Linear Circuit Analysis

1. Introduction
2. Basic Concepts
3. Simple Circuits
4. Nodal and Mesh Analysis
5. Additional Analysis Techniques
6. AC Analysis
7. Magnetically Coupled Circuits
8. Polyphase Systems
9. Operational Amplifiers
10. Laplace Transforms
11. Time-Dependent Circuits
12. Two-Port Networks
Appendix

Power Dissipated

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

$$\begin{equation}P(t)=V(t) I(t)\end{equation}$$

where voltage $V(t)$ and current $I(t)$ are shown in Fig. 1. Notice that we used the same sign convention for $V(t)$ 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.

− + V(t) I(t) Z
Fig. 1. Computing the power dissipated by a 2-terminal component.

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

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

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

Power Generated

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

$$\begin{equation}P_g(t)=-P(t)=-V(t) I(t)\end{equation}$$

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

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

Tellegen's Theorem

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 shows that, if KVL and KCL are satisfied, 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.

Energy Dissipated

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.

− + V(t) I(t) Z
Fig. 2. Computing the energy dissipated by a 2-terminal component. Notice that the current is chosen to flow from the positive to the negative reference of the voltage when computing the power dissipated.

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

The energy dissipated by a component is measured in joules (${\class{mjunit}J}$).

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

Energy Generated

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

Examples of Solved Problems