Rlc Circuit

RLC circuit - Wikipedia, the free encyclopedia
Every RLC circuit consists of two components: a power source and resonator. ... The RLC circuit may be used as a bandpass or band-stop filter by replacing R ...
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Applet
... an applet with a very basic RLC circuit with an external source of alternating current. ... is then the frequency response of the current in this circuit. ...
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RLC circuit - Wikiversity
Every RLC circuit consists of two components: a power source and resonator. ... The RLC circuit may be used as a bandpass or band-stop filter by replacing R ...
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RLC circuit: Definition from Answers.com
tuned circuit Electrically conducting pathway containing both inductance and ... The RLC circuit may be used as a bandpass or band-stop filter by replacing R ...
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Series RLC Circuits
... RLC Circuits. The ... Since this is a series circuit, the current is the same through ... shows normalized values of current through a series RLC circuit at ...
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RLC Series Circuit
... visualize the behavior of the RLC series circuit is with the phasor diagram ... AC behavior of RLC series circuit. Add calculation of voltages and current. Index ...
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RL, RC, and RLC Circuits
AC Circuit Analysis Exercises. RL, RC, and RLC Circuits ... the resonant frequency of a series RLC circuit in terms of and. ... Figure 6: Series RLC circuit. ...
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An RLC circuit (also known as a resonant circuit or a tuner circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel.

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.

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.

Configurations Every RLC circuit consists of two components: a power source and resonator. There are two types of power sources – Thévenin equivalent and Norton equivalent. Likewise, there are two types of resonators – series LC circuit and parallel LC. As a result, there are four configurations of RLC circuits:

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 resonance and the Q factor respectively.

Fundamental Parameters There are two fundamental parameter#Engineering 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.

Resonant frequency The damping resonance of an RLC circuit (in radians per second) is given by

:\omega_o = {1 \over \sqrt{L C-->

In the more familiar unit hertz (or inverse seconds), the natural frequency becomes

:f_o = {\omega_o \over 2 \pi} = {1 \over 2 \pi \sqrt{L C-->

Resonance occurs when the electrical impedance ZLC of the LC resonator becomes zero:

:Z_{LC} = Z_L + Z_C = 0\quad

Both of these impedances are functions of complex angular frequency s:

: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 ωo 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:

:\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.

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

: \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 (physics) 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.

Resonance Damping The damping resonance frequency derives from the natural frequency and the damping factor. If the circuit is underdamped, meaning

\zeta \ < \ \omega_o

then we can define the damped resonance as

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

In an oscillator circuit

\zeta \ \ + {1 \over C} \int_{-\infty}^{t} i(\tau)\, d\tau = 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.

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)

or

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

Frequency Domain The series RLC can be analyzed in the frequency domain using complex number Electrical impedance relations. If the voltage source above produces a complex exponential wave form with amplitude v(s) and angular frequency s = \sigma + i \omega , KVL can be applied:

: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):

: 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 ωo

: 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 Zero (complex analysis) of Y(s) are those values of s such that Y(s) = 0:

: s = 0 and s = \infty

The Pole (complex analysis) of Y(s) are those values of s such that Y(s) = \infty. By the quadratic formula, we find

: s = - \zeta \pm \sqrt{\zeta^2 - \omega_o^2}

Notice that the poles of Y(s) are identical to the roots \lambda_1 and \lambda_2 of the characteristic polynomial.

Sinusoidal Steady State If we now let s = i \omega ....

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_{mag}(\omega) = 1. This is known as the resonant frequency. Solving for this value, we find:

:\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.

{| class="toccolours" align="center" style="float:center; margin: 1em 1em 0 0; width:75%; text-align:left;"| |Parallel RLC Circuit notations: V - the voltage of the power source (measured in volts V) I - the current in the circuit (measured in amperes A) R - the electrical resistance of the resistor (measured in Ohm (unit)s = V/A); L - the inductance of the inductor (measured in henry (inductance) = H = V·second/A) C - the capacitance of the capacitor (measured in farads = F = coulomb/V = A·s/V) |-|}

For a parallel configuration of the same components, where Φ is the magnetic flux in the system

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}{d\tau} + \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.

See also

External links

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

RLC circuit - Wikipedia, the free encyclopedia
An RLC circuit (also known as a resonant circuit, tuned circuit, or LCR circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C ...

the Parallel Resonant Rlc Circuit
The RLC filter that is the subject of this experiment has the property of tending to preferentially allow through frequencies at or near to its Resonant Frequency.

RLC: RLC Circuit Calcuator
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RLC Series Circuit
RLC Series Circuit. The RLC series circuit is a very important example of a resonant circuit. It has a minimum of impedance Z=R at the resonant frequency, and the phase angle is ...

RLC Parallel Circuit
RLC Parallel Circuit. Finding the impedance of a parallel RLC circuit is considerably more difficult than finding the series RLC impedance. This is because each branch has a phase ...

RLC circuit
Java physics applet ... This java applet will show you physics properties of an R-L-C circuit. driven by a harmonic voltage source¡C

Circuit Theory/RLC Circuits - Wikibooks, collection of open-content ...
Z t = Z R + Z L + Z C Z t = R + jωL + 1 / jωC Z t = jωRC + (jω) 2 LC + 1 Z t = (jω) 2 LC + jωRC + 1 Z t = (jω) 2 + jω(R/L) + 1/LC [edit] Frequency Response of the circuit

RLC circuit (AC)
RLC circuit (AC) ... Author: Topic: RLC circuit (AC) (Read 68874 times) 0 Members and 1 Guest are viewing this topic.

Series RLC Circuit
Series RLC Circuit ... Power Up: Chapter 33Alternating Previous: Inductor Circuit. Series RLC Circuit

Series RLC Circuit
Series RLC Circuit