RC time constant

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The RC time constant, also called tau, is the time constant (in seconds) of an RC circuit, is equal to the product of the circuit resistance (in ohms) and the circuit capacitance (in farads), i.e.

 \tau = R * C

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, specifically 1-e^{-1}, more specifically as voltage to charge the capacitor versus time

Charging  V(t) = V_0(1-e^{-t/ \tau}) [1]
Discharging  V(t) = V_0(e^{-t/ \tau})

Cutoff frequency[edit]

The time constant \tau is related to the cutoff frequency fc, an alternative parameter of the RC circuit, by

\tau = RC = \frac{1}{2 \pi f_c}

or, equivalently,

f_c = \frac{1}{2 \pi R C} = \frac{1}{2 \pi \tau}

where resistance in ohms and capacitance in farads yields the time constant in seconds or the frequency in Hz.

Short conditional equations:

fc in Hz = 159155 / τ in µs
τ in µs = 159155 / fc in Hz

Other useful equations are:

rise time (20% to 80%) t_r \approx 1.4 \tau \approx \frac{0.22}{f_c}
rise time (10% to 90%) t_r \approx 2.2 \tau \approx \frac{0.35}{f_c}

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

Organization
 
Time constant \tau
in µs
Cutoff frequency fc
in Hz
RIAA 7958 20
RIAA, NAB 3183 50
1592 100
RIAA 318 500.5
200 796
140 1137
MC 120 1326
NAB 100 1592
MC 90 1768
RIAA, FM 75 2122
FM 50 / 75 2122 / 3183
NAB, PCM 50 3183
DIN 35 4547
25 6366
AES 17.5 9095
PCM 15 10610
12.5 12732
10 15915
Ortofon 3.5 45473
RIAA 3.18 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[edit]

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

Resistive-capacitive delay, or RC delay, hinders the further increasing of speed in microelectronic integrated circuits. When the feature size becomes smaller and smaller to increase the 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.[2] 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.[3][4][5]

See also[edit]

References[edit]

  1. ^ http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capdis.html
  2. ^ Andrew Gray (1908). Lord Kelvin. Dent. 
  3. ^ Ido Yavetz (1995). From Obscurity to Enigma. Birkhäuser. ISBN 3-7643-5180-2. 
  4. ^ Jari Nurmi, Hannu Tenhunen, Jouni Isoaho, and Axel Jantsch (2004). Interconnect-centric Design for Advanced SoC and NoC. Springer. ISBN 1-4020-7835-8. 
  5. ^ Scott Hamilton (2007). An Analog Electronics Companion. Cambridge University Press. ISBN 0-521-68780-2. 

External links[edit]