RC time constant

The RC time constant, denoted $τ$ (lowercase tau), the time constant (in seconds) of a resistor–capacitor circuit (RC circuit), is equal to the product of the circuit resistance (in ohms) and the circuit capacitance (in farads):

\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 $63.2\%\approx 1{-}e^{-1}$ and $36.8\%\approx e^{-1}$ . The following formulae use it, assuming a constant voltage applied across the capacitor and resistor in series, to determine the voltage across the capacitor against time:

Charging toward applied voltage (initially zero voltage across capacitor, constant V₀ across resistor and capacitor together)

V_{0}:\quad V(t)=V_{0}(1-e^{-t/\tau })

^[1]

Discharging toward zero from initial voltage (initially V₀ across capacitor, constant zero voltage across resistor and capacitor together)

V_{0}:\quad V(t)=V_{0}(e^{-t/\tau })

Cutoff frequency

The time constant $\tau$ is related to the RC circuit's cutoff frequency f_c, by

\tau =RC={\frac {1}{2\pi f_{c}}}

or, equivalently,

f_{c}={\frac {1}{2\pi RC}}={\frac {1}{2\pi \tau }}

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

Short conditional equations using the value for $10^{6}/(2\pi )$ :

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}}}

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]

References

^ "Capacitor Discharging".
^ Andrew Gray (1908). Lord Kelvin. Dent. p. 265.
^ Ido Yavetz (1995). From Obscurity to Enigma. Birkhäuser. ISBN 3-7643-5180-2.
^ Jari Nurmi; Hannu Tenhunen; Jouni Isoaho & Axel Jantsch (2004). Interconnect-centric Design for Advanced SoC and NoC. Springer. ISBN 1-4020-7835-8.
^ Scott Hamilton (2007). An Analog Electronics Companion. Cambridge University Press. ISBN 978-0-521-68780-5.

External links

[1] "Capacitor Discharging".

[2] Andrew Gray (1908). Lord Kelvin. Dent. p. 265.

[3] Ido Yavetz (1995). From Obscurity to Enigma. Birkhäuser. ISBN 3-7643-5180-2.

[4] Jari Nurmi; Hannu Tenhunen; Jouni Isoaho & 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 978-0-521-68780-5.

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