 = ("%11.4E"%(f1)).strip()

T = 1/f1
s_2T = ("%11.4E"%(2.0*T)).strip()

l = [
  ('$Vdcby2', s_Vdcby2),
  ('$L', s_L),
  ('$R', s_R),
  ('$f_carrier', s_f_carrier),
  ('$f1', s_f1),
  ('$f2', s_f2),
  ('$m1', s_m1),
  ('$m2', s_m2),
  ('$2T', s_2T),
  ('$dt_min', s_dt_min),
  ('$dt_nrml', s_dt_nrml)
]
calc.replace_strings_1("VSI_3ph_4_orig.in", "VSI_3ph_4.in", l)
print('VSI_3ph_4.in is ready for execution')

VSI_3ph_4.in is ready for execution
Execute the following cell to run GSEIM on VSI_3ph_4.in.
In [4]:
import os
import dos_unix
# uncomment for windows:
#dos_unix.d2u("VSI_3ph_4.in")
os.system('run_gseim VSI_3ph_4.in')

get_lib_elements: filename gseim_aux/xbe.aux
get_lib_elements: filename gseim_aux/ebe.aux
Circuit: filename = VSI_3ph_4.in
main: i_solve = 0
main: calling solve_trns
Transient simulation starts...
i=0
i=1000
i=2000
i=3000
i=4000
i=5000
i=6000
i=7000
i=8000
GSEIM: Program completed.
Out[4]:
0

The circuit file (VSI_3ph_4.in) is created in the same directory as that used for launching Jupyter notebook. The last step (i.e., running GSEIM on VSI_3ph_4.in) creates data files called VSI_3ph_4_1.dat, etc. in the same directory. We can now use the python code below to compute/plot the various quantities of interest.

In [5]:
import numpy as np
import matplotlib.pyplot as plt 
import gseim_calc as calc
from setsize import set_size

slv = calc.slv("VSI_3ph_4.in")

i_slv = 0
i_out = 0
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
u1 = np.loadtxt(filename)
t1 = u1[:, 0]

col_sa = slv.get_index(i_slv,i_out,"sa")
col_sb = slv.get_index(i_slv,i_out,"sb")
col_sc = slv.get_index(i_slv,i_out,"sc")
col_t  = slv.get_index(i_slv,i_out,"t" )
col_g1 = slv.get_index(i_slv,i_out,"g1")
col_g2 = slv.get_index(i_slv,i_out,"g2")
col_g3 = slv.get_index(i_slv,i_out,"g3")
col_g4 = slv.get_index(i_slv,i_out,"g4")
col_g5 = slv.get_index(i_slv,i_out,"g5")
col_g6 = slv.get_index(i_slv,i_out,"g6")

# since we have stored two cycles, we need to divide the last time point
# by 2 to get the period:

T = t1[-1]/2

i_out = 1
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
u2 = np.loadtxt(filename)
t2 = u2[:, 0]

col_v_ab = slv.get_index(i_slv,i_out,"v_ab")
col_v_bc = slv.get_index(i_slv,i_out,"v_bc")
col_v_ca = slv.get_index(i_slv,i_out,"v_ca")
col_v_an = slv.get_index(i_slv,i_out,"v_an")
col_v_bn = slv.get_index(i_slv,i_out,"v_bn")
col_v_cn = slv.get_index(i_slv,i_out,"v_cn")
col_v_a0 = slv.get_index(i_slv,i_out,"v_a0")
col_v_b0 = slv.get_index(i_slv,i_out,"v_b0")
col_v_c0 = slv.get_index(i_slv,i_out,"v_c0")

i_out = 2
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
u3 = np.loadtxt(filename)
t3 = u3[:, 0]

col_IRa  = slv.get_index(i_slv,i_out,"IRa")
col_IRb  = slv.get_index(i_slv,i_out,"IRb")
col_IRc  = slv.get_index(i_slv,i_out,"IRc")
col_IS1  = slv.get_index(i_slv,i_out,"IS1")
col_P_Ra = slv.get_index(i_slv,i_out,"P_Ra")
col_P_Rb = slv.get_index(i_slv,i_out,"P_Rb")
col_P_Rc = slv.get_index(i_slv,i_out,"P_Rc")

l_IS1  = calc.avg_rms_2(t3, u3[:,col_IS1 ], T, 2.0*T, 1.0e-4*T)
l_P_Ra = calc.avg_rms_2(t3, u3[:,col_P_Ra], T, 2.0*T, 1.0e-4*T)

print('average power delivered to load:', "%11.4E"%(3.0*l_P_Ra[1][0]))
print('average DC supply current:', "%11.4E"%l_IS1[1][0])

color1='green'
color2='crimson'
color3='cornflowerblue'
color4='goldenrod'
color5='blue'

fig, ax = plt.subplots(4, sharex=False)
plt.subplots_adjust(wspace=0, hspace=0.0)

set_size(5.5, 8, ax[0])

for i in range(4):
    ax[i].set_xlim(left=T*1e3, right=2.0*T*1e3)
    ax[i].grid(color='#CCCCCC', linestyle='solid', linewidth=0.5)

ax[0].plot(t1*1e3, u1[:,col_t ], color=color4, linewidth=1.0, label="$t$")
ax[0].plot(t1*1e3, u1[:,col_sa], color=color1, linewidth=1.0, label="$sa$")
ax[0].plot(t1*1e3, u1[:,col_sb], color=color2, linewidth=1.0, label="$sb$")
ax[0].plot(t1*1e3, u1[:,col_sc], color=color3, linewidth=1.0, label="$sc$")

ax[1].plot(t2*1e3, u2[:,col_v_an], color=color1, linewidth=0.7, label="$V_{an}$")

ax[2].plot(t2*1e3, u2[:,col_v_ab], color=color2, linewidth=0.7, label="$V_{ab}$")

ax[3].set_ylim(bottom=-400, top=400)
ax[3].plot(t2*1e3, u2[:,col_v_a0], color=color5, linewidth=0.7, label="$V_{a0}$")

ax[3].set_xlabel('time (msec)', fontsize=12)

for i in range(4):
    ax[i].legend(loc = 'lower right',frameon = True, fontsize = 10, title = None,
      markerfirst = True, markerscale = 1.0, labelspacing = 0.5, columnspacing = 2.0,
      prop = {'size' : 12},)

plt.tight_layout()
plt.show()

filename: VSI_3ph_4_1.dat
filename: VSI_3ph_4_2.dat
filename: VSI_3ph_4_3.dat
average power delivered to load:  1.5506E+04
average DC supply current:  2.5823E+01
No description has been provided for this image
In [6]:
import numpy as np
import matplotlib.pyplot as plt 
import gseim_calc as calc
from setsize import set_size

slv = calc.slv("VSI_3ph_4.in")

i_slv = 0
i_out = 0
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
u1 = np.loadtxt(filename)
t1 = u1[:, 0]

col_sa = slv.get_index(i_slv,i_out,"sa")
col_sb = slv.get_index(i_slv,i_out,"sb")
col_sc = slv.get_index(i_slv,i_out,"sc")
col_t  = slv.get_index(i_slv,i_out,"t" )
col_g1 = slv.get_index(i_slv,i_out,"g1")
col_g2 = slv.get_index(i_slv,i_out,"g2")
col_g3 = slv.get_index(i_slv,i_out,"g3")
col_g4 = slv.get_index(i_slv,i_out,"g4")
col_g5 = slv.get_index(i_slv,i_out,"g5")
col_g6 = slv.get_index(i_slv,i_out,"g6")

# since we have stored two cycles, we need to divide the last time point
# by 2 to get the period:

T = t1[-1]/2

i_out = 1
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
u2 = np.loadtxt(filename)
t2 = u2[:, 0]

col_v_ab = slv.get_index(i_slv,i_out,"v_ab")
col_v_bc = slv.get_index(i_slv,i_out,"v_bc")
col_v_ca = slv.get_index(i_slv,i_out,"v_ca")
col_v_an = slv.get_index(i_slv,i_out,"v_an")
col_v_bn = slv.get_index(i_slv,i_out,"v_bn")
col_v_cn = slv.get_index(i_slv,i_out,"v_cn")
col_v_a0 = slv.get_index(i_slv,i_out,"v_a0")
col_v_b0 = slv.get_index(i_slv,i_out,"v_b0")
col_v_c0 = slv.get_index(i_slv,i_out,"v_c0")

i_out = 2
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
u3 = np.loadtxt(filename)
t3 = u3[:, 0]

col_IRa  = slv.get_index(i_slv,i_out,"IRa")
col_IRb  = slv.get_index(i_slv,i_out,"IRb")
col_IRc  = slv.get_index(i_slv,i_out,"IRc")
col_IS1  = slv.get_index(i_slv,i_out,"IS1")
col_P_Ra = slv.get_index(i_slv,i_out,"P_Ra")
col_P_Rb = slv.get_index(i_slv,i_out,"P_Rb")
col_P_Rc = slv.get_index(i_slv,i_out,"P_Rc")

color1='green'
color2='crimson'
color3='cornflowerblue'
color4='goldenrod'
color5='blue'

fig, ax = plt.subplots(2, sharex=False)
plt.subplots_adjust(wspace=0, hspace=0.0)

set_size(5.5, 5, ax[0])

for i in range(2):
    ax[i].set_xlim(left=T*1e3, right=2.0*T*1e3)
    ax[i].grid(color='#CCCCCC', linestyle='solid', linewidth=0.5)

ax[0].plot(t3*1e3, u3[:,col_IRa], color=color1, linewidth=1.0, label="$i_{Ra}$")
ax[0].plot(t3*1e3, u3[:,col_IRb], color=color2, linewidth=1.0, label="$i_{Rb}$")
ax[0].plot(t3*1e3, u3[:,col_IRc], color=color3, linewidth=1.0, label="$i_{Rc}$")

ax[1].plot(t3*1e3, u3[:,col_IS1], color=color5, linewidth=0.7, label="$i_{dc}$")

ax[1].set_xlabel('time (msec)', fontsize=12)

for i in range(2):
    ax[i].legend(loc = 'lower right',frameon = True, fontsize = 10, title = None,
      markerfirst = True, markerscale = 1.0, labelspacing = 0.5, columnspacing = 2.0,
      prop = {'size' : 12},)

plt.tight_layout()
plt.show()

filename: VSI_3ph_4_1.dat
filename: VSI_3ph_4_2.dat
filename: VSI_3ph_4_3.dat
No description has been provided for this image
In [7]:
import numpy as np
import matplotlib.pyplot as plt 
import gseim_calc as calc
from setsize import set_size

slv = calc.slv("VSI_3ph_4.in")

i_slv = 0
i_out = 0
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
u1 = np.loadtxt(filename)
t1 = u1[:, 0]

col_sa = slv.get_index(i_slv,i_out,"sa")
col_sb = slv.get_index(i_slv,i_out,"sb")
col_sc = slv.get_index(i_slv,i_out,"sc")
col_t  = slv.get_index(i_slv,i_out,"t" )
col_g1 = slv.get_index(i_slv,i_out,"g1")
col_g2 = slv.get_index(i_slv,i_out,"g2")
col_g3 = slv.get_index(i_slv,i_out,"g3")
col_g4 = slv.get_index(i_slv,i_out,"g4")
col_g5 = slv.get_index(i_slv,i_out,"g5")
col_g6 = slv.get_index(i_slv,i_out,"g6")

# since we have stored two cycles, we need to divide the last time point
# by 2 to get the period:

T = t1[-1]/2

i_out = 1
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
u2 = np.loadtxt(filename)
t2 = u2[:, 0]

col_v_ab = slv.get_index(i_slv,i_out,"v_ab")
col_v_bc = slv.get_index(i_slv,i_out,"v_bc")
col_v_ca = slv.get_index(i_slv,i_out,"v_ca")
col_v_an = slv.get_index(i_slv,i_out,"v_an")
col_v_bn = slv.get_index(i_slv,i_out,"v_bn")
col_v_cn = slv.get_index(i_slv,i_out,"v_cn")
col_v_a0 = slv.get_index(i_slv,i_out,"v_a0")
col_v_b0 = slv.get_index(i_slv,i_out,"v_b0")
col_v_c0 = slv.get_index(i_slv,i_out,"v_c0")

i_out = 2
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
u3 = np.loadtxt(filename)
t3 = u3[:, 0]

col_IRa  = slv.get_index(i_slv,i_out,"IRa")
col_IRb  = slv.get_index(i_slv,i_out,"IRb")
col_IRc  = slv.get_index(i_slv,i_out,"IRc")
col_IS1  = slv.get_index(i_slv,i_out,"IS1")
col_P_Ra = slv.get_index(i_slv,i_out,"P_Ra")
col_P_Rb = slv.get_index(i_slv,i_out,"P_Rb")
col_P_Rc = slv.get_index(i_slv,i_out,"P_Rc")

# compute Fourier coeffs:

t_start = T 
t_end = 2.0*T

n_fourier = 50

coeff_v_an, thd_v_an = calc.fourier_coeff_1C(t2, u2[:,col_v_an], 
    t_start, t_end, 1.0e-8, n_fourier)

coeff_v_a0, thd_v_a0 = calc.fourier_coeff_1C(t2, u2[:,col_v_a0], 
    t_start, t_end, 1.0e-8, n_fourier)

coeff_IS1, thd_IS1 = calc.fourier_coeff_1C(t3, u3[:,col_IS1], 
    t_start, t_end, 1.0e-8, n_fourier)

coeff_IRa, thd_IRa = calc.fourier_coeff_1C(t3, u3[:,col_IRa], 
    t_start, t_end, 1.0e-8, n_fourier)

#print("THD (v_an): ", "%5.2f"%(thd_v_an*100.0), "%")
#print("THD (IRa): " , "%5.2f"%(thd_IRa*100.0) , "%")

print("I_Ra fundamental: RMS value: ", "%11.4E"%(coeff_IRa[1]/np.sqrt(2.0)))
print("V_an fundamental: RMS value: ", "%11.4E"%(coeff_v_an[1]/np.sqrt(2.0)))
print("V_a0 3rd harmonic: RMS value: ", "%11.4E"%(coeff_v_a0[3]/np.sqrt(2.0)))
print("V_a0 3rd harmonic: amplitude: ", "%11.4E"%(coeff_v_a0[3]))

x = np.linspace(0, n_fourier, n_fourier+1)

y_v_an  = np.array(coeff_v_an)
y_IS1   = np.array(coeff_IS1 )
y_IRa   = np.array(coeff_IRa )

fig, ax = plt.subplots(3, sharex=False)
plt.subplots_adjust(wspace=0, hspace=0.0)
grid_color='#CCCCCC'

set_size(6, 5, ax[0])

delta = 10.0
x_major_ticks = np.arange(0.0, (float(n_fourier+1)), delta)
x_minor_ticks = np.arange(0.0, (float(n_fourier+1)), 1.0)

for i in range(3):
    ax[i].set_xlim(left=-1.0, right=float(n_fourier))
    ax[i].set_xticks(x_major_ticks)
    ax[i].set_xticks(x_minor_ticks, minor=True)
    ax[i].grid(visible=True, which='major', axis='x', color=grid_color, linestyle='-', zorder=0)

ax[0].set_ylabel('$V_{an}$',fontsize=14)
ax[1].set_ylabel('$i_{dc}$',fontsize=14)
ax[2].set_ylabel('$i_{Ra}$',fontsize=14)

ax[2].set_xlabel('N', fontsize=14)

bars1 = ax[0].bar(x, y_v_an, width=0.3, color='red',   label="$V_{an}$", zorder=3)
bars2 = ax[1].bar(x, y_IS1 , width=0.3, color='green', label="$i_{dc}$", zorder=3)
bars3 = ax[2].bar(x, y_IRa , width=0.3, color='blue',  label="$i_{Ra}$", zorder=3)

plt.tight_layout()
plt.show()

filename: VSI_3ph_4_1.dat
filename: VSI_3ph_4_2.dat
filename: VSI_3ph_4_3.dat
I_Ra fundamental: RMS value:   2.2722E+01
V_an fundamental: RMS value:   2.3785E+02
V_a0 3rd harmonic: RMS value:   3.9494E+01
V_a0 3rd harmonic: amplitude:   5.5854E+01
No description has been provided for this image

This notebook was contributed by Prof. Nakul Narayanan K, Govt. Engineering College, Thrissur. He may be contacted at nakul@gectcr.ac.in.