en>AugPi: /* Definition */

2014-01-25T16:07:22Z

Definition

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The '''RC time constant''', also called tau, is the [[time constant]] (in [[second]]s) of an [[RC circuit]], is equal to the product of the circuit resistance(in [[Ohm (unit)|ohms]]) and the circuit [[capacitance]] (in [[farad]]s), i.e.
:<math> \tau = R * C </math>

It is the time required to charge the [[capacitor]], through the [[resistor]], by ≈ 63.2 percent of the difference between the initial value and final value or discharge the capacitor to ≈36.8 percent. This value is derived from the mathematical constant ''[[e (mathematical constant)|e]]'', specifically <math>1-e^{-1}</math>, more specifically as voltage to charge the capacitor versus time

:Charging <math> V(t) = Vo(1-e^{-t/ \tau}) </math><ref>http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capdis.html</ref>
:Discharging <math> V(t) = Vo(e^{-t/ \tau}) </math>

==Cutoff frequency==
The time constant <math>\tau</math> is related to the [[cutoff frequency]] ''f''c, an alternative parameter of the RC circuit, by
:<math>\tau = RC = \frac{1}{2 \pi f_c}</math>
or, equivalently,
:<math>f_c = \frac{1}{2 \pi R C} = \frac{1}{2 \pi \tau}</math>
where resistance in ohms and capacitance in farads yields the time constant in seconds or the frequency in Hz.

Short conditional equations:
:''f''c in Hz = 159155 / τ in µs
:τ in µs = 159155 / ''f''c in Hz

Other useful equations are:
:rise time (20% to 80%) <math>t_r \approx 1.4 \tau \approx \frac{0.22}{f_c}</math>
:rise time (10% to 90%) <math>t_r \approx 2.2 \tau \approx \frac{0.35}{f_c}</math>

'''Standard time constants and cutoff frequencies''' 
'''for pre-emphasis/de-emphasis RC filters:'''

{| class="wikitable"
! Organization  
! Time constant <math>\tau</math> in µs
! Cutoff frequency fc in Hz
|-----
| [[RIAA equalization|RIAA]] || align="center" | 7958 || align="center" | 20
|-----
| RIAA, [[NAB filter|NAB]] || align="center" | 3183 || align="center" | 50
|-----
| — || align="center" | 1592 || align="center" | 100
|-----
| RIAA || align="center" | 318 || align="center" | 500.5
|-----
| — || align="center" | 200 || align="center" | 796
|-----
| — || align="center" | 140 || align="center" | 1137
|-----
| [[MC filter|MC]] || align="center" | 120 || align="center" |1326
|-----
| NAB || align="center" | 100 || align="center" | 1592
|-----
| MC || align="center" | 90 || align="center" | 1768
|-----
| RIAA, FM || align="center" | 75 || align="center" | 2122
|-----
| [[FM filter|FM]] || align="center" | 50 / 75 || align="center" | 2122 / 3183
|-----
| NAB, [[Pulse-code modulation|PCM]] || align="center" | 50 || align="center" | 3183
|-----
| [[DIN]] || align="center" | 35 || align="center" | 4547
|-----
| — || align="center" | 25 || align="center" | 6366
|-----
| [[Audio Engineering Society|AES]] || align="center" | 17.5 || align="center" | 9095
|-----
| PCM || align="center" | 15 || align="center" | 10610
|-----
| — || align="center" | 12.5 || align="center" | 12732
|-----
| — || align="center" | 10 || align="center" | 15915
|-----
| Ortofon || align="center" | 3.5 || align="center" | 45473
|-----
| RIAA || align="center" | 3.18 || align="center" | 50000
|}

In more complicated circuits consisting of more than one resistor and/or capacitor, the [[open-circuit time constant method]] provides a way of approximating the cutoff frequency by computing a sum of several RC time constants.

==Delay==

The signal delay of a wire or other circuit, measured as [[group delay]] or [[phase delay]] or the effective propagation delay of a [[Digital data|digital]] transition, may be dominated by resistive-capacitive effects, depending on the distance and other parameters, or may alternatively be dominated by [[inductance|inductive]], wave, and [[speed of light]] effects in other realms.

Resistive-capacitive delay, or RC delay, hinders the further increasing of speed in [[microelectronics|microelectronic]] [[integrated circuit]]s. When the feature size becomes smaller and smaller to increase the [[clock rate|clock speed]], the RC delay plays an increasingly important role. This delay can be reduced by replacing the [[aluminum]] conducting wire by [[copper]], thus reducing the resistance; it can also be reduced by changing the interlayer [[dielectric]] (typically silicon dioxide) to low-dielectric-constant materials, thus reducing the capacitance.

The typical digital propagation delay of a resistive wire is about half of R times C; since both R and C are proportional to wire length, the delay scales as the square of wire length. Charge spreads by [[diffusion]] in such a wire, as explained by [[Lord Kelvin]] in the mid nineteenth century.<ref>{{cite book | title = Lord Kelvin | author = Andrew Gray | publisher = Dent | year = 1908 | url = http://books.google.com/books?id=nmdBauZWa6sC&pg=PA265&dq=intitle:kelvin+diffusion+%22product+of+the+capacity+and+resistance%22+%22square+of+the+length%22 }}</ref> Until [[Heaviside]] discovered that [[Maxwell's equations]] imply wave propagation when sufficient inductance is in the circuit, this square diffusion relationship was thought to provide a fundamental limit to the improvement of long-distance telegraph cables. That old analysis was superseded in the telegraph domain, but remains relevant for long on-chip interconnects.<ref>{{cite book | title = From Obscurity to Enigma | author = Ido Yavetz | publisher = Birkhäuser | year = 1995 | isbn = 3-7643-5180-2 | url = http://books.google.com/books?id=SQszfj7biVMC&pg=PA245&dq=preece+heaviside+telegraph+square#PPA244,M1 }}</ref><ref>{{cite book | title = Interconnect-centric Design for Advanced SoC and NoC | author = Jari Nurmi, Hannu Tenhunen, Jouni Isoaho, and Axel Jantsch | publisher = Springer | year = 2004 | isbn = 1-4020-7835-8 | url = http://books.google.com/books?id=Uj7RvVE2Ln0C&pg=PA59&dq=vlsi+rc+delay+distributed+diffusion#PPA59,M1 }}</ref><ref>{{cite book | title = An Analog Electronics Companion | author = Scott Hamilton | publisher = Cambridge University Press | year = 2007 | isbn = 0-521-68780-2 | url = http://books.google.com/books?id=2BntAEtXsBMC&pg=PA580&dq=preece+distributed+heaviside+diffusion+thomson#PPA580,M1 }}</ref>

==See also==
* [[Cutoff frequency]] and [[frequency response]]
* [[Emphasis (telecommunications)|Emphasis]], [[preemphasis]], [[deemphasis]]
* [[Exponential decay]]
* [[Filter (signal processing)]] and [[transfer function]]
* [[High-pass filter]], [[low-pass filter]], [[band-pass filter]]
* [[RL circuit]], and [[RLC circuit]]

==References==

<references/>

==External links==
*[http://www.referencedesigner.com/rfcal/cal_05.php RC Time Constant Calculator]
*[http://www.sengpielaudio.com/calculator-timeconstant.htm Conversion time constant <math>\tau</math> to cutoff frequency fc and back]
*[http://www.tpub.com/neets/book2/3d.htm RC time constant]

[[Category:Electronics]]

Subobject - Revision history

en>AugPi: /* Definition */