2.8 Power in Electrical Circuits

Instantaneous and Average Power in Circuits ⚡

In a circuit, power tells you how fast electrical energy is being converted. This conversion might be into heat in a resistor, light in a bulb, mechanical work in a motor, or stored energy in a battery.

Power in electrical circuits is defined from the basic mechanical idea of power as “rate of doing work” or “rate of energy transfer.” If an amount of electrical energy $E$ is delivered over a time $t$, the average electrical power $P_{\text{avg}}$ is
$$
P_{\text{avg}} = \frac{E}{t}.
$$

For circuits, we use instantaneous power most often. At any instant,
$$
p(t) = v(t)\,i(t),
$$
where $v(t)$ is the instantaneous voltage across an element and $i(t)$ is the instantaneous current through it. The unit of power is the watt, written W, equal to one joule per second.

In basic DC circuit work, voltage and current are constant in time, so $p = v\,i$ acts as both instantaneous and average power without distinction.

Whenever you see $P$ in a DC context in this chapter, read it simply as power in watts, based on steady voltage and current.

Source vs Load: Sign Conventions 🔌

To talk sensibly about power, you must fix a sign convention for voltage and current. The same mathematical formula $p = v\,i$ can describe power delivered or power absorbed depending on how you label the directions.

In circuit theory we use the passive sign convention when we talk about element power unless we clearly say otherwise. Under this convention, we mark the positive reference polarity of voltage across an element, then define current $i$ as entering that positive terminal.

If we do this and compute $p = v\,i$ for that element, then:

• If $p > 0$, the element is absorbing power and acting like a load.
• If $p < 0$, the element is delivering power and acting like a source.

For example, if a resistor has $5\ \text{V}$ across it and $2\ \text{A}$ entering the positive terminal, then
$p = 5 \cdot 2 = 10\ \text{W}$ and the resistor absorbs 10 W, converting it to heat.

If instead you analyze a current source that has $10\ \text{V}$ across it and the current enters the terminal that you labeled negative, then with your reference directions it will come out that $p$ is negative. That negative power tells you the source is delivering power to the rest of the circuit.

Power and Energy: From Watts to Joules 🔋

Power is related to energy through time. Energy tells you the total amount transferred, power tells you how fast.

If power is constant in time, the energy transferred over a time interval $t$ is
$$
E = P\,t.
$$

If power varies with time, the energy absorbed between an initial time $t_1$ and a later time $t_2$ is given by the integral
$$
E = \int_{t_1}^{t_2} p(t)\,dt = \int_{t_1}^{t_2} v(t)\,i(t)\,dt.
$$

In everyday electrical usage, energy is often measured in kilowatt hours (kWh). One kilowatt hour is the energy delivered by a power of $1\ \text{kW}$ sustained for $1\ \text{hour}$. In joules, using $1\ \text{W} = 1\ \text{J/s}$ and $1\ \text{h} = 3600\ \text{s}$,
$$
1\ \text{kWh} = 1000\ \text{W} \times 3600\ \text{s} = 3.6\times10^6\ \text{J}.
$$

Table of typical relations:

Power Formulas Using Ohm’s Law 🔁

In resistive DC circuits, voltage, current, and resistance are related by Ohm’s law. Combining this law with $P = V\,I$ lets you express power in several equivalent ways.

Starting from Ohm’s law,
$$
V = I\,R,
$$
and the basic power equation,
$$
P = V\,I,
$$
you can eliminate either $V$ or $I$ to get alternative power formulas.

Substitute $V = I\,R$ into $P = V\,I$:
$$
P = (I\,R)\,I = I^2\,R.
$$

Substitute $I = \dfrac{V}{R}$ into $P = V\,I$:
$$
P = V\left(\frac{V}{R}\right) = \frac{V^2}{R}.
$$

So for a purely resistive element in a DC circuit, you have three interchangeable expressions:

Each form is convenient in different situations.

• Use $P = V\,I$ when both voltage and current are known.
• Use $P = I^2 R$ when current and resistance are known.
• Use $P = \dfrac{V^2}{R}$ when voltage and resistance are known.

These formulas apply to individual resistors or any resistive element operating in DC conditions. They are heavily used when checking component ratings and heating.

Power Ratings and Component Limits 💡

Real components can only handle a limited amount of power before they overheat or fail. Manufacturers specify a power rating that tells you the maximum continuous power the device can safely absorb, at a specified ambient temperature and conditions.

For resistors, you commonly see ratings like 0.25 W, 0.5 W, 1 W, 2 W, and higher for power resistors. If you exceed this rating, the resistor can get very hot, drift in resistance, or burn.

Typical resistor power ratings and physical size relation:

To check whether a resistor is safely rated, you:

1. Determine or calculate the current through it or voltage across it.
2. Compute the power using one of the formulas, for example $P = I^2 R$.
3. Compare the result to the resistor’s rated power.

Suppose a 1 kΩ resistor carries 20 mA in a DC circuit. The power is
$$
P = I^2 R = (0.02)^2 \times 1000 = 0.4\ \text{W}.
$$
A 0.25 W resistor would be under rated, because it would be asked to dissipate 0.4 W. A 0.5 W resistor would be more appropriate.

Other components such as diodes, transistors, and integrated circuits also have maximum power dissipation limits. For them, the power rating connects to thermal limits, packaging, and often requires heat sinks at higher powers. The same basic principle holds, you must ensure that the voltages and currents in your circuit do not cause the device to dissipate more than its allowed power.

Conservation of Power in a Circuit 🔁

In any circuit at any instant in time, the total power delivered by sources equals the total power absorbed by loads. This reflects conservation of energy. Energy can move and change form, but in an ideal circuit it is not created or destroyed.

If you sum the powers of all elements in a circuit, using a consistent sign convention, you will find that the total is zero:
$$
\sum_{\text{all elements}} p_k = 0.
$$

Elements with positive $p_k$ are absorbing power. Elements with negative $p_k$ are delivering power. The negative and positive contributions cancel in the sum.

Consider a simple DC circuit with an ideal voltage source and a resistor. The resistor absorbs power $P_R$. The voltage source delivers the same magnitude of power $P_S = -P_R$. If $P_R = 5\ \text{W}$, the source is delivering 5 W to the resistor, so when you sum them, $P_S + P_R = 0$.

This principle is useful as a check on your circuit calculations. If you compute power in each element and the delivered and absorbed powers do not balance, then some sign or value in your analysis is likely incorrect.

Practical Examples of Circuit Power 📐

Power ideas become clear when you connect them to simple numerical situations in DC circuits.

Consider a 12 V battery connected to a 6 Ω resistor. Assuming an ideal source, the current is
$$
I = \frac{V}{R} = \frac{12}{6} = 2\ \text{A}.
$$
The power in the resistor is
$$
P_R = V\,I = 12 \times 2 = 24\ \text{W}.
$$
You can confirm this with the other forms:
$$
P_R = I^2 R = 2^2 \times 6 = 24\ \text{W},
$$
$$
P_R = \frac{V^2}{R} = \frac{12^2}{6} = 24\ \text{W}.
$$

The resistor absorbs 24 W, which turns into heat. The battery delivers 24 W to the circuit. If the circuit runs for 10 minutes, the energy delivered by the battery is
$$
t = 10\ \text{min} = 600\ \text{s},
$$
$$
E = P\,t = 24 \times 600 = 14{,}400\ \text{J}.
$$

The same reasoning extends to more complex networks of series and parallel resistors. Once you know the current through each resistor or the voltage across it, you can calculate the individual powers and then verify the total balance of delivered and absorbed power for the whole circuit.

Power and Efficiency Concepts ⚙️

In real systems, not all power delivered by a source reaches the desired load. Some power is lost in internal resistances, wiring, and other unintended paths. Efficiency compares useful output power to input power.

If a device or subsystem receives an input power $P_{\text{in}}$ and produces a useful output power $P_{\text{out}}$, the efficiency $\eta$ is defined as
$$
\eta = \frac{P_{\text{out}}}{P_{\text{in}}}.
$$

Often efficiency is expressed as a percentage:
$$
\eta_{\%} = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100\%.
$$

If a circuit draws 10 W from a battery and delivers 8 W to a motor shaft, with 2 W lost as heat, then the efficiency is
$$
\eta = \frac{8}{10} = 0.8,\quad \eta_{\%} = 80\%.
$$

As you work with circuits, power and efficiency become central when choosing power supplies, designing loads, and managing heat dissipation, since any power that is not delivered to useful work will appear as heating in one or more components.