timeconstant

Time constant, often denoted by tau (τ), is a parameter that describes the speed of a transient response in first-order linear time-invariant systems. It is defined as the time required for the system’s response to reach about 63% of its final value after a step input. Equivalently, after one time constant, the exponential approach toward the final state has progressed to approximately 1 − e^−1 ≈ 0.632 of the change.

In electrical circuits, the time constant provides a concrete measure of how quickly signals respond in simple RC and RL networks. For an RC circuit, where a capacitor charges toward a supply voltage, the time constant is τ = RC. The capacitor voltage as a function of time is Vc(t) = V_source (1 − e^(−t/τ)). For discharging, Vc(t) = V_initial e^(−t/τ). In an RL circuit, τ = L/R, and the current or voltage follows analogous exponential forms with the same time-constant concept.

Beyond these circuits, the time constant generalizes to first-order systems described by a single dominant pole. In such cases τ characterizes the speed of the dominant exponential mode; more complex systems may have multiple time constants, with the largest often setting the overall convergence rate. Practical measures, such as settling time (the time to remain within a small percentage of the final value), are frequently expressed as a few to several multiples of τ (commonly 4–5τ).

The time constant is expressed in units of time (seconds in SI) and reflects intrinsic properties of the system, such as resistance and capacitance in electronics, or analogous quantities like thermal resistance and heat capacity in thermal systems. It provides a concise, widely used metric for predicting and comparing transient behavior.