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 depend both on the rise time of input signal and on the characteristics of the system.
Finally, he defines the rise time tr by using the second moment
Rise time of model systems
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,
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 ? = . 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 ? 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: accordingly, the rise time from 0 to 100% of an underdamped 2nd-order system has the following form:
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. 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 (1969, 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).
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, 18, New York: McGraw-Hill., pp. xvii+743.