RC time constant

The RC time constant, also called tau, 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.

$\tau = RC$

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

The time constant $\tau$ is related to the cutoff frequency f_c, 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:

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%) $t_r \approx 1.4 \tau \approx \frac{0.22}{f_c}$

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

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

• Cutoff frequency and frequency response
• 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

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

• RC Time Constant Calculator
• $\tau$
• RC time constant