LC circuit


An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency.
LC circuits are used either for generating signals at a particular frequency, or picking out a signal at a particular frequency from a more complex signal; this function is called a bandpass filter. They are key components in many electronic devices, particularly radio equipment, used in circuits such as oscillators, filters, tuners and frequency mixers.
An LC circuit is an idealized model since it assumes there is no dissipation of energy due to resistance. Any practical implementation of an LC circuit will always include loss resulting from small but non-zero resistance within the components and connecting wires. The purpose of an LC circuit is usually to oscillate with minimal damping, so the resistance is made as low as possible. While no practical circuit is without losses, it is nonetheless instructive to study this ideal form of the circuit to gain understanding and physical intuition. For a circuit model incorporating resistance, see RLC circuit.

Terminology

The two-element LC circuit described above is the simplest type of inductor-capacitor network. It is also referred to as a second order LC circuit to distinguish it from more complicated LC networks with more inductors and capacitors. Such LC networks with more than two reactances may have more than one resonant frequency.
The order of the network is the order of the rational function describing the network in the complex frequency variable. Generally, the order is equal to the number of L and C elements in the circuit and in any event cannot exceed this number.

Operation

An LC circuit, oscillating at its natural resonant frequency, can store electrical energy. See the animation. A capacitor stores energy in the electric field between its plates, depending on the voltage across it, and an inductor stores energy in its magnetic field, depending on the current through it.
If an inductor is connected across a charged capacitor, the voltage across the capacitor will drive a current through the inductor, building up a magnetic field around it. The voltage across the capacitor falls to zero as the charge is used up by the current flow. At this point, the energy stored in the coil's magnetic field induces a voltage across the coil, because inductors oppose changes in current. This induced voltage causes a current to begin to recharge the capacitor with a voltage of opposite polarity to its original charge. Due to Faraday's law, the EMF which drives the current is caused by a decrease in the magnetic field, thus the energy required to charge the capacitor is extracted from the magnetic field. When the magnetic field is completely dissipated the current will stop and the charge will again be stored in the capacitor, with the opposite polarity as before. Then the cycle will begin again, with the current flowing in the opposite direction through the inductor.
The charge flows back and forth between the plates of the capacitor, through the inductor. The energy oscillates back and forth between the capacitor and the inductor until internal resistance makes the oscillations die out. The tuned circuit's action, known mathematically as a harmonic oscillator, is similar to a pendulum swinging back and forth, or water sloshing back and forth in a tank; for this reason the circuit is also called a tank circuit. The natural frequency is determined by the capacitance and inductance values. In most applications the tuned circuit is part of a larger circuit which applies alternating current to it, driving continuous oscillations. If the frequency of the applied current is the circuit's natural resonant frequency, resonance will occur, and a small driving current can excite large amplitude oscillating voltages and currents. In typical tuned circuits in electronic equipment the oscillations are very fast, from thousands to billions of times per second.

Resonance effect

occurs when an LC circuit is driven from an external source at an angular frequency at which the inductive and capacitive reactances are equal in magnitude. The frequency at which this equality holds for the particular circuit is called the resonant frequency. The resonant frequency of the LC circuit is
where is the inductance in henries, and is the capacitance in farads. The angular frequency has units of radians per second.
The equivalent frequency in units of hertz is

Applications

The resonance effect of the LC circuit has many important applications in signal processing and communications systems.
  • The most common application of tank circuits is tuning radio transmitters and receivers. For example, when tuning a radio to a particular station, the LC circuits are set at resonance for that particular carrier frequency.
  • A series resonant circuit provides voltage magnification.
  • A parallel resonant circuit provides current magnification.
  • A parallel resonant circuit can be used as load impedance in output circuits of RF amplifiers. Due to high impedance, the gain of amplifier is maximum at resonant frequency.
  • Both parallel and series resonant circuits are used in induction heating.
LC circuits behave as electronic resonators, which are a key component in many applications:

Kirchhoff's laws

By Kirchhoff's voltage law, the voltage across the capacitor plus the voltage across the inductor must equal zero:
Likewise, by Kirchhoff's current law, the current through the capacitor equals the current through the inductor:
From the constitutive relations for the circuit elements, we also know that

Differential equation

Rearranging and substituting gives the second order differential equation
The parameter, the resonant angular frequency, is defined as
Using this can simplify the differential equation:
The associated Laplace transform is
thus
where is the imaginary unit.

Solution

Thus, the complete solution to the differential equation is
and can be solved for and by considering the initial conditions. Since the exponential is complex, the solution represents a sinusoidal alternating current. Since the electric current is a physical quantity, it must be real-valued. As a result, it can be shown that the constants and must be complex conjugates:
Now let
Therefore,
Next, we can use Euler's formula to obtain a real sinusoid with amplitude, angular frequency, and phase angle.
Thus, the resulting solution becomes

Initial conditions

The initial conditions that would satisfy this result are

Series circuit

In the series configuration of the LC circuit, the inductor and capacitor are connected in series, as shown here. The total voltage across the open terminals is simply the sum of the voltage across the inductor and the voltage across the capacitor. The current into the positive terminal of the circuit is equal to the current through both the capacitor and the inductor.

Resonance

increases as frequency increases, while capacitive reactance decreases with increase in frequency. At one particular frequency, these two reactances are equal and the voltages across them are equal and opposite in sign; that frequency is called the resonant frequency for the given circuit.
Hence, at resonance,
Solving for, we have
which is defined as the resonant angular frequency of the circuit. Converting angular frequency into frequency, one has
and
at.
In a series configuration, and cancel each other out. In real, rather than idealised, components, the current is opposed, mostly by the resistance of the coil windings. Thus, the current supplied to a series resonant circuit is maximal at resonance.
  • In the limit as current is maximal. Circuit impedance is minimal. In this state, a circuit is called an acceptor circuit
  • For  ,   ; hence, the circuit is capacitive.
  • For  ,   ; hence, the circuit is inductive.

    Impedance

In the series configuration, resonance occurs when the complex electrical impedance of the circuit approaches zero.
First consider the impedance of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances:
Writing the inductive impedance as and capacitive impedance as and substituting gives
Writing this expression under a common denominator gives
Finally, defining the natural angular frequency as
the impedance becomes
where gives the reactance of the inductor at resonance.
The numerator implies that in the limit as, the total impedance will be zero and otherwise non-zero. Therefore the series LC circuit, when connected in series with a load, will act as a band-pass filter having zero impedance at the resonant frequency of the LC circuit.

Parallel circuit

When the inductor and capacitor are connected in parallel as shown here, the voltage across the open terminals is equal to both the voltage across the inductor and the voltage across the capacitor. The total current flowing into the positive terminal of the circuit is equal to the sum of the current flowing through the inductor and the current flowing through the capacitor:

Resonance

When equals, the two branch currents are equal and opposite. They cancel each other out to give minimal current in the main line. Since total current in the main line is minimal, in this state the total impedance is maximal. There is also a larger current circulating in the loop formed by the capacitor and inductor. For a finite voltage, this circulating current is finite, with value given by the respective voltage-current relationships of the capacitor and inductor. However, for a finite total current in the main line, in principle, the circulating current would be infinite. In reality, the circulating current in this case is limited by resistance in the circuit, particularly resistance in the inductor windings.
The resonant frequency is given by
Any branch current is not minimal at resonance, but each is given separately by dividing source voltage by reactance. Hence   , as per Ohm's law.
  • At   , the line current is minimal. The total impedance is maximal. In this state a circuit is called a rejector circuit.
  • Below   , the circuit is inductive.
  • Above   , the circuit is capacitive.