Space Vector Representation

The space vector representation of three-phase voltages is given by

${\bf{f}}(t)=\displaystyle\frac{2}{3}\left(v_a(t)+v_b(t)\,e^{j2\pi/3}+ v_c(t)\,e^{j4\pi/3}\right)$.

In a three-phase voltage source,

$v_a(t) = V_m \sin(100\pi t)$,

$v_b(t) = V_m \sin(100\pi t - 120^\circ)$,

$v_c(t) = V_m \sin(100\pi t - 240^\circ)$,

where $V_m=325\,$V. Identify the shape of the space vector. Also, determine the magnitude of the space vector.

In [1]:
import numpy as np
import matplotlib.pyplot as plt 
import gseim_calc as calc
import cmath
from setsize import set_size
from matplotlib.ticker import (MultipleLocator, AutoMinorLocator)
from matplotlib import animation
from matplotlib import ticker
from IPython.display import HTML

N1 = 1
N2 = 0

Vm1 = 325.0
Vm2 = 0.0

T = 20.0e-3
omg = 2.0*np.pi/T

t_total = 2.0*(T/N1)

theta1 = 2.0*np.pi/3

n_div = 100
t = np.linspace(0.0, t_total, (n_div+1))

Va = Vm1*np.sin(N1*omg*t) + \
     Vm2*np.sin(N2*omg*t)
Vb = Vm1*np.sin(N1*(omg*t - theta1)) + \
     Vm2*np.sin(N2*(omg*t - theta1))
Vc = Vm1*np.sin(N1*(omg*t + theta1)) + \
     Vm2*np.sin(N2*(omg*t + theta1))

color_a = "blue"
color_b = "red"
color_c = "green"
color_v = "black"

t_ms = t*1e3

x1a = []
x1b = []
x1c = []

y1a = []
y1b = []
y1c = []

a1 = 5.0*(Vm1+Vm2)

x1a.append(-a1)
x1a.append( a1)

y1a.append(0.0)
y1a.append(0.0)

x1b.append(a1*np.cos(theta1-np.pi))
x1b.append(a1*np.cos(theta1))

y1b.append(a1*np.sin(theta1-np.pi))
y1b.append(a1*np.sin(theta1))

x1c.append(a1*np.cos(-theta1+np.pi))
x1c.append(a1*np.cos(-theta1))

y1c.append(a1*np.sin(-theta1+np.pi))
y1c.append(a1*np.sin(-theta1))

vec_a = []
vec_b = []
vec_c = []
vec2 = []

k1 = 2/3

for i in range(len(Va)):
    vec_a1 = k1*Va[i]*(complex(1.0,0.0))
    vec_b1 = k1*Vb[i]*(cmath.exp(complex(0.0, theta1)))
    vec_c1 = k1*Vc[i]*(cmath.exp(complex(0.0,-theta1)))

    vec1 = vec_a1 + vec_b1 + vec_c1

    vec_a.append(vec_a1)
    vec_b.append(vec_b1)
    vec_c.append(vec_c1)

    vec2.append(vec1)

# axis limits for vector plot:

vec2_real = [x_dummy.real for x_dummy in vec2]
vec2_imag = [x_dummy.imag for x_dummy in vec2]
V         = [abs(x_dummy) for x_dummy in vec2]

vec2_real_min = min(vec2_real)
vec2_real_max = max(vec2_real)
vec2_real_delta = vec2_real_max - vec2_real_min

vec2_imag_min = min(vec2_imag)
vec2_imag_max = max(vec2_imag)
vec2_imag_delta = vec2_imag_max - vec2_imag_min

xmin2 = vec2_real_min - 0.02*vec2_real_delta
xmax2 = vec2_real_max + 0.02*vec2_real_delta
ymin2 = vec2_imag_min - 0.02*vec2_imag_delta
ymax2 = vec2_imag_max + 0.02*vec2_imag_delta

V_min = min(V)
V_max = max(V)

print('Vmin:', "%11.4E"%V_min)
print('Vmax:', "%11.4E"%V_max)

# axis limits for V(t) plot:

xmin1 = 0.0
xmax1 = t_total*1e3

Va_min = min(Va)
Va_max = max(Va)
Va_delta = Va_max - Va_min

ymin1 = Va_min - 0.02*Va_delta
ymax1 = Va_max + 0.02*Va_delta
ymax1 = max(ymax1, ymax2)

fig, ax = plt.subplots(nrows=1, ncols=2, sharex=False)
plt.subplots_adjust(wspace=0, hspace=0.0)

set_size(6.0, 3, ax[0])

ax[1].set_aspect('equal', adjustable='box')

ax[0].set(xlim=[xmin1, xmax1], ylim=[ymin1, ymax1])
ax[1].set(xlim=[xmin2, xmax2], ylim=[ymin2, ymax2])

ax[0].set_xlabel(r'time (msec)', fontsize=12)
ax[0].set_ylabel(r'$V(t)$ (V)', fontsize=12)

ax[1].set_xlabel(r'Re(V)', fontsize=12)
ax[1].set_ylabel(r'Im(V)', fontsize=12)

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

# background V(t) plots (faint):

line1_Va = ax[0].plot(t_ms, Va, color=color_a, linestyle = '-', linewidth=1.0, alpha=0.2)[0]
line1_Vb = ax[0].plot(t_ms, Vb, color=color_b, linestyle = '-', linewidth=1.0, alpha=0.2)[0]
line1_Vc = ax[0].plot(t_ms, Vc, color=color_c, linestyle = '-', linewidth=1.0, alpha=0.2)[0]
line1_V  = ax[0].plot(t_ms, V,  color=color_v, linestyle = '-', linewidth=1.0, alpha=0.2)[0]

line3_Va = ax[1].plot(x1a, y1a, color=color_a, linestyle = '-', linewidth=1.0, alpha=0.2)[0]
line3_Vb = ax[1].plot(x1b, y1b, color=color_b, linestyle = '-', linewidth=1.0, alpha=0.2)[0]
line3_Vc = ax[1].plot(x1c, y1c, color=color_c, linestyle = '-', linewidth=1.0, alpha=0.2)[0]

# live V(t) plots (dark):

line2_Va = ax[0].plot([], [], color=color_a, linestyle = '-', linewidth=1.0)[0]
line2_Vb = ax[0].plot([], [], color=color_b, linestyle = '-', linewidth=1.0)[0]
line2_Vc = ax[0].plot([], [], color=color_c, linestyle = '-', linewidth=1.0)[0]
line2_V  = ax[0].plot([], [], color=color_v, linestyle = '-', linewidth=1.0)[0]

scat_Va = ax[0].scatter(0, 0, color=color_a, marker='o', s=15)
scat_Vb = ax[0].scatter(0, 0, color=color_b, marker='o', s=15)
scat_Vc = ax[0].scatter(0, 0, color=color_c, marker='o', s=15)
scat_V  = ax[0].scatter(0, 0, color=color_v, marker='o', s=15)

# locus traced by vector:

line4 = ax[1].plot([], [], color='grey', linestyle = '--', linewidth=0.8, dashes=(3,3))[0]

# vector diagram:

line5 = []

line5_a = []
line5_b = []
line5_c = []

line5_sum1 = []
line5_sum2 = []

for i in range(3):
    line6 = ax[1].plot([], [], color=color_v, linestyle = '-', linewidth=1.0)[0]

    line6_a = ax[1].plot([], [], color=color_a, linestyle = '-', linewidth=1.0)[0]
    line6_b = ax[1].plot([], [], color=color_b, linestyle = '-', linewidth=1.0)[0]
    line6_c = ax[1].plot([], [], color=color_c, linestyle = '-', linewidth=1.0)[0]

    line5.append(line6)

    line5_a.append(line6_a)
    line5_b.append(line6_b)
    line5_c.append(line6_c)

    line6_sum1 = ax[1].plot([], [], color=color_b, linestyle = '-', alpha=0.5, linewidth=0.8)[0]
    line6_sum2 = ax[1].plot([], [], color=color_c, linestyle = '-', alpha=0.5, linewidth=0.8)[0]

    line5_sum1.append(line6_sum1)
    line5_sum2.append(line6_sum2)

# projections of vector:

line7_Va = ax[1].plot([], [], color=color_a, linestyle = '-', linewidth=1.0)[0]
line7_Vb = ax[1].plot([], [], color=color_b, linestyle = '-', linewidth=1.0)[0]
line7_Vc = ax[1].plot([], [], color=color_c, linestyle = '-', linewidth=1.0)[0]

theta_deg = 20.0
length_arrow = 0.08*Vm1

l1 = []

l1_a = []
l1_b = []
l1_c = []

l1_sum1 = []
l1_sum2 = []

l1_labels = []

l1_a_labels = []
l1_b_labels = []
l1_c_labels = []

l1_sum1_colors = []
l1_sum2_colors = []

def update(frame):
    scat_Va.set_offsets([t_ms[frame], Va[frame]])
    scat_Vb.set_offsets([t_ms[frame], Vb[frame]])
    scat_Vc.set_offsets([t_ms[frame], Vc[frame]])
    scat_V .set_offsets([t_ms[frame], V [frame]])

    x0 = t_ms[:frame+1]
    ya = Va[:frame+1]
    yb = Vb[:frame+1]
    yc = Vc[:frame+1]
    y  = V [:frame+1]

    line2_Va.set_xdata(x0)
    line2_Vb.set_xdata(x0)
    line2_Vc.set_xdata(x0)
    line2_V .set_xdata(x0)

    line2_Va.set_ydata(ya)
    line2_Vb.set_ydata(yb)
    line2_Vc.set_ydata(yc)
    line2_V .set_ydata(y )

    l1.clear()
    l1_labels.clear()

    calc.phasor_append_1(l1, l1_labels, vec2[frame], "$V$")

    l2 = calc.phasor_2(l1, theta_deg, length_arrow, 0.4)

    for k, t in enumerate(l2[0]):
        line5[k].set_xdata(t[0])
        line5[k].set_ydata(t[1])

    l1_a.clear()
    l1_a_labels.clear()

    calc.phasor_append_1(l1_a, l1_a_labels, vec_a[frame], "$V_a$")

    l2_a = calc.phasor_2(l1_a, theta_deg, length_arrow, 0.4)

    for k, t in enumerate(l2_a[0]):
        line5_a[k].set_xdata(t[0])
        line5_a[k].set_ydata(t[1])

    l1_b.clear()
    l1_b_labels.clear()

    calc.phasor_append_1(l1_b, l1_b_labels, vec_b[frame], "$V_b$")

    l2_b = calc.phasor_2(l1_b, theta_deg, length_arrow, 0.4)

    for k, t in enumerate(l2_b[0]):
        line5_b[k].set_xdata(t[0])
        line5_b[k].set_ydata(t[1])

    l1_c.clear()
    l1_c_labels.clear()

    calc.phasor_append_1(l1_c, l1_c_labels, vec_c[frame], "$V_c$")

    l2_c = calc.phasor_2(l1_c, theta_deg, length_arrow, 0.4)

    for k, t in enumerate(l2_c[0]):
        line5_c[k].set_xdata(t[0])
        line5_c[k].set_ydata(t[1])

    l1_sum1.clear()
    l1_sum1_colors.clear()

    calc.phasor_append_2(l1_sum1, l1_sum1_colors, vec_a[frame],
      (vec_a[frame] + vec_b[frame]), color_b)

    l1_sum1_arrow = calc.phasor_2(l1_sum1, theta_deg, length_arrow, 0.4)

    for k, t in enumerate(l1_sum1_arrow[0]):
        line5_sum1[k].set_xdata(t[0])
        line5_sum1[k].set_ydata(t[1])

    l1_sum2.clear()
    l1_sum2_colors.clear()

    calc.phasor_append_2(l1_sum2, l1_sum2_colors,
      (vec_a[frame] + vec_b[frame]),
      (vec_a[frame] + vec_b[frame] + vec_c[frame]), color_c)

    l1_sum2_arrow = calc.phasor_2(l1_sum2, theta_deg, length_arrow, 0.4)

    for k, t in enumerate(l1_sum2_arrow[0]):
        line5_sum2[k].set_xdata(t[0])
        line5_sum2[k].set_ydata(t[1])

    # locus

    rx = []
    ry = []

    for i in range(frame+1):
        rx.append(vec2[i].real)
        ry.append(vec2[i].imag)

    line4.set_xdata(rx)
    line4.set_ydata(ry)

    return

anim = animation.FuncAnimation(
    fig=fig,
    func=update,
    frames=(n_div+1),
    interval=200,
    repeat=False)

plt.tight_layout()
plt.close()

HTML(anim.to_jshtml())

Vmin:  3.2500E+02
Vmax:  3.2500E+02
Out[1]:
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.

In [ ]: