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.^{[1]} These values may be expressed as ratios^{[2]} or, equivalently, as percentages^{[3]} with respect to a given reference value. In analog electronics and digital electronics^{[citation needed]}, these percentages are commonly the 10% and 90% (or equivalently 0.1 and 0.9) of the output step height:^{[4]} however, other values are commonly used.^{[5]} 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 overdamped second order systems, 5% to 95% for critically damped and 10% to 90% for underdamped ones.^{[6]} 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.^{[7]}

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.^{[8]} 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.

For a given system output, its rise time depend both on the rise time of input signal and on the characteristics of the system.^{[9]}

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*.^{[10]}

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:^{[11]} 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.^{[12]} Another definition, introduced by Elmore (1948, p. 57),^{[13]} uses concepts from statistics and probability theory. Considering a step response *V*(*t*), he redefines the delay time *t _{D}* as the first moment of its first derivative

Finally, he defines the rise time *t _{r}* by using the second moment

All notations and assumptions required for the analysis are listed here.

- Following Levine (1996, p. 158, 2011, 9-3 (313)), we define
*x%*as the percentage low value and*y%*the percentage high value respect to a reference value of the signal whose rise time is to be estimated. *t*_{1}is the time at which the output of the system under analysis is at the*x%*of the steady-state value, while*t*_{2}the one at which it is at the*y%*, both measured in seconds.*t*is the rise time of the analysed system, measured in seconds. By definition,_{r}*f*is the lower cutoff frequency (-3 dB point) of the analysed system, measured in hertz._{L}*f*is higher cutoff frequency (-3 dB point) of the analysed system, measured in hertz._{H}*h*(*t*) is the impulse response of the analysed system in the time domain.*H*(*ω*) is the frequency response of the analysed system in the frequency domain.- The bandwidth is defined as and since the lower cutoff frequency
*f*is usually several decades lower than the higher cutoff frequency_{L}*f*,_{H} - All systems analyzed here have a frequency response which extends to 0 (low-pass systems), thus exactly.
- For the sake of simplicity, all systems analysed in the "Simple examples of calculation of rise time" section are unity gain electrical networks, and all signals are thought as voltages: the input is a step function of
*V*_{0}volts, and this implies that *ζ*is the damping ratio and*ω*_{0}is the natural frequency of a given second order system.

The aim of this section is the calculation of rise time of step response for some simple systems:

A system is said to have a *Gaussian response* if it is characterized by the following frequency response

where *σ* > 0 is a constant,^{[14]} related to the high cutoff frequency by the following relation:

Even if this kind frequency response is not realizable by a causal filter,^{[15]} its usefulness lies in the fact that behaviour of a cascade connection of first order low pass filters approaches the behaviour of this system more closely as the number of cascaded stages asymptotically rises to infinity.^{[16]} The corresponding impulse response can be calculated using the inverse Fourier transform of the shown frequency response

Applying directly the definition of step response,

To determine the 10% to 90% rise time of the system it is necessary to solve for time the two following equations:

By using known properties of the error function, the value *t* = −*t*_{1} = *t*_{2} is found: since *t _{r}* =

and finally

^{[17]}

For a simple one-stage low-pass RC network,^{[18]} the 10% to 90% rise time is proportional to the network time constant *τ* = *RC*:

The proportionality constant can be derived from the knowledge of the step response of the network to a unit step function input signal of *V*_{0} amplitude:

Solving for time

and finally,

Since *t*_{1} and *t*_{2} are such that

solving these equations we find the analytical expression for *t*_{1} and *t*_{2}:

The rise time is therefore proportional to the time constant:^{[19]}

Now, noting that

^{[20]}

then

and since the high frequency cutoff is equal to the bandwidth,

^{[17]}

Finally note that, if the 20% to 80% rise time is considered instead, *t _{r}* becomes:

Even for a simple one-stage low-pass RL network, the 10% to 90% rise time is proportional to the network time constant *τ* = *L*⁄*R*. The formal proof of this assertion proceed exactly as shown in the previous section: the only difference between the final expressions for the rise time is due to the difference in the expressions for the time constant *τ* of the two different circuits, leading in the present case to the following result

According to Levine (1996, p. 158), for underdamped systems used in control theory rise time is commonly defined as the time for a waveform to go from 0% to 100% of its final value:^{[6]} accordingly, the rise time from 0 to 100% of an underdamped 2nd-order system has the following form:^{[21]}

The quadratic approximation for normalized rise time for a 2nd-order system, step response, no zeros is:

where *ζ* is the damping ratio and *ω*_{0} is the natural frequency of the network.

Consider a system composed by *n* cascaded non interacting blocks, each having a rise time *t _{ri}*,

According to Valley & Wallman (1948, pp. 77–78), this result is a consequence of the central limit theorem and was proved by Wallman (1950):^{[23]}^{[24]} however, a detailed analysis of the problem is presented by Petitt & McWhorter (1961, §4–9, pp. 107–115),^{[25]} who also credit Elmore (1948) as the first one to prove the previous formula on a somewhat rigorous basis.^{[26]}