RC time constant
Encyclopedia
In an RC circuit
, the value of the time constant
(in second
s) is equal to the product of the circuit resistance (in ohms) and the circuit capacitance
(in farad
s), i.e.
= R × C. It is the time required to charge the capacitor
, through the resistor
, to 63.2 (≈ 63) percent of full charge; or to discharge it to 36.8 (≈ 37) percent of its initial voltage. These values are derived from the mathematical constant e
, specifically
and 
respectively.
is related to the cutoff frequency
fc, an alternative parameter of the RC circuit, by
.
or, equivalently,
Short conditional equations:
Other useful equations are:
Standard time constants and cutoff frequencies
for pre-emphasis/de-emphasis RC filters:
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.
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 circuit
s. 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. 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 That old analysis was superseded in the telegraph domain, but remains relevant for long on-chip interconnects.
RC circuit
A resistor–capacitor circuit ', or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source...
, the value of the time constant
Time constant
In physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...
(in second
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
s) is equal to the product of the circuit resistance (in ohms) and the circuit capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
(in farad
Farad
The farad is the SI unit of capacitance. The unit is named after the English physicist Michael Faraday.- Definition :A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second...
s), i.e.
= R × C. It is the time required to charge the capacitorCapacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
, through the resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
, to 63.2 (≈ 63) percent of full charge; or to discharge it to 36.8 (≈ 37) percent of its initial voltage. These values are derived from the mathematical constant e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
, specifically
and respectively.Cutoff frequency
The time constant is related to the cutoff frequencyCutoff frequency
In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.Typically in electronic systems such as filters and...
fc, an alternative parameter of the RC circuit, by
.or, equivalently,
Short conditional equations:
- fc in Hz = 159155 / τ in µs
- τ in µs = 159155 / fc in Hz
Other useful equations are:
- rise time (20% to 80%)
- rise time (10% to 90%)
Standard time constants and cutoff frequencies
for pre-emphasis/de-emphasis RC filters:
| Organization |
Time constant  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 |
| — | 100 | 1592 |
| MC | 90 | 1768 |
| RIAA | 75 | 2122 |
| FM | 70 | 2274 |
| NAB, PCM | 50 | 3183 |
| DIN Din DIN or Din or din can have several meanings:* A din is a loud noise.* Dīn, an Arabic term meaning "religion" or "way of life".* Din is one of the ten aspects of the Ein Sof in Kabbalah .... |
35 | 4547 |
| — | 25 | 6366 |
| AES Audio Engineering Society Established in 1948, the Audio Engineering Society draws its membership from amongst engineers, scientists, other individuals with an interest or involvement in the professional audio industry. The membership largely comprises engineers developing devices or products for audio, and persons working... |
17.5 | 9095 |
| PCM | 15 | 10610 |
| RIAA | 3.18 | 50000 |
In more complicated circuits consisting of more than one resistor and/or capacitor, the open-circuit time constant method
Open-circuit time constant method
The open-circuit time constant method is an approximate analysis technique used in electronic circuit design to determine the corner frequency of complex circuits. It also is known as the zero-value time constant technique...
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 delayGroup delay
Group delay is a measure of the time delay of the amplitude envelopes of the various sinusoidal components of a signal through a device under test, and is a function of frequency for each component...
or phase delay or the effective propagation delay of a digital
Digital
A digital system is a data technology that uses discrete values. By contrast, non-digital systems use a continuous range of values to represent information...
transition, may be dominated by resistive-capacitive effects, depending on the distance and other parameters, or may alternatively be dominated by inductive
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
, wave, and speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
effects in other realms.
Resistive-capacitive delay, or RC delay, hinders the further increasing of speed in microelectronic
Microelectronics
Microelectronics is a subfield of electronics. As the name suggests, microelectronics relates to the study and manufacture of very small electronic components. Usually, but not always, this means micrometre-scale or smaller,. These devices are made from semiconductors...
integrated circuit
Integrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...
s. When the feature size becomes smaller and smaller to increase the clock speed
Clock rate
The clock rate typically refers to the frequency that a CPU is running at.For example, a crystal oscillator frequency reference typically is synonymous with a fixed sinusoidal waveform, a clock rate is that frequency reference translated by electronic circuitry into a corresponding square wave...
, the RC delay plays an increasingly important role. This delay can be reduced by replacing the aluminum conducting wire by copper
Copper
Copper is a chemical element with the symbol Cu and atomic number 29. It is a ductile metal with very high thermal and electrical conductivity. Pure copper is soft and malleable; an exposed surface has a reddish-orange tarnish...
, thus reducing the resistance; it can also be reduced by changing the interlayer dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing 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
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
in such a wire, as explained by Lord Kelvin in the mid nineteenth century. Until Heaviside discovered that Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.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.
See also
- Time constantTime constantIn physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...
and exponential decay - RC circuitRC circuitA resistor–capacitor circuit ', or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source...
, RL circuitRL circuitA resistor-inductor circuit ', or RL filter or RL network, is one of the simplest analogue infinite impulse response electronic filters. It consists of a resistor and an inductor, either in series or in parallel, driven by a voltage source.-Introduction:The fundamental passive linear circuit...
, and RLC circuitRLC circuitAn RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively... - Filter (signal processing)Filter (signal processing)In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...
and transfer functionTransfer functionA transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e... - Cutoff frequencyCutoff frequencyIn physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.Typically in electronic systems such as filters and...
and frequency responseFrequency responseFrequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input... - EmphasisEmphasisEmphasis or emphatic may refer to:* Emphasis , intentional alteration of the amplitude-vs.-frequency characteristics of the signal to reduce adverse effects of noise...
, preemphasisPreemphasisIn processing electronic audio signals, pre-emphasis refers to a system process designed to increase the magnitude of some frequencies with respect to the magnitude of other frequencies in order to improve the overall signal-to-noise ratio by minimizing the adverse effects of such phenomena as...
, deemphasisDeemphasisIn telecommunication, de-emphasis is the complement of pre-emphasis, in the antinoise system called emphasis. Emphasis is a system process designed to decrease, , the magnitude of some frequencies with respect to the magnitude of other frequencies in order to improve the overall signal-to-noise... - High-pass filterHigh-pass filterA high-pass filter is a device that passes high frequencies and attenuates frequencies lower than its cutoff frequency. A high-pass filter is usually modeled as a linear time-invariant system...
, low-pass filterLow-pass filterA low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
, band-pass filterBand-pass filterA band-pass filter is a device that passes frequencies within a certain range and rejects frequencies outside that range.Optical band-pass filters are of common usage....
to cutoff frequency fc and back