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**Self-Inductance and Circuits**

RLC circuits

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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

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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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**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):

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Solution

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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.

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**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

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**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.

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**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

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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 ωdt ωd 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 - R<Rc (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 > 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 >RC

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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?

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Solution

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**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?

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Solution

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**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.)

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Solution

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