Methods for Analyzing Linear Circuits

Definition

A branch consists of one or more elements connected in series and carrying the same current.
A linear resistor obeys Ohm’s law: the voltage across it is proportional to the current through it.

U = R I

A branch consists of a circuit element between two nodes.
A node is the junction of at least three conductors.
A mesh is a set of branches forming a closed circuit.
A loop in a circuit consists of a closed path without passing through the same mesh twice.

(See Figure 2, Step 2)
In a linear circuit, the values of sources and elements are assumed to be known. The goal is to calculate voltages and currents in the circuit.

\sum_k U_k = 0

E_1 + R_1 I_1 - R_2 I_2 - R_3 I_3 - E_3 = 0

If a circuit has n nodes and b branches:
- There are (n-1) independent node equations.
- There are (b - (n-1)) independent mesh equations.
(Example: See Figure 3, Step 2)

Objective: TP n°5 (unspecified)

A linear circuit containing several sources obeys the principle of superposition. The current or voltage produced by the sources acts independently.

(Figure 4)

(Figure 5)

I = I_{\alpha} + I_{\beta}

Calculation of $I_\alpha$ (Figure 6)

I_{\alpha} = \frac{R_2}{R_1 + R_2} \times I_1

Calculation of $I_\beta$

I_{\beta} = \frac{R_2}{R_1 + R_2} \times I_2

Conclusion for $I$

I = I_{\alpha} + I_{\beta} = \frac{R_2}{R_1 + R_2} \times (I_1 + I_2)

(Figure 7)

Any linear circuit located between points A and B can be modeled by an equivalent generator $(E_{th}, R_{th})$ , where:

$E_{th}$ is the open-circuit voltage between A and B
$R_{th}$ is the equivalent resistance seen between A and B when the sources are deactivated

(Figure 8 → Figure 9)