11.3. Three-phase circuits#

11.3.1. Star-star network#

11.3.1.1. General solution#

Tension \(v_{AB}\) between the centers of the stars \(A\), \(B\)

\[v_{AB} = \dfrac{\sum_{g=1}^{3} Y_g e_g}{\sum_{i=1}^{4} Y_i} \ .\]
Proof.

PSCE is used on the linear network, leaving only one tension generator on at a time, and then combining the results.

Tension generator \(e_1\) on, \(e_2 = e_3 = 0\) off. Leaving \(e_1\) on, and switching off \(e_2 = e_3 = 0\), tension generator sees an equivalent impedance

\[\begin{split}\begin{aligned} Z_{eq,1} & = Z_1 + (Z_2 \parallel Z_3 \parallel Z_4) \\ & = \dfrac{1}{Y_1} + \dfrac{1}{Y_2 + Y_3 + Y_4} = \dfrac{Y_{1234}}{Y_1 Y_{234}} \ , \end{aligned}\end{split}\]

so that:

  • the current through the generator reads

    \[i_{1,1} = \dfrac{e_1}{Z_{eq,1}} = \frac{Y_1 Y_{234}}{Y_{1234}} e_1\]

  • the currents through the other sides (acting as current dividers are):

    \[\begin{split}\begin{aligned} i_{2,1} & = - \frac{Y_2}{Y_{234}} \, i_{1,1} = - \frac{Y_1 Y_2}{Y_{1234}} e_1 \\ i_{3,1} & = - \frac{Y_3}{Y_{234}} \, i_{1,1} = - \frac{Y_1 Y_3}{Y_{1234}} e_1 \\ i_{4,1} & = \ \ \ \frac{Y_4}{Y_{234}} \, i_{1,1} = \ \ \ \frac{Y_1 Y_4}{Y_{1234}} e_1 \\ \end{aligned}\end{split}\]

  • tension \(v_{AB}\)

    \[v_{AB,1} = e_1 - Z_1 i_{1,1} = \left( 1 - \frac{Y_{234}}{Y_{1234}} \right) e_1 = \frac{Y_1 e_1}{\sum_{k=1}^4 Y_k} \ . \]

PSCE. Exploiting the PSCE and the symmetry of the system, the expressions of currents in the phases, in the neutral and the center-center voltage seamlessly follow

\[\begin{split}\begin{aligned} i_1 & = \frac{Y_1 Y_{234}}{Y_{1234}} e_1 - \frac{Y_1 Y_2}{Y_{1234}} e_2 - \frac{Y_1 Y_3}{Y_{1234}} e_3 = \\ & = Y_1 e_1 - \frac{Y_1}{Y_{1234}} \sum_{g=1}^{3} Y_g \, e_g \\ i_2 & = Y_2 e_2 - \frac{Y_2}{Y_{1234}} \sum_{g=1}^{3} Y_g \, e_g \\ i_3 & = Y_3 e_3 - \frac{Y_3}{Y_{1234}} \sum_{g=1}^{3} Y_g \, e_g \\ i_4 & = \frac{Y_4}{Y_{1234}} \sum_{g=1}^{3} Y_g \, e_g \\ v_{AB} & = \frac{\sum_{g=1}^{3} Y_g \, e_g}{\sum_{k=1}^4 Y_k} \\ \end{aligned}\end{split}\]

11.3.1.2. Equilibrated generation and loads#

11.3.1.3. Extra connections#

11.3.1.3.1. Phase-neutral connections#

Connections of a phase with the neutral result in parallel impedence with the generators and/or the loads

../_images/star-star-parallel-connections.png

11.3.1.3.2. Phase-phase connections#

Phase-phase connections don’t influence the voltage \(v_{AB}\) between the centers \(A\), \(B\).

todo Write the proof.

../_images/star-star-extra-connections.png