RC delay

The RC time constant, denoted ' (lowercase tau), the time constant of a resistor–capacitor circuit (RC circuit), is equal to the product of the circuit resistance and the circuit capacitance:
:$\tau = RC \, .$

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 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'', where $63.2\% \approx 1e^$ and $36.8\% \approx e^$. When using the International System of Units, is in ohms, is in farads, and is in seconds.

Discharging a capacitor through a series resistor to zero volts from an initial voltage of results in the capacitor having the following exponentially-decaying voltage curve:

:$V_\text(t) = V_0 \cdot (e^)$

Charging an uncharged capacitor through a series resistor to an applied constant input voltage results in the capacitor having the following voltage curve over time:
:$V_\text(t) = V_0 \cdot (1-e^)$
which is a vertical mirror image of the charging curve.

Cutoff frequency

The time constant $\tau$ is related to the RC circuit's cutoff frequency ''f''_c, by
:$\tau = RC = \frac \approx \frac,$
or, equivalently,
:$f_c = \frac = \frac \approx \frac.$
Using resistance in ohms and capacitance in farads yields a time constant in seconds and cutoff frequency in hertz (Hz). The cutoff frequency when expressed as an angular frequency $( \omega_c 2 \pi f_c )$ is simply the reciprocal of the time constant.

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.

A rise time that depends primarily on an RC circuit will be proportional to the time constant:
:rise time (20% to 80%) $t_r \approx 1.4 \tau \approx \frac$
:rise time (10% to 90%) $t_r \approx 2.2 \tau \approx \frac$

Calculator

For instance, of resistance with of capacitance produces a time constant of approximately This corresponds to a cutoff frequency of approximately or If the capacitor has an initial voltage of , then after (approximately or the capacitor's voltage will discharge to approximately

 (''τ'') ≈ % of  

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 (RC delay) hinders microelectronic integrated circuit (IC) speed improvements. As semiconductor 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. 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.{{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

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
* Rise time

References

External links

RC Time Constant Calculator
* ttp://www.tpub.com/neets/book2/3d.htm RC time constantbr>Calibration of RC time constant in CMOS IC analog filters

Analog circuits
Time