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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 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 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.
Additional recommended knowledge
First order LTI systems are characterized by the differential equation
where represents the exponential decay constant and V is a function of time t
The time constant is related to the exponential decay constant by
The general solution to the differential equation is
is the initial value of V.
The diagram below depicts the exponential function y = Aeat in the specific case where a < 0, otherwise referred to as a "decaying" exponential function:
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
where R is the resistance (in ohms) and L is the inductance (in henries).
Similarly, in an RC circuit, the time constant τ (in seconds) is:
where R is the resistance (in ohms) and C is the capacitance (in farads).
Thermal time constant
See discussion page.
Time constants in neurobiology
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
and the fall is described by
Where voltage is in millivolts, time is in seconds, and τ is in seconds.
Vmax is defined as the maximum voltage attained in the action potential, where
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 half-life THL of a radioactive isotope is related to the exponential time constant τ by
Step Response with Non-Zero Initial Conditions
If the motor is initially spinning at a constant speed(expressed by voltage), the time constant τ is 63% of Vinfinity minus Vo.
can be used where the initial and final voltages, respectively, are:
Step 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:
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Time_constant". A list of authors is available in Wikipedia.