3-phase VSI (sine-triangle PWM)
A three-phase voltage source inverter is supplied from a $400\,$V DC source and is operated using sinusoidal pulse width modulation technique as shown in the figure below. The modulating signals for inverter legs a, b, c are $m_a=m\,\sin(100\,\pi \,t)$, $m_b=m\,\sin(100\,\pi \,t-2\,\pi/3)$, $m_c=m\,\sin(100\,\pi \,t-4\,\pi/3)$, respectively. The switching frequency of the inverter is $2\,$kHz. The inverter is connected to a balanced star-connected series $RL$ load with $R=20\,\Omega$, $L=5\,$mH. The modulation index $m=0.8$. What is the amplitude of the 50-Hz component in the output phase voltage $v_{an}$ in steady state?In [1]:
from IPython.display import Image
Image(filename =r'pwm_4_fig_1.png', width=600)
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In [2]:
# run this cell to view the circuit file.
%pycat pwm_4_orig.in
We now replace the strings such as \$Vdc, \$L, with the values of our choice by running the python script given below. It takes an existing circuit file pwm_4_orig.in and produces a new circuit file pwm_4.in, after replacing \$Vdc, \$L, etc. with values of our choice.
In [3]:
import gseim_calc as calc
s_Vdc = "400"
s_L = "5e-3"
s_R = "20"
s_f_carrier = "2e3"
s_M = "0.8"
s_dt_min = "0.1e-6"
s_dt_nrml = "5e-6"
s_f_sin = "50.0"
f_sin = float(s_f_sin)
T = 1/f_sin
s_2T = ("%11.4E"%(2.0*T)).strip()
l = [
('$Vdc', s_Vdc),
('$L', s_L),
('$R', s_R),
('$f_carrier', s_f_carrier),
('$f_sin', s_f_sin),
('$2T', s_2T),
('$M', s_M),
('$dt_min', s_dt_min),
('$dt_nrml', s_dt_nrml)
]
calc.replace_strings_1("pwm_4_orig.in", "pwm_4.in", l)
print('pwm_4.in is ready for execution')
pwm_4.in is ready for execution
Execute the following cell to run GSEIM on pwm_4.in.
In [4]:
import os
import dos_unix
# uncomment for windows:
#dos_unix.d2u("pwm_4.in")
os.system('run_gseim pwm_4.in')
get_lib_elements: filename gseim_aux/xbe.aux get_lib_elements: filename gseim_aux/ebe.aux Circuit: filename = pwm_4.in Circuit: n_xbeu_vr = 10 Circuit: n_ebeu_nd = 9 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 solve_trns_exc completed. GSEIM: Program completed.
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0
The circuit file (pwm_4.in) is created in the same directory as that used for launching Jupyter notebook. The last step (i.e., running GSEIM on pwm_4.in) creates data files called pwm_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("pwm_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")
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_IS = slv.get_index(i_slv,i_out,"IS")
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_IS = calc.avg_rms_2(t3, u3[:,col_IS ], 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_IS[1][0])
color1='green'
color2='crimson'
color3='cornflowerblue'
color4='goldenrod'
color5='blue'
fig, ax = plt.subplots(3, sharex=False)
plt.subplots_adjust(wspace=0, hspace=0.0)
set_size(5.5, 7, ax[0])
for i in range(3):
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=1.0, label="$V_{an}$")
ax[2].plot(t2*1e3, u2[:,col_v_ab], color=color2, linewidth=1.0, label="$V_{ab}$")
ax[2].set_xlabel('time (msec)', fontsize=12)
for i in range(3):
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: pwm_4_1.dat filename: pwm_4_2.dat filename: pwm_4_3.dat average power delivered to load: 1.9528E+03 average DC supply current: 4.8836E+00
In [6]:
import numpy as np
import matplotlib.pyplot as plt
import gseim_calc as calc
from setsize import set_size
slv = calc.slv("pwm_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")
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_IS = slv.get_index(i_slv,i_out,"IS")
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_IS ], color=color5, linewidth=1.0, 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: pwm_4_1.dat filename: pwm_4_2.dat filename: pwm_4_3.dat
In [7]:
import numpy as np
import matplotlib.pyplot as plt
import gseim_calc as calc
from setsize import set_size
slv = calc.slv("pwm_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")
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_IS = slv.get_index(i_slv,i_out,"IS")
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_IS, thd_IS = calc.fourier_coeff_1C(t3, u3[:,col_IS],
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_an fundamental: peak value: ", "%11.4E"%(coeff_v_an[1]))
x = np.linspace(0, n_fourier, n_fourier+1)
y_v_an = np.array(coeff_v_an)
y_IS = np.array(coeff_IS )
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)
bars1 = ax[1].bar(x, y_IS , width=0.3, color='green', label="$i_{dc}$", zorder=3)
bars2 = ax[2].bar(x, y_IRa , width=0.3, color='blue', label="$i_{Ra}$", zorder=3)
plt.tight_layout()
plt.show()
filename: pwm_4_1.dat filename: pwm_4_2.dat filename: pwm_4_3.dat THD (v_an): 91.51 % THD (IRa): 15.35 % I_Ra fundamental: RMS value: 5.6389E+00 V_an fundamental: RMS value: 1.1314E+02 V_an fundamental: peak value: 1.6000E+02
This notebook was contributed by Prof. Nakul Narayanan K, Govt. Engineering College, Thrissur. He may be contacted at nakul@gectcr.ac.in.
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