Below we present four typical connections between a three-phase source and a three-phase load encountered in practical applications. The first connection, which is also somewhat easier to study, is the wye-wye connection. Although it is possible to analyze the other connections independently, it is often easier to transform them into a wye-wye connection, compute the currents and voltages, and then transform them back to the original connection to find the sought variables. To simplify the calculations, we will assume that the network is balanced.

The circuit shown in Fig. 1 represents a standard balanced wye-wye connected system. In general, the voltages of the sources are given by $$\begin{equation}V_{an}(t) = V \cos(\omega t)\end{equation}$$ $$\begin{equation}V_{bn}(t) = V \cos(\omega t-120^\circ)\end{equation}$$ $$\begin{equation}V_{cn}(t) = V \cos(\omega t+120^\circ)\end{equation}$$ In a normal problem (not a design problem), the line impedances are the same because they all have the same length (from the power generation facility to the consumer) and go through similar environmental conditions. Since here we focus on balanced systems, the loads are also identical, and we need to compute the current in the lines and the powers consumed or generated by the different components.

You can look at this solved problem for an example on how to analyze wye-wye systems.

The circuit shown in Fig. 2 represents a balanced delta-wye connected system. The voltages of the sources are again given by equations similar to the previous ones, and one usually needs to find the currents in the lines or the currents going through the sources. In this case, it is easier to first transform the sources from the delta to the wye configuration, then compute the line currents in the same way as in the previous case. To compute the currents going through the original sources, we can make one more current transformation from a wye to a delta configuration.

You can look at this solved problem for an example on how to analyze delta-wye systems.

The circuit shown in Fig. 3 represents a balanced wye-delta connected system. The voltages of the sources are given by the same equations as in the wye-wye case and one needs to compute the currents in the lines or the currents going through the load impedances. In this case it is easier to first transform the loads from the delta to the wye configuration, then compute the line currents. The currents going though the original sources can be computed by making one more current transformation like in this solved problem.

Finally, the circuit shown in Fig. 4 represents a balanced delta-delta connected system. To compute the currents in the lines or the currents going through the delta branches (either at the source or at the load), it is again easier to first transform the sources and loads from the delta to wye configurations, then compute the line currents. As in the previous case, the currents going though the original sources or load impedances can be computed by making an additional current transformation like in this solved problem.

• Wye-wye connections
  Three-phase (Y-Y) system with given source voltages, line and load impedances (2 questions)
  Three-phase (Y-Y) system in which total power, line voltage and pf are given
  
• Delta-wye connections
  Three-phase (D-Y) system with given source voltages, line and load impedances (2 questions)
  
• Wye-delta connections
  Three-phase (D-Y) system with given source voltages, line and load impedances (3 questions)
  
• Delta-delta connections
  Three-phase (D-D) system with given source voltages, line and load impedances (4 questions)