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.

Kirchhoff’s Laws

Node Law (KCL - Kirchhoff’s Current Law)

• The sum of currents at a node is zero.

Mesh Law (KVL - Kirchhoff’s Voltage Law)

• The sum of voltage drops around a closed mesh is zero.

$$\sum_k U_k = 0$$

• Example equation:

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

Writing Equations Using Kirchhoff’s Laws

• 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)

Node Potential Method

Objective: TP n°5 (unspecified)

Superposition Theorem

Statement

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

Rules for Source Elimination

(Figure 4)

Example

(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 $I$

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

Thévenin’s Theorem

Objective

(Figure 7)

Statement

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

Example

• Isolate the network (i.e., remove all elements that are not part of the subnetwork for which you want to determine the Thévenin equivalent generator).
• Replace voltage sources with short circuits and current sources with open circuits.
• Calculate the Thévenin resistance (the equivalent resistance of the circuit).
• Reconnect the sources (undo step 2).
• Calculate the Thévenin voltage (the equivalent voltage between the two terminals of the network for which you are finding the Thévenin generator).

(Figure 8 → Figure 9)

Norton’s Theorem

Any subnetwork of a circuit can be replaced by a current source in parallel with a resistor.