Description

• The electrical circuit shown here is a simple series oscillating circuit. It consists of an inductor L, a capacitor C, a resistor R, a voltage generator E that can be connected to the circuit, and a voltmeter V for measuring voltages.
• The oscillating character of the circuit is due to the simultaneous presence of the elements L and C which, connected to each other, can bounce back and forth between them the electrical energy they receive from the generator E. These two elements are entirely analogous to a mass and a spring which, when coupled, "pass" between them the energy coming from an external push.
• The resistor R does not contribute to the oscillation, but is included for completeness, because every real electrical circuit has a resistive component. In the mechanical analogue it corresponds to a friction term, which dissipates as heat part of the energy it receives.

Operation

• The circuit shown above, called RLC after its components, is characterised by a natural frequency of oscillation ${\displaystyle f_0}$: that is, it can produce an oscillating current in response to an external impulse, such as quickly closing and opening the switch.
• The natural frequency ${\displaystyle f_0}$ is the resonance of the circuit: the circuit can absorb the maximum amount of energy from an external alternating voltage generator E when the voltage supplied by the generator oscillates at a frequency very close to ${\displaystyle f_0}$ (see also the specific sections on resonance and impedance).

Let us try to grasp these features through a few examples, following step by step how the oscillation takes place in each case.

Free oscillations

We illustrate how oscillations are produced in the RLC circuit by means of the mechanical analogy with the mass-spring system

As already noted, only two parameters enter the definition of the natural frequency of the system: the inertia of the oscillator (mass M, or inductance L), and the constant of the restoring force (spring constant k, or the inverse of the capacitance C). The balance between these two quantities produces the oscillatory behaviour, under suitable initial conditions.

Oscillation and energy transformation

From another point of view, we could describe the oscillation as the periodic transformation of energy from one form into another.

Forced oscillations

Now let us study what happens when, instead of letting the circuit oscillate freely, we force it to oscillate from outside, starting from the equilibrium condition and connecting the circuit to a sinusoidal current generator at a frequency f.

• Essentially, what we are doing is channelling energy from the external generator into the oscillator. But energy in what form? This depends, instant by instant, on the state of the generator, and therefore on its frequency f.
• However, the oscillating system reacts according to its own dynamics, that is, over time it stores energy in a form that depends on its own natural frequency ${\displaystyle f_0}$. Indeed, since the system has its own "inertia", it takes a certain time for the energy to turn from potential into kinetic (or from electric into magnetic), and for the oscillation to be completed.
• So the transfer of energy from the generator to the oscillator is more effective the closer the generator frequency is to the natural frequency of the oscillator, because only in this way do the generator and the circuit not hinder each other.
• When ${\displaystyle f=f_0}$ the generator and the oscillator are said to be in resonance.
• In this condition the generator supplies energy to the circuit at every oscillation. Without resistance, the amplitude of the oscillations would become infinite. In reality, even though most of the energy of the generator goes into sustaining the oscillations, part of it is dissipated, and the oscillations reach a finite maximum value. The effect of the resistance is therefore to make the resonance weaker, but also to soften the resonance condition: if the system dissipates energy, even frequencies not exactly equal to f₀, but within an interval f₀± Δf, will be "in resonance".
• When the frequency of the generator is very different from the natural frequency of the system, the action of the generator hinders the free oscillation of the circuit, and the resulting oscillations are small. The circuit absorbs little energy from the generator: what it receives in one part of the cycle it gives back in the other.

Note:

1. If the current produced by the generator oscillates like a sinusoid, the current flowing through the circuit also oscillates like a sinusoid. This is a fundamental property of harmonic motion.
2. In general, the oscillations of the system and of the energy source (the generator) do not have the same phase: for example, the maximum amplitude of the oscillation is not reached at the same instant. The size of the phase shift depends on the frequency of the oscillator.

Remember that the two properties of the system that determine the oscillatory behaviour actually compete with each other to "win" the energy. Since the energy available at each instant is finite, when the energy is at a maximum in one form it must be at a minimum in the other. This explains qualitatively why energy must be supplied from outside in a "targeted" way, that is, at the right moment. It is easy to understand how to reinforce or hinder the oscillation if we consider a particular periodic generator which, instead of supplying energy like a sinusoid, supplies pulses at regular intervals. Here is an example

Remarks and further details

• The model represented by this circuit can be applied to describe the most diverse physical systems through the tool of physical analogy. For example, the same model describes very well, from a macroscopic point of view (that is, without resorting to an atomic theory), the behaviour of piezoelectric crystals, which are used as precision oscillators in "quartz" watches, or in the synchronisation circuitry of modern television sets, with frequencies between 5 kHz and 50 MHz.
• With the advent of digital electronics, many analogue oscillating circuits like the one described above have been replaced by circuits based on transistors or operational amplifiers. Moreover, the oscillation used is often a square wave rather than a sinusoid. Such is, for example, the CPU clock in microprocessors. However, the general ideas about the physics of oscillation are already fully contained in the analogue prototype described here, which for decades was the basic component of radio receivers and transmitters.
• From a technological point of view, oscillating circuits have many applications, especially as filters in hi-fi equipment, and as tuned circuits in receivers (radio, TV, etc.). Indeed, it is clear that when an RLC-type circuit is driven by a complex oscillation, that is, not by a single sinusoidal oscillation of frequency f but by the superposition of several oscillations, the effect of resonance is to select the frequencies close to ${\displaystyle f_0}$, attenuating all the others.
• Exploiting the fact that the resonance frequency of the circuit can be adjusted by changing one of its parameters, such as the capacitance C, we get an idea of how a radio tuner works (the knob that selects the station when turned is usually a variable capacitor).