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. 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):

5. Solution

6. 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.

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 coefficient
F=ma give:
For weak damping (small b), the solution is: x t
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. 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

10. 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

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
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?

17. Solution

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.)

19. Solution