Electric Potential

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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
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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 usELEREC0
R
Which can be written as (remember, PVII2R)
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Solution
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SHM and Damping
SHM x(t) A cos ?t 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.
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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
Fma give
For weak damping (small b), the solution is
A e-(b/2m)t
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No damping angular frequency for spring is
With damping
The type of damping depends on the difference
between ?o and (b/2m) in this case.
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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
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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 ?dt
• ?d is the angular frequency of oscillation for
  the circuit and

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Damped RLC Circuit, Graph

• The maximum value of Q decreases after each
  oscillation- RltRc (critical value)
• This is analogous to the amplitude of a damped
  spring-mass system

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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 gt RC, the
  circuit is said to be overdamped
• - If R RC, the circuit is said to be
  critically damped

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Overdamped RLC Circuit, Graph

• The oscillations damp out very rapidly
• Values of R gtRC

14
Example Electrical oscillations are initiated in
a series circuit containing a capacitance C,
inductance L, and resistance R. a) If R ltlt
(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?
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Solution
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Example In the figure below, let R 7.60 O, 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?
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Solution
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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 3.14 108 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.)
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Solution
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