Editing RC time constant

{{short description|Time constant of an RC circuit}}

The '''RC time constant''', denoted ''{{mvar|τ}}'' (lowercase [[tau]]), the [[time constant]] of a [[resistor–capacitor circuit]] (RC circuit), is equal to the product of the circuit [[Electrical resistance and conductance|resistance]] and the circuit [[capacitance]]: 
:<math> \tau = RC \, . </math>

[[File:RC Series Filter (with V&I Labels).svg|thumb|right|When the capacitance {{Mvar|C}} in this [[series and parallel circuits#Series circuits|series]] [[RC circuit]] is charged or discharged through the resistance {{Mvar|R}}, the capacitor's voltage {{Mvar|V<sub>C</sub>}} is an [[Exponential decay|exponentially-decaying]] function of time scaled by the RC time constant.|269x269px]]

It is the [[time]] required to charge the [[capacitor]], through the [[resistor]], from an initial charge voltage of zero to approximately 63.2% of the value of an applied [[Direct current|DC voltage]], or to discharge the capacitor through the same resistor to approximately 36.8% of its initial charge voltage. These values are derived from the mathematical constant ''[[e (mathematical constant)|e]]'', where <math>63.2\% \approx 1{-}e^{-1}</math> and <math>36.8\% \approx e^{-1}</math>. When using the [[International System of Units]], {{mvar|R}} is in [[Ohm|ohms]], {{mvar|C}} is in [[Farad|farads]], and {{mvar|τ}} is in [[Second|seconds]].

Discharging a capacitor through a [[in series|series]] resistor to zero volts from an initial voltage of {{Mvar|V<sub>0</sub>}} results in the capacitor having the following [[Exponential decay|exponentially-decaying]] voltage curve:

:<math>V_\text{C}(t) = V_0 \cdot (e^{-t/ \tau}) </math>

Charging an uncharged capacitor through a series resistor to an applied constant input voltage {{Mvar|V<sub>0</sub>}} results in the capacitor having the following voltage curve over time:
:<math>V_\text{C}(t) = V_0 \cdot (1-e^{-t/ \tau}) </math>
which is a vertical [[mirror image]] of the charging curve.

==Cutoff frequency==
The time constant <math>\tau</math> is related to the RC circuit's [[cutoff frequency]] ''f''<sub>c</sub>, 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 [[Ohm|ohms]] and capacitance in [[farads]] yields the time constant in [[Second|seconds]] or the cutoff frequency in [[hertz]] (Hz). The cutoff frequency when expressed as an [[angular frequency]] <math>( \omega_c {=} 2 \pi f_c )</math> is simply the [[Multiplicative inverse|reciprocal]] of the time constant.

Short conditional equations using the value for <math>10^6 / (2 \pi)</math>:
:''f''<sub>c</sub> in Hz = 159155 / &tau; in [[Microsecond|μs]]
:&tau; in μs = 159155 / ''f''<sub>c</sub> 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>

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.

== Calculator ==

{{calculator
	|id=capacitance
	|formula=capacitance_number*pow(10,capacitance_unitlog10)
	|type=hidden
	|default=.000001
}}

{{calculator
	|id=resistance
	|formula=resistance_number*pow(10,resistance_unitlog10)
	|type=hidden
	|default=1000000
}}

{{calculator
	|id=tau
	|formula=resistance*capacitance
	|type=hidden
	|default=1
}}

{{calculator
	|id=tau_unitlog10div3floor
	|formula=min(max(floor(log10(tau)/3),-5),5)
	|type=hidden
	|default=1
}}

{{calculator
	|id=tau_sigdigits
	|formula=round(tau/pow(10,tau_unitlog10div3floor*3),3)
	|type=hidden
	|default=1
}}

{{calculator
	|id=initialvoltage
	|formula=initialvoltage_number*pow(10,initialvoltage_unitlog10)
	|type=hidden
	|default=1
}}

{{calculator
	|id=decaytime
	|formula=decaytime_in_tau*tau
	|type=hidden
	|default=1
}}

{{calculator
	|id=decaytime_unitlog10div3floor
	|formula=min(max(ifzero(decaytime,0,floor(log10(abs(decaytime))/3)),-5),5)
	|type=hidden
	|default=1
}}

{{calculator
	|id=decaytime_sigdigits
	|formula=round(decaytime/pow(10,decaytime_unitlog10div3floor*3),3)
	|type=hidden
	|default=1
}}

{{calculator
	|id=fraction_of_initialvoltage
	|default=.368
	|type=hidden
	|formula=exp(-decaytime_in_tau)
}}

{{calculator
	|id=percent_of_initialvoltage
	|default=36.8
	|type=hidden
	|formula=round(fraction_of_initialvoltage*100,1)
}}

{{calculator
	|id=initialvoltage
	|type=hidden
	|default=1
	|formula=initialvoltage_number*pow(10,initialvoltage_unitlog10)
}}

{{calculator
	|id=finalvoltage
	|type=hidden
	|size=4
	|NaN-text=?
	|formula=initialvoltage*fraction_of_initialvoltage|default=0.368}}

{{calculator
	|id=finalvoltage_unitlog10div3floor
	|formula=min(max(ifzero(decaytime,0,floor(log10(abs(finalvoltage))/3)),-5),5)
	|type=hidden
	|default=1
}}
{{calculator
	|id=finalvoltage_sigdigits
	|formula=round(finalvoltage/pow(10,finalvoltage_unitlog10div3floor*3),3)
	|type=hidden
	|default=1
}}

{{calculator
	|id=xpix
	|type=hidden
	|formula=87.5+562*max(min(decaytime_in_tau,5.1),-0.1)/5
}}

{{calculator
	|id=ypix
	|type=hidden
	|formula=12+434.5*(1-max(min(percent_of_initialvoltage,101),0)/100)
}}

{{calculator
	|id=cutoff_angular
	|type=hidden
	|size=4
	|NaN-text=?
	|formula=1/tau
	|default=1
}}

{{calculator
	|id=cutoff_ordinal
	|type=hidden
	|size=4
	|NaN-text=?
	|formula=cutoff_angular/(2*PI)
	|default=0.159
}}

{{calculator
	|id=cutoff_angular_unitlog10div3floor
	|formula=min(max(floor(log10(cutoff_angular)/3),-4),4)
	|type=hidden
	|default=1
}}

{{calculator
	|id=cutoff_angular_sigdigits
	|formula=round(cutoff_angular/pow(10,cutoff_angular_unitlog10div3floor*3),3)
	|type=hidden
	|default=1
}}

{{calculator
	|id=cutoff_ordinal_unitlog10div3floor
	|formula=min(max(floor(log10(cutoff_ordinal)/3),-4),4)
	|type=hidden
	|default=1
}}

{{calculator
	|id=cutoff_ordinal_sigdigits
	|formula=round(cutoff_ordinal/pow(10,cutoff_ordinal_unitlog10div3floor*3),3)
	|type=hidden
	|default=1
}}

For instance, {{nowrap|{{calculator|id=resistance_number|type=number|min=1|size=3|default=1|max=999|step=.1}}&nbsp;{{calculator|id=resistance_unitlog10|type=select|class=cdx-select|mapping={ "teraohms":12, "gigaohms":9, "megaohms":6, "kiloohms":3, "ohms":0, "milliohms":-3 }|style=min-width:1ch|value=6}}}} of resistance with {{nowrap|{{calculator|id=capacitance_number|type=number|min=1|size=3|default=1|max=999|step=.1}}&nbsp;{{calculator|id=capacitance_unitlog10|type=select|class=cdx-select|mapping={ "farads":0, "millifarads":-3, "microfarads":-6, "nanofarads":-9, "picofarads":-12}|style=min-width:1ch|value=-6}}}} of capacitance produces a time constant of approximately {{nowrap|{{calculator
	|id=tau_sigdigits_label
	|type=plain
	|size=4
	|NaN-text=?
	|formula=tau_sigdigits
	|default=1
}} {{calculator
	|id=tau_unit
	|formula=tau_unitlog10div3floor
	|type=plain
	|default=seconds
	|mapping={
		"[unknown time unit]": "default",
		"petaseconds": 5,
		"teraseconds": 4,
		"gigaseconds": 3,
		"megaseconds": 2,
		"kiloseconds": 1,
		"seconds": 0,
		"milliseconds": -1,
		"microseconds": -2,
		"nanoseconds": -3,
		"picoseconds": -4,
		"femtoseconds": -5
	}
}}.}} This {{Mvar|τ}} corresponds to a [[cutoff frequency]] of approximately {{nowrap|{{calculator
	|id=cutoff_ordinal_sigdigits_label
	|type=plain
	|size=4
	|NaN-text=?
	|formula=cutoff_ordinal_sigdigits
	|default=159
}} {{calculator
	|id=cutoff_ordinal_unit
	|formula=cutoff_ordinal_unitlog10div3floor
	|type=plain
	|default=millihertz
	|mapping={ "[unknown frequency unit]":"default", "terahertz":4, "gigahertz":3, "megahertz":2, "kilohertz":1, "hertz":0, "millihertz":-1, "microhertz":-2, "nanohertz":-3, "picohertz":-4}
}}}} or {{nowrap|{{calculator
	|id=cutoff_angular_sigdigits_label
	|type=plain
	|size=4
	|NaN-text=?
	|formula=cutoff_angular_sigdigits
	|default=1
}} {{calculator
	|id=cutoff_angular_unit
	|formula=cutoff_angular_unitlog10div3floor
	|type=plain
	|default=radians per second
	|mapping={ "[unknown frequency unit]":"default", "teraradians per second":4, "gigaradians per second":3, "megaradians per second":2, "kiloradians per second":1, "radians per second":0, "milliradians per second":-1, "microradians per second":-2, "nanoradians per second":-3, "picoradians per second":-4}
}}.}} If the capacitor has an initial voltage {{Math|''V''<sub>0</sub>}} of {{nowrap|{{calculator
	|id=initialvoltage_number
	|type=number
	|min=-999
	|size=3
	|default=1
	|max=999
	|step=.1
}} {{calculator
	|id=initialvoltage_unitlog10
	|type=select
	|class=cdx-select
	|mapping={ "teravolts":12, "gigavolts":9, "megavolts":6, "kilovolts":3, "volts":0, "millivolts":-3, "microvolts":-6, "nanovolts":-9, "picovolts":-12}
	|style=min-width:1ch
	|value=0
}}}}, then after {{nowrap|{{calculator
	|id=decaytime_in_tau
	|default=1
	|type=number
	|size=5
	|step=.1
	|min=0
}}&nbsp;{{Mvar|τ}}}} (approximately {{nowrap|{{calculator
	|id=decaytime_sigdigits_label
	|type=plain
	|size=4
	|NaN-text=?
	|formula=decaytime_sigdigits
	|default=1
}} {{calculator
	|id=decaytime_unit
	|formula=decaytime_unitlog10div3floor
	|type=plain
	|default=seconds
	|mapping={
		"[unknown time unit]": "default",
		"petaseconds": 5,
		"teraseconds": 4,
		"gigaseconds": 3,
		"megaseconds": 2,
		"kiloseconds": 1,
		"seconds": 0,
		"milliseconds": -1,
		"microseconds": -2,
		"nanoseconds": -3,
		"picoseconds": -4,
		"femtoseconds": -5
	}
}}}} or {{nowrap|{{calculator
	|id=decaytime_in_halflives
	|type=plain
	|size=4
	|NaN-text=?
	|formula=round(decaytime_in_tau/log(2),3)
	|default=1.443
}}&nbsp;[[half-lives]]),}} the capacitor's voltage will discharge to approximately {{nowrap|{{calculator
	|id=finalvoltage_sigdigits_label
	|type=plain
	|size=4
	|NaN-text=?
	|formula=finalvoltage_sigdigits
	|default=368
}} {{calculator
	|id=finalvoltage_unit
	|formula=finalvoltage_unitlog10div3floor
	|type=plain
	|default=millivolts
	|mapping={
		"[unknown voltage unit]": "default",
		"petavolts": 5,
		"teravolts": 4,
		"gigavolts": 3,
		"megavolts": 2,
		"kilovolts": 1,
		"volts": 0,
		"millivolts": -1,
		"microvolts": -2,
		"nanovolts": -3,
		"picovolts": -4,
		"femtovolts": -5
	}
}}:}}

<div class="floatnone noresize" style="position: relative; width: 900px; height: 550px;">

[[File:Series_RC_resistor_voltage_relabeled.svg|alt=|frameless|720x500px|class=skin-invert-image]]

<div style="position: absolute; left: calc( var( --calculator-xpix, 200)*1px ); top: calc( var( --calculator-ypix, 286.6)*1px ); padding: 0;">[[Image:Blue pog.svg|10px|link=|alt=|class=|]]</div>

<div style="position: absolute; left: calc( (var( --calculator-xpix, 200) + 10)*1px); top: calc( (var( --calculator-ypix, 286.6) - 10)*1px ); padding: 0; background-color: white; border: 3px solid blue; color: blue">&nbsp;{{Math|''V''<sub>C</sub>}}({{calculator|id=decaytime_in_tau_label|default=1|type=plain|formula=round(decaytime_in_tau,1)}}''τ'')&nbsp;≈&nbsp;{{calculator|default=36.8|type=plain|formula=percent_of_initialvoltage}}%&nbsp;of&nbsp;{{Math|''V''<sub>0</sub>}}&nbsp;</div>

</div>

==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 (RC delay) hinders [[microelectronics|microelectronic]] [[integrated circuit]] (IC) speed improvements. As semiconductor [[Semiconductor device fabrication#Feature size|feature size]] becomes smaller and smaller to increase the [[clock rate]], the RC delay plays an increasingly important role. This delay can be reduced by replacing the [[aluminum]] conducting wire by [[copper]] to reduce resistance or by changing the interlayer [[dielectric]] (typically [[silicon dioxide]]) to low-dielectric-constant materials to reduce 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 = https://archive.org/details/lordkelvinanacc01graygoog | page = [https://archive.org/details/lordkelvinanacc01graygoog/page/n291 265] }}</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 = https://books.google.com/books?id=SQszfj7biVMC&dq=preece+heaviside+telegraph+square&pg=PA245 }}</ref><ref>{{cite book | title = Interconnect-centric Design for Advanced SoC and NoC |author1=Jari Nurmi |author2=Hannu Tenhunen |author3=Jouni Isoaho |author4=Axel Jantsch  |name-list-style=amp | publisher = Springer | year = 2004 | isbn = 1-4020-7835-8 | url = https://books.google.com/books?id=Uj7RvVE2Ln0C&dq=vlsi+rc+delay+distributed+diffusion&pg=PA59 }}</ref><ref>{{cite book | title = An Analog Electronics Companion | author = Scott Hamilton | publisher = Cambridge University Press | year = 2007 | isbn = 978-0-521-68780-5 | url = https://books.google.com/books?id=2BntAEtXsBMC&dq=preece+distributed+heaviside+diffusion+thomson&pg=PA580 }}</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]]
* [[Rise time]]

==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 f<sub>c</sub> and back]
*[http://www.tpub.com/neets/book2/3d.htm RC time constant]
*[https://www.researchgate.net/publication/391861381_Low-Complexity_Autonomous_RC_Time_Constant_Calibration_for_Accuracy_Improvement_of_CMOS_Integrated_Analog_Frequency_Filters#fullTextFileContent Calibration of RC time constant in CMOS IC analog filters]

[[Category:Analog circuits]]
[[Category:Time]]