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It is the time required to charge the capacitor, through the resistor, from an initial charge voltage of zero to approximately 63.2% of the value of an applied DC voltage, or to discharge the capacitor through the same resistor to approximately 36.8% of its initial charge voltage.
This means that the time constant is the time elapsed after 63% of V max has been reached Setting for t = 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 a neuron.
The article states this as a definition of RC time constant: It is the time required to charge the capacitor, through the resistor, by ≈ 63.2 percent of the difference between the initial value and final value or discharge the capacitor to ≈36.8 percent.
Prouhet–Thue–Morse constant [63] ... Constant related to the asymptotic behavior of Lebesgue constants [65] 0.98943 12738 31146 95174 [Mw 48] [OEIS ...
The general time- and transfer-constants (TTC) analysis [1] is the generalized version of the Cochran-Grabel (CG) method, [2] which itself is the generalized version of zero-value time-constants (ZVT), which in turn is the generalization of the open-circuit time constant method (OCT). [3]
That is, τ is the time it takes V L to reach V( 1 / e ) and V R to reach V(1 − 1 / e ). The rate of change is a fractional 1 − 1 / e per τ. Thus, in going from t = Nτ to t = (N + 1)τ, the voltage will have moved about 63% of the way from its level at t = Nτ toward its final value.
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For applications in control theory, according to Levine (1996, p. 158), rise time is defined as "the time required for the response to rise from x% to y% of its final value", with 0% to 100% rise time common for underdamped second order systems, 5% to 95% for critically damped and 10% to 90% for overdamped ones. [6]