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Cutoff frequency

The time constant τ {\displaystyle \tau } is related to the RC circuit's cutoff frequency fc, by

τ = R C = 1 2 π f c {\displaystyle \tau =RC={\frac {1}{2\pi f_{c}}}}

or, equivalently,

f c = 1 2 π R C = 1 2 π τ {\displaystyle 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 ( ω c = 2 π f c ) {\displaystyle (\omega _{c}{=}2\pi f_{c})} is simply the reciprocal of the time constant.

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

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 ≈ 1.4 τ ≈ 0.22 f c {\displaystyle t_{r}\approx 1.4\tau \approx {\frac {0.22}{f_{c}}}} rise time (10% to 90%) t r ≈ 2.2 τ ≈ 0.35 f c {\displaystyle 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.

Calculator

.000001

1000000

1

1

1

1

1

1

1

.368

36.8

1

0.368

1 1

1

0.159

1

1

1

1

For instance, 1  of resistance with 1  of capacitance produces a time constant of approximately 1 seconds. This τ corresponds to a cutoff frequency of approximately 159 millihertz or 1 radians. If the capacitor has an initial voltage V0 of 1 , then after 1 τ (approximately 1 seconds or 1.443 half-lives), the capacitor's voltage will discharge to approximately 368 millivolts:

 VC(1τ) ≈ 36.8% of V0 

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.¹ 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.²³⁴

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

External links

• RC Time Constant Calculator
• Conversion time constant τ {\displaystyle \tau } to cutoff frequency fc and back
• RC time constant

References

1. Andrew Gray (1908). Lord Kelvin. Dent. p. 265. https://archive.org/details/lordkelvinanacc01graygoog ↩

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

3. Jari Nurmi; Hannu Tenhunen; Jouni Isoaho & Axel Jantsch (2004). Interconnect-centric Design for Advanced SoC and NoC. Springer. ISBN 1-4020-7835-8. 1-4020-7835-8 ↩

4. Scott Hamilton (2007). An Analog Electronics Companion. Cambridge University Press. ISBN 978-0-521-68780-5. 978-0-521-68780-5 ↩