Circuit RLC Forcé - Série 9 - PDF

Summary

These are lecture notes on forced RLC circuits. They detail the characteristics of capacitive, inductive, and resistive circuits, including calculations for maximum voltage, current, and impedance. Also including diagrams for Fresnel and resonance. The notes include analysis methods, formulae and exercises to solve related problems.

Full Transcript

# 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.

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