2 Recall, for LC CircuitsIn actual circuits, there is always some resistanceTherefore, there is some energy transformed to internal energyThe total energy in the circuit continuously decreases as a result of these processes
3 RLC circuitsA circuit containing a resistor, an inductor and a capacitor is called an RLC CircuitAssume the resistor represents the total resistance of the circuitThe 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-LR
4 C RLC circuits The switch is closed at t =0; I Find I (t). + - L Looking at the energy loss in eachcomponent of the circuit gives us: EL+ER+EC=0RWhich can be written as (remember, P=VI=I2R):
6 SHM and DampingtxSHM: x(t) = A cos ωt Motion continues indefinitely. Only conservative forces act, so the mechanical energy is constant.xtDamped 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 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 coefficientF=ma give:For weak damping (small b), the solution is:xtA 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 x(t) t : “Underdamped”, oscillations with decreasing amplitude : “Critically damped”: “Overdamped”, no oscillationx(t)overdampedcritical dampingCritical damping provides the fastest dissipation of energy.tunderdamped
10 RLC Circuit Compared to Damped Oscillators When R is small:The RLC circuit is analogous to light damping in a mechanical oscillatorQ = Qmax e -Rt/2L cos ωdtωd 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 CircuitWhen 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 rapidlyValues 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?
16 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?
18 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.)