RLC circuit and Pequannock: Difference between pages

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{{mergeto|Pequannock Township, New Jersey}}

An '''RLC circuit''' (sometimes known as [[resonant]] or [[tuner|tuned]] circuit) is an [[electrical circuit]] consisting of a [[resistor]] (R), an [[inductor]] (L), and a [[capacitor]] (C), connected in series or in parallel.

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]].

'''Pequannock''' is a small [[suburban]] township located in [[Morris County, New Jersey|Morris County]] northern [[New Jersey]]. It is primarily a bedroom community to nearby [[New York City]] and home to roughly 4,661 residents. What the town lacks in entertainment or commerce it makes up for in historical significance.

== Fundamental Parameters ==

There are two fundamental parameters that describe the behavior of ''RLC circuits'': the resonant frequency and the damping factor. In addition, there are several other parameters that can be derived from these first two (see next section).

== Linguistic Significance ==

Pequannock is thought to have been derived from the Lenni Lenape "Paquettahhnuake", meaning, "cleared land ready or being readied for cultivation". Pompton has been cited by some sources to mean "a place where they catch soft fish".

===Resonant frequency ===

The [[damping|undamped]] [[resonance|resonance or natural frequency]] of an ''RLC circuit'' (in [[radians]] per second) is:

== Historic Pequannock ==

::<math>\omega_o = {1 \over \sqrt{L C}}</math>

Incorporated in [[1740]] as one of the largest townships in the region, this 6.96 square mile bedroom community composed of [[Pompton Plains]] in its northern portion and old Pequannock in its southern was once a vast 176 square mile region of [[rural]] [[farmland]] settled by the [[Netherlands|Dutch]] after its purchase by [[Arent Schuyler]] and associates in the late [[1690]]'s.

During the [[Revolutionary War]], [[George Washington]]'s troops camped on what is now the site of the Pequannock Valley Middle School. Washington himself of course made sure to get a room in the nearby Mandeville Inn.

In the more familiar unit [[hertz]], the natural frequency becomes

During the Civil War, Pequannock was a stop on the underground railroad. The Giles Mandeville House, a field and quarrystone structure located at 515 Turnpike, which served as a waypoint for many runaway slaves, still stands today in use as the Manse of the adjacent First Reformed Church since 1953.

::<math>f_o = {\omega_o \over 2 \pi} = {1 \over 2 \pi \sqrt{L C}}</math>

== Other Pequannock Information ==

Resonance occurs when the [[impedance | complex impedance]] ''Z_LC'' of the LC resonator becomes zero:

People in Pequannock do lots of hardxcore drugs and get trashed nearly every night. Hooray Beer!

Pequannock people are usually white. Notar needs to leave town.

== Current statistics==

::<math>Z_{LC} = Z_L + Z_C = 0</math>

*Population ([[2000]] Census): 4,661

Both of these impedances are functions of complex [[angular frequency]] ''s'':

*Housing Units: 1,675

*Land Area: 1.67 square miles

*Water Area: 0.07 square miles

*Zip Codes: 07440, 07444

*Area Code: (973)

*County: Morris

*State: New Jersey

[http://www.hometownlocator.com/ZCTA.cfm?ZIPCode=07440 2000 Census Info]

== External links ==

::<math>Z_C = { 1 \over Cs }</math>

::<math>Z_L = Ls </math>

*[http://www.pequannocktownship.org/ Pequannock Township Official Website]

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

*[http://www.pequannock.org/ Pequannock School District]

*[http://www.hometownlocator.com/ZCTA.cfm?ZIPCode=07440 2000 Census Information]

::<math> s = \pm j \omega_o = \pm j {1 \over \sqrt{L C}}</math>

*[http://www.pequannocklacrosse.org/ Pequannock Lacrosse Club]

*[http://www.rootsweb.com/~genepool/njpequa.htm Revolutionary Petition of Patriots (May 1776)]

where the resonance frequency &omega;_o is given in the expression above.

=== Damping factor ===

The [[damping|damping factor]] of the circuit (in [[radians]] per second) is:

::<math> \zeta = {R \over 2L}</math>

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. In this case, the ''RLC circuit'' becomes a good approximation to an ideal [[LC circuit]], which is not realizable in practice (even if the resistor is removed from the circuit, there is always a small but non-zero amount of resistance in the wiring and interconnects between the other circuit elements that can never be eliminated entirely).

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, and the [[bandwidth]] (in radians per second) is

::<math> \delta \omega = 2 \zeta = { R \over L}</math>

Alternatively, the bandwidth in hertz is

::<math>BW = { \delta f \over 2 \pi } = { \zeta \over \pi }= { R \over 2 \pi L}</math>

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)|power]] is proportional to the square of the circuit voltage (or current), the frequency response will drop to <math> { 1 \over \sqrt{2} } </math> at the half-power frequencies.

=== Quality or Q factor ===

The Quality of the tuned circuit, or [[Q factor]], is calculated as the ratio of the resonance frequency <math>\omega_o</math> to the bandwidth <math>\beta</math> (in radians per second):

::<math>Q = {\omega_o \over \delta \omega } = {\omega_o \over 2\zeta } = {L \over R \sqrt{LC}} = {1 \over R} \sqrt{L \over C}</math>

Or in hertz:

::<math>Q = {f_o \over delta f} = {2 \pi f_o L \over R} = {1 \over \sqrt{R^2 C / L}} = {1 \over R} \sqrt{L \over C}</math>

=== Damped Resonance ===

The [[damping|damped]] resonant frequency derives from the natural frequency and the damping factor:

:<math> \omega_d = \sqrt{ \omega_o^2 - \alpha^2 } </math>

In an oscillator circuit

:<math> \zeta \ \ << \ \ \omega_o </math>.

As a result

:<math> \omega_d \ \ = \ \ \omega_o \ \ </math> (approx).

==Configurations==

Every RLC circuit consists of two components: a ''power source'' and ''resonator''. There are two types of power sources &ndash; [[Thevenin equivalent|Th&eacute;venin]] and [[Norton equivalent|Norton]]. Likewise, there are two types of resonators &ndash; series [[LC circuit|LC]] and parallel LC. As a result, there are four configurations of RLC circuits:

*Series LC with Th&eacute;venin power source

*Series LC with Norton power source

*Parallel LC with Th&eacute;venin power source

*Parallel LC with Norton power source.

==Circuit Analysis==

===Series RLC with Th&eacute;venin 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_L(0), and the capacitor voltage V_C(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 &alpha; and &omega;₀, 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 &alpha; and &omega;₀, 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>

===== 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 &lambda;₁ and &lambda;₂ 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 &alpha; and &omega;_o

::<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 &omega;

::<math> | I( i \omega ) | = | Y(i \omega) | \times | V(i \omega) |</math>

If we choose 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 &omega; (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 [[resonance 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 &Phi; 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]]

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