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In addition to the revision notes for Alternating Current. LC Circuits on this page, you can also access the following Magnetism learning resources for Alternating Current. LC Circuits
| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
|---|---|---|---|---|---|---|
| 16.14 | Alternating Current. LC Circuits |
In these revision notes for Alternating Current. LC Circuits, we cover the following key points:
The current produced by cells and batteries is known as direct current (DC) because it is obtained through direct contact between battery and the conducting wire. However, most energy in use is transferred not as DC but rather, as alternating current (AC), which is a kind of electromagnetic oscillation propagating through the conductor in a sinusoidal way. Such oscillations are made possible through the induced current, which take place in LC circuits with inductance L and capacitance C.
The current and potential difference in a LC circuit does not increase or decrease exponentially but in a sinusoidal fashion instead. The behavior of electric-related quantities here, is similar to all kinds of waves (especially electromagnetic ones). The oscillations produced in the capacitor (resulting in the change of capacitor's electric field) and in the inductor (resulting in the change in the inductor's magnetic field) are known as electromagnetic oscillations. Therefore, the energy is a LC circuit is always the sum of electric and magnetic energy stored in the capacitor and inductor respectively, i.e.
or
When the capacitor is fully charged, the inductor does not store any energy in its magnetic field. As a result, the above equation becomes
On the other hand, when the capacitor is discharged, the entire energy of the circuit is stored in the magnetic field of the inductor. Hence, the equation representing the total energy in the circuit becomes
There is no direct source in LC circuits, so we cannot assign a positive or negative direction to the current flow. As a result, the plates of capacitor are charged oppositely in equal time intervals.
The current in LC circuits changes direction during its flow through the circuit. The same thing can be said for the magnetic field of the inductor as well. Since the current changes direction, the magnetic field lines change direction as well.
From the analogy between block-and-spring systems and LC circuits, we obtain the following correspondences:
the following correspondences:
The current i flowing in a LC circuit, the potential difference ΔVC between the capacitor plates, the charge Q stored in the capacitor plates, and the potential difference ΔVR due to the resistance of the circuit - are all sinusoidal.
The angular frequency of a LC circuit is
The general equation of block-and-spring oscillating systems is
The general solution of this equation is
When applied for a LC oscillator, we obtain
This is the fundamental equation of a LC oscillator, the general solution of which, is
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,
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:
As for the electrical and magnetic oscillations, we can write for the electrical energy stored in a LC circuit at a given time
while for the magnetic energy stored in the magnetic field of inductor at a given time, we have
Substituting ω = 1/√L ∙ C, we obtain for the magnetic energy stored in a LC circuit
Since potential difference is proportional to current, we use the same approach as for the current to find the potential difference in a LC circuit. Furthermore, potential difference is in phase with the current because they are related to a positive constant (resistance) to each other. Hence, we have
Multiplying both sides by R, we obtain
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