Subsection 4.1.7 Power
Electrical power is the rate at which electrical energy is transferred or converted from one form to another in an electrical circuit. Electrical power can be converted to useful forms such as mechanical power, light, or heat, with a suitable device such as a motor, lightbulb, or resistor. Electrical power is measured in watts (W) and is represented by the symbol P.
Definition 4.1.11.
The watt (Unit symbol: W) is the unit of power the International System of Units (SI). One watt is the power delivered when one ampere of current is pushed through an electric circuit by one volt of electromotive force, or equivalently 1 joule per second.
The watt is a relatively small quantity, so kilowatts (kW) and megawatts (MW) are commonly used. One kW is equal to 1000 W, and one MW equals 1000 kW.
To convert between SI and US customary power units, use these conversion factors:
\begin{align*}
1 \text{ hp} \amp= 746 \text{ W} \amp \amp = 0.746 \text{ kW} \\
1 \text{ kW}\amp = 1.340 \text{ hp} \amp \amp= 3412 \text{ Btu/hr}
\end{align*}
DC Circuits.
In a DC circuits, where the voltage (V) is constant, electrical power can be calculated simply as the product of voltage and current
\begin{equation}
P = EI\text{,}\tag{4.1.2}
\end{equation}
or as the product of resistance and current squared.
\begin{equation}
P = I^2 R\tag{4.1.3}
\end{equation}
Where:
\begin{align*}
P \amp = \text{Electrical power, in watts.}\\
I \amp = \text{Current, in amperes.}\\
V \amp = \text{Voltage, in volts.}
\end{align*}
The second version of this equation uses the Ohm’s law relation \(E = IR\) to eliminate \(E\) and express power in terms of current and resistance. This equation indicates that the power dissipated by resistance is proportional to the square of the current, and that energy is lost whenever current flows through a resistor.
AC Circuits.
In AC circuits the power cannot be calculated as easily as DC power for two reasons. First, the sinusoidal current and voltage are continuously changing, which means that the power is too. Second, any reactive elements in the circuit shift the current out of phase with the voltage, as shown in
Figure 4.1.10.
The first issue is solved by using RMS average values for current, voltage and power, however simply applying
(4.1.2) and multiplying the RMS voltage by the RMS current does not give the correct RMS power because of the second issue, the phase shift between the current and voltage waveforms.
The correct formula for calculating power in a single-phase AC circuit is
\begin{equation}
P = E I \cos\theta\tag{4.1.4}
\end{equation}
where:
\begin{align*}
P \amp = \text{Real electrical power (RMS, in watts)}\\
I \amp = \text{Current (RMS, in amperes)}\\
V \amp = \text{Voltage (RMS, in volts)}\\
\theta \amp = \text{phase angle between current and voltage}
\end{align*}
The \(\cos\theta\) term is called the power factor. Angle \(\theta\) is the amount of phase shift, and the power factor the power factor reduces the apparent power \(S = EI\text{,}\) to the true power value, sometimes called the real power. The power factor gives an indication of how reactive elements affect the system. Note that since the power factor is the cosine of angle \(\theta\text{,}\) its value is always between one and negative one.
Three phase AC Circuits.
Power in a three phase circuit is \(\sqrt{3}\) times larger than the power in a single phase circuit.
The correct formula for calculating power in a three-phase AC circuit is
\begin{equation}
P = \sqrt{3} E I \cos\theta\tag{4.1.5}
\end{equation}
where:
\begin{align*}
P \amp = \text{Real electrical power (RMS, in watts)}\\
I \amp = \text{Current (RMS, in amperes)}\\
V \amp = \text{Voltage (RMS, in volts)}\\
\theta \amp = \text{phase angle}
\end{align*}
