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Physics Lesson 16.14.3 - LC Oscillations - A Quantitative Approach

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Welcome to our Physics lesson on LC Oscillations - A Quantitative Approach, this is the third lesson of our suite of physics lessons covering the topic of Alternating Current. LC Circuits, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

LC Oscillations - A Quantitative Approach

Before deriving the equations of the physical quantities involved in a LC oscillator, we must recall the concept of derivative as a rate of function's change. For example, in kinematics, the instantaneous velocity is the rate of position change in a very narrow interval, so we can write

v = dx/dt

We say, "velocity is the first derivative of position to the time". Likewise, acceleration is the first derivative of change in velocity to the time or the second derivative of the change in position to the time (derivative of derivative). We have

a = dv/dt = d2 x/dt2

and so on.

In addition, we will use again the block-and-spring and LC circuit analogy to describe the quantities involved in LC oscillation systems in a more comprehensive way. For, example, we can write for the total energy of a block-and-spring system of oscillations:

Etot = KEblock + EPEspring
= m ∙ v2/2 + k ∙ x2/2

If we ignore any friction, then the total energy is conserved and such oscillations are known as harmonic oscillations. Since the total energy does not change with time but remains constant instead, we can write

dEtot/dt = 0

or

dEtot/dt = d/dt (m ∙ v2/2 + k ∙ x2/2)
= m ∙ v ∙ dv/dt + k ∙ x ∙ dx/dt
= 0

Substituting v = dx/dt and dv/dt = d2x/dt2 as discussed above, we obtain

m ∙ d2 x/dt2 + k ∙ x = 0

This is the fundamental equation of block-and-spring oscillator, the general solution of which (as discussed in the tutorial 11.2), is

x(t) = xmax ∙ cos⁡(ωt + φ)

where xmax is the amplitude of oscillations (the maximum displacement from the equilibrium position, ω is the angular frequency, x(t) is the position of the block aa a given instant and φ is the initial phase of oscillations (not necessarily the block must be at the equilibrium position initially). Physics Tutorials: This image provides visual information for the physics tutorial Alternating Current. LC Circuits

Similarly, if we substitute the above values with the corresponding ones for a LC system of oscillations given in the table of the previous paragraph, we obtain for the total energy stored in a LC system of oscillations

Wtot = WM + We
= L ∙ i2/2 + Q2/2C

Neglecting the resistance of the conductor, we obtain a harmonic LC oscillator in which the energy remains constant with time. Again, we can write

dWtot/dt = 0

or

dWtot/dt = d/dt (L ∙ i2/2 + Q2/2C)
= L ∙ i ∙ di/dt + Q/C ∙ dQ/dt
= 0

Again, substituting i = dQ/dt and di/dt = d2Q/dt2, we obtain

L ∙ d2 Q/dt2 + 1/C ∙ Q = 0

This is the fundamental equation of a LC oscillator, the general solution of which, is

Q(t) = Qmax ∙ cos⁡(ω ∙ t + φ)

where Q(t) gives the charge stored in the capacitor at a given instant, Qmax is the maximum charge variation in the capacitor, ω is the angular frequency of LC oscillations and φ is the initial phase of oscillations.

Taking the first derivative of the last equation, we obtain the equation of current in a LC circuit. Thus,

i(t) = dQ/dt = -ω ∙ Qmax ∙ sin⁡(ω ∙ t + φ)

where imax = ω ∙ t is the amplitude of the current in this kind of circuit. Hence, we can write the equation of current in a LC circuit:

i(t) = -imax ∙ sin⁡(ω ∙ t + φ)

Example 3

A current is flowing through a LC circuit according the sinusoidal function shown in the i vs t graph below.

Physics Tutorials: This image provides visual information for the physics tutorial Alternating Current. LC Circuits

Calculate:

  1. The frequency of current in the circuit
  2. The angular frequency
  3. The initial phase
  4. The equation of current in this circuit
  5. The current in the circuit after 2 s
  6. The current in the circuit after 3.079 s

Solution 3

  1. From the graph, we can see that one cycle is completed in 0.02 s. This value represents the period T of oscillations in the LC circuit. Since frequency f is the inverse of period, we obtain
    f = 1/T
    = 1/0.02 s
    = 50 Hz
  2. The angular frequency ω of the LC circuit is
    ω = 2 ∙ π ∙ f
    = 2 ∙ 3.14 ∙ 50 Hz
    = 314 rad/s
  3. From the graph we can see that the current at the beginning is zero, despite its amplitude imax is 0.5 A. Therefore, we obtain for the initial phase φ:
    i(t) = -imax ∙ sin(ω ∙ t + φ)
    i(0) = -imax ∙ sin(ω ∙ 0 + φ)
    0 = -0.5 ∙ sin(314 ∙ 0 + φ)
    0 = -0.5 ∙ sinφ
    Thus, sin φ must be zero to fit the values. This means φ = 0 because sin 0 = 0.
  4. The general equation of current in this LC circuit, therefore is
    i(t) = -(0.5 A) ∙ sin⁡(100π ∙ t)
    or
    i(t) = -(0.5 A) ∙ sin⁡(314 ∙ t)
  5. Substituting t = 2, we obtain for the current in the circuit after 2 s:
    i(2) = -(0.5 A) ∙ sin(100 ∙ π ∙ 2)
    = -(0.5 A) ∙ sin[100 ∙ (2π)]
    = -(0.5 A) ∙ sin[100 ∙ 0]
    = -(0.5 A) ∙ sin0
    = 0
    (We know that 2π rad = 3600, so sin (N ∙ 2π) = sin 2π = sin 0 = 0. Here we have N = 100.)
  6. We must substitute t = 3.079 in the equation of current to find the current in this instant. Thus,
    i(3.079) = -(0.5 A) ∙ sin (100 ∙ π ∙ 3.079)
    = -(0.5 A) ∙ sin (307.9 ∙ π)
    = -(0.5 A) ∙ sin (306 ∙ π + 1.9 ∙ π)
    Since 306π is a multiple of 2π (306π = 153 ∙ 2π) and giving that sin 2π = 0, we write
    i(3.079) = -(0.5 A) ∙ sin (1.9 ∙ π)
    = -(0.5 A) ∙ (-0.309)
    = 0.1545 A

You have reached the end of Physics lesson 16.14.3 LC Oscillations - A Quantitative Approach. There are 5 lessons in this physics tutorial covering Alternating Current. LC Circuits, you can access all the lessons from this tutorial below.

More Alternating Current. LC Circuits Lessons and Learning Resources

Magnetism Learning Material
Tutorial IDPhysics Tutorial TitleTutorialVideo
Tutorial		Revision
Notes		Revision
Questions
16.14Alternating Current. LC Circuits
Lesson IDPhysics Lesson TitleLessonVideo
Lesson
16.14.1LC Oscillations
16.14.2Electrical to Mechanical Analogy between Two Oscillating Systems
16.14.3LC Oscillations - A Quantitative Approach
16.14.4Angular Frequency
16.14.5Potential Difference in a LC Circuit

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