3-phase rectifier

The three-phase diode rectifier given below is connected to a three-phase ac voltage source through inductor $L=5\,$mH. The inductor Ld is large enough so that the current through it is ripple free.
  1. What is the average value of the DC voltage ($V_{pm}$) and the DC current ($i_{Ld}$)?
  2. What is the power delivered to the voltage source?
  3. What is the duration for which a diode conducts in one fundamental cycle of the input voltage?
  4. Which are the dominant harmonics in the ac line current?
In [1]:
from IPython.display import Image
Image(filename =r'rectifier_3ph_9_fig_1.png', width=450)
Out[1]:
No description has been provided for this image
In [2]:
# run this cell to view the circuit file.
%pycat rectifier_3ph_9_orig.in

We now replace the strings such as \$Ls with the values of our choice by running the python script given below. It takes an existing circuit file rectifier_3ph_9_orig.in and produces a new circuit file rectifier_3ph_9.in, after replacing \$Ls (etc) with values of our choice.

In [3]:
import gseim_calc as calc
import numpy as np

s_Ls = "5e-3"
s_Vdc = "400"

VL = 400.0
A_sin = VL*np.sqrt(2/3)

s_A_sin = ("%11.4E"%(A_sin)).strip()

l = [
  ('$Ls', s_Ls),
  ('$Vdc', s_Vdc),
  ('$A_sin', s_A_sin),
]
calc.replace_strings_1("rectifier_3ph_9_orig.in", "rectifier_3ph_9.in", l)
print('rectifier_3ph_9.in is ready for execution')

rectifier_3ph_9.in is ready for execution
Execute the following cell to run GSEIM on rectifier_3ph_9.in.
In [4]:
import os
import dos_unix
# uncomment for windows:
#dos_unix.d2u("rectifier_3ph_9.in")
os.system('run_gseim rectifier_3ph_9.in')

get_lib_elements: filename gseim_aux/xbe.aux
get_lib_elements: filename gseim_aux/ebe.aux
Circuit: filename = rectifier_3ph_9.in
Circuit: n_xbeu_vr = 0
Circuit: n_ebeu_nd = 11
main: i_solve = 0
ssw_allocate_1 (2): n_statevar=4
main: calling solve_trns
mat_ssw_1_e: n_statevar: 4
mat_ssw_1_e0: cct.n_ebeu: 16
Transient simulation starts...
i=0
i=1000
solve_ssw_e: ssw_iter_newton=0, rhs_ssw_norm=4.6680e+00
Transient simulation starts...
i=0
i=1000
solve_ssw_e: ssw_iter_newton=1, rhs_ssw_norm=4.8733e+01
Transient simulation starts...
i=0
i=1000
solve_ssw_e: ssw_iter_newton=2, rhs_ssw_norm=5.2612e-04
Transient simulation starts...
i=0
i=1000
solve_ssw_e: ssw_iter_newton=3, rhs_ssw_norm=3.1229e-12
solve_ssw_e: calling solve_ssw_1_e for one more trns step
Transient simulation starts...
i=0
i=1000
solve_ssw_1_e over (after trns step for output)
solve_ssw_e ends, slv.ssw_iter_newton=3
GSEIM: Program completed.
Out[4]:
0

The circuit file (rectifier_3ph_9.in) is created in the same directory as that used for launching Jupyter notebook. The last step (i.e., running GSEIM on rectifier_3ph_9.in) creates data files called rectifier_3ph_9_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

f_hz = 50.0
T = 1.0/f_hz

slv = calc.slv("rectifier_3ph_9.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_v_o = slv.get_index(i_slv,i_out,"v_o")
col_ID1 = slv.get_index(i_slv,i_out,"ID1")
col_ID2 = slv.get_index(i_slv,i_out,"ID2")
col_ID3 = slv.get_index(i_slv,i_out,"ID3")
col_ID4 = slv.get_index(i_slv,i_out,"ID4")
col_ID5 = slv.get_index(i_slv,i_out,"ID5")
col_ID6 = slv.get_index(i_slv,i_out,"ID6")
col_ISa = slv.get_index(i_slv,i_out,"ISa")
col_ISb = slv.get_index(i_slv,i_out,"ISb")
col_ISc = slv.get_index(i_slv,i_out,"ISc")
col_ILd = slv.get_index(i_slv,i_out,"ILd")

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

col_P_Vdc = slv.get_index(i_slv,i_out,"P_Vdc")
col_P_Vsa = slv.get_index(i_slv,i_out,"P_Vsa")
col_P_Vsb = slv.get_index(i_slv,i_out,"P_Vsb")
col_P_Vsc = slv.get_index(i_slv,i_out,"P_Vsc")

l_v_o   = calc.avg_rms_2(t1, u1[:,col_v_o  ], T, 2.0*T, 1.0e-5*T)
l_P_Vdc = calc.avg_rms_2(t2, u2[:,col_P_Vdc], T, 2.0*T, 1.0e-5*T)
l_P_Vsa = calc.avg_rms_2(t2, u2[:,col_P_Vsa], T, 2.0*T, 1.0e-5*T)
l_ID1   = calc.avg_rms_2(t1, u1[:,col_ID1  ], T, 2.0*T, 1.0e-5*T)
l_ILd   = calc.avg_rms_2(t1, u1[:,col_ILd  ], T, 2.0*T, 1.0e-5*T)
l_ISa   = calc.avg_rms_2(t1, u1[:,col_ISa  ], T, 2.0*T, 1.0e-5*T)

print('average value of v_pm:'         , "%11.4E"%( l_v_o  [1][0]))
print('rms value of v_pm:'             , "%11.4E"%( l_v_o  [2][0]))
print('rms value of ID1:'              , "%11.4E"%( l_ID1  [2][0]))
print('rms value of ISa:'              , "%11.4E"%( l_ISa  [2][0]))
print('average value of ILd:'          , "%11.4E"%( l_ILd  [1][0]))
print('average power delivered to Vdc:', "%11.4E"%(-l_P_Vdc[1][0]))
print('average power delivered by Vsa:', "%11.4E"%( l_P_Vsa[1][0]))

# time instants when the diode current crosses 1.0e-6:
l_cross_1_ID1, l_cross_2_ID1 = calc.cross_over_points_1(t1, u1[:,col_ID1], 0.0, T, 1.0e-6)
print('zero-crossing points of D1 current (positive slope):')
for tx in l_cross_1_ID1:
    print("  ", "%11.4E"%tx)
print('zero-crossing points of D1 current (negative slope):')
for tx in l_cross_2_ID1:
    print("  ", "%11.4E"%tx)
theta = (l_cross_2_ID1[0]-l_cross_1_ID1[0])*360.0/T
print('angle of conduction for D1:', "%11.4E"%theta)

color1='red'
color2='goldenrod'
color3='blue'
color4='green'
color5='crimson'
color6='cornflowerblue'

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

set_size(5.5, 6, ax[0])

for i in range(3):
    ax[i].set_xlim(left=0.0, right=2.0*T*1e3)
    ax[i].grid(color='#CCCCCC', linestyle='solid', linewidth=0.5)

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

ax[0].plot(t1*1e3, u1[:,col_v_o], color=color4, linewidth=1.0, label="$v_o$")

ax[1].plot(t1*1e3, u1[:,col_ID1], color=color1, linewidth=1.0, label="$I_{D1}$")
ax[1].plot(t1*1e3, u1[:,col_ID3], color=color2, linewidth=1.0, label="$I_{D3}$")
ax[1].plot(t1*1e3, u1[:,col_ID5], color=color3, linewidth=1.0, label="$I_{D5}$")

ax[2].plot(t1*1e3, u1[:,col_ISa], color=color1, linewidth=1.0, label="$I_{sa}$")
ax[2].plot(t1*1e3, u1[:,col_ISb], color=color2, linewidth=1.0, label="$I_{sb}$")
ax[2].plot(t1*1e3, u1[:,col_ISc], color=color3, linewidth=1.0, label="$I_{sc}$")

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: rectifier_3ph_9_1.dat
filename: rectifier_3ph_9_2.dat
average value of v_pm:  4.0009E+02
rms value of v_pm:  4.0716E+02
rms value of ID1:  5.0050E+01
rms value of ISa:  7.0781E+01
average value of ILd:  9.3357E+01
average power delivered to Vdc:  3.7343E+04
average power delivered by Vsa:  1.2483E+04
zero-crossing points of D1 current (positive slope):
    1.7400E-03
zero-crossing points of D1 current (negative slope):
    1.1740E-02
angle of conduction for D1:  1.8000E+02
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In [6]:
import numpy as np
import matplotlib.pyplot as plt 
import gseim_calc as calc
from setsize import set_size

f_hz = 50.0
T = 1.0/f_hz

slv = calc.slv("rectifier_3ph_9.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_v_o = slv.get_index(i_slv,i_out,"v_o")
col_ID1 = slv.get_index(i_slv,i_out,"ID1")
col_ID2 = slv.get_index(i_slv,i_out,"ID2")
col_ID3 = slv.get_index(i_slv,i_out,"ID3")
col_ID4 = slv.get_index(i_slv,i_out,"ID4")
col_ID5 = slv.get_index(i_slv,i_out,"ID5")
col_ID6 = slv.get_index(i_slv,i_out,"ID6")
col_ISa = slv.get_index(i_slv,i_out,"ISa")
col_ISb = slv.get_index(i_slv,i_out,"ISb")
col_ISc = slv.get_index(i_slv,i_out,"ISc")
col_ILd = slv.get_index(i_slv,i_out,"ILd")

# compute Fourier coeffs:

t_start = T 
t_end = 2.0*T

n_fourier = 20

coeff_ISa, thd_ISa = calc.fourier_coeff_1C(t1, u1[:,col_ISa], 
    t_start, t_end, 1.0e-8, n_fourier)

coeff_v_o, thd_v_o = calc.fourier_coeff_1C(t1, u1[:,col_v_o], 
    t_start, t_end, 1.0e-8, n_fourier)

print('THD in ac line current:', "%5.2f"%(thd_ISa*100.0), "%")
print("ac line current fundamental: RMS value: ", "%11.4E"%(coeff_ISa[1]/np.sqrt(2.0)))

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

y_v_o = np.array(coeff_v_o)
y_ISa = np.array(coeff_ISa)

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

set_size(6, 4, 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)), 1.0)

for i in range(2):
    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('$V_{out}$',fontsize=14)
ax[1].set_ylabel('$i_{sa}$',fontsize=14)

bars1 = ax[0].bar(x, y_v_o, width=0.3, color='red',   label="$v_o$", zorder=3)
bars2 = ax[1].bar(x, y_ISa  , width=0.3, color='green', label="$i_{sa}$" , zorder=3)

plt.tight_layout()
plt.show()

filename: rectifier_3ph_9_1.dat
THD in ac line current:  9.37 %
ac line current fundamental: RMS value:   7.0472E+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.