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 $\frac{2\pi}{3}$:
$$\begin{aligned} V_{L1 - N} = V_{pp} \cos \left( \omega t \right) \\ V_{L2 - N} = V_{pp} \cos \left( \omega t - \frac{2\pi}{3} \right) \\ V_{L3 - N} = V_{pp} \cos \left( \omega t - \frac{4\pi}{3} \right) \end{aligned}$$We rewrite them in complex notation:
$$\begin{aligned} V_{L1 - N} = V_{pp} e^{j\omega t} \\ V_{L2 - N} = V_{pp} e^{j \left(\omega t - \frac{2\pi}{3} \right) } \\ V_{L3 - N} = V_{pp} e^{j \left(\omega t - \frac{4\pi}{3} \right) } \end{aligned}$$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 :
$$\begin{aligned} V_{L1 - L2} = V_{L1} \sqrt{3} e^{j \frac{\pi}{6} } \\ V_{L2 - L3} = V_{L2} \sqrt{3} e^{j \frac{\pi}{6} } \\ V_{L3 - L1} = V_{L3} \sqrt{3} e^{j \frac{\pi}{6} } \end{aligned}$$In comparison to the Y configuration, the voltages in delta configuration are magnified by a factor $\sqrt{3}$ and dephased of $\frac{\pi}{6}$.
Finally we rewrite them in temporal notation:
$$\begin{aligned} V_{L1 - L2} = V_{pp} \sqrt{3} \cos \left( \omega t + \frac{\pi}{6} \right) \\ V_{L2 - L3} = V_{pp} \sqrt{3} \cos \left( \omega t - \frac{\pi}{2} \right) \\ V_{L3 - L1} = V_{pp} \sqrt{3} \cos \left( \omega t - \frac{7\pi}{6} \right) \end{aligned}$$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')
<matplotlib.lines.Line2D at 0x7fedfe40b820>