Circuit RLC Forcé - Série 9 - PDF

# Résumé de cours pour la série n°9 - Circuit RLC forcé ## Résumé **Professeur :** Youssef Allouche **Numéro de téléphone:** 98 416 018 ## Introduction * **Excitateur:** * Pulsation: **ω = 2πN** (réglable) * **Résonateur:** * Pulsation: **ω₀ = 2πN₀ = 1 / √LC** (constante) ## Trois...

# Résumé de cours pour la série n°9 - Circuit RLC forcé ## Résumé **Professeur :** Youssef Allouche **Numéro de téléphone:** 98 416 018 ## Introduction * **Excitateur:** * Pulsation: **ω = 2πN** (réglable) * **Résonateur:** * Pulsation: **ω₀ = 2πN₀ = 1 / √LC** (constante) ## Trois circuits à considérer * **Circuit capacitif:** ω < ω₀, N < N₀, Φu < Φ₁ * **Circuit inductif:** ω > ω₀, N > N₀, Φu > Φ₁ * **Circuit résistif:** ω = ω₀, N = N₀, Φu = Φ₁ ## Diagrammes et Formules **Tableau comparant les trois types de circuits :** | Circuit Capacitif | Circuit Inducitf | Circuit Résistif | | ------------- | ------------- | ------------- | | ω < ω₀ et Φu < Φ₁ | ω > ω₀ et Φu > Φ₁ | ω = ω₀ et Φu = Φ₁ | | u(t) est en retard de phase par rapport à i(t) | u(t) est en avance de phase par rapport à i(t) | u(t) et i(t) sont en phase | ## Diagramme de Fresnel * **Fresnel for a capacitive circuit:** * LwIm < Cw * Δφ = Φu - Φ₁ > 0 * **Calculs:** * Im = Um / √(R+r)² + (Lw/Cw - 1)² * Z = √(R+r)² + (Lw/Cw - 1)² * tg(Φu - Φ₁) = (Lw/Cw - 1) / (R+r) * sin(Φu - Φ₁) = (Lw/Cw - 1) / Z * cos(Φu - Φ₁) = (R+r) / Z * **Fresnel for an inductive circuit:** * LwIm > Cw * Δφ = Φu - Φ₁ > 0 * **Calculs:** * Im = Um / √(R+r)² + (Lw/Cw - 1)² * Z = √(R+r)² + (Lw/Cw - 1)² * tg(Φu - Φ₁) = (Lw/Cw - 1) / (R+r) * sin(Φu - Φ₁) = (Lw/Cw - 1) / Z * cos(Φu - Φ₁) = (R+r) / Z * **Fresnel for a resistive circuit:** * LwIm = Cw * Δφ = Φu - Φ₁ = 0 * **Calculs:** * Im = Um / (R+r) * Z = R+r ## Impédances * Impédance du circuit : **Z = Um / Im = √(R+r)² + (Lw/Cw - 1)²** * Impédance du condensateur: **Zc = Uc / Im = 1 / (Cw)** * Impédance de la bobine: * **Without resistance:** ZL = Uв / Im = Lw * **With resistance:** ZL = Uв / Im = √(r² + L²w²) * Impédance de l'ensemble bobine-condensateur: **ZL,C = Uв / Im = √(r² + L²w²) / (Cw)** * Impédance de l'ensemble résistor-bobine: **ZR,L = UB,Rm / Im = √(R+r)² + L²w²** * Impédance de l'ensemble résistor-condensateur: **ZR,C = URC / Im = √(R+r)² + (1 / (Cw))²** ## Résonance d'intensité * **Definition:** * When the circuit is **resistive**, ω = ω₀, **Im** reaches its maximum value. * **Im = f(ω) = Um / √(R+r)² + (Lw/Cw - 1)²** ## Phénomène de surtension * **Definition:** * The voltage across the capacitor can exceed **Um** at resonance. * **Q Factor (Quality Factor):** * **Q = ω₀L / (R+r) = 1 / (RCω₀)** * **Observations:** * **Q ≤ 1:** No overvoltage * **Q > 1:** Overvoltage; Higher **Q** means a more pronounced resonance. * **Q >> 1:** Risky, can lead to capacitor breakdown and sparking in the coils. ## Puissance moyenne consommée par le dipôle RLC * **Formula:** **P = Um * Im * cos(Φu - Φ₁) / 2** * **Observations:** * P ≥ 0 for passive circuits. * W = P * Δt is the energy consumed by the circuit. * Power losses are due to Joule's effect and are minimised when cos(Φu - Φ₁) is maximized. ## Résonance de charge * **Definition:** When the circuit is **capacitive**, ω = ωr, **Qm** reaches its maximum value. * **Formula:** **ωr² = (R+r)² / (2L²)** ## Influence of damping on the resonance of charge * **Small damping (R+r) is small:** Sharp resonance (curve ①) * **Large damping (R+r) is large:** Wide resonance (curve ②) ## Exercice n°1 * **Circuit:** * Resistor: R = 15 Ω * Inductor: L (with internal resistance r) * Capacitor: C = 1.6 * 10^-4 F * GBF: u(t) = Um sin(wt) * **Goals:** * Determine the nature of the circuit for w < w₀ and w > w₀. * Analyze the Fresnel diagram and determine the maximum current, impedance, and phase angle. * Determine the inductor's inductance (L) and internal resistance (r). ## Exercice n°2 * **Circuit:** * Resistor: R = 50 Ω * Inductor: L (with internal resistance r) * Capacitor: C = 8 μF * GBF: u(t) = Um sin (2πNt) * **Goals:** * Each group studies the circuit at a fixed frequency (N1 = 366 Hz and N2 = 199 Hz). * Compare the measured values of L and r. * Assess the risk of overvoltage. ## Exercice n°3 * **Circuit:** * Resistor: R = 25 Ω * Inductor: L * Capacitor: C * GBF: u(t) = Um sin (2πNt) * **Goals:** * Analyze the circuit in resonance. * Determine the phase angle, maximum voltage, and current. * Determine the internal resistance (r) of the inductance. ## Exercice n°4 * **Circuit:** * Resistor: R = 30 Ω * Inductor: L = 25,6 * 10^-3 H * Capacitor: C * GBF: u(t) = 30 / √2 sin(2πNt) * **Goals:** * Analyze the circuit at resonance, determining the current, voltage, and inductance. ## Exercice n°5 * **Circuit:** * Resistor: R * Inductor: L * Capacitor: C * GBF: u(t) = Um sin (2πNt + π/6) * **Goals:** * Analyze the circuit at resonance and determine the value of the Q factor. ## Exercice n°6 * **Circuit:** * Resistor: R = 20 Ω * Inductor: L * Capacitor: C = 4 * 10^-6 F * GBF: u(t) = Um sin(ωt), Um = 10V * **Goals:** * Analyze the circuit at resonance. * Determine the influence of the resistance on the resonance. Please note that some images are too blurry to accurately transcribe, so the text might be incomplete and inaccurate.