Rise time

Template:Short description 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.<ref name="Std1037C">Template:Citation</ref> These values may be expressed as ratios<ref name="10-90">See for example Template:Harv, Template:Harv and Template:Harv.</ref> or, equivalently, as percentages<ref>See for example Template:Harvtxt, Template:Harv and Template:Harv.</ref> with respect to a given reference value. In analog electronics and digital electronics,Template:Citation needed these percentages are commonly the 10% and 90% (or equivalently Template:Math and Template:Math) of the output step height:<ref>See for example Template:Harv, Template:Harv and Template:Harv.</ref> however, other values are commonly used.<ref>For example Template:Harvtxt 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.".</ref> For applications in control theory, according to Template:Harvtxt, rise time is defined as "the time required for the response to rise from Template:Math to Template:Math 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.<ref name="risedef">Precisely, Template:Harvtxt 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 Template:Harv.</ref> According to Template:Harvtxt, the term "rise time" applies to either positive or negative step response, even if a displayed negative excursion is popularly termed fall time.<ref>Again according to Template:Harvtxt.</ref>

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.<ref>According to Template:Harvtxt, "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 Template:Harvtxt, "The two most significant parameters in the square-wave response of an amplifier are its rise time and percentage tilt".</ref> 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.<ref>See Template:Harv and the "Rise time of cascaded blocks" section.</ref>

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 Template:Math.<ref>See for example Template:Harv, Template:Harv or the "One-stage low-pass RC network" section.</ref>

Alternative definitions

Other definitions of rise time, apart from the one given by the [[#Template:Harvid|Federal Standard 1037C (1997]], p. R-22) and its slight generalization given by Template:Harvtxt, are occasionally used:<ref>See Template:Harv and Template:Harv.</ref> 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.<ref>See Template:Harv and Template:Harv.</ref> Another definition, introduced by Template:Harvtxt,<ref>See also Template:Harv.</ref> uses concepts from statistics and probability theory. Considering a step response Template:Math, he redefines the delay time Template:Math as the first moment of its first derivative Template:Math, i.e.

<math>t_D = \frac{\int_0^{+\infty}t V^\prime(t)\mathrm{d}t}{\int_0^{+\infty} V^\prime(t)\mathrm{d}t}.</math>

Finally, he defines the rise time Template:Math by using the second moment

<math>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}}</math>

Rise time of model systems

Notation

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

• Following Template:Harvs, we define Template:Math as the percentage low value and Template:Math the percentage high value respect to a reference value of the signal whose rise time is to be estimated.
• Template:Math is the time at which the output of the system under analysis is at the Template:Math of the steady-state value, while Template:Math the one at which it is at the Template:Math, both measured in seconds.
• Template:Math is the rise time of the analysed system, measured in seconds. By definition, <math display="block">t_r = t_2 - t_1.</math>
• Template:Math is the lower cutoff frequency (-3 dB point) of the analysed system, measured in hertz.
• Template:Math is higher cutoff frequency (-3 dB point) of the analysed system, measured in hertz.
• Template:Math is the impulse response of the analysed system in the time domain.
• Template:Math is the frequency response of the analysed system in the frequency domain.
• The bandwidth is defined as <math display="block">BW = f_{H} - f_{L}</math> and since the lower cutoff frequency Template:Math is usually several decades lower than the higher cutoff frequency Template:Math, <math display="block">BW\cong f_H</math>
• All systems analyzed here have a frequency response which extends to Template:Math (low-pass systems), thus <math display="block">f_L=0\,\Longleftrightarrow\,f_H=BW</math> 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 Template:Math volts, and this implies that <math display="block">\frac{V(t_1)}{V_0}=\frac{x\%}{100} \qquad \frac{V(t_2)}{V_0}=\frac{y\%}{100}</math>
• Template:Math is the damping ratio and Template:Math 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

<math>|H(\omega)|=e^{-\frac{\omega^2}{\sigma^2}} </math>

where Template:Math is a constant,<ref>See Template:Harv and Template:Harv.</ref> related to the high cutoff frequency by the following relation:

<math>f_H = \frac{\sigma}{2\pi} \sqrt{\frac{3}{20}\ln 10} \cong 0.0935 \sigma.</math>

Even if this kind frequency response is not realizable by a causal filter,<ref>By the Paley-Wiener criterion: see for example Template:Harv. Also Template:Harvtxt briefly recall this fact.</ref> 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.<ref>See Template:Harv, Template:Harv and Template:Harv.</ref> The corresponding impulse response can be calculated using the inverse Fourier transform of the shown frequency response

<math>\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^2t^2}</math>

Applying directly the definition of step response,

<math>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].</math>

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

<math>\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_2)}{V_0} = 0.9= \frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\sigma t_2}{2}\right)\right],</math>

By using known properties of the error function, the value Template:Math is found: since Template:Math,

<math>t_r=\frac{4}{\sigma}{\operatorname{erf}^{-1}(0.8)}\cong\frac{0.3394}{f_H},</math>

and finally

<math>t_r\cong\frac{0.34}{BW}\quad\Longleftrightarrow\quad BW\cdot t_r\cong 0.34.</math><ref name="Orwp30">Compare with Template:Harv.</ref>

One-stage low-pass RC network

For a simple one-stage low-pass RC network,<ref>Called also "single-pole filter". See Template:Harv.</ref> the 10% to 90% rise time is proportional to the network time constant Template:Math:

<math>t_r\cong 2.197\tau</math>

The proportionality constant can be derived from the knowledge of the step response of the network to a unit step function input signal of Template:Math amplitude:

<math>V(t) = V_0 \left(1-e^{-\frac{t}{\tau}} \right)</math>

Solving for time

<math>\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}},</math>

and finally,

<math>\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)</math>

Since Template:Math and Template:Math are such that

<math>\frac{V(t_1)}{V_0}=0.1 \qquad \frac{V(t_2)}{V_0}=0.9,</math>

solving these equations we find the analytical expression for Template:Math and Template:Math:

<math> 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})</math>

<math>t_2=\tau\ln{10}</math>

The rise time is therefore proportional to the time constant:<ref>Compare with Template:Harv, Template:Harv or Template:Harv.</ref>

<math>t_r = t_2-t_1 = \tau\cdot\ln 9\cong\tau\cdot 2.197</math>

Now, noting that

<math>\tau = RC = \frac{1}{2\pi f_H},</math><ref>See the section "Relation of time constant to bandwidth" section of the "Time constant" entry for a formal proof of this relation.</ref>

then

<math>t_r=\frac{2\ln3}{2\pi f_H}=\frac{\ln3}{\pi f_H}\cong\frac{0.349}{f_H},</math>

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

<math>t_r\cong\frac{0.35}{BW}\quad\Longleftrightarrow\quad BW\cdot t_r\cong 0.35.</math><ref name="Orwp30" />

Finally note that, if the 20% to 80% rise time is considered instead, Template:Math becomes:

<math>t_r = \tau\cdot\ln\frac{8}{2}=(2\ln2)\tau

\cong 1.386\tau\quad\Longleftrightarrow\quad t_r=\frac{\ln2}{\pi BW}\cong\frac{0.22}{BW}</math>

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 Template:Math. 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 Template:Math of the two different circuits, leading in the present case to the following result

<math>t_r=\tau\cdot\ln 9 = \frac{L}{R}\cdot\ln 9\cong \frac{L}{R} \cdot 2.197</math>

Rise time of damped second order systems

According to Template:Harvtxt, 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:<ref name="risedef"/> accordingly, the rise time from 0 to 100% of an underdamped 2nd-order system has the following form:<ref>See Template:Harv.</ref>

<math> t_r \cdot\omega_0= \frac{1}{\sqrt{1-\zeta^2}}\left [ \pi - \tan^{-1}\left ( {\frac{\sqrt{1-\zeta^2}}{\zeta}} \right) \right ]</math>

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

<math> t_r \cdot\omega_0= 2.230\zeta^2-0.078\zeta+1.12</math>

where Template:Math is the damping ratio and Template:Math is the natural frequency of the network.

Rise time of cascaded blocks

Consider a system composed by Template:Math cascaded non interacting blocks, each having a rise time Template:Math, Template:Math, and no overshoot in their step response: suppose also that the input signal of the first block has a rise time whose value is Template:Math.<ref>"Template:Math" stands for "source", to be understood as current or voltage source.</ref> Afterwards, its output signal has a rise time Template:Math equal to

<math>t_{r_O} = \sqrt{t_{r_S}^2+t_{r_1}^2+\dots+t_{r_n}^2}</math>

According to Template:Harvtxt, this result is a consequence of the central limit theorem and was proved by Template:Harvtxt:<ref>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"Template:Harv.</ref><ref>See also Template:Harv and Template:Harv.</ref> however, a detailed analysis of the problem is presented by Template:Harvtxt,<ref>Cited by Template:Harv.</ref> who also credit Template:Harvtxt as the first one to prove the previous formula on a somewhat rigorous basis.<ref>See Template:Harv.</ref>

See also

• Fall time
• Frequency response
• Impulse response
• Step response
• Settling time

References

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