James Watt
Wikimedia Commons

The power dissipated (also known as absorbed or consumed power) by a two-terminal component is given by

where voltage $V(t)$ and current $I(t)$ are shown in Fig. 1. Notice that we used the same sign convention for defining $V(t)$ and $I(t)$ as in Ohm's law. In general, power dissipation is time-dependent and varies with the instantaneous values of the voltage and current.

In general, the sign of the power dissipated by a component depends on the direction of current flow relative to the voltage polarity:

• If current flows from the positive terminal to the negative, it is supplying power (positive).
• If current flows from the negative terminal to the positive, it is absorbing power (negative).

In the case of resistors (that will be discussed in the next chapter), the power dissipated becomes $$\begin{equation}P(t)=R~\left[I(t)\right]^2=\frac{\left[V(t)\right]^2}{R}\end{equation}$$ where we used Ohm's law. Notice that, the power dissipated by a resistor is always non-negative (assuming that the resistance is positive). In the case of DC circuits, at steady-state, the power dissipated does not depend on time and $$\begin{equation}P=RI^2=\frac{V^2}{R}\end{equation}$$

The power dissipated by a component is measured in watts (${\class{mjunit}W}$). The name comes from the Scottish engineer James Watt (1736–1819), who made major improvements to the steam engine and helped usher in the Industrial Revolution.

Note: For the power absorbed by inductors and capacitors see Inductors and Capacitors.

The power generated (also known as produced power)) by a two-terminal component is defined as the negative of the power dissipated by that component

where voltage $V(t)$ and current $I(t)$ are shown in Fig. 1.

The power generated by a component is measured in watts (${\class{mjunit}W}$).

Consider an arbitrary lumped network that has $b$ branches and $n$ nodes. Suppose that to each branch we assign arbitrarily a branch potential difference $V_k(t)$ and a branch current $I_k(t)$ for $k=1,...,b$ that are measured with respect to arbitrarily picked associated reference directions. Tellegen's theorem states that $$\begin{equation}\sum_{k=1}^{b}{V_k(t) I_k(t)}=0\end{equation}$$

Tellegen's theorem can be proved using KVL and KCL and shows that the algebraic sum of the powers dissipated by an isolated electric circuit is equal to 0. In other words, the total power dissipated in a circuit is equal to the total power generated in the circuit. The theorem is a direct application of the law of conservation of energy and applies to any circuit with arbitrary elements, including linear, nonlinear, or time-varying components.

James Prescott Joule
Wikimedia Commons

The energy dissipated (also called absorbed or consumed) by a two terminal component from $t=0$ until $t=T$ is equal to $$\begin{equation}W(T)=\int_{0}^{T} \, V(t)I(t) \, dt\end{equation}$$

where voltage $V(t)$ and current $I(t)$ are shown in Fig. 2. Notice that, when we compute the power dissipated by a component, the current is defined as going from the positive to the negative terminal of the voltage definition.

At steady-state (in the case of DC circuits), the energy dissipated becomes $W=IVT$. Notice that the energy dissipated increases linearly with time.

The energy dissipated by a component is measured in joules (${\class{mjunit}J}$). The name comes from the English physicist James Prescott Joule (1818–1889), who investigated the relationship between heat, work, and mechanical energy.

Note: For the energy absorbed by (or stored in) inductors and capacitors see Inductors and Capacitors.

The energy generated (also called produced) by a two terminal component is defined as the negative of the energy dissipated by that component $$\begin{equation}W_g=-W\end{equation}$$ and is also measured in joules (${\class{mjunit}J}$).