Time constant

In physics and engineering, the time constant usually denoted by the Greek letter "$au$", (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.

Time constants are also used in 'lumped capacity method' analysis of thermal systems, for example when object is cooled down under the influence of convective cooling.

Physically, the constant represents the time it takes the system's step response to reach approximately 63% of its final (asymptotic) 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:$au\; =\; \{\; 1\; over\; alpha\; \}\; ,$

General Solution

The general solution to the differential equation is:$V(t)\; =\; V\_o\; e^\{-alpha\; t\}\; =\; V\_o\; e^\{-t\; /\; au\}\; ,$

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 , otherwise referred to as a "decaying" exponential function:

Suppose:$y=Ae^\{-at\}\; =\; Ae^\{-\{t\; over\; au$

then:$au=\{\; 1\; over\; a\}$

The term $au$ (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).

pecific cases

:1). Let $t=0$, hence $y=Ae^0$, and so $y=A$

:2). Let $t=\; au$, hence $y=Ae^\{-1\}$, ≈ $0.37\; A$:3). Let $y=f(t)=Ae^\{-\{t\; over\; au$, and so $lim\_\{t\; o\; infty\}f(t)\; =\; 0$

:4). Let $t=5\; au$, hence $y=Ae^\{-5\}$, ≈ $0.0067\; A$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 "$au$" (in seconds) is :$au\; =\; \{\; L\; over\; R\; \}\; ,$

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

Similarly, in an RC circuit, the time constant "$au$" (in seconds) is::$au\; =\; R\; C\; ,$

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 "$au$" is:$au\; =\; r\_\{m\}\; c\_\{m\}\; ,$

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:$V(t)\; =\; V\_\{max\}\; (1\; -\; e^\{-t\; /\; au\})\; ,$

and the fall is described by:$V(t)\; =\; V\_\{max\}\; e^\{-t\; /\; au\}\; ,$

Where voltage is in millivolts, time is in seconds, and "$au$" is in seconds.

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" = "$au$" 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" = "$au$" 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 neuron. A long time constant can result in temporal summation, or the algebraic summation of repeated potentials.

Radioactive half-life

The half-life "T"_"HL" of a radioactive isotope is related to the exponential time constant "$au$" by:$T\_\{HL\}\; =\; au\; cdot\; mathrm\{ln2\}\; ,$

tep Response with Non-Zero Initial Conditions

If the motor is initially spinning at a constant speed (expressed by voltage), the time constant $au$ is 63% of $V\_infty$ minus V_"o".

Therefore,:$V(t)\; =\; V\_o\; +\; (V\_\{infty\}\; -\; V\_o)\; cdot\; (1\; -\; e^\{-t\; /\; au\})\; ,$

can be used where the initial and final voltages, respectively, are:

:$V\_o\; =\; V(t=0)$and:$V\_\{infty\}\; =\; V(t=infty)$

tep Response from Rest

From rest, the voltage equation is a simplification of the case with non-zero ICs. With an initial velocity of zero, V₀ drops out and the resulting equation is::$V(t)\; =\; V\_\{infty\}\; cdot\; (1\; -\; e^\{-t\; /\; au\; \})\; ,$

The time constant will remain the same for the same system regardless of the starting conditions. For example, if an electric motor reaches 63% of its final speed from rest in ⅛ of a second, it will also take ⅛ of a second for the motor to reach 63% of its final speed when started with some non-zero initial speed. Simply stated, a system will require a certain amount of time to reach its final, steady-state situation regardless of how close it is to that value when started.

ee also

*RC time constant
*Cutoff frequency
*EQ filter
*Exponential decay
*Length constant

External links

* [http://www.sengpielaudio.com/calculator-timeconstant.htm Conversion of time constant τ to cutoff frequency fc and vice versa]
* [http://www.allaboutcircuits.com/vol_1/chpt_16/4.html All about circuits - Voltage and current calculations]