Rise time

Encyclopedia

In electronics

, when describing a voltage

or current step function

,

of a system

is characterized also by fall time

: both parameters depend on rise and fall times of input signal and on the characteristics of the system

.

, since it is a measure of the ability of a circuit to respond to fast input signals. 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 digital signals apt to the control of analog ones, where a longer rise time means lower capacitive feedthrough, and thus lower coupling noise

.

for some simple systems: all notations and assumptions required for the following analysis are listed here.

response if it is characterized by the following frequency response

where is a constant, related to the high frequency cutoff by the following relation:

The corresponding impulse response

can be calculated using the inverse Fourier transform

of the shown frequency response

Applying directly the definition of step response

Solving for ts the two following equations by using known properties of the error function

the value is then known and since

and then

The proportionality constant can be derived by using the output response of the network to a step function

input Signal (electrical engineering)

of amplitude, aka its step response

:

Solving for ts the two equations

the times and to 10% and 90% of steady-state value thus known

Subtracting from

which is the rise time. Therefore rise time is proportional to the time constant:

Now, noting that

then

and since the high frequency cutoff is equal to the bandwidth

This formula implies that if the bandwidth of an oscilloscope

is 350 MHz

, its 10% to 90% risetime is 1 nanosecond.

in their step responseThe step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from...

: suppose also that the input signal of the first block has a rise time whose value is . Then its output signal has a rise time equal to

This result is a consequence of the central limit theorem

, as reported in and proved by Henry Wallman

in .

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%), as shown above, is approximately equal to .

The quadratic

approximation

for normalizedThe step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from...

, no zeros is:

where ζ is the damping ratio

and ω

However, the proper calculation for rise time from 0 to 100% of an under-damped 2nd-order system is:

where ζ is the damping ratio and ω

Electronics

Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...

, when describing a voltage

Voltage

Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

or current step function

Step function

In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals...

,

**rise time**refers to the time required for a signal to change from a specified low value to a specified high value. Typically, in analog electronics, these values are 10% and 90% of the step height: in control theory applications, according to , rise time is defined as "the time required for the response to rise from x% to y% of its final value", with 0%-100% rise time common for underdamped second order systems, 5%-95% for critically damped and 10%-90% for overdamped. The output signalSignal (electrical engineering)

In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity....

of a system

System

System is a set of interacting or interdependent components forming an integrated whole....

is characterized also by fall time

Fall time

In electronics, fall time \scriptstyle t_f\, is the time required for the amplitude of a pulse to decrease from a specified value to another specified value...

: both parameters depend on rise and fall times of input signal and on the characteristics of the system

System

System is a set of interacting or interdependent components forming an integrated whole....

.

## Overview

Rise time is an analog parameter of fundamental importance in high speed electronicsElectronics

Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...

, since it is a measure of the ability of a circuit to respond to fast input signals. 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

Electronics

Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...

, long (compared to the attainable state of the art) rise times are sometimes desirable: examples are the dimming

Dimmer

Dimmers are devices used to vary the brightness of a light. By decreasing or increasing the RMS voltage and, hence, the mean power to the lamp, it is possible to vary the intensity of the light output...

of a light, where a longer rise-time results, amongst other things, in a longer life for the bulb, or digital signals apt to the control of analog ones, where a longer rise time means lower capacitive feedthrough, and thus lower coupling noise

Noise

In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...

.

## Simple examples of calculation of rise time

The aim of this section is the calculation of rise time of step responseStep response

The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from...

for some simple systems: all notations and assumptions required for the following analysis are listed here.

- is the
**rise time**of the analyzed system, measured in secondSecondThe second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

s.

- is the
*low frequency cutoff*(-3 dB point) of the analyzed system, measured in hertzHertzThe hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

.

- is
*high frequency cutoff*(-3 dB point) of the analyzed system, measured in hertz.

- is the
*impulse response*of the analyzed system in the time domain.Impulse responseIn signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

- is the
*frequency response*of the analyzed system in the frequency domain.Frequency responseFrequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...

- The bandwidth is defined as

- and since the low frequency cutoff is usually several decades lower than the high frequency cutoff ,

- All systems analyzed here have a frequency responseFrequency responseFrequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...

which extends to 0 (low-pass systems), thus exactly. - All systems analyzed are thought as electrical networkElectrical networkAn electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...

s and all the signals are thought as voltageVoltageVoltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

s for the sake of simplicity: the input is a step functionStep functionIn mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals...

of voltVoltThe volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...

s.

### Gaussian response system

A system is said to have a GaussianGAUSSIAN

Gaussian is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70. It has been continuously updated since then...

response if it is characterized by the following frequency response

where is a constant, related to the high frequency cutoff by the following relation:

The corresponding impulse response

Impulse response

In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

can be calculated using the inverse Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

of the shown frequency response

Frequency response

Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...

Applying directly the definition of step response

Step response

The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from...

Solving for ts the two following equations by using known properties of the error function

Error function

In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...

the value is then known and since

and then

### One stage low pass RC network

For a simple one stage low pass network, rise time is proportional to the network time constant :The proportionality constant can be derived by using the output response of the network to a step function

Step function

In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals...

input Signal (electrical engineering)

Signal (electrical engineering)

In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity....

of amplitude, aka its step response

Step response

The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from...

:

Solving for ts the two equations

the times and to 10% and 90% of steady-state value thus known

Subtracting from

which is the rise time. Therefore rise time is proportional to the time constant:

Now, noting that

then

and since the high frequency cutoff is equal to the bandwidth

This formula implies that if the bandwidth of an oscilloscope

Oscilloscope

An oscilloscope is a type of electronic test instrument that allows observation of constantly varying signal voltages, usually as a two-dimensional graph of one or more electrical potential differences using the vertical or 'Y' axis, plotted as a function of time,...

is 350 MHz

Hertz

The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

, its 10% to 90% risetime is 1 nanosecond.

### Rise time of cascaded blocks

Consider a system composed by cascaded non interacting blocks, each having a rise time and no overshootOvershoot (signal)

In signal processing, control theory, electronics, and mathematics, overshoot is when a signal or function exceeds its target. It arises especially in the step response of bandlimited systems such as low-pass filters...

in their step response

Step response

: suppose also that the input signal of the first block has a rise time whose value is . Then its output signal has a rise time equal to

This result is a consequence of the central limit theorem

Central limit theorem

In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...

, as reported in and proved by Henry Wallman

Henry Wallman

Henry Wallman was an American mathematician, known for his work in lattice theory, dimension theory, topology, and electronic circuit design.A native of Brooklyn and a 1933 graduate of Brooklyn College, Wallman received his Ph.D...

in .

## Factors affecting rise time

Rise time values in a resistive circuit are primarily due to stray capacitanceCapacitance

In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

and inductance

Inductance

In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

in the circuit. Because every circuit

Electrical network

An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...

has not only resistance

Electrical resistance

The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

, but also capacitance

Capacitance

In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

and inductance

Inductance

In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

, a delay in voltage and/or current at the load is apparent until the steady state

Steady State theory

In cosmology, the Steady State theory is a model developed in 1948 by Fred Hoyle, Thomas Gold, Hermann Bondi and others as an alternative to the Big Bang theory...

is reached. In a pure RC circuit

RC circuit

A resistor–capacitor circuit ', or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source...

, the output risetime (10% to 90%), as shown above, is approximately equal to .

## Rise time in control applications

In control theory, rise time is commonly defined as the time for a waveform to go from 10% to 90% of its final value.The quadratic

Quadratic

In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms...

approximation

Approximation

An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...

for normalized

**rise time**for a 2nd-order system, step responseStep response

, no zeros is:

where ζ is the damping ratio

Damping ratio

[[Image:Damped spring.gif|right|frame|Underdamped [[spring–mass system]] with ζ 1 , and is referred to as overdamped.*Underdamped:If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like \exp...

and ω

_{0}is the natural frequency of the network.However, the proper calculation for rise time from 0 to 100% of an under-damped 2nd-order system is:

where ζ is the damping ratio and ω

_{0}is the natural frequency of the network.## See also

- Fall timeFall timeIn electronics, fall time \scriptstyle t_f\, is the time required for the amplitude of a pulse to decrease from a specified value to another specified value...
- Frequency responseFrequency response
- Impulse responseImpulse responseIn signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...
- Step responseStep response
- Transition timeTransition timeThe transition time \scriptstyle t_t\, is the time a dynamical system needs to switch between two different stable states, when responding to a stable input signal. In a logic circuit undergoing a change of state, it identifies the rise time or the fall time of the output voltage...
- Settling timeSettling timeThe settling time of an amplifier or other output device is the time elapsed from the application of an ideal instantaneous step input to the time at which the amplifier output has entered and remained within a specified error band, usually symmetrical about the final value.Settling time includes a...