Time Constant

Time constant - Wikipedia, the free encyclopedia
In physics and engineering, the time constant usually denoted by the Greek ... 3.3 Time constants in neurobiology. 3.4 Radioactive half-life ...
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time constant: Definition from Answers.com
time constant ( ?t?m ?känst?nt ) ( physics ) The time required for a physical quantity to rise from zero to 1-1/ e (that is, 63.2%) of its final
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RC time constant - Wikipedia, the free encyclopedia
The time constant ? is related to the cutoff frequency fc, an alternative ... Time constant and exponential decay. RC circuit and RL circuit ...
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System Dynamics - Time Constants
Introduction to system dynamics for first and second order ... time-constant behavior. ... have measured the time constant that determines how much ...
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RC Time Constant
The value of one time constant is expressed mathematically as t = RC. ... If the time constant and the initial or final voltage for the circuit in ...
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In physics and engineering, the time constant usually denoted by the Greek language letter \tau, (tau), characterizes the frequency response of a first-order, LTI system theory (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 radio receivers, record cutting and replay equipment, and digital filters – which can be modelled or approximated by first-order LTI systems.

Other examples include time constant used in control systems for integral and derivative action controllers, which are often pneumatic, rather than electrical.

Physically, the time 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.

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 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=Ae^{at} in the specific case where a

then \tau={ 1 \over a}

The term \tau (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=Ae^0, and so y=A

2). Let t= \tau, 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=5 \tau, 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 as a rule of thumb.

Examples of time constants Time constants in electrical circuits In an RL circuit, the time constant \tau (in seconds) is \tau \ = \ { L \over R } \,

where R is the resistance (in ohms) and L is the inductance (in henry (inductance)).

Similarly, in an RC circuit, the time constant \tau (in seconds) is: \tau \ = \ R C \,

where R is the resistance (in ohms) and C is the capacitance (in farads).

===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 \tau is \tau \ = \ r_{m} c_{m} \,

where rm is the resistance across the membrane and cm is the capacitance 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 time constant is used to describe the rise and fall of the action potential, where the rise is described by 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 \tau is in seconds.

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 = \tau 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 = \tau 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.

Radioactive half-life The half-life THL of a radioactive decay isotope is related to the exponential time constant \tau by T_{HL} = \tau \cdot \mathrm{ln2} \,

See also

RC time constant
Cutoff frequency
EQ filter
Exponential decay
Length constant

External links

Conversion of time constant τ to cutoff frequency fc and vice versa
All about circuits - Voltage and current calculations

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