PWM

In the circuit given below, the switches are operated using the pulse width modulation technique. The modulation voltage is $m(t)$, and the triangular carrier voltage is $v_c(t)$. The switch S1 is turned on when $m(t) > v_c(t)$, and the switch S2 is turned on otherwise. The parameter values are $R=12\,\Omega$, $L=0.1\,$H, $f_c=1\,$kHz, and $V_{dc}=200\,$V. If the modulation voltage is $m(t)=2\,\sin (100\,\pi\,t)$, what is the RMS value of the $50\,$Hz component of the load current $i_L$?
In [1]:
from IPython.display import Image
Image(filename =r'pwm_6_fig_1.png', width=500)
Out[1]:
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In [2]:
# run this cell to view the circuit file.
%pycat pwm_6_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_6_orig.in and produces a new circuit file pwm_6.in, after replacing \$Vdc, \$L, etc. with values of our choice.

In [3]:
import gseim_calc as calc
s_Vdc = "200"
s_L = "0.1"
s_R = "12"
s_f_carrier = "1e3"
s_M = "2"
s_dt_min = "1e-6"
s_dt_nrml = "10e-6"

s_f_sin = "50"
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_6_orig.in", "pwm_6.in", l)
print('pwm_6.in is ready for execution')

pwm_6.in is ready for execution
Execute the following cell to run GSEIM on pwm_6.in.
In [4]:
import os
import dos_unix
# uncomment for windows:
#dos_unix.d2u("pwm_6.in")
os.system('run_gseim pwm_6.in')

get_lib_elements: filename gseim_aux/xbe.aux
get_lib_elements: filename gseim_aux/ebe.aux
Circuit: filename = pwm_6.in
Circuit: n_xbeu_vr = 4
Circuit: n_ebeu_nd = 5
main: i_solve = 0
main: calling solve_trns
Transient simulation starts...
i=0
i=1000
i=2000
i=3000
i=4000
solve_trns_exc completed.
GSEIM: Program completed.
Out[4]:
0

The circuit file (pwm_6.in) is created in the same directory as that used for launching Jupyter notebook. The last step (i.e., running GSEIM on pwm_6.in) creates a data file called pwm_6.datin 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_6.in")

i_slv = 0
i_out = 0
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
u = np.loadtxt(filename)
t = u[:, 0]

col_v_out = slv.get_index(i_slv,i_out,"v_out")
col_IR    = slv.get_index(i_slv,i_out,"IR"   )
col_ISrc1 = slv.get_index(i_slv,i_out,"ISrc1")
col_ISrc2 = slv.get_index(i_slv,i_out,"ISrc2")
col_P_R   = slv.get_index(i_slv,i_out,"P_R"  )
col_g1    = slv.get_index(i_slv,i_out,"g1"   )
col_g2    = slv.get_index(i_slv,i_out,"g2"   )
col_s     = slv.get_index(i_slv,i_out,"s"    )
col_t     = slv.get_index(i_slv,i_out,"t"    )

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

T = t[-1]/2

l_IR    = calc.avg_rms_2(t, u[:,col_IR],    T, 2.0*T, 1.0e-4*T)
l_v_out = calc.avg_rms_2(t, u[:,col_v_out], T, 2.0*T, 1.0e-4*T)
l_P_R   = calc.avg_rms_2(t, u[:,col_P_R],   T, 2.0*T, 1.0e-4*T)
l_ISrc1 = calc.avg_rms_2(t, u[:,col_ISrc1], T, 2.0*T, 1.0e-4*T)

print('rms load voltage:', "%11.4E"%l_v_out[2][0])
print('average source current:', "%11.4E"%l_ISrc1[1][0])
print('average power delivered to load:', "%11.4E"%l_P_R[1][0])

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

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

set_size(7, 7, ax[0])

for i in range(4):
    ax[i].set_xlim(left=0, right=T*1e3)

ax[0].grid(color='#CCCCCC', linestyle='solid', linewidth=0.5)
ax[1].grid(color='#CCCCCC', linestyle='solid', linewidth=0.5)
ax[2].grid(color='#CCCCCC', linestyle='solid', linewidth=0.5)
ax[3].grid(color='#CCCCCC', linestyle='solid', linewidth=0.5)

ax[0].plot((t-T)*1e3, u[:,col_t    ], color=color5, linewidth=1.0, label="$t$")
ax[0].plot((t-T)*1e3, u[:,col_s    ], color=color4, linewidth=1.0, label="$s$")
ax[1].plot((t-T)*1e3, u[:,col_v_out], color=color1, linewidth=1.0, label="$V_{out}$")
ax[2].plot((t-T)*1e3, u[:,col_ISrc1], color=color3, linewidth=1.0, label="$i_1$")
ax[3].plot((t-T)*1e3, u[:,col_IR   ], color=color2, linewidth=1.0, label="$i_L$")

ax[1].set_ylabel(r'$V_{out}$', fontsize=14)
ax[2].set_ylabel(r'$i_1$',     fontsize=14)
ax[3].set_ylabel(r'$i_L$',     fontsize=14)

ax[3].set_xlabel('time (msec)', fontsize=14)

ax[0].legend(loc = 'lower right',frameon = True, fontsize = 10, title = None,
   markerfirst = True, markerscale = 1.0, labelspacing = 0.5, columnspacing = 2.0,
   prop = {'size' : 12},)

ax[0].tick_params(labelbottom=False)
ax[1].tick_params(labelbottom=False)
ax[2].tick_params(labelbottom=False)

#plt.tight_layout()
plt.show()

filename: pwm_6.dat
rms load voltage:  2.0000E+02
average source current:  1.9189E-01
average power delivered to load:  5.3964E+01
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In [6]:
import numpy as np
import matplotlib.pyplot as plt 
from matplotlib.ticker import (MultipleLocator, AutoMinorLocator)
import gseim_calc as calc
from setsize import set_size

slv = calc.slv("pwm_6.in")

i_slv = 0
i_out = 0
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
u = np.loadtxt(filename)
t = u[:, 0]

col_v_out = slv.get_index(i_slv,i_out,"v_out")
col_IR    = slv.get_index(i_slv,i_out,"IR"   )
col_ISrc1 = slv.get_index(i_slv,i_out,"ISrc1")
col_ISrc2 = slv.get_index(i_slv,i_out,"ISrc2")
col_P_R   = slv.get_index(i_slv,i_out,"P_R"  )
col_g1    = slv.get_index(i_slv,i_out,"g1"   )
col_g2    = slv.get_index(i_slv,i_out,"g2"   )
col_s     = slv.get_index(i_slv,i_out,"s"    )
col_t     = slv.get_index(i_slv,i_out,"t"    )

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

T = t[-1]/2

# compute Fourier coeffs:

t_start = T 
t_end = 2.0*T

n_fourier = 45

coeff_IR, thd_IR = calc.fourier_coeff_1C(t, u[:,col_IR], 
    t_start, t_end, 1.0e-8, n_fourier)

coeff_v_out, thd_v_out = calc.fourier_coeff_1C(t, u[:,col_v_out], 
    t_start, t_end, 1.0e-8, n_fourier)

coeff_ISrc1, thd_ISrc1 = calc.fourier_coeff_1C(t, u[:,col_ISrc1], 
    t_start, t_end, 1.0e-8, n_fourier)

print("THD (load current): ", "%5.2f"%(thd_IR*100.0), "%")
print("load current fundamental: RMS value: ", "%11.4E"%(coeff_IR[1]/np.sqrt(2.0)))
print("THD (load voltage): ", "%5.2f"%(thd_v_out*100.0), "%")
print("load voltage fundamental: RMS value: ", "%11.4E"%(coeff_v_out[1]/np.sqrt(2.0)))
print("DC source current fundamental: RMS value: ", "%11.4E"%(coeff_ISrc1[1]/np.sqrt(2.0)))
print("DC source current 1st harmonic: RMS value: ", "%11.4E"%(coeff_ISrc1[2]/np.sqrt(2.0)))

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

y_IR    = np.array(coeff_IR)
y_v_out = np.array(coeff_v_out)
y_ISrc1 = np.array(coeff_ISrc1)

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

set_size(7, 6, ax[0])

delta = 5.0
x_major_ticks = np.arange(0.0, (float(n_fourier+1)), delta)
x_minor_ticks = np.arange(0.0, (float(n_fourier+1)), delta/5)

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[i].grid(visible=True, which='minor', axis='x', color=grid_color, linestyle='-', zorder=0)

ax[0].set_ylabel('$i_{load}$',fontsize=14)
ax[1].set_ylabel('$v_{out}$', fontsize=14)
ax[2].set_ylabel('$i_1$',     fontsize=14)

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

bars1 = ax[0].bar(x, y_IR,    width=0.3, color='red',   label="$i_{load}$", zorder=3)
bars2 = ax[1].bar(x, y_v_out, width=0.3, color='blue',  label="$V_{out}$",  zorder=3)
bars3 = ax[2].bar(x, y_ISrc1, width=0.3, color='green', label="$i_1$",      zorder=3)

plt.tight_layout()
plt.show()

filename: pwm_6.dat
THD (load current):  12.37 %
load current fundamental: RMS value:   2.1024E+00
THD (load voltage):  264.56 %
load voltage fundamental: RMS value:   7.0715E+01
DC source current fundamental: RMS value:   1.0546E+00
DC source current 1st harmonic: RMS value:   2.5060E-01
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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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