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