Time constant

In <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Engineering/RFVg8H3I">engineering</a>, the time constant, usually denoted by the <a href="/facts/Greek_language/pzDd6eIz">Greek</a> letter τ (tau), is the <a href="/facts/Parameter/zFvVFMv9">parameter</a> characterizing the response to a step input of a first-order, <a href="/facts/LTI_system_theory/kEmj0QDC">linear time-invariant</a> (LTI) system. The time constant is the main <a href="/facts/Characteristic_unit/f2r2yqvb">characteristic unit</a> of a first-order LTI system. It gives speed of the response.
In the time domain, the usual choice to explore the time response is through the <a href="/facts/Step_response/7pq4VyNt">step response</a> to a <a href="/facts/Heaviside_step_function/bgsUtxgP">step input</a>, or the <a href="/facts/Impulse_response/l4vaE5er">impulse response</a> to a <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> input. In the frequency domain (for example, looking at the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of the step response, or using an input that is a simple sinusoidal function of time) the time constant also determines the <a href="/facts/Bandwidth_(signal_processing)/7eMDzfJI">bandwidth</a> of a first-order time-invariant system, that is, the frequency at which the output signal power drops to half the value it has at low frequencies.
The time constant is also used to characterize the frequency response of various <a href="/facts/Signal_processing/Npaja6zb">signal processing</a> systems – <a href="/facts/Magnetic_tape/wj6IDTR1">magnetic tapes</a>, <a href="/facts/Radio_transmitter/nFYghtKv">radio transmitters</a> and <a href="/facts/Radio_receiver/5olDTxkg">receivers</a>, record cutting and replay equipment, and <a href="/facts/Digital_filter/2wcxCRdh">digital filters</a> – which can be modelled or approximated by first-order LTI systems. Other examples include time constant used in <a href="/facts/Control_system/8CEjIDlS">control systems</a> for integral and derivative action controllers, which are often <a href="/facts/Pneumatic/QzkvCCpk">pneumatic</a>, rather than electrical.
Time constants are a feature of the <a href="/facts/Lumped_system_analysis/Nqtt18v5">lumped system analysis</a> (lumped capacity analysis method) for thermal systems, used when objects cool or warm uniformly under the influence of convective cooling or warming.
Physically, the time constant represents the elapsed time required for the system response to decay to zero if the system had continued to decay at the initial rate, because of the progressive change in the rate of decay the response will have actually decreased in value to 1 / e ≈ 36.8% in this time (say from a step decrease). In an increasing system, the time constant is the time for the system's <a href="/facts/Step_response/7pq4VyNt">step response</a> to reach 1 − 1 / e ≈ 63.2% of its final (asymptotic) value (say from a step increase). In radioactive decay the time constant is related to the <a href="/facts/Decay_constant/wLB1zYDP">decay constant</a> (λ), and it represents both the mean lifetime of a decaying system (such as an atom) before it decays, or the time it takes for all but 36.8% of the atoms to decay. For this reason, the time constant is longer than the <a href="/facts/Half-life/kGVH8BAB">half-life</a>, which is the time for only 50% of the atoms to decay.

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Time constant