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

Voltage Division

Voltage division is a technique that can be used to compute the voltage across each resistor of a series combination of resistors when the voltage across all the resistors is known. For instance, if we have $n$ resistors connected in series and the total voltage across them is $V$, the voltage across resistor $R_i$ is equal to $$\begin{equation}V_i=V\frac{R_i}{R_1+R_2+...+R_n}\end{equation}$$

In the case of only two resistors, the previous equation gives $$\begin{equation}V_1=V\frac{R_1}{R_1+R_2}\end{equation}$$ $$\begin{equation}V_2=V\frac{R_2}{R_1+R_2}\end{equation}$$

Voltages $V_1$, $V_2$,... are defined such that their polarity is the same as the direction of voltage $V$. For instance, consider the circuit shown in Fig. 1.

20V − + V1 2Ω − + V2 5Ω − + V3 3Ω
Fig. 1. Voltage division when in a circuit with three resistors connected in series. The moving dots represent the flow of the current; the color of the dots is related to the electrostatic potential of the wires (nodes): notice that this color changes as electrons travel from node to node.

Applying voltage division we obtain $$V_{1}=20\ V \times\frac{2}{2+5+3}=4\ {\class{mjunit}{V}}$$ $$V_{2}=20\ V \times\frac{5}{2+5+3}=10\ {\class{mjunit}{V}}$$ $$V_{3}=-20\ V \times\frac{3}{2+5+3}=-6\ {\class{mjunit}{V}}$$ Notice that, in the last equation, $V_3$ has a negative sign because its defined polarity is opposite to the reference direction of the $20\ V$ source.

Examples of Solved Problems
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Voltage divider