NationMaster - Encyclopedia: RLC circuit

An RLC circuit (also known as a resonant circuit or a tuned circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. This article is about resonance in physics. ... A tuner is a device to adjust the resonant frequency of an antenna or transmission line to work most efficiently at one frequency or band of frequencies. ... An electrical network or electrical circuit is an interconnection of analog electrical elements such as resistors, inductors, capacitors, diodes, switches and transistors. ... Resistor symbols (non-European) Resistor symbols (Europe, IEC) Axial-lead resistors on tape. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ... Capacitors: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. ...

Tuned circuits have many applications particularly for oscillating circuits and in radio and communication engineering. They can be used to select a certain narrow range of frequencies from the total spectrum of ambient radio waves. For example, AM/FM radios with analog tuners typically use an RLC circuit to tune a radio frequency. Most commonly a variable capacitor is attached to the tuning knob, which allows you to change the value of C in the circuit and tune to stations on different frequencies. In most modern usages of the word spectrum, there is a unifying theme of between extremes at either end. ...

An RLC circuit is called a second-order circuit as any voltage or current in the circuit can be described by a second-order differential equation for circuit analysis. A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...

1 Configurations
2 Similarities and differences between series and parallel circuits
3 Fundamental Parameters
- 3.1 Resonant frequency
- 3.2 Damping factor
4 Derived Parameters
- 4.1 Bandwidth
- 4.2 Resonance Damping
5 Circuit Analysis
- 5.1 Series RLC with Thévenin power source
  - 5.1.1 Frequency Domain
- 5.2 Parallel RLC circuit
6 External links

Configurations

Every RLC circuit consists of two components: a power source and resonator. There are two types of power sources – Thévenin and Norton. Likewise, there are two types of resonators – series LC and parallel LC. As a result, there are four configurations of RLC circuits: It has been suggested that this article or section be merged with Thevenins theorem. ... It has been suggested that this article or section be merged with Nortons theorem. ... An LC circuit consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electrical current can alternate between them at an angular frequency of where L is the inductance in henries, and C is the capacitance in farads. ...

Series LC with Thévenin power source
Series LC with Norton power source
Parallel LC with Thévenin power source
Parallel LC with Norton power source.

Similarities and differences between series and parallel circuits

The expressions for the bandwidth in the series and parallel configuration are inverses of each other. This is particularly useful for determining whether a series or parallel configuration is to be used for a particular circuit design. However, in circuit analysis, usually the reciprocal of the latter two variables is used to characterize the system instead. They are known as the resonant frequency and the Q factor respectively. This article is about resonance in physics. ... This article or section does not cite any references or sources. ...

Fundamental Parameters

There are two fundamental parameters that describe the behavior of RLC circuits: the resonant frequency and the damping factor. In addition, other parameters derived from these first two are discussed below. The factual accuracy of this article is disputed. ...

Resonant frequency

The undamped resonance or natural frequency of an RLC circuit (in radians per second) is given by Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. ... This article is about resonance in physics. ... In mathematics and physics, the radian is a unit of angle measure. ...

$omega_o = {1 over sqrt{L C}}$

In the more familiar unit hertz (or inverse seconds), the natural frequency becomes MHZ redirects here. ...

$f_o = {omega_o over 2 pi} = {1 over 2 pi sqrt{L C}}$

Resonance occurs when the complex impedance Z_LC of the LC resonator becomes zero: Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ...

$Z_{LC} = Z_L + Z_C = 0quad$

Both of these impedances are functions of complex angular frequency s: It has been suggested that this article or section be merged into Angular velocity. ...

$Z_C = { 1 over Cs }$

$Z_L = Ls quad$

Setting these expressions equal to one another and solving for s, we find:

$s = pm j omega_o = pm j {1 over sqrt{L C}}$

where the resonance frequency ω_o is given in the expression above.

$omega_o = {1 over sqrt{L C}}$

Damping factor

The damping factor of the circuit (in radians per second) is: In audio system terminology the damping factor gives the ratio of the rated impedance of the loudspeaker to the source impedance. ... In mathematics and physics, the radian is a unit of angle measure. ...

$zeta = {R over 2L}$

for a series RLC circuit, and:

$zeta = {1 over 2RC}$

for a parallel RLC circuit.

For applications in oscillator circuits, it is generally desirable to make the damping factor as small as possible, or equivalently, to increase the quality factor (Q) as much as possible. In practice, this requires decreasing the resistance R in the circuit to as small as physically possible for a series circuit, and increasing R to as large a value as possible for a parallel circuit. In this case, the RLC circuit becomes a good approximation to an ideal LC circuit. An LC circuit consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electrical current can alternate between them at an angular frequency of where L is the inductance in henries, and C is the capacitance in farads. ...

Alternatively, for applications in bandpass filters, the value of the damping factor is chosen based on the desired bandwidth of the filter. For a wider bandwidth, a larger value of the damping factor is required (and vice versa). In practice, this requires adjusting the relative values of the resistor R and the inductor L in the circuit.

Derived Parameters

The derived parameters include Bandwidth, Q factor, and damped resonance frequency.

Bandwidth

The RLC circuit may be used as a bandpass or band-stop filter by replacing R with a receiving device with the same input resistance, and the bandwidth (in radians per second) is The frequency axis of this symbolic diagram would be logarithmically scaled. ... In electronics, a band-stop filter is a filter that attenuates, usually to very low levels, all frequencies between two non-zero, finite limits and passes all frequencies not within the limits. ... This article does not cite any references or sources. ...

$Delta omega = 2 zeta = { R over L}$

Alternatively, the bandwidth in hertz is

$Delta f = { Delta omega over 2 pi } = { zeta over pi } = { R over 2 pi L }$

The bandwidth is a measure of the width of the frequency response at the two half-power frequencies. As a result, this measure of bandwidth is sometimes called the full-width at half-power. Since electrical power is proportional to the square of the circuit voltage (or current), the frequency response will drop to ${ 1 over sqrt{2} }$ at the half-power frequencies. In physics, power (symbol: P) is the rate at which work is performed or energy is transferred. ...

Resonance Damping

The damped resonance frequency derives from the natural frequency and the damping factor. If the circuit is underdamped, meaning Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. ...

$zeta < omega_o$

then we can define the damped resonance as

$omega_d = sqrt{ omega_o^2 - zeta^2 }$

In an oscillator circuit

$zeta << omega_o$ .

As a result

$omega_d = omega_o$ (approx).

See discussion of underdamping, overdamping, and critical damping, below.

Circuit Analysis

Series RLC with Thévenin power source

In this circuit, the three components are all in series with the voltage source. It has been suggested that this article or section be merged with Current source. ...

Series RLC Circuit notations: Wikipedia does not have an article with this exact name. ...

v - the voltage of the power source (measured in volts V)

i - the current in the circuit (measured in amperes A)

R - the resistance of the resistor (measured in ohms = V/A);

L - the inductance of the inductor (measured in henrys = H = V·s/A)

C - the capacitance of the capacitor (measured in farads = F = C/V = A·s/V)

q - the charge across the capacitor (measured in coulombs C)

Given the parameters v, R, L, and C, the solution for the current q using Kirchhoff's voltage law (KVL) gives Josephson junction array chip developed by NIST as a standard volt. ... Current can be measured by a galvanometer, via the deflection of a magnetic needle in the magnetic field created by the current. ... Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... The ohm (symbol: Î©) is the SI unit of electric resistance. ... Inductance (or electric inductance) is a measure of the amount of magnetic flux produced for a given electric current. ... An inductor. ... Look up second in Wiktionary, the free dictionary. ... Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ... Examples of various types of capacitors. ... The coulomb (symbol: C) is the SI unit of electric charge. ... The coulomb (symbol: C) is the SI unit of electric charge. ... Kirchhoffs circuit laws are a pair of laws that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff. ...

${v_R+v_L+v_C=v} ,$

For a time-changing voltage v(t), this becomes

$Ri(t) + L { {di} over {dt}} + {1 over C} int_{-infty}^{t} i(tau), dtau = v(t)$

Using the relationship between charge and current:

$i(t) = {{dq} over {dt}}$

The above expression can be expressed in terms of charge across the capacitor:

$L {{d^2 q} over {dt^2}} +{R} {{dq} over {dt}} + {1 over {C}} q(t) = v(t)$

Dividing by L gives the following second order differential equation:

${{d^2 q} over {dt^2}} +{R over L} {{dq} over {dt}} + {1 over {LC}} q(t) = {1 over L} v(t)$

We now define two key parameters:

$zeta = {R over 2L}$

and

$omega_0 = { 1 over sqrt{LC}}$

both of which are measured as radians per second. In mathematics and physics, the radian is a unit of angle measure. ...

Substituting these parameters into the differential equation, we obtain:

${{d^2 q} over {dt^2}} + 2 zeta {{dq} over {dt}} + omega_0^2 q(t) = {1 over L} v(t)$

$q''+2zeta q' + omega_0^2 q = {1 over L} v(t)$

Frequency Domain

The series RLC can be analyzed in the frequency domain using complex impedance relations. If the voltage source above produces a complex exponential wave form with amplitude v(s) and angular frequency $s = σ + i ω$ , KVL can be applied: Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... It has been suggested that this article or section be merged into Angular velocity. ... Kirchhoffs circuit laws are a pair of laws that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff. ...

$v(s) = i(s) left ( R + Ls + frac{1}{Cs} right )$

where i(s) is the complex current through all components. Solving for i:

$i(s) = frac{1}{ R + Ls + frac{1}{Cs} } v(s)$

And rearranging, we have

$i(s) = frac{s}{ L left ( s^2 + {R over L}s + frac{1}{LC} right ) } v(s)$

Complex Admittance

Next, we solve for the complex admittance Y(s): In electrical engineering, the admittance (Y) is the inverse of the impedance (Z). ...

$Y(s) = { i(s) over v(s) } = frac{s}{ L left ( s^2 + {R over L}s + frac{1}{LC} right ) }$

Finally, we simplify using parameters ζ and ω_o

$Y(s) = { i(s) over v(s) } = frac{s}{ L left ( s^2 + 2 zeta s + omega_o^2 right ) }$

Notice that this expression for Y(s) is the same as the one we found for the Zero State Response.

Poles and Zeros

The zeros of Y(s) are those values of s such that $Y (s) = 0$ : In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ...

s = 0

and $s = infty$

The poles of Y(s) are those values of s such that $Y(s) = infty$ . By the quadratic formula, we find In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

$s = - zeta pm sqrt{zeta^2 - omega_o^2}$

Notice that the poles of Y(s) are identical to the roots $λ 1$ and $λ 2$ of the characteristic polynomial.

Sinusoidal Steady State

If we now let $s = i ω$ ....

Taking the magnitude of the above equation:

$| Y(s=i omega) | = frac{1}{sqrt{ R^2 + left ( omega L - frac{1}{omega C} right )^2 }}$

Next, we find the magnitude of current as a function of ω

$| I( i omega ) | = | Y(i omega) | | V(i omega) |,$

If we choose values where R = 1 ohm, C = 1 farad, L = 1 henry, and V = 1.0 volt, then the graph of magnitude of the current i (in amperes) as a function of ω (in radians per second) is:

Sinusoidal steady-state analysis

Note that there is a peak at $i m a g (ω) = 1$ . This is known as the resonant frequency. Solving for this value, we find: Plot of current magnitude of series RLC circuit This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... This article is about resonance in physics. ...

$omega_o = frac{1}{sqrt{L C}}$

Parallel RLC circuit

A much more elegant way of recovering the circuit properties of an RLC circuit is through the use of nondimensionalization. Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ...

The remainder of this article may require cleanup to meet Wikipedia's quality standards.
Please discuss this issue on the talk page, or replace this tag with a more specific message. This article has been tagged since August 2006.

Parallel RLC Circuit notations: Image File history File links Broom_icon. ... Wikipedia does not have an article with this exact name. ...

V - the voltage of the power source (measured in volts V)

I - the current in the circuit (measured in amperes A)

R - the resistance of the resistor (measured in ohms = V/A);

L - the inductance of the inductor (measured in henrys = H = V·s/A)

C - the capacitance of the capacitor (measured in farads = F = C/V = A·s/V)

For a parallel configuration of the same components, where Φ is the magnetic flux in the system Josephson junction array chip developed by NIST as a standard volt. ... Current can be measured by a galvanometer, via the deflection of a magnetic needle in the magnetic field created by the current. ... Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... The ohm (symbol: Î©) is the SI unit of electric resistance. ... Inductance (or electric inductance) is a measure of the amount of magnetic flux produced for a given electric current. ... An inductor. ... Look up second in Wiktionary, the free dictionary. ... Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ... Examples of various types of capacitors. ... The coulomb (symbol: C) is the SI unit of electric charge. ...

$C frac{d^2 Phi}{dt^2} + frac{1}{R} frac{d Phi}{dt} + frac{1}{L} Phi = i_0 cos(omega t) Rightarrow frac{d^2 chi}{d tau^2} + 2 zeta frac{d chi}{dtau} + chi = cos(Omega tau)$

with substitutions

$Phi = chi x_c, t = tau t_c, x_c = L i_0, t_c = sqrt{LC}, 2 zeta = frac{1}{R} sqrt{frac{L}{C}}, Omega = omega t_c .$

The first variable corresponds to the maximum magnetic flux stored in the circuit. The second corresponds to the period of resonant oscillations in the circuit.

External links

a treatment that starts with the mechanical analogy
An interactive simulation on series RCL circuit

Categories: Cleanup from August 2006 | Analog circuits

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