# LC circuit

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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.

An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil around a core.

A capacitor is a device that stores electrical energy in an electric field. It is a passive electronic component with two terminals.

A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a resonator can be either electromagnetic or mechanical. Resonators are used to either generate waves of specific frequencies or to select specific frequencies from a signal. Musical instruments use acoustic resonators that produce sound waves of specific tones. Another example is quartz crystals used in electronic devices such as radio transmitters and quartz watches to produce oscillations of very precise frequency.

## Contents

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 electronic oscillator is an electronic circuit that produces a periodic, oscillating electronic signal, often a sine wave or a square wave. Oscillators convert direct current (DC) from a power supply to an alternating current (AC) signal. They are widely used in many electronic devices ranging from simplest clock generators to digital instruments and complex computers and peripherals etc. Common examples of signals generated by oscillators include signals broadcast by radio and television transmitters, clock signals that regulate computers and quartz clocks, and the sounds produced by electronic beepers and video games.

Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributed-element filters. That is, using components and interconnections that, in analysis, can be considered to exist at a single point. These components can be in discrete packages or part of an integrated circuit.

In electronics, a mixer, or frequency mixer, is a nonlinear electrical circuit that creates new frequencies from two signals applied to it. In its most common application, two signals are applied to a mixer, and it produces new signals at the sum and difference of the original frequencies. Other frequency components may also be produced in a practical frequency mixer.

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.

An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

## Terminology

The two-element LC circuit described above is the simplest type of inductor-capacitor network (or LC network). It is also referred to as a second order LC circuit to distinguish it from more complicated (higher order) 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 s. Generally, the order is equal to the number of L and C elements in the circuit and in any event cannot exceed this number.

In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L.

## 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 (E) between its plates, depending on the voltage across it, and an inductor stores energy in its magnetic field (B), depending on the current through it.

Electrical energy is energy derived from electric potential energy or kinetic energy. When used loosely, electrical energy refers to energy that has been converted from electric potential energy. This energy is supplied by the combination of electric current and electric potential that is delivered by an electrical circuit. At the point that this electric potential energy has been converted to another type of energy, it ceases to be electric potential energy. Thus, all electrical energy is potential energy before it is delivered to the end-use. Once converted from potential energy, electrical energy can always be called another type of energy.

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named volt. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for volt uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by V, but more often simply as V, for instance in the context of Ohm's or Kirchhoff's circuit laws.

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 (if not replenished from an external circuit) 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. [1] The natural frequency (that is, the frequency at which it will oscillate when isolated from any other system, as described above) 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 (natural frequency ${\displaystyle f_{0}\,}$ below ), 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

Resonance occurs when an LC circuit is driven from an external source at an angular frequency ω0 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

${\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}}}$

where L is the inductance in henrys, and C is the capacitance in farads. The angular frequency ω0 has units of radians per second.

The equivalent frequency in units of hertz is

${\displaystyle f_{0}={\frac {\omega _{0}}{2\pi }}={\frac {1}{2\pi {\sqrt {LC}}}}.}$

## 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 we tune 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:

## Time domain solution

### Kirchhoff's laws

By Kirchhoff's voltage law, the voltage across the capacitor, VC, plus the voltage across the inductor, VL must equal zero:

${\displaystyle V_{C}+V_{L}=0\,.}$

Likewise, by Kirchhoff's current law, the current through the capacitor equals the current through the inductor:

${\displaystyle I_{C}=I_{L}\,.}$

From the constitutive relations for the circuit elements, we also know that

{\displaystyle {\begin{aligned}V_{L}(t)&=L{\frac {\mathrm {d} I_{L}}{\mathrm {d} t}}\,,\\I_{C}(t)&=C{\frac {\mathrm {d} V_{C}}{\mathrm {d} t}}\,.\end{aligned}}}

### Differential equation

Rearranging and substituting gives the second order differential equation

${\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+{\frac {1}{LC}}I(t)=0\,.}$

The parameter ω0, the resonant angular frequency, is defined as:

${\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}}\,.}$

Using this can simplify the differential equation

${\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+\omega _{0}^{2}I(t)=0\,.}$

The associated Laplace transform is

${\displaystyle s^{2}+\omega _{0}^{2}=0\,;}$

thus,

${\displaystyle s=\pm j\omega _{0}\,,}$

where j is the imaginary unit.

### Solution

Thus, the complete solution to the differential equation is

${\displaystyle I(t)=Ae^{+j\omega _{0}t}+Be^{-j\omega _{0}t}\,}$

and can be solved for A and B by considering the initial conditions. Since the exponential is complex, the solution represents a sinusoidal alternating current. Since the electric current I is a physical quantity, it must be real-valued. As a result, it can be shown that the constants A and B must be complex conjugates:

${\displaystyle A=B^{*}}$

Now, let

${\displaystyle A={\frac {I_{0}}{2}}e^{+j\phi }}$

Therefore,

${\displaystyle B={\frac {I_{0}}{2}}e^{-j\phi }}$

Next, we can use Euler's formula to obtain a real sinusoid with amplitude I0, angular frequency ω0 = 1/LC, and phase angle φ.

Thus, the resulting solution becomes:

${\displaystyle I(t)=I_{0}\cos \left(\omega _{0}t+\phi \right)\,.}$

and

${\displaystyle V(t)=L{\frac {\mathrm {d} I}{\mathrm {d} t}}=-\omega _{0}LI_{0}\sin \left(\omega _{0}t+\phi \right)\,.}$

### Initial conditions

The initial conditions that would satisfy this result are:

${\displaystyle I(0)=I_{0}\cos \phi \,.}$

and

${\displaystyle V(0)=L{\frac {\mathrm {d} I}{\mathrm {d} t}}{\Bigg |}_{t=0}=-\omega _{0}LI_{0}\sin \phi \,.}$

## Series circuit

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

{\displaystyle {\begin{aligned}V&=V_{L}+V_{C}\\I&=I_{L}=I_{C}\,.\end{aligned}}}

### Resonance

Inductive reactance magnitude XL increases as frequency increases while capacitive reactance magnitude XC decreases with the increase in frequency. At one particular frequency, these two reactances are equal in magnitude but opposite in sign; that frequency is called the resonant frequency f0 for the given circuit.

Hence, at resonance:

{\displaystyle {\begin{aligned}X_{L}&=X_{C}\\\omega L&={\frac {1}{\omega C}}\,.\end{aligned}}}

Solving for ω, we have

${\displaystyle \omega =\omega _{0}={\frac {1}{\sqrt {LC}}}\,,}$

which is defined as the resonant angular frequency of the circuit. Converting angular frequency (in radians per second) into frequency (in hertz), one has

${\displaystyle f_{0}={\frac {\omega _{0}}{2\pi }}={\frac {1}{2\pi {\sqrt {LC}}}}\,.}$

In a series configuration, XC and XL 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 a maximum at resonance.

• In the limit as ff0 current is maximum. Circuit impedance is minimum. In this state, a circuit is called an acceptor circuit [2]
• For f < f0, XL ≪ −XC. Hence, the circuit is capacitive.
• For f > f0, XL ≫ −XC. 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:

${\displaystyle Z=Z_{L}+Z_{C}}$

Writing the inductive impedance as ZL = jωL and capacitive impedance as ZC = 1/jωC and substituting gives

${\displaystyle Z(\omega )=j\omega L+{\frac {1}{j\omega C}}\,.}$

Writing this expression under a common denominator gives

${\displaystyle Z(\omega )=j\left({\frac {\omega ^{2}LC-1}{\omega C}}\right)\,.}$

Finally, defining the natural angular frequency as

${\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}}\,,}$

the impedance becomes

${\displaystyle Z(\omega )=jL\left({\frac {\omega ^{2}-\omega _{0}^{2}}{\omega }}\right)\,.}$

The numerator implies that in the limit as ω → ±ω0, the total impedance Z 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

In the parallel configuration, the inductor L and capacitor C are connected in parallel, as shown here. The voltage V across the open terminals is equal to both the voltage across the inductor and the voltage across the capacitor. The total current I 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:

{\displaystyle {\begin{aligned}V&=V_{L}=V_{C}\\I&=I_{L}+I_{C}\,.\end{aligned}}}

### Resonance

When XL equals XC, the reactive branch currents are equal and opposite. Hence they cancel out each other to give minimum current in the main line. Since total current is minimum, in this state the total impedance is maximum.

The resonant frequency is given by

${\displaystyle f_{0}={\frac {\omega _{0}}{2\pi }}={\frac {1}{2\pi {\sqrt {LC}}}}\,.}$

Note that any reactive branch current is not minimum at resonance, but each is given separately by dividing source voltage (V) by reactance (Z). Hence I = V/Z, as per Ohm's law.

• At f0, the line current is minimum. The total impedance is at the maximum. In this state a circuit is called a rejector circuit. [3]
• Below f0, the circuit is inductive.
• Above f0, the circuit is capacitive.

### Impedance

The same analysis may be applied to the parallel LC circuit. The total impedance is then given by:

${\displaystyle Z={\frac {Z_{L}Z_{C}}{Z_{L}+Z_{C}}}\,,}$

and after substitution of ZL = jωL and ZC = 1/jωC and simplification, gives

${\displaystyle Z(\omega )=-j\cdot {\frac {\omega L}{\omega ^{2}LC-1}}}$

Using

${\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}}\,,}$

it further simplifies to

${\displaystyle Z(\omega )=-j\left({\frac {1}{C}}\right)\left({\frac {\omega }{\omega ^{2}-\omega _{0}^{2}}}\right)\,.}$

Note that

${\displaystyle \lim _{\omega \to \omega _{0}}Z(\omega )=\infty }$

but for all other values of ω the impedance is finite. The parallel LC circuit connected in series with a load will act as band-stop filter having infinite impedance at the resonant frequency of the LC circuit. The parallel LC circuit connected in parallel with a load will act as band-pass filter.

## Laplace solution

The LC circuit can be solved by Laplace transform.

Let the general equation be:

${\displaystyle v_{C}(t)=v(t)}$
${\displaystyle i(t)=C{\frac {\mathrm {d} v_{C}}{\mathrm {d} t}}}$
${\displaystyle v_{L}(t)=L{\frac {\mathrm {d} i}{\mathrm {d} t}}}$

Let the differential equation of LC series be:

${\displaystyle v_{in}(t)=v_{L}(t)+v_{C}(t)=L{\frac {\mathrm {d} i}{\mathrm {d} t}}+v=LC{\frac {\mathrm {d} ^{2}v}{\mathrm {d} t^{2}}}+v}$

With initial condition:

${\displaystyle {\begin{cases}v(0)=v_{0}\\i(0)=i_{0}=C\cdot v'(0)=C\cdot v'_{0}\end{cases}}}$

Let define:

${\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}}}$
${\displaystyle f(t)=\omega _{0}^{2}v_{in}(t)}$

Gives:

${\displaystyle f(t)={\frac {\mathrm {d} ^{2}v}{\mathrm {d} t^{2}}}+\omega _{0}^{2}v}$

Transform with Laplace:

${\displaystyle {\mathcal {L}}\left[f(t)\right]={\mathcal {L}}\left[{\frac {\mathrm {d} ^{2}v}{\mathrm {d} t^{2}}}+\omega _{0}^{2}v\right]}$
${\displaystyle F(s)=s^{2}V(s)-sv_{0}-v'_{0}+\omega _{0}^{2}V(s)}$
${\displaystyle V(s)={\frac {sv_{0}+v'_{0}+F(s)}{s^{2}+\omega _{0}^{2}}}}$

Then antitransform:

${\displaystyle v(t)=v_{0}\cos(\omega _{0}t)+{\frac {v'_{0}}{\omega _{0}}}\sin(\omega _{0}t)+{\mathcal {L}}^{-1}\left[{\frac {F(s)}{s^{2}+\omega _{0}^{2}}}\right]}$

In case input voltage is Heaviside step function:

${\displaystyle v_{in}(t)=Mu(t)}$
${\displaystyle {\mathcal {L}}^{-1}\left[\omega _{0}^{2}{\frac {V_{in}(s)}{s^{2}+\omega _{0}^{2}}}\right]={\mathcal {L}}^{-1}\left[\omega _{0}^{2}M{\frac {1}{s(s^{2}+\omega _{0}^{2})}}\right]=M(1-\cos(\omega _{0}t))}$
${\displaystyle v(t)=v_{0}\cos(\omega _{0}t)+{\frac {v'_{0}}{\omega _{0}}}\sin(\omega _{0}t)+M(1-\cos(\omega _{0}t))}$

In case input voltage is sinusoidal function:

${\displaystyle v_{in}(t)=U\sin(\omega _{f}t)\Rightarrow V_{in}(s)={\frac {U\omega _{f}}{s^{2}+\omega _{f}^{2}}}}$
${\displaystyle {\mathcal {L}}^{-1}\left[\omega _{0}^{2}{\frac {1}{s^{2}+\omega _{0}^{2}}}{\frac {U\omega _{f}}{s^{2}+\omega _{f}^{2}}}\right]={\mathcal {L}}^{-1}\left[{\frac {\omega _{0}^{2}U\omega _{f}}{\omega _{f}^{2}-\omega _{0}^{2}}}\left({\frac {1}{s^{2}+\omega _{0}^{2}}}-{\frac {1}{s^{2}+\omega _{f}^{2}}}\right)\right]={\frac {\omega _{0}^{2}U\omega _{f}}{\omega _{f}^{2}-\omega _{0}^{2}}}\left({\frac {1}{\omega _{0}}}\sin(\omega _{0}t)-{\frac {1}{\omega _{f}}}\sin(\omega _{f}t)\right)}$
${\displaystyle v(t)=v_{0}\cos(\omega _{0}t)+{\frac {v'_{0}}{\omega _{0}}}\sin(\omega _{0}t)+{\frac {\omega _{0}^{2}U\omega _{f}}{\omega _{f}^{2}-\omega _{0}^{2}}}\left({\frac {1}{\omega _{0}}}\sin(\omega _{0}t)-{\frac {1}{\omega _{f}}}\sin(\omega _{f}t)\right)}$

## History

The first evidence that a capacitor and inductor could produce electrical oscillations was discovered in 1826 by French scientist Felix Savary. [4] [5] He found that when a Leyden jar was discharged through a wire wound around an iron needle, sometimes the needle was left magnetized in one direction and sometimes in the opposite direction. He correctly deduced that this was caused by a damped oscillating discharge current in the wire, which reversed the magnetization of the needle back and forth until it was too small to have an effect, leaving the needle magnetized in a random direction. American physicist Joseph Henry repeated Savary's experiment in 1842 and came to the same conclusion, apparently independently. [6] [7] British scientist William Thomson (Lord Kelvin) in 1853 showed mathematically that the discharge of a Leyden jar through an inductance should be oscillatory, and derived its resonant frequency. [4] [6] [7] British radio researcher Oliver Lodge, by discharging a large battery of Leyden jars through a long wire, created a tuned circuit with its resonant frequency in the audio range, which produced a musical tone from the spark when it was discharged. [6] In 1857, German physicist Berend Wilhelm Feddersen photographed the spark produced by a resonant Leyden jar circuit in a rotating mirror, providing visible evidence of the oscillations. [4] [6] [7] In 1868, Scottish physicist James Clerk Maxwell calculated the effect of applying an alternating current to a circuit with inductance and capacitance, showing that the response is maximum at the resonant frequency. [4] The first example of an electrical resonance curve was published in 1887 by German physicist Heinrich Hertz in his pioneering paper on the discovery of radio waves, showing the length of spark obtainable from his spark-gap LC resonator detectors as a function of frequency. [4]

One of the first demonstrations of resonance between tuned circuits was Lodge's "syntonic jars" experiment around 1889. [4] [6] He placed two resonant circuits next to each other, each consisting of a Leyden jar connected to an adjustable one-turn coil with a spark gap. When a high voltage from an induction coil was applied to one tuned circuit, creating sparks and thus oscillating currents, sparks were excited in the other tuned circuit only when the circuits were adjusted to resonance. Lodge and some English scientists preferred the term "syntony" for this effect, but the term "resonance" eventually stuck. [4] The first practical use for LC circuits was in the 1890s in spark-gap radio transmitters to allow the receiver and transmitter to be tuned to the same frequency. The first patent for a radio system that allowed tuning was filed by Lodge in 1897, although the first practical systems were invented in 1900 by Italian radio pioneer Guglielmo Marconi. [4]

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## References

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2. "rejector circuit | Definition of rejector circuit in English by Oxford Dictionaries". Oxford Dictionaries | English. Retrieved 2018-09-20.
3. Blanchard, Julian (October 1941). "The History of Electrical Resonance". Bell System Technical Journal. U.S.: American Telephone & Telegraph Co. 20 (4): 415. doi:10.1002/j.1538-7305.1941.tb03608.x . Retrieved 2011-03-29.
4. Savary, Felix (1827). "Memoirs sur l'Aimentation". Annales de Chimie et de Physique. Paris: Masson. 34: 5–37.
5. Kimball, Arthur Lalanne (1917). A College Text-book of Physics (2nd ed.). New York: Henry Hold. pp. 516–517.
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