Consider the component shown in Fig. 1 and assume that $$\begin{equation}V(t) = V_M \cos(\omega t+\phi_V)\end{equation}$$ $$\begin{equation}I(t) = I_M \cos(\omega t+\phi_I)\end{equation}$$ where $V_M$, $I_M$, $\phi_V$, and $\phi_I$ are the magnitudes and phases of the voltage and current, respectively.

The instantaneous power dissipated by the component is defined as $$\begin{equation}P(t)=V(t) I(t)=\frac{V_M I_M}{2} [\cos(\phi_V-\phi_I) + \cos(2\omega t + \phi_V +\phi_I)]\end{equation}$$

while the average power dissipated by the component is defined as $$\begin{equation}P=\frac{1}{2}V_M I_M \cos(\phi_V-\phi_I)\end{equation}$$ Notice that in a purely resistive load $P=\frac{V_M I_M}{2}$, while for ideal inductors and capacitors $P=0$.

We can also define the instantaneous power generated by the component as $-P(t)$ and the average power generated by it as $-P$.

The apparent power is defined as $P=\frac{V_M I_M}{2}$, while the power factor (pf) is ration of the average power to the apparent power $$\begin{equation}pf=\cos(\phi_V-\phi_I)\end{equation}$$ The power factor is a measure of the energy efficiency of the an AC equipment. The lower the power factor, the less efficient the power usage is. If an equipment is running with $pf<0.95$ is considered inefficient because of the ohmic losses in transmission lines.

Sometimes, it is convenient to write the powers in terms of the root-mean square (rms) value of the voltage and current, which are defined as $V_{rms}=\frac{V_M}{\sqrt{2}}$ and $I_{rms}=\frac{I_M}{\sqrt{2}}$ (notice that $V_{rms}$ and $I_{rms}$ are real numbers). With these notations, the average power dissipated by the component is $$\begin{equation}P=V_{rms}I_{rms}\cos(\phi_V-\phi_I)\end{equation}$$

The average power, the apparent power and the power factor of a component can be computed relatively easy by using the formalism of complex analysis. To do so, it is instrumental to define the complex power absorbed by the compoenent as $$\begin{equation}S=\frac{1}{2}V I^*\end{equation}$$ where $V$ is the complex voltage, $I$ is the complex current, and $I^*$ denotes the complex conjugate of $I$. If the component is a complex impedance $Z$ the complex power can be computed by introducing Ohm's law in the previous equation, $S=\frac{1}{2} |I|^2 Z=\frac{|V|^2}{2 Z^*}$. Notice that, in the case of resistors, the complex power becomes a real number. The complex power supplied by the component is defined as the negative of the absorbed complex power.

The complex power can be written in a slighlty simpler form as $\begin{equation}S=V_{rms} I_{rms}^*\end{equation}$, where $V_{rms}=\frac{V}{\sqrt(2)}$ and $I_{rms}=\frac{I}{\sqrt(2)}$ are the rms values of the complex voltage and current, respectively. Notice that, using these notations, $V_{rms}$ and $I_{rms}$ are complex numbers and they should not be confused with rms value of defined in the previous section.

If the complex power is known, one can compute

• The average power $$P=Re(S)$$
• The reactive power $$Q=Im(S)$$
• The apparent power $$|S|=Abs(S)$$
• The power factor $$pf=\frac{Re(S)}{|S|}$$

where $|S|$ donotes the absolut value of $S$. The power factor is called lagging if $Im(S)>0$ (i.e. the element is inductive) and leading if $Im(S)<0$ (i.e. the element is capacitive). Table 1 gives a summary of the AC power quantities described in this section.

Table 1. Power-related quantities and their units.

Name	Notation	Units
Complex power	$S$	volt-amperes, $\: \textcolor{gray}{VA}$
Apparent power	$|S|$	volt-amperes, $\: \textcolor{gray}{VA}$
Average (real) power	$P$	watts, $\: \textcolor{gray}{W}$
Reactive power	$Q$	volt-amperes reactive, $\: \textcolor{gray}{VAR}$
Power factor	$pf$

Consider the circuit in Fig. 3. If $V_{Th}$ and $Z_{Th}$ are specified and one can change the value of the load impedance, the maxium power transfer occurs when $$\begin{equation}Z_L=Z_{Th}^*\end{equation}$$ When this condition is satisfied $\cos(\phi_V-\phi_I)=1$ and the average power becomes maximum.

In general, if the $V_{Th} - Z_{Th}$ circuit contains multiple impedances, voltage sources and current sources, it can always be replaced by its Thévenin equivalent circuit and we can still use the above equation to calculate the load impedance for which the average power transferred is maximum.

• The power triangle states that $$\begin{equation}\tan(\phi_V-\phi_I)=\frac{Q}{P}\end{equation}$$ which can be obtained from the definition of real and reactive powers.
• The rms value of a signal can be defined in general for any periodic waveform of the voltage or current as the DC voltage or current that dissipates the same amount of power as the average power dissipated by the time-varying voltage or current.