Self-Inductance and Circuits RLC circuits

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

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

C RLC circuits The switch is closed at t =0; I Find I (t). + - 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):

Solution

SHM and Damping t x SHM: x(t) = A cos ωt Motion continues indefinitely. Only conservative forces act, so the mechanical energy is constant. x t 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.

f = -bv where b is a constant damping coefficient 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 where b is a constant damping coefficient F=ma give: For weak damping (small b), the solution is: x t A e-(b/2m)t

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.

x(t) t : “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

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

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

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

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

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?

Solution

Example: In the figure below, let R = 7. 60 Ω, L = 2. 20 mH, and C = 1 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 circuit b) What is the critical resistance?

Solution

Example: The resistance of a superconductor 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.)

Solution