RC time constant

Cutoff frequency

The time constant \tau is related to the RC circuit's cutoff frequency fc, by :\tau = RC = \frac{1}{2 \pi f_c} \approx \frac{0.159}{f_c}, or, equivalently, :f_c = \frac{1}{2 \pi R C} = \frac{1}{2 \pi \tau} \approx \frac{0.159}{\tau}. 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{0.22}{f_c} :rise time (10% to 90%) t_r \approx 2.2 \tau \approx \frac{0.35}{f_c}

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VC(τ) ≈ % of V0

The tangent of the voltage V(t) hits the zero axis at a time t + \tau.

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.

References

References

1. Andrew Gray. (1908). "Lord Kelvin". Dent.
2. Ido Yavetz. (1995). "From Obscurity to Enigma". Birkhäuser.
3. (2004). "Interconnect-centric Design for Advanced SoC and NoC". Springer.
4. Scott Hamilton. (2007). "An Analog Electronics Companion". Cambridge University Press.