Figure E.1: General first-order circuit response showing initial value, final
value, and half-life time.
The general first-order circuit response takes the form
x(t) = x(∞) + [x(0) - x(∞)]e-t∕τ, t ≥ 0,
where x(0) is the initial value at the start of the transient, x(∞) is the final value, and τ is the time constant. Figure E.1 plots this equation for the case of a final value lower than the initial value. This figure also shows the half-life time THL, defined as the time interval from the onset of the transient at time t = 0 to the time at which x(t) reaches the midpoint of the initial and final values. The half-life is easy to measure on an oscilloscope from which the time constant follows as
τ = THL/ln2.