Power

Power is energy per unit time.

$$P=\frac{E}{t}=\frac{dw}{dt}=\text{F}\cdot \text{v}$$

Relating to Ohm’s law:

$$P(t)=v(t)i(t)=R{i}^2(t)=\frac{v^2(t)}{R}=G{v}^2(t)$$

We can extend further with the concept of efficiency. This outputs a percentage value.

$$e=\frac{\text{useful power}}{\text{input power}}$$

If $P>0$ across a circuit element, then power is absorbed. For $P<0$, it is supplied. Tellegen’s theorem tells us that energy is conserved in a circuit, i.e., power supplied is exactly equal to the power absorbed.

Devices (diodes, resistors) and loads are often rated for a certain power, which motivates the choice of what component to use.

Complex power

When discussing the power grid, it’s convenient to use RMS phasors. This way, we can define the complex power $S$ (in units of volt-amperes, VA) as:

$$S=\text{V}{\text{I}}^{\ast }$$

where ${\text{I}}^{\ast }$ is the conjugate of $\text{I}$. The complex power is made up of two components: a real component, called the average power $P$ (also the real power or active power) and the imaginary component, called the reactive power $Q$.

$$S=|S|\angle \theta =P+jQ=|S|\cos \theta +|S|\sin \theta$$

The power factor angle is the same one as in the power triangle, given by:

$$\theta =\arccos (\text{pf})$$ $$\text{pf}=\frac{P}{|S|}$$

We also call $|S|$ the apparent power, in units of VA. This measure is useful because if we drive a certain load, the power is $P$. But we actually have to drive more, because $\text{pf}<1$.

$$|S|=\sqrt{P^2+Q^2}=V_{RMS}{I}_{RMS}=\frac{P}{\text{pf}}$$

Electromagnetism

To determine the Ohmic power dissipated, we integrate over the entire volume. By Joule’s law:

$$P={\iiint }_v\vec{E}\cdot \vec{J}dv={\iiint }_v\sigma |\vec{E}{|}^{2}dv$$

This ends up giving us an expression that is equivalent to the Ohm’s law expressions above.

See also

• Energy production