In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, the time constant, usually denoted by the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
letter (tau), is the
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
characterizing the response to a step input of a first-order,
linear time-invariant
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defin ...
(LTI) system.
[Concretely, a first-order LTI system is a system that can be modeled by a single ]first order differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
in time. Examples include the simplest single-stage electrical RC circuit
A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC c ...
s and RL circuit
A resistor–inductor circuit (RL circuit), or RL filter or RL network, is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor, eithe ...
s. The time constant is the main
characteristic unit
Nondimensionalization is the partial or full removal of physical dimensions from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units a ...
of a first-order LTI system.
In the time domain, the usual choice to explore the time response is through the
step response
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the out ...
to a
step input, or the
impulse response to a
Dirac delta function input.
In the frequency domain (for example, looking at the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the step response, or using an input that is a simple sinusoidal function of time) the time constant also determines the
bandwidth
Bandwidth commonly refers to:
* Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range
* Bandwidth (computing), the rate of data transfer, bit rate or thr ...
of a first-order time-invariant system, that is, the frequency at which the output signal power drops to half the value it has at low frequencies.
The time constant is also used to characterize the frequency response of various
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
systems –
magnetic tape
Magnetic tape is a medium for magnetic storage made of a thin, magnetizable coating on a long, narrow strip of plastic film. It was developed in Germany in 1928, based on the earlier magnetic wire recording from Denmark. Devices that use magne ...
s,
radio transmitter
In electronics and telecommunications, a radio transmitter or just transmitter is an electronic device which produces radio waves with an antenna. The transmitter itself generates a radio frequency alternating current, which is applied to the ...
s and
receivers, record cutting and replay equipment, and
digital filter
In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, t ...
s – which can be modelled or approximated by first-order LTI systems. Other examples include time constant used in
control system
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
s for integral and derivative action controllers, which are often
pneumatic, rather than electrical.
Time constants are a feature of the
lumped system analysis
The lumped-element model (also called lumped-parameter model, or lumped-component model) simplifies the description of the behaviour of spatially distributed physical systems, such as electrical circuits, into a topology consisting of discrete e ...
(lumped capacity analysis method) for thermal systems, used when objects cool or warm uniformly under the influence of convective cooling or warming.
Physically, the time constant represents the elapsed time required for the system response to decay to zero if the system had continued to decay at the initial rate, because of the progressive change in the rate of decay the response will have actually decreased in value to in this time (say from a step decrease). In an increasing system, the time constant is the time for the system's
step response
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the out ...
to reach of its final (asymptotic) value (say from a step increase). In radioactive decay the time constant is related to the
decay constant
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
(λ), and it represents both the mean lifetime of a decaying system (such as an atom) before it decays, or the time it takes for all but 36.8% of the atoms to decay. For this reason, the time constant is longer than the
half-life
Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable ato ...
, which is the time for only 50% of the atoms to decay.
Differential equation
First order LTI systems are characterized by the differential equation
:
where represents the
exponential decay constant and is a function of time
:
The right-hand side is the ''forcing function'' describing an external driving function of time, which can be regarded as the system ''input'', to which is the ''response'', or system output. Classical examples for are:
The
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...
, often denoted by :
:
the
impulse function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
, often denoted by , and also the sinusoidal input function:
:
or
:
where is the amplitude of the forcing function, is the frequency in Hertz, and is the frequency in radians per second.
Example solution
An example solution to the differential equation with initial value and no forcing function is
:
where
:
is the initial value of . Thus, the response is an exponential decay with time constant .
Discussion
Suppose
This behavior is referred to as a "decaying" exponential function. The time (tau) is referred to as the "time constant" and can be used (as in this case) to indicate how rapidly an exponential function decays.
Here:
* is time (generally in control engineering)
* is the initial value (see "specific cases" below).
Specific cases
# Let
; then
, and so
# Let
; then
# Let
, and so
# Let
; then
After a period of one time constant the function reaches = approximately 37% of its initial value. In case 4, after five time constants the function reaches a value less than 1% of its original. In most cases this 1% threshold is considered sufficient to assume that the function has decayed to zero – as a rule of thumb, in control engineering a stable system is one that exhibits such an overall damped behavior.
Relation of time constant to bandwidth
![Single-pole frequency response](https://upload.wikimedia.org/wikipedia/commons/d/d1/Single-pole_frequency_response.JPG)
Suppose the forcing function is chosen as sinusoidal so:
:
(Response to a real cosine or sine wave input can be obtained by taking the real or imaginary part of the final result by virtue of
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
.) The general solution to this equation for times , assuming is:
:
For long times the decaying exponentials become negligible and the
steady-state solution or long-time solution is:
:
The magnitude of this response is:
:
By convention, the bandwidth of this system is the frequency where drops to half-value, or where . This is the usual
bandwidth
Bandwidth commonly refers to:
* Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range
* Bandwidth (computing), the rate of data transfer, bit rate or thr ...
convention, defined as the frequency range where power drops by less than half (at most −3 dB). Using the frequency in hertz, rather than radians/s ():
:
The notation stems from the expression of power in
decibels
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a Power, root-power, and field quantities, power or root-power quantity on a logarithmic scale. Two signals whose ...
and the observation that half-power corresponds to a drop in the value of by a factor of 1/2 or by 3 decibels.
Thus, the time constant determines the bandwidth of this system.
Step response with arbitrary initial conditions
![Single-pole step response](https://upload.wikimedia.org/wikipedia/commons/7/75/Single-pole_step_response.JPG)
Suppose the forcing function is chosen as a step input so:
:
with the Heaviside step function. The general solution to this equation for times , assuming is:
:
(It may be observed that this response is the limit of the above response to a sinusoidal input.)
The long-time solution is time independent and independent of initial conditions:
:
The time constant remains the same for the same system regardless of the starting conditions. Simply stated, a system approaches its final, steady-state situation at a constant rate, regardless of how close it is to that value at any arbitrary starting point.
For example, consider an electric motor whose startup is well modelled by a first-order LTI system. Suppose that when started from rest, the motor takes of a second to reach 63% of its nominal speed of 100 RPM, or 63 RPM—a shortfall of 37 RPM. Then it will be found that after the next of a second, the motor has sped up an additional 23 RPM, which equals 63% of that 37 RPM difference. This brings it to 86 RPM—still 14 RPM low. After a third of a second, the motor will have gained an additional 9 RPM (63% of that 14 RPM difference), putting it at 95 RPM.
In fact, given ''any'' initial speed of a second later this particular motor will have gained an additional
Examples
Time constants in electrical circuits
![Series RC resistor voltage](https://upload.wikimedia.org/wikipedia/commons/3/39/Series_RC_resistor_voltage.svg)
In an
RL circuit
A resistor–inductor circuit (RL circuit), or RL filter or RL network, is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor, eithe ...
composed of a single resistor and inductor, the time constant ''
'' (in
second
The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ...
s) is
:
where ''R'' is the
resistance (in
ohm
Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm.
Ohm or OHM may also refer to:
People
* Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm''
* Germán Ohm (born 1936), Mexican boxer
* Jörg Ohm (b ...
s) and ''L'' is the
inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
(in
henrys).
Similarly, in an
RC circuit
A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC c ...
composed of a single resistor and capacitor, the time constant
(in seconds) is:
:
where ''R'' is the resistance (in
ohm
Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm.
Ohm or OHM may also refer to:
People
* Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm''
* Germán Ohm (born 1936), Mexican boxer
* Jörg Ohm (b ...
s) and ''C'' is the
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
(in
farad
The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units (SI). It is named after the English physicist Michael Faraday (1791–1867). In SI base unit ...
s).
Electrical circuits are often more complex than these examples, and may exhibit multiple time constants (See
Step response
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the out ...
and
Pole splitting Pole splitting is a phenomenon exploited in some forms of frequency compensation used in an electronic amplifier. When a capacitor is introduced between the input and output sides of the amplifier with the intention of moving the pole lowest in fre ...
for some examples.) In the case where
feedback
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
is present, a system may exhibit unstable, increasing oscillations. In addition, physical electrical circuits are seldom truly linear systems except for very low amplitude excitations; however, the approximation of linearity is widely used.
In digital electronic circuits another measure, the
FO4
In digital electronics, Fan-out of 4 is a measure of time used in digital CMOS technologies: the gate delay of a component with a fan-out of 4.
Fan out = Cload / Cin, where
:Cload = total MOS gate capacitance driven by the logic gate under con ...
is often used. This can be converted to time constant units via the equation
.
Thermal time constant
Time constants are a feature of the
lumped system analysis
The lumped-element model (also called lumped-parameter model, or lumped-component model) simplifies the description of the behaviour of spatially distributed physical systems, such as electrical circuits, into a topology consisting of discrete e ...
(lumped capacity analysis method) for thermal systems, used when objects cool or warm uniformly under the influence of
convective cooling or warming. In this case, the heat transfer from the body to the ambient at a given time is proportional to the temperature difference between the body and the ambient:
:
where ''h'' is the
heat transfer coefficient
In thermodynamics, the heat transfer coefficient or film coefficient, or film effectiveness, is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ). ...
, and ''A''
s is the surface area, ''T'' is the temperature function, i.e., ''T''(''t'') is the body temperature at time ''t'', and ''T''
a is the constant ambient temperature. The positive sign indicates the convention that ''F'' is positive when heat is ''leaving'' the body because its temperature is higher than the ambient temperature (''F'' is an outward flux). If heat is lost to the ambient, this heat transfer leads to a drop in temperature of the body given by:
[
:
where ''ρ'' = density, ''c''p = ]specific heat
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
and ''V'' is the body volume. The negative sign indicates the temperature drops when the heat transfer is ''outward'' from the body (that is, when ''F'' > 0). Equating these two expressions for the heat transfer,
:
Evidently, this is a first-order LTI system that can be cast in the form:
:
with
:
In other words, larger masses ''ρV'' with higher heat capacities ''c''p lead to slower changes in temperature (longer time constant ''τ''), while larger surface areas ''A''s with higher heat transfer ''h'' lead to more rapid temperature change (shorter time constant ''τ'').
Comparison with the introductory differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
suggests the possible generalization to time-varying ambient temperatures ''T''a. However, retaining the simple constant ambient example, by substituting the variable Δ''T'' ≡ (''T − T''a), one finds:
:
Systems for which cooling satisfies the above exponential equation are said to satisfy Newton's law of cooling
In the study of heat transfer, Newton's law of cooling is a physical law which states that
The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment.
The law is frequently q ...
. The solution to this equation suggests that, in such systems, the difference between the temperature of the system and its surroundings Δ''T'' as a function of time ''t'', is given by:
:
where Δ''T''0 is the initial temperature difference, at time ''t'' = 0. In words, the body assumes the same temperature as the ambient at an exponentially slow rate determined by the time constant.
Time constants in Biophysics
In an excitable cell such as a muscle
Skeletal muscles (commonly referred to as muscles) are organs of the vertebrate muscular system and typically are attached by tendons to bones of a skeleton. The muscle cells of skeletal muscles are much longer than in the other types of muscl ...
or neuron
A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa. N ...
, the time constant of the membrane potential is
:
where ''r''m is the resistance across the membrane and ''c''m is the capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
of the membrane.
The resistance across the membrane is a function of the number of open ion channels and the capacitance is a function of the properties of the lipid bilayer
The lipid bilayer (or phospholipid bilayer) is a thin polar membrane made of two layers of lipid molecules. These membranes are flat sheets that form a continuous barrier around all cells. The cell membranes of almost all organisms and many vir ...
.
The time constant is used to describe the rise and fall of membrane voltage, where the rise is described by
:
and the fall is described by
:
where voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
is in millivolts, time is in seconds, and is in seconds.
''V''max is defined as the maximum voltage change from the resting potential A relatively static membrane potential which is usually referred to as the ground value for trans-membrane voltage.
The relatively static membrane potential of quiescent cells is called the resting membrane potential (or resting voltage), as opp ...
, where
:
where ''r''m is the resistance across the membrane and ''I'' is the membrane current.
Setting for ''t'' = for the rise sets ''V''(''t'') equal to 0.63''V''max. This means that the time constant is the time elapsed after 63% of ''V''max has been reached
Setting for ''t'' = for the fall sets ''V''(''t'') equal to 0.37''V''max, meaning that the time constant is the time elapsed after it has fallen to 37% of ''V''max.
The larger a time constant is, the slower the rise or fall of the potential of a neuron. A long time constant can result in temporal summation
Summation, which includes both spatial summation and temporal summation, is the process that determines whether or not an action potential will be generated by the combined effects of excitatory and inhibitory signals, both from multiple simultane ...
, or the algebraic summation of repeated potentials. A short time constant rather produces a coincidence detector through spatial summation
Summation, which includes both spatial summation and temporal summation, is the process that determines whether or not an action potential will be generated by the combined effects of excitatory and inhibitory signals, both from multiple simultane ...
.
Exponential decay
In exponential decay, such as of a radioactive
Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consi ...
isotope, the time constant can be interpreted as the mean lifetime
A quantity is subject to exponential decay if it decreases at a rate Proportionality (mathematics), proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and ...
. The half-life
Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable ato ...
''T''''HL'' is related to the exponential time constant by
:
The reciprocal of the time constant is called the decay constant
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
, and is denoted
Meteorological sensors
A time constant is the amount of time it takes for a meteorological sensor to respond to a rapid change in a measure, and until it is measuring values within the accuracy tolerance usually expected of the sensor.
This most often applies to measurements of temperature, dew-point temperature, humidity and air pressure. Radiosonde
A radiosonde is a battery-powered telemetry instrument carried into the atmosphere usually by a weather balloon that measures various atmospheric parameters and transmits them by radio to a ground receiver. Modern radiosondes measure or calcula ...
s are especially affected due to their rapid increase in altitude.
See also
*RC time constant
The RC time constant, also called tau, the time constant (in seconds) of an RC circuit, is equal to the product of the circuit resistance (in ohms) and the circuit capacitance (in farads), i.e.
: \tau = RC econds
It is the time required to c ...
*Cutoff 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 ( attenuated or reflected) rather tha ...
* Exponential decay
* Lead-lag compensator
* Length constant
*Rise time In electronics, when describing a voltage or current step function, rise time is the time taken by a signal to change from a specified low value to a specified high value. These values may be expressed as ratiosSee for example , and . or, equivale ...
*Fall time
In electronics, fall time (pulse decay time) t_f is the time taken for the amplitude of a pulse to decrease (fall) from a specified value (usually 90% of the peak value exclusive of overshoot or undershoot) to another specified value (usually 10 ...
*Frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...
* Impulse response
*Step response
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the out ...
*Transition time In electronics, when describing a voltage or current step function, rise time is the time taken by a signal to change from a specified low value to a specified high value. These values may be expressed as ratiosSee for example , and . or, equivalen ...
*Settling time
In control theory the settling time of a dynamical system such as an amplifier or other output device is the time elapsed from the application of an ideal instantaneous step input to the time at which the amplifier output has entered and remained ...
Notes
References
External links
Conversion of time constant τ to cutoff frequency fc and vice versa
Energy and Thermal Time Constant of Buildings
{{Authority control
Physical constants
Neuroscience