We have already seen in the Introduction that the total instanteneous power delivered by a balanced three-phase system is constant in time $P(t)= \frac{3VI}{2} \cos\psi$, where $\psi$ is the phase difference between current and voltage. This fact, together with the smaller number of wires necessary to transport the same amount of power from generator to consumer, are the the main advantages of using a three-phase system vs two-phase and one-phase systems for power delivery.

The complex power formalism introduced in the AC power section, can also be used to compute the average real and reactive powers generated or consumed by components in polyphase circuits. In three-phase circuits we are usually interested in the total power generated by the sources and in the total power consumed by the load. Since in this circuits we have three sources and 3 loads and assuming they are balanced, we need to compute the complex power in only one phase and multiply by 3 to obtain the total power.

In AC systems and particularly in three-phase systems, it is often more practical to express the currents and voltage in terms of their rms values, instead of their magnitude. All the linear equations that we were writting before for the complex currents and voltages can also be written using their rms values. The only difference appears when we write equations for the complex power, because these equations are nonlinear. For instance, the complex power generated by the $a$-phase voltage source in a three-phase wye-wye network is $$\begin{equation}S = \frac{1}{2} V_{an}I_{a}^* \end{equation}$$ where the complex voltage and current are expressed in terms of their magnitude (like we have done throughout this Webbook). However, if we want to express the complex voltage and current are expressed in terms of their rms values, the complex power geneated by a single source is $$\begin{equation}S = V_{an}^{rms}I_a^{rms*} \end{equation}$$ where $V_{an}^{rms}=\frac{V_{an}}{\sqrt{2}}$ and $I_a^{rms*}=\frac{I_{a}}{\sqrt{2}}$ are the complex rms values of current and voltage of the source. The total complex power generated by the three sources is $$\begin{equation}S = \frac{3}{2} V_{an}I_{a^*} \end{equation}$$ or, in terms of the complex rms values $$\begin{equation}S = 3 V_{an}^{rms}I_a^{rms*} \end{equation}$$ In the case of load impedances, the total complex power consumed is $$\begin{equation}S = \frac{3}{2} \frac{|V|^2}{Z_L} = \frac{3}{2} |I|^2 Z_L \end{equation}$$ or, in terms of the complex rms values of the current and voltage $$\begin{equation}S = 3 \frac{|V^{rms}|^2}{Z_L} = |I^{rms}|^2 Z_L \end{equation}$$

For an unbalanced system, the combined power factor of source of load is computed by first calculating the total complex power of the source of load ($S$) which is equal to the sum of the complex powers of the three sources or three loads. Then, the combined power factor is $$\begin{equation}pf = \frac{Im(S)}{Re(S)} \end{equation}$$ For a balanced system, the power factors computed for the different lines are equal to each other and $pf = \frac{Im(S)}{Re(S)} = \cos(\phi_V-\phi_I)$.

The following examples use the concept of total complex power generated when the voltages are given in terms of their magnitude or rms values.

• Examples related to power and power factor
  Three-phase (Y-Y) system with given source voltages, line and load impedances (4 questions)
  Three-phase (D-D) system with given source voltages, line and load impedances (6 questions)