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LC circuit: principles, behavior, and common applications

An LC circuit pairs an inductor (L) and a capacitor (C) to form a resonant electrical network used in tuning, filtering, and oscillators; ideal LC is lossless and exchanges energy between fields.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: May 30, 2026

en.wikipedia.org · Public domain

An LC circuit is an electrical network composed of an inductor (L) and a capacitor (C) connected in series or in parallel. In these simple arrangements the two components exchange energy: the capacitor stores energy in an electric field and the inductor stores energy in a magnetic field. The interplay between the stored electric and magnetic energies produces oscillations at a characteristic resonant frequency. Practical descriptions and basic component roles are often presented together with component datasheets and tutorials; see an introductory discussion of the basic parts at components overview.

Basic behavior and defining formulas

When isolated from resistive losses, an LC circuit behaves as an almost ideal oscillator. The angular resonant frequency ω0 equals 1 divided by the square root of the product L·C, written as ω0 = 1/√(LC). In ordinary frequency units the resonance f0 = ω0/(2π) = 1/(2π√(LC)). At resonance the energy oscillates back and forth: charge on the capacitor produces a voltage and a current that builds the inductor's magnetic field; when the capacitor discharges the inductor returns energy to the capacitor, and the cycle repeats. A concise primer on oscillation dynamics can be found at oscillation fundamentals.

\omega ={\sqrt {1 \over LC}}

Series and parallel arrangements

Two common topologies are series LC and parallel LC. In a series connection the impedance of the inductor and capacitor add; at resonance their reactive impedances cancel and the series network has minimal impedance. This property is used to pass a narrow band of frequencies and is the basis of series resonant circuits in filters and tuners. Conversely, in a parallel connection the branch admittances combine so that at resonance the parallel network presents a very large impedance, acting to reject the resonant frequency from a node. Practical circuit analysis and formulas for impedances are covered in many electronics texts and online guides such as impedance and resonance.

{U}={q^{2} \over 2C}

Energy storage and loss

Energy in an LC circuit is shared between the capacitor and the inductor. For the capacitor the instantaneous stored energy equals (q^2)/(2C) where q is the instantaneous charge; for the inductor it equals (L i^2)/2 where i is the current. In an idealized LC network with no resistance these energies sum to a constant total and no net energy is dissipated. Real circuits include resistance in wires and components; that resistance converts part of the oscillating energy to heat and causes the amplitude to decay unless energy is replenished by an active element. For a practical discussion about damping and quality factor see damping and Q.

{U}={Li^{2} \over 2}

History and development

The concept of electrical resonance emerged in the late 19th century as investigators of electromagnetic waves and telegraphy explored tuned circuits. Early experimental demonstrations of electromagnetic resonance and radio-frequency phenomena by researchers such as Heinrich Hertz provided the basis for tuned transmitters and receivers that used coils and capacitors to select frequencies. Subsequent refinements of component manufacturing and theory helped turn resonant LC circuits into fundamental building blocks for radio, radar and later electronics; historical summaries and timeline overviews are available at historical context.

U={q^{2} \over 2C}+{Li^{2} \over 2}

Applications and examples

LC circuits are central to many practical applications: they form the frequency-selective stages in radio receivers (tuning coils and variable capacitors), they are used in filters to pass or reject bands of frequencies, they act as the frequency-determining network in many oscillators, and they appear in impedance matching networks for antennas and amplifiers. Small examples include tuning a single radio station with a variable capacitor, while larger designs use coupled LC networks for multi-stage band-pass filters. Many application notes and tutorials covering design methods and examples are collected at practical LC designs.

{dU \over dt}=0

Notable properties and distinctions

Lossless ideality: An ideal LC circuit has no resistance and thus no energy loss; this is a theoretical limit useful for analysis.
Quality factor (Q): The sharpness of resonance depends on losses; higher Q gives narrower bandwidth and less energy loss per cycle. Design references explain Q for series and parallel variants at quality factor.
Filter behavior: Series LC behaves like a band-pass in series signal paths, while parallel LC tends to act as a band-stop (notch) at its resonant frequency when placed across a signal node; general filtering principles are discussed at filter types.

{Ld^{2}q \over dt^{2}}+{q \over C}=0

Z=Z_{L}+Z_{C}

Z_{L}=j\omega L

Z_{C}={\frac {1}{j{\omega C}}}

Z=j\omega L+{\frac {1}{j{\omega C}}}

Z={\frac {Z_{L}Z_{C}}{Z_{L}+Z_{C}}}

Z_{L}

Z_{C}

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

Z={\frac {-L\omega j}{\omega ^{2}LC-1}}

Because LC circuits are simple yet versatile, they remain a core topic in electronics education and design. Whether used in a hobby radio, an RF front end, or a precision laboratory oscillator, understanding how inductance and capacitance interact lets engineers control frequency response, bandwidth and stability in a wide variety of systems.

General oscillating circuit, representation with circuit symbol according to EN 60617-4:1996

Free oscillations in an ideal oscillating circuit

For an externally closed circuit of ideal (lossless) components, which contain a certain energy, a periodic process results. For the description, the state at an arbitrarily chosen point in time is defined as the initial state.

At first, the coil is without magnetic flux. The capacitor is charged and the entire energy of the oscillating circuit is stored in its electric field. No current is flowing through the coil yet. (Figure 1)
Due to the voltage at the capacitor, which also drops at the coil, current flow starts, but not abruptly increasing. According to Lenz's rule, a change in the current flow induces a voltage that counteracts its change. Thus, the current intensity and the magnetic flux increase only slowly (initially linearly with time). As the current increases, charge is removed from the capacitor over time, causing its voltage to decrease at the same time. As the voltage decreases, the increase in current flow decreases.
When the voltage has dropped to zero, the current no longer increases and thus reaches its maximum. At this point, the magnetic field strength of the coil is also at its maximum and the capacitor is completely discharged. All the energy is now stored in the magnetic field of the coil. (Figure 2)
When the coil is voltage-free, the current continues to flow steadily, since it cannot change abruptly - just like the magnetic flux. The current begins to charge the capacitor in the opposite direction. This causes a reverse voltage to build up in it (initially linear with time). This voltage, which increases with a negative sign, is equal to a voltage in the coil, which, according to the rules of induction, reduces the magnetic flux over time, which simultaneously reduces the current strength. With the reduction of the current flow, the charging of the capacitor and the increase of its negative voltage slows down.
When the current is reduced to zero, the magnitude of the voltage no longer increases and thus reaches its maximum. The capacitor regains its original charge, but with the opposite polarity. All the magnetic field energy has been converted back into electric field energy. (Figure 3)
These processes continue in the opposite direction. (Picture 4, then again picture 1)

With continuous repetition, the voltage curve is set according to the cosine function; the current curve follows the sine function. The transition from Fig. 1 to Fig. 2 corresponds in the functions to the range x = 0 ... π/2; the transition from Fig. 2 to Fig. 3 runs as in the range x = π/2 ... π, from Fig. 3 via Fig. 4 to Fig. 1 as in x = π ... 2π.

Voltage curve (blue dashed line) and current curve (red line) in the oscillating circuit

Free oscillations in the real series oscillating circuit

As a first approximation, the losses occurring in the real resonant circuit can be represented by an ohmic resistor R, which is in series with the inductance L. Based on the mesh theorem and the behaviour of the three components (and the assumption that current and voltage arrows all have the same direction of circulation), such an RLC series resonant circuit can be described by the following (linear) differential equation system (in state form with the capacitor voltage uC and the coil current i as state variables):

$L\cdot {\frac {di}{dt}}=-u_{C}-R\cdot i$

$C\cdot {\frac {du_{C}}{dt}}=i$

If one is only interested in the current in the oscillating circuit, then one can (by eliminating uC) transform this DGL system into a single linear differential equation of second order:

$LC\cdot {\frac {d^{2}i}{dt^{2}}}+RC\cdot {\frac {di}{dt}}+i=0$

If, for the sake of simplification and generalization, the "abbreviations" for the (ideal) resonant circuit frequency

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

and the decay constant

$\delta ={\frac {R}{2L}}$

one obtains the differential equation

${\frac {d^{2}i}{dt^{2}}}+2\delta \cdot {\frac {di}{dt}}+\omega _{0}^{2}\cdot i=0$

The differential equation for the capacitor voltage has the same form. For the two initial conditions required for the unambiguous solution, it is usually assumed that at time t=0 the capacitor is charged with a voltage _UC0 and the current through the inductance is 0.

Real resonant circuit

In general, a real resonant circuit can be described by the damped harmonic oscillator model. If one assumes that the losses in the oscillating circuit are small, specifically that δ $\delta <\omega _{0}{\text{ oder }}R<2{\sqrt {L/C}}$ , and still introduces the natural circuit frequency

$\omega _{e}={\sqrt {\omega _{0}^{2}-\delta ^{2}}}$

then, using the classical methods for solving a linear homogeneous differential equation, using the Laplace transformation, or using some other operator calculus, one obtains the solution functions for the two state variables

$i(t)=-{\frac {U_{C0}}{\omega _{e}L}}\cdot e^{-\delta t}\cdot \sin {\omega _{e}t}$

$u_{C}(t)=U_{C0}\cdot e^{-\delta t}\cdot \left(\cos {\omega _{e}t+{\frac {\delta }{\omega _{e}}}\cdot \sin {\omega _{e}t}}\right)=U_{C0}\cdot {\frac {\omega _{0}}{\omega _{e}}}\cdot e^{-\delta t}\cdot \cos \left({\omega _{e}t-\varphi }\right)$

Where φ $\varphi =\arctan {\frac {\delta }{\omega _{e}}}$ . The minus sign in front of the current comes from the direction of the current during discharge. The correctness of the solutions can be checked by plugging them into the differential equations and checking the initial state.

In this "normal case of practice", current and capacitor voltage are $e^{{-\delta t}}$ weakly damped by the factor and are not exactly 90° out of phase with each other. The natural circuit frequency ωe is below the ideal resonant circuit frequency ω0 due to damping, and it becomes lower and lower as losses become more severe.

Ideal resonant circuit

For the ideal case of a resonant circuit without losses, δ $\delta =0$ yields the solution of undamped harmonic oscillations (phase-shifted by 90°) described graphically above.

$i(t)=-{\frac {U_{C0}}{\omega _{0}L}}\cdot \sin {\omega _{0}t}$

$u_{C}(t)=U_{C0}\cdot \cos {\omega _{0}t}$

Aperiodic limiting case

If the losses are larger, then in the special case δ $\delta =\omega _{0}{\text{ oder }}R=2{\sqrt {L/C}}$ "without overshoot" the resting state is reached again most quickly. This behavior is called the aperiodic limit case. Then one obtains

$i(t)=-{\frac {U_{C0}}{L}}\cdot t\cdot e^{-\delta t}$

$u_{C}(t)=U_{C0}\cdot \left(1+\delta t\right)\cdot e^{-\delta t}$

Creep

If δ $\delta >\omega _{0}$ holds, then no oscillation occurs. The greater the damping, the slower the current and voltage creep towards 0. This behavior is called (aperiodic) creep. If one introduces the "creep constant"