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In addition to the revision notes for Introduction to RLC Circuits on this page, you can also access the following Magnetism learning resources for Introduction to RLC Circuits
| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
|---|---|---|---|---|---|---|
| 16.15 | Introduction to RLC Circuits |
In these revision notes for Introduction to RLC Circuits, we cover the following key points:
A RLC circuit is a circuit that contains at least a resistor, a capacitor and an inductor. The simplest RLC circuit is the series RLC circuit.
The differential equation for the damped oscillations in a RLC circuit, is
The charge decay in such a circuit is calculated through an expression, which is a combination of exponential and sinusoidal equation, as occurs in all types of damped oscillations. Thus, the charge left in a RLC circuit after a given time t of operation is found by:
where
is the angular frequency of damped oscillations and
is the angular frequency of undamped oscillations.
The equation for the energy decay in a damped oscillation system of electricity is
This means the energy of the electric field in a RLC circuit oscillates in a cos2 fashion while the amplitude decreases exponentially with time.
A RLC circuit needs an external sustainable source of emf to make it operate for a long time at steady values. This is made possible by connecting the circuit to an AC power supply, which on the other hand is supplied by an AC generator. Such a system produces forced oscillations in the circuit, which makes it operate for a long time with constant periodic values.
The equation of this induced emf in an AC generator is
where εmax is the amplitude of the induced emf in the generator (usually the amplitude is the initial induced emf), while ωd is called the "driving angular frequency" because it drives an induced current in the circuit, the equation of which is given by
in circuits with no or very small resistance, the induced current (and emf) oscillate at angular frequency
which is known as the "natural angular frequency". Such oscillations are known as free oscillations.
On the other hand, when a resistor is present is the circuit, the oscillations are known as damped (when no external source is present) or forced (when an external source is needed to keep the values of the induced current and emf constant). The following rule is true for the forced oscillations:
"The induced current and emf in a circuit always occur according the angular frequency of the forced oscillations, regardless the value of the natural angular frequency."
The resistive load as a function of time in a circuit containing only an AC power source and a resistor is
and the current in this circuit as a function of time is
Giving that in this type of circuit the current is in phase with the potential difference, we have the initial phase φ = 0.
When considering a circuit supplied by an AC source, which contains only a capacitor C, we have
for the potential difference at any instant across the capacitor:
and for the current at any instant t:
The quantity
is called capacitive reactance of capacitor. It has the unit of resistance (Ohm).
From experiments, it results that current leads by one quarter of a cycle the voltage in such a circuit. If we replace the cos ωd ∙ t term with a phase-shifted sine expression of + π/2 rad, we obtain
Hence, we obtain for the current in the circuit:
In addition, we have for the maximum potential difference in the circuit
The current is π/2 (a quarter of a cycle) in advantage to potential difference. Therefore, we say: "the current leads the voltage by π/2".
Similarly, in a circuit containing only an AC power source and an inductor, the potential difference across the inductor is
The quantity
is called inductive reactance and is measured in Ohms, similarly to capacitive reactance.
The current flowing in a circuit containing an inductive load is
From this equation, we can see that for a purely inductive load, the current is delayed (is out of phase) by π/2 (a quarter of a cycle) to the potential difference.
To make the graph plotting easier and its understanding more digestible, we use arrows similar to vectors even though neither current nor voltage are vectors. Such diagrams are known as phasor diagrams. The angle formed by the arrows and the horizontal (time) axis gives the ωt term.
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