Time constant

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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 modeled 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 about 63.2% of its final value. See exponential decay.

1 Differential equation
- 1.1 General Solution
2 Examples of time constants
3 See also
4 External links

[edit] 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

$\tau = { 1 \over \alpha }$

[edit] 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.

[edit] Examples of time constants

[edit] Time constants in electrical circuits

In an RL circuit, the time constant τ (in seconds) is

$\tau \ = \ { L \over R }$

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

Similarly, in an RC circuit, the time constant τ is:

$\tau \ = \ R C$

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

[edit] Time constants in neurobiology

In an action potential in a neuron, the constant τ is

$\tau \ = \ r_{m} c_{m}$

where r_m is the resistance across the membrane and c_m 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 τ is in millimeters.

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

$V_{max} \ = \ r_{m}I$

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

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

Setting for t = τ for the fall sets V(t) equal to 0.37V_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 neuron. A long time constant can result in temporal summation, or the algebraic summation of repeated potentials.