In electronics, when describing a voltage or current step function, rise time is the time taken by a signal to change from a specified low value to a specified high value. These values may be expressed as ratios or, equivalently, as percentages with respect to a given reference value. In analog electronics and digital electronics, these percentages are commonly the 10% and 90% (or equivalently 0.1 and 0.9) of the output step height: however, other values are commonly used. 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. Similarly, fall time (pulse decay time) t f {\displaystyle t_{f}} is the time taken for the amplitude of a pulse to decrease (fall) from a specified value (usually 90% of the peak value exclusive of overshoot or undershoot) to another specified value (usually 10% of the maximum value exclusive of overshoot or undershoot). Limits on undershoot and oscillation (also known as ringing and hunting) are sometimes additionally stated when specifying fall time limits. According to Orwiler (1969, p. 22), the term "rise time" applies to either positive or negative step response, even if a displayed negative excursion is popularly termed fall time.
Overview Rise time is an analog parameter of fundamental importance in high speed electronics, since it is a measure of the ability of a circuit to respond to fast input signals. There have been many efforts to reduce the rise times of circuits, generators, and data measuring and transmission equipment. These reductions tend to stem from research on faster electron devices and from techniques of reduction in stray circuit parameters (mainly capacitances and inductances). For applications outside the realm of high speed electronics, long (compared to the attainable state of the art) rise times are sometimes desirable: examples are the dimming of a light, where a longer rise-time results, amongst other things, in a longer life for the bulb, or in the control of analog signals by digital ones by means of an analog switch, where a longer rise time means lower capacitive feedthrough, and thus lower coupling noise to the controlled analog signal lines.
Factors affecting rise time For a given system output, its rise time depends both on the rise time of input signal and on the characteristics of the system. For example, rise time values in a resistive circuit are primarily due to stray capacitance and inductance. Since every circuit has not only resistance, but also capacitance and inductance, a delay in voltage and/or current at the load is apparent until the steady state is reached. In a pure RC circuit, the output risetime (10% to 90%) is approximately equal to 2.2 RC.
Alternative definitions Other definitions of rise time, apart from the one given by the Federal Standard 1037C (1997, p. R-22) and its slight generalization given by Levine (1996, p. 158), are occasionally used: these alternative definitions differ from the standard not only for the reference levels considered. For example, the time interval graphically corresponding to the intercept points of the tangent drawn through the 50% point of the step function response is occasionally used. Another definition, introduced by Elmore (1948, p. 57), uses concepts from statistics and probability theory. Considering a step response V(t), Elmore redefines the delay time tD as the first moment of its first derivative V′(t), i.e.
t D = ∫ 0 + ∞ t V ′ ( t ) d t ∫ 0 + ∞ V ′ ( t ) d t . {\displaystyle t_{D}={\frac {\int _{0}^{+\infty }tV^{\prime }(t)\,\mathrm {d} t}{\int _{0}^{+\infty }V^{\prime }(t)\,\mathrm {d} t}}.}
Finally, he defines the rise time tr by using the second moment
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