Équation différentielle ordre 1 - Exercice corrigé et décharge d'un condensateur

Page 2: Mathematical Solution and Graphical Analysis

This page delves into the mathematical resolution of the équation différentielle ordre 1 homogène and presents the graphical representation of capacitor discharge.

Definition: The time constant τ = RC determines the rate at which the capacitor discharges.

Example: The voltage equation Uc = Ae^$-t/RC$ represents the general solution to the differential equation.

Highlight: At t = τ, the capacitor voltage drops to approximately 37% of its initial value.

Quote: "La tangente à l'origine coupe l'axe des abscisses à t = τ" - This indicates the relationship between the initial slope and the time constant.

Page 3: Current Analysis and Final Equations

This page concludes with a detailed analysis of current behavior during discharge and the final mathematical relationships in the circuit rc parallèle.

Definition: The current during discharge is given by i = C(dUc/dt), representing the rate of voltage change scaled by capacitance.

Example: The current equation i = -(E/R)e^$-t/RC$ shows how current decreases exponentially during discharge.

Highlight: The negative sign in the current equation indicates that current flows in the opposite direction during discharge compared to charging.

Vocabulary: The term "e^$-t/RC$" represents the exponential decay factor that characterizes the discharge process.

Page 1: Introduction to RC Circuit Discharge Analysis

This page establishes the foundational concepts of RC circuit discharge analysis. The content focuses on deriving the équation différentielle linéaire that governs the behavior of the circuit.

Definition: An RC circuit consists of a resistor and capacitor connected in series, with the discharge behavior governed by a first-order differential equation.

Vocabulary: The term "dq/dt" represents the rate of change of charge with respect to time, which is equal to the current in the circuit.

Example: The fundamental equation -Uc + Ri = 0 demonstrates the application of Kirchhoff's voltage law in the RC circuit.

Highlight: The resulting differential equation dUc/dt + Uc/RC = 0 is a homogeneous first-order linear differential equation.