time constant

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Time constant

A characteristic time that governs the approach of an exponential function to a steady-state value. When a physical quantity is varying as a decreasing exponential function of time as in Eq. (1), or as an increasing exponential function as in Eq. (2), the approach to the steady-state value achieved after a long time is governed by a characteristic time T as given in Eq. (3). This time T is called the time constant. (1) 

(3) When time t is zero, f(t) in Eq. (1) has the magnitude 1, and when t equals T the magnirude is 1/e. Here e is the transcendental number whose value is approximately 2.71828, and the change in magnitude is 1 - (1/e) = 0.63212. The function has moved 63.2% of the way to its final value. The same factor also holds for Eq. (2).

The initial rate of change of both the increasing and decreasing functions is equal to the maximum amplitude of the function divided by the time constant.

The concept of time constant is useful when evaluating the presence of transient phenomena.

time constant

Symbol: τ. A measure of the speed with which an electronic system can respond to a sudden change at its input.

Time Constant


a generalized parameter that describes the dynamic properties (time lag) of a given entity and that has the dimension of time. Any complex physical process can be represented as a set of simpler processes, each of which can be mathematically described by a first- or second-order linear differential equation. These simple processes are called basic components in the theory of automatic control. For example, a first-order aperiodic basic component is described by the equation

where x is the input coordinate, y the output coordinate, k a proportionality factor, and T the time constant.

Time constants are widely used to calculate the dynamics of various entities or processes under study. Thus, the heating of a substance within a closed container at constant ambient temperature is described by the equation

where m is the mass and c the heat capacity of the substance, a is the coefficient of heat transfer into the medium surrounding the container, F is the reduced cooling surface, Θ is the ambient temperature, Θ0 is the initial temperature of the substance, and p is the heat energy suddenly imparted to the substance from a heater at the initial moment of time t = 0. The change in the temperature of the substance is determined by the equation

where T = mc/ α F is the time constant. The greater the time constant, the less rapidly the heating proceeds.

The time constant of transient phenomena in electric circuits characterizes the rate at which current or voltage varies in the circuit. For example, when a capacitor with capacitance C is charged from a DC source with electromotive force E. through a resistance r that is much greater than the internal resistance of the current source, the voltage across the capacitor plates varies according to the law

uc = E(1 - e-1/T)

where T = rC is the time constant that determines the rate at which the charging process occurs. The time constant of an electric circuit with inductance L is L/r.


time constant

[′tīm ‚kän·stənt]
The time required for a physical quantity to rise from zero to 1-1/ e (that is, 63.2%) of its final steady value when it varies with time t as 1 -e -kt .
The time required for a physical quantity to fall to 1/ e (that is, 36.8%) of its initial value when it varies with time t as e -kt .
Generally, the time required for an instrument to indicate a given percentage of the final reading resulting from an input signal. Also known as lag coefficient.
References in periodicals archive ?
The discrete time one step explicit method [25] is employed for estimating the unknown parameters in (14) and then to retrieve the unknown time constant ([T.
To assess the impact of the value of the electromagnetic Time constant of the dynamics of the positioning of Trees and shrubs, consider the nature of the transients during Acceleration and fixing position of the rotor with a Predetermined moment of inertia for different inductance Values (Fig.
When a step change in the measured environment is applied, the response can be determined as a function of the time constant, x.
It is interesting to see how the time constant varies depending on the operating conditions of the heat pump.
f] is adjusted to match the both time constants (equation (4)), current [I.
In [10, 11] the value of tachometric flow meters' time constant is considered as an unchangeable during the step response process.
This threshold viscosity is high enough to demarcate the gel region from the smaller molecule region using the sigmoidal curve and allow for objective interpretations for the time constants.
Boiler thermal time constant CB was identified when a special heat to boiler signal (steam flow signal subtracted from fuel flow signal in per units) is provided to input while boiler steam pressure variation--to the output.
si] can be physically interpreted as either angle ratios or time constant divided by the period of exposure for the regenerator side and supply side, respectively.
Treated wood is a solid electrolyte, and the baseline impedance spectra of wood has at least one time constant in the frequency range of interest to corrosion scientists (Zelinka 2006).
The droop setting value and rotor time constant have been optimized [8].
The Hench system saves energy by calculating the thermal time constant of products.

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