Rise time
In
Overview
Rise time is an analog parameter of fundamental importance in
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
Alternative definitions
Other definitions of rise time, apart from the one given by the
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.
- The bandwidth is defined as and since the lower cutoff frequency fL is usually several decades lower than the higher cutoff frequency fH,
- 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 V0 volts, and this implies that
- ζ is the second order system.
Simple examples of calculation of rise time
The aim of this section is the calculation of rise time of step response for some simple systems:
Gaussian response system
A system is said to have a
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
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 = −t1 = t2 is found: since tr = t2 - t1 = 2t,
and finally
One-stage low-pass RC network
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
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]
Now, noting that
then
and since the high frequency cutoff is equal to the bandwidth,
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]
The quadratic approximation for normalized rise time for a 2nd-order system, step response, no zeros is:
where ζ is the
Rise time of cascaded blocks
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
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]
See also
Notes
- ^ "rise time", Federal Standard 1037C, August 7, 1996
- ^ See for example (Cherry & Hooper 1968, p.6 and p.306), (Millman & Taub 1965, p. 44) and (Nise 2011, p. 167).
- ^ See for example Levine (1996, p. 158), (Ogata 2010, p. 170) and (Valley & Wallman 1948, p. 72).
- ^ See for example (Cherry & Hooper 1968, p. 6 and p. 306), (Millman & Taub 1965, p. 44) and (Valley & Wallman 1948, p. 72).
- ^ 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.".
- ^ a b Precisely, 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)).
- ^ Again according to Orwiler (1969, p. 22).
- ^ 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".
- ^ See (Orwiler 1969, pp. 27–29) and the "Rise time of cascaded blocks" section.
- ^ See for example (Valley & Wallman 1948, p. 73), (Orwiler 1969, p. 22 and p. 30) or the "One-stage low-pass RC network" section.
- ^ See (Valley & Wallman 1948, p. 72, footnote 1) and (Elmore 1948, p. 56).
- ^ See (Valley & Wallman 1948, p. 72, footnote 1) and (Elmore 1948, p. 56 and p. 57, fig. 2a).
- ^ See also (Petitt & McWhorter 1961, pp. 109–111).
- ^ See (Valley & Wallman 1948, p. 724) and (Petitt & McWhorter 1961, p. 122).
- Paley-Wiener criterion: see for example (Valley & Wallman 1948, p. 721 and p. 724). Also Petitt & McWhorter (1961, p. 122) briefly recall this fact.
- ^ See (Valley & Wallman 1948, p. 724), (Petitt & McWhorter 1961, p. 111, including footnote 1, and p.) and (Orwiler 1969, p. 30).
- ^ a b Compare with (Orwiler 1969, p. 30).
- ^ Called also "single-pole filter". See (Cherry & Hooper 1968, p. 639).
- ^ Compare with (Valley & Wallman 1948, p. 72, formula (2)), (Cherry & Hooper 1968, p. 639, formula (13.3)) or (Orwiler 1969, p. 22 and p. 30).
- ^ See the section "Relation of time constant to bandwidth" section of the "Time constant" entry for a formal proof of this relation.
- ^ See (Ogata 2010, p. 171).
- ^ "S" stands for "source", to be understood as current or voltage source.
- 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).
- ^ See also (Cherry & Hooper 1968, p. 656) and (Orwiler 1969, pp. 27–28).
- ^ Cited by (Cherry & Hooper 1968, p. 656).
- ^ See (Petitt & McWhorter 1961, p. 109).
References
- John Wiley & Sons, pp. xxxii+1036.
- .
- Levine, William S. (1996), The Control Handbook, ISBN 0-8493-8570-9.
- Levine, William S. (2011) [1996], The Control Handbook: Control Systems Fundamentals (2nd ed.), ISBN 978-1-4200-7362-1.
- Millman, Jacob; Taub, Herbert (1965), Pulse, digital and switching waveforms, McGraw-Hill, pp. xiv+958.
- 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.
- Nise, Norman S. (2011), Control Systems Engineering (6th ed.), New York: ISBN 978-0470-91769-5.
- Ogata, Katsuhiko (2010) [1970], Modern Control Engineering (5th ed.), ISBN 978-0-13-615673-4.
- Orwiler, Bob (December 1969), Vertical Amplifier Circuits (PDF), Circuit Concepts, vol. 062-1145-00 (1st ed.), Beaverton, OR: Tektronix, p. 461.
- McGraw-Hill, pp. xiii+325.
- Valley, George E. Jr.; McGraw-Hill., pp. xvii+743.
- Zbl 0035.08102.