Electrical Power & Energy

In electrical engineering, moving charge (current) through a potential difference (voltage) requires energy. The rate at which this energy is transferred or dissipated is known as electrical power. Understanding power and energy is fundamental to sizing components, determining battery life, and analyzing circuit efficiency.

Electrical Power ($P$)

Power is defined as the time rate of expending or absorbing energy. In a basic Direct Current (DC) circuit, the electrical power delivered to or dissipated by a component is the product of the voltage across it and the current flowing through it.

$P = V I$

where:

• $P$ = electrical power [Watts, W]
• $V$ = voltage across the component [Volts, V]
• $I$ = current through the component [Amperes, A]

One Watt (W) is defined as the transfer of one Joule of energy per second (1 W = 1 J/s).

Equivalent Forms for Resistive Loads

Using Ohm's Law ($V = IR$), we can substitute voltage or current to derive two highly useful equivalent forms of the power equation. These are specifically valid for ideal resistive components (like heating elements or basic resistors).

Substituting $V = IR$: $P = (IR) I = I^2 R$

Substituting $I = \frac{V}{R}$: $P = V \left(\frac{V}{R}\right) = \frac{V^2}{R}$

The formula $P = I^2 R$ is particularly valuable for calculating the power lost as heat in transmission lines, commonly referred to as Joule heating or "$I^2 R$ losses."

Electrical Energy ($W$)

While power measures how fast energy is being transferred, energy is the total accumulated amount of work done over a period of time.

For a system where the electrical power $P$ remains constant over a time interval $t$, the total energy $W$ is simply the product of power and time:

$W = P t$

where:

• $W$ = electrical energy [Joules, J]
• $P$ = constant electrical power [Watts, W]
• $t$ = time duration [seconds, s]

If power varies with time, the energy must be found by integrating the power function over the time interval ($W = \int P(t) dt$).

In practical and commercial applications, electrical energy is often measured in kilowatt-hours (kWh) rather than Joules. One kWh is the energy consumed by a 1,000 W appliance operating constantly for one hour ($1 \text{ kWh} = 3.6 \times 10^6 \text{ J}$).

Worked Example

An electric space heater operates on a 120 V DC supply and draws a steady current of 10 A. Calculate the power dissipated by the heater and the total energy consumed if it runs for 2 hours.

1. Calculate Power: $P = V I = (120 \text{ V})(10 \text{ A}) = 1200 \text{ W} = 1.2 \text{ kW}$

2. Calculate Energy (in Joules): Time in seconds = $2 \text{ hours} \times 3600 \text{ s/hour} = 7200 \text{ s}$ $W = P t = (1200 \text{ W})(7200 \text{ s}) = 8,640,000 \text{ J} = 8.64 \text{ MJ}$

3. Calculate Energy (in kWh): $W = (1.2 \text{ kW})(2 \text{ hours}) = 2.4 \text{ kWh}$

Engineering Check

The equations $P = VI$ and $W = Pt$ presented here are fundamental for DC circuits. When dealing with Alternating Current (AC) circuits containing capacitors or inductors, the voltage and current are time-varying and may be out of phase. In such cases, concepts like real power, reactive power, and power factor must be introduced, and the simple $P = VI$ formula must be modified for time-average power.