8.5.1. Three-phased Current: Y and Delta configurations

This examples shows the computation of the voltage for the Y and Delta configurations.

import math

import numpy as np
import matplotlib.pyplot as plt


from PySpice.Unit import *

Let use an European 230 V / 50 Hz electric network.

frequency = 50@u_Hz
w = frequency.pulsation
period = frequency.period

rms_mono = 230
amplitude_mono = rms_mono * math.sqrt(2)

The phase voltages in Y configuration are dephased of 2π3:

VL1N=Vppcos(ωt)VL2N=Vppcos(ωt2π3)VL3N=Vppcos(ωt4π3)

We rewrite them in complex notation:

VL1N=VppejωtVL2N=Vppej(ωt2π3)VL3N=Vppej(ωt4π3)
t = np.linspace(0, 3*float(period), 1000)
L1 = amplitude_mono * np.cos(t*w)
L2 = amplitude_mono * np.cos(t*w - 2*math.pi/3)
L3 = amplitude_mono * np.cos(t*w - 4*math.pi/3)

From these expressions, we compute the voltage in delta configuration using trigonometric identities :

VL1L2=VL13ejπ6VL2L3=VL23ejπ6VL3L1=VL33ejπ6

In comparison to the Y configuration, the voltages in delta configuration are magnified by a factor 3 and dephased of π6.

Finally we rewrite them in temporal notation:

VL1L2=Vpp3cos(ωt+π6)VL2L3=Vpp3cos(ωtπ2)VL3L1=Vpp3cos(ωt7π6)
rms_tri = math.sqrt(3) * rms_mono
amplitude_tri = rms_tri * math.sqrt(2)

L12 = amplitude_tri * np.cos(t*w + math.pi/6)
L23 = amplitude_tri * np.cos(t*w - math.pi/2)
L31 = amplitude_tri * np.cos(t*w - 7*math.pi/6)

Now we plot the waveforms:

figure, ax = plt.subplots(figsize=(20, 10))
ax.plot(
    t, L1, t, L2, t, L3,
    t, L12, t, L23, t, L31,
    # t, L1-L2, t, L2-L3, t, L3-L1,
)
ax.grid()
ax.set_title('Three-phase electric power: Y and Delta configurations (230V Mono/400V Tri 50Hz Europe)')
ax.legend(
    ('L1-N', 'L2-N', 'L3-N',
     'L1-L2', 'L2-L3', 'L3-L1'),
    loc=(.7,.5),
)
ax.set_xlabel('t [s]')
ax.set_ylabel('[V]')
ax.axhline(y=rms_mono, color='blue')
ax.axhline(y=-rms_mono, color='blue')
ax.axhline(y=rms_tri, color='blue')
ax.axhline(y=-rms_tri, color='blue')

plt.show()
../../_images/three-phase.png