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Self-Inductance and CircuitsSelf-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 circuitsRLC circuits
C L
R
I
+
-
•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)
RLC circuitsRLC circuits
The switch is closed at t =0; Find I (t).
C L
R
I
Which can be written as (remember, P=VI=I2R):
+
-Looking at the energy loss in eachcomponent of the circuit gives us:
EL+ER+EC=0
0
02
C
QIR
dt
dIL
IC
QRI
dt
dILI
Solution
t
x
x
t
SHM: x(t) = A cos ωt Motion 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.
SHM and Damping
)cos()( 2
tAetxt
m
b
For weak damping (small b), the solution is:
f = bv where b is a constant damping coefficient
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.
A e-(b/2m)t
2
2
dt
xdm
dt
dxbkx F=ma give:
No damping: angular frequency for spring is:
22
0
2
22
mb
mb
mk
mk0
With damping:
The type of damping depends on the difference between ωo and (b/2m) in this case.
02 mb
x(t)
t
overdamped
critical damping
underdamped
: “Overdamped”, no oscillation
: “Underdamped”, oscillations with decreasing amplitude
: “Critically damped”
Critical damping provides the fastest dissipation of energy.
02 mb
02 mb
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
12 2
1
2d
Rω
LC L
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
4 /CR L C
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
4 /CR L C
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?
CL /4
Solution
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. 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
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