In
electronics, when describing a
voltage or
currentstep 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]
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.[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.
Factors affecting rise time
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]
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:[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 responseV(t), he redefines the
delay timetD as the
first moment of its
first derivativeV′(t), i.e.
Finally, he defines the rise time tr by using the second moment
Rise time of model systems
Notation
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.
t1 is the time at which the output of the system under analysis is at the x% of the steady-state value, while t2 the one at which it is at the y%, both measured in
seconds.
tr is the rise time of the analysed system, measured in seconds. By definition,
fL is the lower
cutoff frequency (-3 dB point) of the analysed system, measured in
hertz.
fH is higher cutoff frequency (-3 dB point) of the analysed system, measured in hertz.
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.
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 V0 amplitude:
Solving for time
and finally,
Since t1 and t2 are such that
solving these equations we find the analytical expression for t1 and t2:
The rise time is therefore proportional to the time constant:[19]
Finally note that, if the 20% to 80% rise time is considered instead, tr becomes:
One-stage low-pass LR network
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
Rise time of damped second order systems
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]
Consider a system composed by n cascaded non interacting blocks, each having a rise time tri, i = 1,…,n, and no
overshoot in their
step response: suppose also that the input signal of the first block has a rise time whose value is trS.[22] Afterwards, its output signal has a rise time tr0 equal to
^For example
Valley & Wallman (1948, p. 72, footnote 1) state that "For some applications it is desirable to measure rise time between the 5 and 95 per cent points or the 1 and 99 per cent points.".
^
abPrecisely,
Levine (1996, p. 158) states: "The rise time is the time required for the response to rise from x% to y% of its final value. For overdamped
second order systems, the 0% to 100% rise time is normally used, and for underdamped systems(...)the 10% to 90% rise time is commonly used". However, this statement is incorrect since the 0%–100% rise time for an overdamped 2nd order control system is infinite, similarly to the one of an
RC network: this statement is repeated also in the second edition of the book (
Levine 2011, p. 9-3 (313)).
^According to
Valley & Wallman (1948, p. 72), "The most important characteristics of the reproduction of a leading edge of a rectangular pulse or step function are the rise time, usually measured from 10 to 90 per cent, and the "
overshoot"". And according to
Cherry & Hooper (1968, p. 306), "The two most significant parameters in the square-wave response of an
amplifier are its rise time and percentage tilt".
^This beautiful one-page paper does not contain any calculation.
Henry Wallman simply sets up a table he calls "
dictionary", paralleling concepts from
electronics engineering and
probability theory: the key of the process is the use of
Laplace transform. Then he notes, following the correspondence of concepts established by the "
dictionary", that the
step response of a cascade of blocks corresponds to the
central limit theorem and states that: "This has important practical consequences, among them the fact that if a network is free of overshoot its time-of-response inevitably increases rapidly upon cascading, namely as the square-root of the number of cascaded network"(
Wallman 1950, p. 91).
National Communication Systems, Technology and Standards Division (1 March 1997), Federal Standard 1037C. Telecommunications: Glossary of Telecommunications Terms, FSC TELE, vol. FED–STD–1037, Washington: General Service Administration Information Technology Service, p. 488.
Petitt, Joseph Mayo; McWhorter, Malcolm Myers (1961), Electronic Amplifier Circuits. Theory and Design, McGraw-Hill Electrical and Electronics Series, New York–Toronto–London:
McGraw-Hill, pp. xiii+325.
Valley, George E. Jr.;
Wallman, Henry (1948), "§2 of chapter 2 and §1–7 of chapter 7", Vacuum Tube Amplifiers, MIT Radiation Laboratory Series, vol. 18,
New York:
McGraw-Hill., pp. xvii+743.