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===[[Muhammed Sonny Mercan]]===
An '''RLC circuit''' (sometimes known as [[resonant]] or [[tuner|tuned]] circuit) is an [[electrical circuit]] comprising a [[resistor]] (R), an [[inductor]] (L), and a [[capacitor]] (C), connected in series or in parallel.
{{REMOVE THIS TEMPLATE WHEN CLOSING THIS AfD|B}}
A 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]].
:{{la|Muhammed Sonny Mercan}} – <includeonly>([[Wikipedia:Articles for deletion/Muhammed Sonny Mercan|View AfD]])</includeonly><noinclude>([[Wikipedia:Articles for deletion/Log/2007 June 23#{{anchorencode:Muhammed Sonny Mercan}}|View log]])</noinclude>
== Fundamental Parameters ==
This activist's name does not appear anywhere on Google except in Wikipedia and similar websites. The article seems to promote his art, so I suspect a [[WP:COI]] violation. [[User:Vizjim|Vizjim]] 15:22, 23 June 2007 (UTC)
There are two fundamental parameters that describe the behavior of ''RLC circuits'': Resonant frequency and Q factor. The secondary parameters, bandwidth and damping factor can be derived from the previous two.
*'''Comment''' You may want to check out [[Special:Contributions/Akkanat]] --[[User:Wafulz|Wafulz]] 16:17, 23 June 2007 (UTC)
*<small>'''Note''': This debate has been included in the [[Wikipedia:WikiProject Deletion sorting/Turkey|list of Turkey-related deletions]]. </small> <small>-- [[User:Jayvdb|John Vandenberg]] 17:12, 23 June 2007 (UTC)</small>
'''Delete'''' Deleted [http://en.wikipedia.org/w/index.php?title=Special:Log&page=Muhammed_Sonny_Mercan twice]. Person lacks notability. Salt the page? [[User:Corpx|Corpx]] 17:16, 23 June 2007 (UTC)
*'''Comment''' This is a little off. This page seems to start off as an autobio, then it's edited by Akkanat, who also has an article, edited by Muhammed Sonny Mercan. What's going on here? Unless there are reliable sources, I think both this and the Akkanat article should be deleted. Sounds like massive COI.--[[User:Ispy1981|Ispy1981]] 17:22, 23 June 2007 (UTC)
===Resonant frequency ===
The [[resonant frequency|resonant or natural frequency]] of an ''RLC circuit'' (in [[radians]] per second) is:
::<math>\omega_o = {1 \over \sqrt{L C}}</math>
In the more familiar unit [[hertz]], the natural frequency becomes
::<math>f_o = {\omega_o \over 2 \pi} = {1 \over 2 \pi \sqrt{L C}}</math>
Resonance occurs when the [[impedance | complex impedance]] ''Z<sub>LC</sub>'' of the LC resonator becomes zero:
::<math>Z_{LC} = Z_L + Z_C = 0</math>
Both of these impedances are functions of complex [[angular frequency]] ''s'':
::<math>Z_C = { 1 \over Cs }</math>
::<math>Z_L = Ls </math>
Setting these expressions equal to one another and solving for ''s'', we find:
::<math> s = \pm j \omega_o = \pm j {1 \over \sqrt{L C}}</math>
where the resonant frequency ω<sub>o</sub> is given in the expression above.
=== Damping factor ===
The damping factor of the circuit (in [[radians]] per second) is:
::<math> \alpha = {R \over 2L}</math>
=== Bandwidth ===
The ''RLC circuit'' may be used as a [[bandpass]] or [[band-stop]] filter, and the [[bandwidth]] (in radians per second) is
::<math> \beta = 2 \alpha = { R \over L}</math>
Alternatively, the bandwidth in hertz is
::<math>BW = { \beta \over 2 \pi } = { 2 \alpha \over 2 \pi } = { \alpha \over \pi }= { R \over 2 \pi L}</math>
=== Quality or Q factor ===
The Quality of the filter or [[Q factor]] is calculated as the ratio of the resonant frequency <math>\omega_o</math> over the bandwidth <math>\beta</math> (in radians per second):
::<math>Q = {\omega_o \over \beta } = {\omega_o \over 2\alpha } = {L \over R \sqrt{LC}} = {1 \over R} \sqrt{L \over C}</math>
Or in hertz:
::<math>Q = {f_o \over BW} = {2 \pi f_o L \over R} = {1 \over \sqrt{R^2 C / L}} = {1 \over R} \sqrt{L \over C}</math>
==Configurations==
Every RLC circuit consists of two components: a ''power source'' and ''resonator''. There are two types of power sources – [[Thevenin equivalent|Thévenin]] and [[Norton equivalent|Norton]]. Likewise, there are two types of resonators – series [[LC circuit|LC]] 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.
==Circuit Analysis==
===Series RLC with Thévenin power source===
In this circuit, the three components are all in series with the [[voltage source]].
{| class="toccolours" align="center" style="float:center; margin: 1em 1em 0 0; width:75%; text-align:left;"
| [[Image:RLC series circuit.png|center|RLC series circuit]]
|
Series RLC Circuit notations:
: '''V''' - the voltage of the power source (measured in [[volt]]s V)
: '''I''' - the current in the circuit (measured in [[ampere]]s A)
: '''R''' - the [[electrical resistance|resistance]] of the resistor (measured in [[ohm]]s = V/A);
: '''L''' - the [[inductance]] of the inductor (measured in [[henry_(inductance)|henries]] = H = V·[[second|s]]/A)
: '''C''' - the [[capacitance]] of the capacitor (measured in [[farad]]s = F = [[coulomb|C]]/V = A·s/V)
|-
|}
Given the parameters V, R, L, and C, the solution for the current (I) using [[Kirchoff's voltage law]] is:
<center>
<math>
{V_R+V_L+V_C=V} \,
</math></center>
For a time-changing voltage ''V(t)'', this becomes
<center>
:<math>
RI(t) + L { {dI} \over {dt}} + {1 \over C} \int_{-\infty}^{t} I(\tau)\, d\tau = V(t)
</math></center>
Rearranging the equation gives the following second order differential equation:
<center>
:<math>
{{d^2 I} \over {dt^2}} +{R \over L} {{dI} \over {dt}} + {1 \over {LC}} I(t) = {1 \over L} {{dV} \over {dt}}
</math></center>
We now define two key parameters:
::<math> \alpha = {R \over 2L} </math>
:and
::<math>\omega_0 = { 1 \over \sqrt{LC}} </math>
both of which are measured as [[radians]] per second.
Substituting these parameters into the differential equation, we obtain:
:<math>
{{d^2 I} \over {dt^2}} + 2 \alpha {{dI} \over {dt}} + \omega_0^2 I(t) = {1 \over L} {{dV} \over {dt}}
</math>
====The [[Zero Input Response]] (ZIR) solution====
Setting the input (voltage sources) to zero, we have:
::<math>
{{d^2 I} \over {dt^2}} +{R \over L} {{dI} \over {dt}} + {1 \over {LC}} I(t) = 0
</math>
with the initial conditions for the inductor current, I<sub>L</sub>(0), and the capacitor voltage V<sub>C</sub>(0). In order to solve the equation properly, the initial conditions needed are I(0) and I'(0).
The first one we already have since the current in the main branch is also the current in the inductor, therefore
::<math>
I(0)=I_L(0) \,
</math>
The second one is obtained employing KVL again:
::<math>
V_R(0)+V_L(0)+V_C(0)=0 \,
</math>
::<math>
\Rightarrow I(0)R+I'(0)L+V_C(0)=0 \,
</math>
::<math>
\Rightarrow I'(0)={1 \over L}\left[-V_C(0)-I(0)R \right]
</math>
We have now a [[homogeneous]] second order differential equation with two initial conditions. Substituting the two parameters α and ω<sub>0</sub>, we have
::<math>
I''+2\alpha I' + \omega_0^2 I = 0
</math>
We now convert the form of this equation to its [[characteristic polynomial]]
::<math>\lambda^2 + 2 \alpha \lambda + \omega_0^2 = 0 </math>
Using the quadratic formula, we find the roots as
::<math> \lambda = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2} </math>
Depending on the values of α and ω<sub>0</sub>, there are three possible cases:
=====Over-damping=====
:[[Image:RLC-serial-Over_Damping.PNG|frame|RLC series Over Damped Response]]
:<math>
\alpha>\omega_0 \Rightarrow RC>4 { L \over R} \,
</math>
In this case, the characteristic polynomial's solutions are both negative real numbers. This is called "over damping".
Two negative real roots, the solutions are:
:<math>
I(t)=A e^{\lambda_1 t} + B e^{\lambda_2 t}
</math>
<br style="clear:both;">
=====Critical damping=====
:[[Image:RLC-serial-Critical_Damping.PNG|frame|RLC series Critically Damped]]
:<math>
\alpha=\omega_0 \Rightarrow RC=4 { L \over R } \,
</math>
In this case, the characteristic polynomial's solutions are identical negative real numbers. This is called "critical damping".
The two roots are identical (<math> \lambda_1=\lambda_2=\lambda </math>), the solutions are:
:<math>I(t)=(A+Bt) e^{\lambda t}</math>
:for arbitrary constants ''A'' and ''B''
<br style="clear:both;">
=====Under-damping=====
[[Image:RLC-serial-Under_Damping.PNG|frame|RLC series Under Damped]]
:<math>
\alpha<\omega_0 \Rightarrow RC<4 { L \over R } \,
</math>
In this case, the characteristic polynomial's solutions are [[complex conjugate]] and have negative real part. This is called "under damping" and results in oscillations or ''ringing'' in the circuit.
The solution consists of two conjugate roots
::<math>\lambda_1 = -\alpha + i\omega_c</math>
and
::<math>\lambda_2 = -\alpha - i\omega_c</math>
where
::<math>\omega_c = \sqrt{\omega_o^2 - \alpha^2}</math>
The solutions are:
::<math>I(t) = Ae^{-\alpha + i \omega_c} + Be^{-\alpha - i \omega_c} </math>
::for arbitrary constants ''A'' and ''B''.
Using [[Euler's formula]], we can simplify the solution as
::<math>I(t)=e^{-\alpha t} \left[ C \sin(\omega_c t) + D \cos(\omega_c t) \right]</math>
::for arbitrary constants ''C'' and ''D''.
These solutions are characterized by ''exponentially decaying sinusoidal response''. The time required for the oscillations to "die out" depends on the Quality of the circuit, or [[Q factor]]. The higher the Quality, the longer it takes for the oscillations to decay.
<br style="clear:both;">
==== The [[Zero State Response]] (ZSR) solution====
This time we set the initial conditions to zero and use the following equation:
::<math>
\left\{\begin{matrix} {{d^2 I} \over {dt^2}} +{R \over L} {{dI} \over {dt}} + {1 \over {LC}} I(t) = {1 \over L}{{dV} \over {dt}} \\ \\ I(0^{-})=I'(0^{-})=0 \end{matrix}\right.
</math>
::<math>{{d^2 I} \over {dt^2}} +{2 \alpha } {{dI} \over {dt}} + {\omega_o} I(t) = {1 \over L}{{dV} \over {dt}} </math>
There are two approaches we can take to finding the ZSR: (1) the [[Laplace Transform]], and (2) the [[convolution | Convolution Integral]].
===== Laplace Transform =====
We first take the Laplace transform of the second order differential equation:
:: <math> (s^2 + 2\alpha s + \omega_o^2) I(s) = {s \over L } V(s) </math>
::where ''V(s)'' is the Laplace Transform of the input signal:
::::<math>V(s) = \mathcal{L} \left\{ v(t) \right\} </math>
We then solve for the complex admittance ''Y(s)'' (in [[Siemens (unit)|siemens]]):
::<math> Y(s) = { I(s) \over V(s) } = { s \over L (s^2 + 2\alpha s + \omega_o^2) } </math>
We can then use the admittance ''Y(s)'' and the Laplace transform of the input voltage ''V(s)'' to find the complex electrical current ''I(s)'':
::<math> I(s) = Y(s) \times V(s) </math>
Finally, we can find the electrical current in the time ___domain by taking the inverse Laplace Transform:
::<math>i(t) = \mathcal{L}^{-1} \left\{ I(s) \right\} </math>
<i>Example:</i>
Suppose <math>v(t) = Au(t) </math>
:: where ''u(t)'' is the [[Heaviside]] [[Heaviside step function|step function]].
Then
:: <math> V(s) = { A \over s }</math>
::<math> I(s) = { A \over L (s^2 + 2\alpha s + \omega_o^2) } </math>
::.... to be continued ....
===== Convolution Integral =====
A separate solution for every possible function for V(t) is impossible. However, there is a way to find a formula for I(t) using [[convolution]]. In order to do that, we need a solution for a basic input - the [[Dirac]] [[delta function]].
In order to find the solution more easily we will start solving for the [[Heaviside step function]] and then using the fact that our circuit is a [[linear system]], its derivative will be the solution for the delta function.
The equation will be therefore, for t>0:
::<math>
\left\{\begin{matrix} {{d^2 I_u} \over {dt^2}} +{R \over L} {{dI_u} \over {dt}} + {1 \over {LC}} I_u(t) = 0 \\ I(0^{+})=0 \qquad I'(0^{+})={1 \over L} \end{matrix}\right.
</math>
Assuming λ<sub>1</sub> and λ<sub>2</sub> are the roots of
::<math>
P(\lambda)= \lambda^2+2 \alpha \lambda + \omega_o^2
</math>
then as in the ZIR solution, we have 3 cases here:
===== Over-damping =====
Two negative real roots, the solution is:
:: <math>
I_u(t)={1 \over {L(\lambda_1-\lambda_2)}} \left[ e^{\lambda_1 t}-e^{\lambda_2 t} \right]
</math>
::<math>
\Rightarrow I_{\delta}(t)={1 \over {L(\lambda_1-\lambda_2)}} \left[ \lambda_1 e^{\lambda_1 t}-\lambda_2 e^{\lambda_2 t} \right]
</math>
===== Critical damping =====
The two roots are identical (<math> \lambda_1=\lambda_2=\lambda </math>), the solution is:
:: <math>
I_u(t)={1 \over L} t e^{\lambda t}
</math>
:: <math>
\Rightarrow I_{\delta}(t)={1 \over L} (\lambda t+1) e^{\lambda t}
</math>
===== Under-damping =====
Two conjugate roots (<math>\lambda_1 = \bar \lambda_2 = \alpha + i\omega_c</math>), the solution is:
::<math>
I_u(t)={1 \over {\omega_c L}} e^{\alpha t} \sin(\omega_c t)
</math>
::<math>
\Rightarrow I_{\delta}(t)={1 \over {\omega_c L}} e^{\alpha t} \left[ \alpha \sin(\omega_c t) + \omega_c \cos(\omega_c t) \right]
</math>
(to be continued...)
==== Frequency Domain ====
The series RLC can be analyzed in the [[frequency ___domain]] using [[complex number|complex]] [[impedance]] relations. If the voltage source above produces a complex exponential wave form with amplitude V(s) and [[angular frequency]] <math> s = \sigma + i \omega</math> , [[KVL]] can be applied:
::<math>V(s) = I(s) \left ( R + Ls + \frac{1}{Cs} \right ) </math>
where I(s) is the complex current through all components. Solving for I:
::<math>I(s) = \frac{1}{ R + Ls + \frac{1}{Cs} } V(s) </math>
And rearranging, we have
::<math>I(s) = \frac{s}{ L \left ( s^2 + {R \over L}s + \frac{1}{LC} \right ) } V(s)</math>
===== Complex Admittance =====
Next, we solve for the complex [[admittance]] Y(s):
::<math> Y(s) = { I(s) \over V(s) } = \frac{s}{ L \left ( s^2 + {R \over L}s + \frac{1}{LC} \right ) } </math>
Finally, we simplify using parameters α and ω<sub>o</sub>
::<math> Y(s) = { I(s) \over V(s) } = \frac{s}{ L \left ( s^2 + 2 \alpha s + \omega_o^2 \right ) } </math>
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) | zeros]] of ''Y(s)'' are those values of ''s'' such that <math>Y(s) = 0</math>:
::<math> s = 0 </math> and <math> s = \infty </math>
The [[Pole (complex analysis) | poles]] of ''Y(s)'' are those values of ''s'' such that <math> Y(s) = \infty</math>:
:: <math> s = - \alpha \pm \sqrt{\alpha^2 - \omega_o^2} </math>
Notice that the poles of ''Y(s)'' are identical to the roots <math>\lambda_1</math> and <math>\lambda_2</math> of the characteristic polynomial.
===== Sinusoidal Steady State =====
If we now let <math> s = i \omega </math>....
Taking the magnitude of the above equation:
::<math> | Y(s=i \omega) | = \frac{1}{\sqrt{ R^2 + \left ( \omega L - \frac{1}{\omega C} \right )^2 }} </math>
Next, we find the magnitude of current as a function of ω
::<math> | I( i \omega ) | = | Y(i \omega) | \times | V(i \omega) |</math>
If we choose trivial values where ''R'' = 1 ohm, ''C'' = 1 farad, ''L'' = 1 henry, and ''V'' = 1 volt, then the graph of magnitude of the current ''I'' (in amperes) as a function of ω (in radians per second) is:
<div style="float: center; text-align: center; margin: 1em 1em 1em 1em;">[[Image:RLC_series_imag.png]]<br>''Sinusoidal steady-state analysis''</div>
Note that there is a peak at <math>I_{mag}(\omega) = 1</math>. This is known as the [[resonant frequency]]. Solving for this value, we find:
::<math>\omega_o = \frac{1}{\sqrt{L C}} </math>
===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;"
| [[Image:RLC parallel circuit.png|Center|RLC Parallel circuit]]
|
Parallel RLC Circuit notations:
: '''V''' - the voltage of the power source (measured in [[volt]]s V)
: '''I''' - the current in the circuit (measured in [[ampere]]s A)
: '''R''' - the [[electrical resistance|resistance]] of the resistor (measured in [[ohm]]s = V/A);
: '''L''' - the [[inductance]] of the inductor (measured in [[henry_(inductance)|henries]] = H = V·[[second|s]]/A)
: '''C''' - the [[capacitance]] of the capacitor (measured in [[farad]]s = F = [[coulomb|C]]/V = A·s/V)
|-
|}
For a parallel configuration of the same components, where Φ is the magnetic flux in the system
<center> <math> 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) </math></center>
with substitutions
<center> <math>\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 . </math></center>
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.
== 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 are used to characterize the system instead. They are known as the [[resonant frequency]] and the [[Q factor]] respectively.
== Applications of tuned circuits ==
There are many applications for tuned circuits especially in radio and communication systems. They can be used to select a certain narrow range of frequencies from the total [[spectrum]] of radio waves.
==See also==
*[[Resonant frequency]]
*[[Electronic oscillator]]
*[[LC circuit]]
*[[Bandwidth]]
*[[Bandpass filter]]
*[[Quality factor]]
*[[Oliver Heaviside]]
[[Category:Electronic circuits]]
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