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Encyclopedia > Time constant

In physics and engineering, the time constant usually denoted by the Greek letter τ, (tau), characterizes the frequency response of a first-order, linear time-invariant (LTI) system. Examples include electrical RC circuits and RL circuits. It is also used to characterize the frequency response of various signal processing systems – magnetic tapes, radio transmitters and receivers, record cutting and replay equipment, and digital filters – which can be modelled or approximated by first-order LTI systems. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Engineering is the discipline of acquiring and applying knowledge of design, analysis, and/or construction of works for practical purposes. ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ... In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ... resistor-capacitor circuit (RC circuit), or RC filter or RC network, is one of the simplest analogue electronic filters. ... This article does not cite any references or sources. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... Compact audio cassette Magnetic tape is a non-volatile storage medium consisting of a magnetic coating on a thin plastic strip. ... In communications and information processing, a transmitter (sometimes abbreviated XMTR) is an object (source) which sends information to an observer (receiver). ... For the device which is a tuner (radio) and a amplifier and/or loudspeaker, see receiver (home stereo). ... An FIR filter In electronics,nirali a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. ...


Other examples include time constant used in control systems for integral and derivative action controllers, which are often pneumatic, rather than electrical. A control system is a device or set of devices to manage, command, direct or regulate the behaviour of other devices or systems. ... Pneumatics, from the Greek πνευματικός (pneumatikos, coming from the wind) is the use of pressurized air in science and technology. ...


Physically, the constant represents the time it takes the system's step response to reach approximately 63% of its final (asymptotic) value, ie about 37% below its final value. In control theory the unit step response is the response of a dynamic system to the Heaviside step function. ...

Contents

Differential equation

First order LTI systems are characterized by the differential equation

 {dV over dt} = - alpha V ,

where   alpha represents the exponential decay constant and V is a function of time t A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

 V  =  V(t) ,

The time constant is related to the exponential decay constant by

 tau =  { 1 over alpha } ,

General Solution

The general solution to the differential equation is

 V(t)  =  V_o e^{-alpha t}  =  V_o e^{-t / tau} ,

where

 V_o  =  V(t=0) ,

is the initial value of V.


Control Engineering

The diagram below depicts the exponential function y = Aeat in the specific case where a < 0, otherwise referred to as a "decaying" exponential function:


Image:Exponential_function_showing_time_constant.jpg Image File history File links No higher resolution available. ...


Suppose

y=Ae^{-at}  =  Ae^{-{t over tau}}


then

tau={ 1 over a}


The term τ (tau) is referred to as the "time constant" and can be used (as in this case) to indicate how rapidly an exponential function decays.


Where:

t = time (generally always t > 0 in control engineering)
A = initial value (see "specific cases" below).

Specific cases

1). Let t = 0, hence y = Ae0, and so y = A
2). Let t = τ, hence y = Ae − 1, ≈ 0.37A
3). Let y=f(t)=Ae^{-{t over tau}}, and so  lim_{t to infty}f(t) = 0
4). Let t = 5τ, hence y = Ae − 5, ≈ 0.0067A

After a period of one time constant the function reaches e-1 = 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 - Hence in control engineering a stable system is mostly assumed to have settled after five time constants.


Examples of time constants

Time constants in electrical circuits

In an RL circuit, the time constant τ (in seconds) is This article does not cite any references or sources. ... This article is about the unit of time. ...

 tau  =  { L over R } ,

where R is the resistance (in ohms) and L is the inductance (in henries). A multimeter can be used to measure resistance in ohms. ... An electric current i flowing around a circuit produces a magnetic field and hence a magnetic flux Φ through the circuit. ... An inductor. ...


Similarly, in an RC circuit, the time constant τ (in seconds) is: resistor-capacitor circuit (RC circuit), or RC filter or RC network, is one of the simplest analogue electronic filters. ...

 tau  =  R C ,

where R is the resistance (in ohms) and C is the capacitance (in farads). Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ... Examples of various types of capacitors. ...


Thermal time constant

See discussion page.


Time constants in neurobiology

In an action potential (or even in a passive spread of signal) in a neuron, the time constant τ is A. A schematic view of an idealized action potential illustrates its various phases as the action potential passes a point on a cell membrane. ... This article is about cells in the nervous system. ...

 tau  =  r_{m} c_{m} ,

where rm is the resistance across the membrane and cm is the capacitance of the membrane. Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ...


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. Ion channels are present in the membranes that surround all biological cells. ... This fluid lipid bilayer cross section is made up entirely of phosphatidylcholine. ...


The time constant is used to describe the rise and fall of the action potential, where the rise is described by A. A schematic view of an idealized action potential illustrates its various phases as the action potential passes a point on a cell membrane. ...

 V(t)  =  V_{max} (1 - e^{-t /tau}) ,

and the fall is described by

 V(t)  =  V_{max} e^{-t /tau} ,

Where voltage is in millivolts, time is in seconds, and τ is in seconds. International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ...


Vmax is defined as the maximum voltage attained in the action potential, where

 V_{max}  =  r_{m}I ,

where rm is the resistance across the membrane and I is the current flow.


Setting for t = τ for the rise sets V(t) equal to 0.63Vmax. This means that the time constant is the time elapsed after 63% of Vmax has been reached.


Setting for t = τ for the fall sets V(t) equal to 0.37Vmax, meaning that the time constant is the time elapsed after it has fallen to 37% of Vmax.


The larger a time constant is, the slower the rise or fall of the potential of neuron. A long time constant can result in temporal summation, or the algebraic summation of repeated potentials. The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...


Radioactive half-life

The half-life THL of a radioactive isotope is related to the exponential time constant τ by Half-Life For a quantity subject to exponential decay, the half-life is the time required for the quantity to fall to half of its initial value. ... Radioactive decay is the process in which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. ...

 T_{HL} = tau cdot mathrm{ln2} ,

See also

The RC time constant, usually denoted by the Greek letter &#964; (tau), is a parameter that characterizes the frequency response of a resistance-capacitance (RC) circuit. ... A bode plot of the Butterworth filters frequency response, with corner frequency labeled. ... An equalization(EQ) filter is a filter, usually adjustable, chiefly meant to compensate for the unequal frequency response of some other signal processing circuit or system. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... Length constant is a constant used in neurobiology signified by the Greek letter lambda (λ). In an action potential in a neuron, the constant λ is where rm is the resistance across the membrane, ri is the resistance inside the membrane, and ro is the resistance outside the membrane. ...

External links


  Results from FactBites:
 
Time constant - Wikipedia, the free encyclopedia (454 words)
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 time constant is used to describe the rise and fall of the action potential, where the rise is described by
The larger a time constant is, the slower the rise or fall of the potential of neuron.
Archived -- Prisoners Of Time (1928 words)
As time runs out on the teacher, perceptive students are left to wonder about the integrity of an instructional system that behaves, year-in and year-out, as though the last chapters of their textbooks are not important.
By relying on time as the metric for school organization and curriculum, we have built a learning enterprise on a foundation of sand, on five premises educators know to be false.
Time, the missing element in the school reform debate, is also the overlooked solution to the standards problem.
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