Understanding RLC Circuits: Energy Transformation and Damping in Electronics

1. Self-Inductance and Circuits • RLC circuits

2. Recall, for LC Circuits • In actual circuits, there is always some resistance • Therefore, there is some energy transformed to internal energy • The total energy in the circuit continuously decreases as a result of these processes

3. RLC circuits • A circuit containing a resistor, an inductor and a capacitor is called an RLC Circuit • Assume the resistor represents the total resistance of the circuit • The total energy is not constant, since there is a transformation to internal energy in the resistor at the rate of dU/dt = -I2R (power loss) I + C - L R

4. RLC circuits The switch is closed at t =0; Find I (t). I + C - L Looking at the energy loss in each component of the circuit gives us:EL+ER+EC=0 R Which can be written as (remember, P=VI=I2R):

5. Solution

6. x t x t SHM and Damping SHM: x(t) = A cosωtMotion continues indefinitely. Only conservative forces act, so the mechanical energy is constant. Damped oscillator: dissipative forces (friction, air resistance, etc.) remove energy from the oscillator, and the amplitude decreases with time. In this case, the resistor removes the energy.

7. x t A damped oscillator has external nonconservative force(s) acting on the system. A common example in mechanics is a force that is proportional to the velocity. f= -bv wherebis a constant damping coefficient F=ma give: For weak damping (small b), the solution is: A e-(b/2m)t

8. No damping: angular frequency for spring is: With damping: The type of damping depends on the difference between ωo and (b/2m) in this case.

9. : “Underdamped”, oscillations with decreasing amplitude : “Critically damped” : “Overdamped”, no oscillation x(t) overdamped critical damping Critical damping provides the fastest dissipation of energy. t underdamped

10. RLC Circuit Compared to Damped Oscillators • When R is small: • The RLC circuit is analogous to light damping in a mechanical oscillator • Q = Qmaxe -Rt/2L cos ωdt • ωd is the angular frequency of oscillation for the circuit and

11. Damped RLC Circuit, Graph • The maximum value of Q decreases after each oscillation- R<Rc (critical value) • This is analogous to the amplitude of a damped spring-mass system

12. Damped RLC Circuit • When R is very large- the oscillations damp out very rapidly - there is a critical value of R above which no oscillations occur:- When R > RC, the circuit is said to be overdamped - If R = RC, the circuit is said to be critically damped

13. Overdamped RLC Circuit, Graph • The oscillations damp out very rapidly • Values of R >RC

14. Example: Electrical oscillations are initiated in a series circuit containing a capacitance C, inductance L, and resistance R. a) If R << (weak damping), how much time elapses before the amplitude of the current oscillation falls off to 50.0% of its initial value? b) How long does it take the energy to decrease to 50.0% of its initial value?

15. Solution

16. Example: In the figure below, let R = 7.60 Ω, L = 2.20 mH, and C = 1.80 μF. a) Calculate the frequency of the damped oscillation of the circuitb) What is the critical resistance?

17. Solution

18. Example: The resistance of a superconductor. In an experiment carried out by S. C. Collins between 1955 and 1958, a current was maintained in a superconducting lead ring for 2.50 yr with no observed loss. If the inductance of the ring was 3.14 × 10–8 H, and the sensitivity of the experiment was 1 part in 109, what was the maximum resistance of the ring? (Suggestion: Treat this as a decaying current in an RL circuit, and recall that e– x ≈ 1 – x for small x.)

19. Solution