Linear Circuit Analysis

1. Introduction
2. Basic Concepts
3. Simple Circuits
4. Nodal and Mesh Analysis
5. Additional Analysis Techniques
6. AC Analysis
7. Operational Amplifiers
8. Laplace Transforms
9. Time-Dependent Circuits
10. Two-port networks

Power Dissipated

The power dissipated by a two terminal component is equal to

$$\begin{equation}P_d(t)=V(t) I(t)\end{equation}$$
− + V(t) I(t) Z
Fig. 1. Computing the power dissipated by a 2-terminal component.

where voltage $V(t)$ and current $I(t)$ are shown in Fig. 2. Notice that we used the same sign convention for $V(t)$ and $I(t)$ as in Ohm's law. In general, the power dissipated by a component is a time-dependent quantity and depends on the instanteneous values of the voltage and current.

In the case of resistors, the power dissipated becomes $$\begin{equation}P_d(t)=R~I(t)^2=\frac{V(t)^2}{R}\end{equation}$$where we used Ohm's law. Notice that, since we square both the current and voltage, the power dissipated by a resistors 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 $P_d=RI^2=\frac{V^2}{R}$.

The power dissipated by a component is measured in watts (W).

Power Generated

The power generated by a two terminal component is negative the power dissipated by that component

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

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

The power generated by a component is measured in watts (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 $V_k$ and a branch current $I_k$ for $k=1,...,b$ and suppose that they are measured with respect to arbitrarily picked associated reference directions. Tellegen's theorem states that $$\begin{equation}\sum_{k=1}^{b}{V_k I_k}=0\end{equation}$$

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 by a two terminal component from moment $t_0$ until $t$ is equal to $$\begin{equation}E_d=\int_{t_0}^{t} \, V(t)I(t) \, dt\end{equation}$$

− + V(t) I(t) Z
Fig. 2. Applying Ohm's law to a resistor.

where voltage $V$ and current $I$ are shown in Fig. 3. Notice that we used the same sign convention for $V$ and $I$ as in Ohm's law.

At steady-state (in the case of DC circuits), the energy dissipated dobecomes $E_d=IV(t-t_0)$. Notice that the energy dissipated increases linearly as times passes.

The energy dissipated by a component is measured in joules (J).

Energy Generated

The energy generated by a two terminal component is negative the energy dissipated by that component $$\begin{equation}E_g=-E_d\end{equation}$$ and is measured in joules (J).

Sample Solved Problems