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. 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. 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 . Alternative definitions Other definitions of rise time, apart from the one given by the Federal Standard 1037C (1997, p. R-22) and its slight generalization given by , are occasionally used: 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. Another definition, introduced by , uses concepts from statistics and probability theory. Considering a step response , he redefines the delay time as the first moment of its first derivative , i.e. :t_D = \frac{\int_0^{+\infty}t V^\prime(t) \, \mathrm{d}t}{\int_0^{+\infty} V^\prime(t) \, \mathrm{d}t}. Finally, he defines the rise time 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 , we define as the percentage low value and the percentage high value respect to a reference value of the signal whose rise time is to be estimated. • is the time at which the output of the system under analysis is at the of the steady-state value, while the one at which it is at the , both measured in seconds. • is the rise time of the analysed system, measured in seconds. By definition, t_r = t_2 - t_1. • is the lower cutoff frequency (-3 dB point) of the analysed system, measured in hertz. • is higher cutoff frequency (-3 dB point) of the analysed system, measured in hertz. • is the impulse response of the analysed system in the time domain. • 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 is usually several decades lower than the higher cutoff frequency , BW\cong f_H • All systems analyzed here have a frequency response which extends to (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 volts, and this implies that \frac{V(t_1)}{V_0}=\frac{x\%}{100} \qquad \frac{V(t_2)}{V_0}=\frac{y\%}{100} • is the damping ratio and 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^{-\omega^2/\sigma^2} where is a constant, 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 of frequency response is not realizable by a causal filter, 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. 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^{-\omega^2/\sigma^2} e^{i\omega t}} \, d\omega = \frac{\sigma}{2\sqrt{\pi}} e^{-\frac14 \sigma^2t^2} Applying directly the definition of step response, :V(t) = V_0{H*h}(t) = \frac{V_0}{\sqrt\pi} \int\limits_{-\infty}^{\sigma t/2}e^{-\tau^2} \, d\tau = \frac{V_0}{2} \left[1+\operatorname{erf}\left(\frac{\sigma t}{2}\right)\right] \quad \Longleftrightarrow \quad \frac{V(t)}{V_0} = \frac{1}{2} \left[1+\operatorname{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+\operatorname{erf}\left(\frac{\sigma t_1}{2}\right)\right] \qquad \frac{V(t_2)}{V_0} = 0.9= \frac{1}{2}\left[1+\operatorname{erf}\left(\frac{\sigma t_2}{2}\right)\right], By using known properties of the error function, the value is found: since , :t_r=\frac{4}{\sigma} \operatorname{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. One-stage low-pass RC network For a simple one-stage low-pass RC network, the 10% to 90% rise time is proportional to the network time constant : :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 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 and 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 and : : 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: :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}, then :t_r=\frac{2\ln3}{2\pi f_H}=\frac{\ln3}{\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. : 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 is the damping ratio and is the natural frequency of the network. Rise time of cascaded blocks Consider a system composed by cascaded non interacting blocks, each having a rise time , , and no overshoot in their step response: suppose also that the input signal of the first block has a rise time whose value is . Afterwards, its output signal has a rise time equal to :t_{r_O} = \sqrt{t_{r_S}^2+t_{r_1}^2+\dots+t_{r_n}^2} According to , this result is a consequence of the central limit theorem and was proved by : however, a detailed analysis of the problem is presented by , who also credit as the first one to prove the previous formula on a somewhat rigorous basis. == See also ==