Rise time

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 or digital electronics, 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 underdamped second order systems, 5% to 95% for critically damped and 10% to 90% for overdamped 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] Many efforts over the years have been made to reduce the rise times of generators, analog and digital circuits, measuring and data transmission equipment, focused on the research of faster electron devices and on techniques of reduction of 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 in the circuit. Because 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 by , p. R-22) and its slight generalization given by Levine (1996, p. 158), are occasionally used:^[11] these alternative definitions differ respect to the standard one 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 some 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 $V' (t)$ , i.e.

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 $t r$ by using the second moment

t_{r}^{2}={\frac {\int _{0}^{+\infty }(t-t_{D})^{2}V^{\prime }(t)\mathrm {d} t}{\int _{0}^{+\infty }V^{\prime }(t)\mathrm {d} t}}\quad \Longleftrightarrow \quad t_{r}={\sqrt {\frac {\int _{0}^{+\infty }(t-t_{D})^{2}V^{\prime }(t)\mathrm {d} t}{\int _{0}^{+\infty }V^{\prime }(t)\mathrm {d} t}}}

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.
$t 1$ 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.
$t r$ is the rise time of the analysed system, measured in seconds. By definition,

t_{r}=t_{2}-t_{1}.

$f L$ is the lower cutoff frequency (-3 dB point) of the analysed system, measured in hertz.
$f H$ 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

BW=f_{H}-f_{L}\,

and since the lower cutoff frequency

f L

is usually several decades lower than the higher cutoff frequency

f H

BW\cong f_{H}\,

All systems analyzed here have a frequency response which extends to $0$ (low-pass systems), thus

f_{L}=0\,\Longleftrightarrow \,f_{H}=BW

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

{\frac {V(t_{1})}{V_{0}}}={\frac {x\%}{100}}\qquad {\frac {V(t_{2})}{V_{0}}}={\frac {y\%}{100}}

$ζ$ is the damping ratio and $ω 0$ is the natural frequency of a given 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 Gaussian response if it is characterized by the following frequency response

|H(\omega )|=e^{-{\frac {\omega ^{2}}{\sigma ^{2}}}}

where $σ = 0$ is a constant,^[14] related to the high cutoff frequency by the following relation:

f_{H}={\frac {\sigma }{2\pi }}{\sqrt {{\frac {3}{20}}\ln 10}}\cong 0.0935\sigma .

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 and 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

{\mathcal {F}}^{-1}\{H\}(t)=h(t)={\frac {1}{2\pi }}\int \limits _{-\infty }^{+\infty }{e^{-{\frac {\omega ^{2}}{\sigma ^{2}}}}e^{i\omega t}}d\omega ={\frac {\sigma }{2{\sqrt {\pi }}}}e^{-{\frac {1}{4}}\sigma ^{2}t^{2}}

Applying directly the definition of step response,

V(t)=V_{0}{H*h}(t)={\frac {V_{0}}{\sqrt {\pi }}}\int \limits _{-\infty }^{\frac {\sigma t}{2}}e^{-\tau ^{2}}d\tau ={\frac {V_{0}}{2}}\left[1+\mathrm {erf} \left({\frac {\sigma t}{2}}\right)\right]\quad \Longleftrightarrow \quad {\frac {V(t)}{V_{0}}}={\frac {1}{2}}\left[1+\mathrm {erf} \left({\frac {\sigma t}{2}}\right)\right].

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

{\frac {V(t_{1})}{V_{0}}}=0.1={\frac {1}{2}}\left[1+\mathrm {erf} \left({\frac {\sigma t_{1}}{2}}\right)\right]\qquad {\frac {V(t_{1})}{V_{0}}}=0.9={\frac {1}{2}}\left[1+\mathrm {erf} \left({\frac {\sigma t_{2}}{2}}\right)\right],

By using known properties of the error function, the value $t = - t 1 = t 2$ is found: since $t r = t 2 - t 1 = 2 t$ ,

t_{r}={\frac {4}{\sigma }}{\mathrm {erf} ^{-1}(0.8)}\cong {\frac {0.3394}{f_{H}}},

and finally

t_{r}\cong {\frac {0.34}{BW}}\quad \Longleftrightarrow \quad BW\cdot t_{r}\cong 0.34.

^[17]

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 $τ = RC$ :

t_{r}\cong 2.197\tau \,

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:

V(t)=V_{0}\left(1-e^{-{\frac {t}{\tau }}}\right)

Solving for time

{\frac {V(t)}{V_{0}}}=\left(1-e^{-{\frac {t}{\tau }}}\right)\quad \Longleftrightarrow \quad {\frac {V(t)}{V_{0}}}-1=-e^{-{\frac {t}{\tau }}}\quad \Longleftrightarrow \quad 1-{\frac {V(t)}{V_{0}}}=e^{-{\frac {t}{\tau }}},

and finally,

\ln \left(1-{\frac {V(t)}{V_{0}}}\right)=-{\frac {t}{\tau }}\quad \Longleftrightarrow \quad t=-\tau \;\ln \left(1-{\frac {V(t)}{V_{0}}}\right)

Since $t 1$ and $t 2$ are such that

{\frac {V(t_{1})}{V_{0}}}=0.1\qquad {\frac {V(t_{2})}{V_{0}}}=0.9,

solving these equations we find the analytical expression for $t 1$ and $t 2$ :

t_{1}=-\tau \;\ln \left(1-0.1\right)=-\tau \;\ln \left(0.9\right)=-\tau \;\ln \left({\frac {9}{10}}\right)=\tau \;\ln \left({\frac {10}{9}}\right)=\tau ({\ln 10}-{\ln 9})

t_{2}=\tau \ln {10}\,

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

t_{r}=t_{2}-t_{1}=\tau \cdot \ln 9\cong \tau \cdot 2.197

Now, noting that

\tau =RC={\frac {1}{2\pi f_{H}}},

^[20]

then

t_{r}\cong {\frac {2.197}{2\pi f_{H}}}\cong {\frac {0.349}{f_{H}}},

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

t_{r}\cong {\frac {0.35}{BW}}\quad \Longleftrightarrow \quad BW\cdot t_{r}\cong 0.35.

^[17]

Finally note that, if the 20% to 80% rise time is considered instead, $t r$ becomes:

t_{r}\cong 1.386\tau \quad \Longleftrightarrow \quad t_{r}\cong {\frac {0.22}{BW}}

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

t_{r}=\tau \cdot \ln 9={\frac {L}{R}}\cdot \ln 9\cong {\frac {L}{R}}\cdot 2.197

Rise time of damped second order systems

According to Levine (1996, p. 158), for overdamped 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 overdamped 2nd-order system has the following form:^[21]

t_{r}\cdot \omega _{0}={\frac {1}{\sqrt {1-\zeta ^{2}}}}\left[\pi -\tan ^{-1}\left({\frac {\sqrt {1-\zeta ^{2}}}{\zeta }}\right)\right]

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

t_{r}\cdot \omega _{0}=2.230\zeta ^{2}-0.078\zeta +1.12\,

where $ζ$ is the damping ratio and $ω 0$ is the natural frequency of the network.

Rise time of cascaded blocks

Consider a system composed by $n$ cascaded non interacting blocks, each having a rise time $t r i$ , $i = 1,..., n$ , and no overshoot in their step response: suppose also that the input signal of the first block has a rise time whose value is $t r S$ .^[22] Then its output signal has a rise time $t r 0$ equal to

t_{r_{O}}={\sqrt {t_{r_{S}}^{2}+t_{r_{1}}^{2}+\dots +t_{r_{n}}^{2}}}

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]

Notes

↑ According to the Federal Standard 1037C (1997, p. R-22).
↑ 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 poins.".
1 2 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 (1969, 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).
↑ By the 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).
1 2 Compare with (Orwiler 1969, p. 30).
↑ Called also "single-pole filter". See (Cherry & Hooper 1969, p. 639).
↑ Compare with (Valley & Wallman 1948, p. 72, formula (2)), (Cherry & Hooper 1969, 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.
↑ 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).
↑ See also (Cherry & Hooper 1969, p. 656) and (Orwiler 1969, pp. 27–28).
↑ Cited by (Cherry & Hooper 1969, p. 656).
↑ See (Petitt & McWhorter 1961, p. 109).

References

Cherry, E. M.; Hooper, D. E. (1968), Amplifying Devices and Low-pass Amplifier Design, New York–London–Sidney: John Wiley & Sons, pp. xxxii+1036 .
Elmore, William C. (January 1948), "The Transient Response of Damped Linear Networks with Particular Regard to Wideband Amplifiers", Journal of Applied Physics, 19 (1): 55–63, doi:10.1063/1.1697872 .
Levine, William S. (1996), The Control Handbook, Boca Raton, FL: CRC Press, pp. xvi+1548, ISBN 0-8493-8570-9 .
Levine, William S. (2011) [1996], The Control Handbook: Control Systems Fundamentals (2nd ed.), Boca Raton, FL: CRC Press, pp. xx+766, ISBN 978-1-4200-7362-1 .
Millman, Jacob; Taub, Herbert (1965), Pulse, digital and switching waveforms, New York–St. Louis–San Francisco–Toronto–London–Sydney: McGraw-Hill, pp. xiv+958 .
National Communication Systems, Technology and Standards Division (1 March 1997), Federal Standard 1037C. Telecomunications: Glossary of Telecommunications Terms, FSC TELE, FED–STD–1037, Washington: General Service Administration Information Technology Service, p. 488 .
Nise, Norman S. (2011), Control Systems Engineering (6th ed.), New York: John Wiley & Sons, pp. xviii+928, ISBN 978-0470-91769-5 .
Ogata, Katsuhiko (2010) [1970], Modern Control Engineering (5th ed.), Englewood Cliffs, NJ: Prentice Hall, pp. x+894, ISBN 978-0-13-615673-4 .
Orwiler, Bob (December 1969), Vertical Amplifier Circuits (PDF), Circuit Concepts, 062-1145-00 (1st ed.), Beaverton, OR: Tektronix, p. 461 .
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 .
Wallman, Henry (1950), "Transient response and the central limit theorem of probability", in Taub, A. H., Electromagnetic Theory (Massachusetts Institute of Technology, July 29–31 1948), Proceedings of Symposia in Applied Mathematics, 2, Providence: American Mathematical Society., p. 91, MR 0034250, Zbl 0035.08102 .

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