rise time

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.According to {{harvtxt|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 {{harvtxt|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". 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.

For a given system output, its rise time depend both on the rise time of input signal and on the characteristics of the system.See {{harv|Orwiler|1969|pp=27–29}} and the "Rise time of cascaded blocks" section.

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 {{math|2.2 RC}}.See for example {{harv|Valley|Wallman|1948|p=73}}, {{harv|Orwiler|1969|loc=p. 22 and p. 30}} or the "One-stage low-pass RC network" section.

Other definitions of rise time, apart from the one given by the #{{harvid, p. R-22) and its slight generalization given by {{harvtxt|Levine|1996|p=158}}, are occasionally used:See {{harv|Valley|Wallman|1948|loc=p. 72, footnote 1}} and {{Harv|Elmore|1948|p=56}}. 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.See {{harv|Valley|Wallman|1948|loc=p. 72, footnote 1}} and {{Harv|Elmore|1948|loc=p. 56 and p. 57, fig. 2a}}. Another definition, introduced by {{Harvtxt|Elmore|1948|p=57}},See also {{harv|Petitt|McWhorter|1961|pp=109–111}}. uses concepts from statistics and probability theory. Considering a step response {{math|V(t)}}, he redefines the delay time {{math|t_D}} as the first moment of its first derivative {{math|V′(t)}}, i.e.

:$t\_D\; =\; \backslash frac\{\backslash int\_0^\{+\backslash infty\}t\; V^\backslash prime(t)\backslash mathrm\{d\}t\}\{\backslash int\_0^\{+\backslash infty\}\; V^\backslash prime(t)\backslash mathrm\{d\}t\}.$

Finally, he defines the rise time {{math|t_r}} by using the second moment

:$t\_r^2\; =\; \backslash frac\{\backslash int\_0^\{+\backslash infty\}(t\; -t\_D)^2\; V^\backslash prime(t)\backslash mathrm\{d\}t\}\{\backslash int\_0^\{+\backslash infty\}\; V^\backslash prime(t)\backslash mathrm\{d\}t\}\; \backslash quad$

\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}}

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

• Following {{harvs|txt|last=Levine | year1=1996 | year2=2011 |loc1=p. 158| loc2= 9-3 (313)}}, we define {{math|x%}} as the percentage low value and {{math|y%}} the percentage high value respect to a reference value of the signal whose rise time is to be estimated.
• {{math|t₁}} is the time at which the output of the system under analysis is at the {{math|x%}} of the steady-state value, while {{math|t₂}} the one at which it is at the {{math|y%}}, both measured in seconds.
• {{math|t_r}} is the rise time of the analysed system, measured in seconds. By definition, $$t\_r\; =\; t\_2\; -\; t\_1.$$
• {{math|f_L}} is the lower cutoff frequency (-3 dB point) of the analysed system, measured in hertz.
• {{math|f_H}} is higher cutoff frequency (-3 dB point) of the analysed system, measured in hertz.
• {{math|h(t)}} is the impulse response of the analysed system in the time domain.
• {{math|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 {{math|f_L}} is usually several decades lower than the higher cutoff frequency {{math|f_H}}, $$BW\backslash cong\; f\_H$$
• All systems analyzed here have a frequency response which extends to {{math|0}} (low-pass systems), thus $$f\_L=0\backslash ,\backslash Longleftrightarrow\backslash ,f\_H=BW$$ 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 {{math|V₀}} volts, and this implies that $$\backslash frac\{V(t\_1)\}\{V\_0\}=\backslash frac\{x\backslash \%\}\{100\}\; \backslash qquad\; \backslash frac\{V(t\_2)\}\{V\_0\}=\backslash frac\{y\backslash \%\}\{100\}$$
• {{math|ζ}} is the damping ratio and {{math|ω₀}} is the natural frequency of a given second order system.

The aim of this section is the calculation of rise time of step response for some simple systems:

A system is said to have a Gaussian response if it is characterized by the following frequency response

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

where {{math|σ > 0}} is a constant,See {{harv|Valley|Wallman|1948|p=724}} and {{harv|Petitt|McWhorter|1961|p=122}}. related to the high cutoff frequency by the following relation:

:$f\_H\; =\; \backslash frac\{\backslash sigma\}\{2\backslash pi\}\; \backslash sqrt\{\backslash frac\{3\}\{20\}\backslash ln\; 10\}\; \backslash cong\; 0.0935\; \backslash sigma.$

Even if this kind frequency response is not realizable by a causal filter,By the Paley-Wiener criterion: see for example {{harv|Valley|Wallman|1948|loc=p. 721 and p. 724}}. Also {{harvtxt|Petitt|McWhorter|1961|p=122}} briefly recall this fact. 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.See {{harv|Valley|Wallman|1948|p=724}}, {{harv|Petitt|McWhorter|1961|loc=p. 111, including footnote 1, and p.}} and {{harv|Orwiler|1969|p=30}}. The corresponding impulse response can be calculated using the inverse Fourier transform of the shown frequency response

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

Applying directly the definition of step response,

:$V(t)\; =\; V\_0\{H*h\}(t)\; =\; \backslash frac\{V\_0\}\{\backslash sqrt\{\backslash pi\}\}\backslash int\backslash limits\_\{-\backslash infty\}^\{\backslash frac\{\backslash sigma\; t\}\{2\}\}e^\{-\backslash tau^2\}d\backslash tau\; =\; \backslash frac\{V\_0\}\{2\}\backslash left[1+\backslash mathrm\{erf\}\backslash left(\backslash frac\{\backslash sigma\; t\}\{2\}\backslash right)\backslash right]\; \backslash quad\; \backslash Longleftrightarrow\; \backslash quad\; \backslash frac\{V(t)\}\{V\_0\}\; =\; \backslash frac\{1\}\{2\}\backslash left[1+\backslash mathrm\{erf\}\backslash left(\backslash frac\{\backslash sigma\; t\}\{2\}\backslash right)\backslash right].$

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

:$\backslash frac\{V(t\_1)\}\{V\_0\}\; =\; 0.1\; =\; \backslash frac\{1\}\{2\}\backslash left[1+\backslash mathrm\{erf\}\backslash left(\backslash frac\{\backslash sigma\; t\_1\}\{2\}\backslash right)\backslash right]$

\qquad \frac{V(t_2)}{V_0} = 0.9= \frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\sigma t_2}{2}\right)\right],

By using known properties of the error function, the value {{math|1=t = −t₁ = t₂}} is found: since {{math|1=t_r = t₂ - t₁ = 2t}},

:$t\_r=\backslash frac\{4\}\{\backslash sigma\}\{\backslash operatorname\{erf\}^\{-1\}(0.8)\}\backslash cong\backslash frac\{0.3394\}\{f\_H\},$

and finally

:$t\_r\backslash cong\backslash frac\{0.34\}\{BW\}\backslash quad\backslash Longleftrightarrow\backslash quad\; BW\backslash cdot\; t\_r\backslash cong\; 0.34.$Compare with {{harv|Orwiler|1969|p=30}}.

For a simple one-stage low-pass RC network,Called also "single-pole filter". See {{harv|Cherry|Hooper|1968|p=639}}. the 10% to 90% rise time is proportional to the network time constant {{math|1=τ = RC}}:

:$t\_r\backslash cong\; 2.197\backslash tau$

The proportionality constant can be derived from the knowledge of the step response of the network to a unit step function input signal of {{math|V₀}} amplitude:

:$V(t)\; =\; V\_0\; \backslash left(1-e^\{-\backslash frac\{t\}\{\backslash tau\}\}\; \backslash right)$

Solving for time

:$\backslash frac\{V(t)\}\{V\_0\}=\backslash left(1-e^\{-\backslash frac\{t\}\{\backslash tau\}\}\backslash right)\; \backslash quad\; \backslash Longleftrightarrow\; \backslash quad\; \backslash frac\{V(t)\}\{V\_0\}-1=-e^\{-\backslash frac\{t\}\{\backslash tau\}\}\; \backslash quad\; \backslash Longleftrightarrow\; \backslash quad\; 1-\backslash frac\{V(t)\}\{V\_0\}=e^\{-\backslash frac\{t\}\{\backslash tau\}\},$

and finally,

:$\backslash ln\backslash left(1-\backslash frac\{V(t)\}\{V\_0\}\backslash right)=-\backslash frac\{t\}\{\backslash tau\}\; \backslash quad\; \backslash Longleftrightarrow\; \backslash quad\; t\; =\; -\backslash tau\; \backslash ;\; \backslash ln\backslash left(1-\backslash frac\{V(t)\}\{V\_0\}\backslash right)$

Since {{math|t₁}} and {{math|t₂}} are such that

:$\backslash frac\{V(t\_1)\}\{V\_0\}=0.1\; \backslash qquad\; \backslash frac\{V(t\_2)\}\{V\_0\}=0.9,$

solving these equations we find the analytical expression for {{math|t₁}} and {{math|t₂}}:

:$t\_1\; =\; -\backslash tau\backslash ;\backslash ln\backslash left(1-0.1\backslash right)\; =\; -\backslash tau\; \backslash ;\; \backslash ln\backslash left(0.9\backslash right)\; =\; -\backslash tau\backslash ;\backslash ln\backslash left(\backslash frac\{9\}\{10\}\backslash right)\; =\; \backslash tau\backslash ;\backslash ln\backslash left(\backslash frac\{10\}\{9\}\backslash right)\; =\; \backslash tau(\{\backslash ln\; 10\}-\{\backslash ln\; 9\})$

:$t\_2=\backslash tau\backslash ln\{10\}$

The rise time is therefore proportional to the time constant:Compare with {{harv|Valley|Wallman|1948|loc=p. 72, formula (2)}}, {{harv|Cherry|Hooper|1968|loc=p. 639, formula (13.3)}} or {{harv|Orwiler|1969|loc=p. 22 and p. 30}}.

:$t\_r\; =\; t\_2-t\_1\; =\; \backslash tau\backslash cdot\backslash ln\; 9\backslash cong\backslash tau\backslash cdot\; 2.197$

Now, noting that

:$\backslash tau\; =\; RC\; =\; \backslash frac\{1\}\{2\backslash pi\; f\_H\},$See the section "Relation of time constant to bandwidth" section of the "Time constant" entry for a formal proof of this relation.

then

:$t\_r=\backslash frac\{2\backslash ln3\}\{2\backslash pi\; f\_H\}=\backslash frac\{\backslash ln3\}\{\backslash pi\; f\_H\}\backslash cong\backslash frac\{0.349\}\{f\_H\},$

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

:$t\_r\backslash cong\backslash frac\{0.35\}\{BW\}\backslash quad\backslash Longleftrightarrow\backslash quad\; BW\backslash cdot\; t\_r\backslash cong\; 0.35.$

Finally note that, if the 20% to 80% rise time is considered instead, {{math|t_r}} becomes:

:$t\_r\; =\; \backslash tau\backslash cdot\backslash ln\backslash frac\{8\}\{2\}=(2\backslash ln2)\backslash tau$

\cong 1.386\tau\quad\Longleftrightarrow\quad t_r=\frac{\ln2}{\pi BW}\cong\frac{0.22}{BW}

Even for a simple one-stage low-pass RL network, the 10% to 90% rise time is proportional to the network time constant {{math|1=τ = {{frac|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 {{math|τ}} of the two different circuits, leading in the present case to the following result

:$t\_r=\backslash tau\backslash cdot\backslash ln\; 9\; =\; \backslash frac\{L\}\{R\}\backslash cdot\backslash ln\; 9\backslash cong\; \backslash frac\{L\}\{R\}\; \backslash cdot\; 2.197$

According to {{harvtxt|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:See {{harv|Ogata|2010|p=171}}.

:$t\_r\; \backslash cdot\backslash omega\_0=\; \backslash frac\{1\}\{\backslash sqrt\{1-\backslash zeta^2\}\}\backslash left\; [\; \backslash pi\; -\; \backslash tan^\{-1\}\backslash left\; (\; \{\backslash frac\{\backslash sqrt\{1-\backslash zeta^2\}\}\{\backslash zeta\}\}\; \backslash right)\; \backslash right\; ]$

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

:$t\_r\; \backslash cdot\backslash omega\_0=\; 2.230\backslash zeta^2-0.078\backslash zeta+1.12$

where {{math|ζ}} is the damping ratio and {{math|ω₀}} is the natural frequency of the network.

Consider a system composed by {{math|n}} cascaded non interacting blocks, each having a rise time {{math|t_ri}}, {{math|1=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 {{math|t_rS}}."{{math|S}}" stands for "source", to be understood as current or voltage source. Afterwards, its output signal has a rise time {{math|t_r0}} equal to

:$t\_\{r\_O\}\; =\; \backslash sqrt\{t\_\{r\_S\}^2+t\_\{r\_1\}^2+\backslash dots+t\_\{r\_n\}^2\}$

According to {{Harvtxt|Valley|Wallman|1948|pp=77–78}}, this result is a consequence of the central limit theorem and was proved by {{Harvtxt|Wallman|1950}}: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"{{harv|Wallman|1950|p=91}}.See also {{harv|Cherry|Hooper|1968|p=656}} and {{harv|Orwiler|1969|pp=27–28}}. however, a detailed analysis of the problem is presented by {{harvtxt|Petitt|McWhorter|1961|loc=§4–9, pp. 107–115}},Cited by {{harv|Cherry|Hooper|1968|p=656}}. who also credit {{harvtxt|Elmore|1948}} as the first one to prove the previous formula on a somewhat rigorous basis.See {{harv|Petitt|McWhorter|1961|p=109}}.

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

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Category:Control theory

Category:Control engineering

Category:Computational mathematics

Category:Transient response characteristics