.. include:: /project-links.txt
.. include:: /abbreviation.txt



.. getthecode:: three-phase.py
    :language: python3
    :hidden:


=================================================
 Three-phased Current: Y and Delta configurations
=================================================

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


.. code-block:: py3

    
    
    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.

.. code-block:: py3

    
    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 :math:`\frac{2\pi}{3}`:

.. math::
 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)

We rewrite them in complex notation:

.. math::
 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) }

.. code-block:: py3

    
    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 :

.. math::
  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} }

In comparison to the Y configuration, the voltages in delta configuration are magnified by
a factor :math:`\sqrt{3}` and dephased of :math:`\frac{\pi}{6}`.

Finally we rewrite them in temporal notation:

.. math::
 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)

.. code-block:: py3

    
    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:

.. code-block:: py3

    figure = plt.figure(1, (20, 10))
    plt.plot(t, L1, t, L2, t, L3,
             t, L12, t, L23, t, L31,
             # t, L1-L2, t, L2-L3, t, L3-L1,
    )
    plt.grid()
    plt.title('Three-phase electric power: Y and Delta configurations (230V Mono/400V Tri 50Hz Europe)')
    plt.legend(('L1-N', 'L2-N', 'L3-N',
                'L1-L2', 'L2-L3', 'L3-L1'),
               loc=(.7,.5))
    plt.xlabel('t [s]')
    plt.ylabel('[V]')
    plt.axhline(y=rms_mono, color='blue')
    plt.axhline(y=-rms_mono, color='blue')
    plt.axhline(y=rms_tri, color='blue')
    plt.axhline(y=-rms_tri, color='blue')
    plt.show()
    


.. image:: three-phase.png
    :align: center