In
electronicsElectronics is a branch of science and technology that deals with the controlled flow of electrons. The ability to control electron flow is usually applied to information handling or device control. Electronics is distinct from electrical science and technology, which deals with the generation,...
, when describing a
voltageVoltage is commonly used as a short name for electrical potential difference. Its corresponding SI unit is the volt...
or current
step 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...
,
rise time (also
risetime) refers to the time required for a signal to change from a specified low value to a specified high value. Typically, these values are 10% and 90% of the step height. The output
signalIn the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity....
of a
systemSystem is a set of interacting or interdependent entities forming an integrated whole....
is characterized also by
fall timeIn electronics, fall time 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
systemSystem is a set of interacting or interdependent entities forming an integrated whole....
.
Rise time is an analog parameter of fundamental importance in
high speed electronicsElectronics is a branch of science and technology that deals with the controlled flow of electrons. The ability to control electron flow is usually applied to information handling or device control. Electronics is distinct from electrical science and technology, which deals with the generation,...
, since it is a measure of the ability of a circuit to respond to fast input signals.
In
electronicsElectronics is a branch of science and technology that deals with the controlled flow of electrons. The ability to control electron flow is usually applied to information handling or device control. Electronics is distinct from electrical science and technology, which deals with the generation,...
, when describing a
voltageVoltage is commonly used as a short name for electrical potential difference. Its corresponding SI unit is the volt...
or current
step 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...
,
rise time (also
risetime) refers to the time required for a signal to change from a specified low value to a specified high value. Typically, these values are 10% and 90% of the step height. The output
signalIn the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity....
of a
systemSystem is a set of interacting or interdependent entities forming an integrated whole....
is characterized also by
fall timeIn electronics, fall time 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
systemSystem is a set of interacting or interdependent entities forming an integrated whole....
.
Overview
Rise time is an analog parameter of fundamental importance in
high speed electronicsElectronics is a branch of science and technology that deals with the controlled flow of electrons. The ability to control electron flow is usually applied to information handling or device control. Electronics is distinct from electrical science and technology, which deals with the generation,...
, 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
electronicsElectronics is a branch of science and technology that deals with the controlled flow of electrons. The ability to control electron flow is usually applied to information handling or device control. Electronics is distinct from electrical science and technology, which deals with the generation,...
, long (compared to the attainable state of the art) rise times are sometimes desirable: examples are the
dimmingDimmers 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 feedthough, and thus lower coupling
noiseIn common use, the word noise means unwanted sound or noise pollution. In both analog and digital electronics, noise or signal noise is an unwanted random addition to a wanted signal; it is called noise as a generalisation of the audible noise heard when listening to a weak radio transmission...
.
Simple examples of calculation of rise time
The aim of this section is the calculation of rise time of
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...
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 second
The second , sometimes abbreviated sec., is the name of a unit of time, and is the International System of Units base unit of time...
s.
- is the low frequency cutoff (-3 dB point) of the analyzed system, measured in hertz
The hertz is a unit of frequency. It is defined as the number of complete cycles per second. It is the basic unit of frequency in the International System of Units , and is used worldwide in both general-purpose and scientific contexts...
.
- is high frequency cutoff (-3 dB point) of the analyzed system, measured in hertz.
- is the 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...
of the analyzed system in the time domain.
- is the frequency response
Frequency response is the measure of any system's output spectrum in response to an input signal. In the audible range it is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cables,...
of the analyzed system in the frequency domain.
- 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 response
Frequency response is the measure of any system's output spectrum in response to an input signal. In the audible range it is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cables,...
which extends to 0 (low-pass systems), thus exactly.
- All systems analyzed are thought as electrical network
An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources, and switches....
s and all the signals are thought as voltageVoltage is commonly used as a short name for electrical potential difference. Its corresponding SI unit is the volt...
s for the sake of simplicity: the input is a step 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 voltThe volt is the SI derived unit of electromotive force, commonly called "voltage". It is also the unit for the related but slightly different quantity electric potential difference...
s.
Gaussian response system
A system is said to have a
GaussianGAUSSIAN is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70...
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 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...
can be calculated using the inverse
Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
of the shown
frequency responseFrequency response is the measure of any system's output spectrum in response to an input signal. In the audible range it is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cables,...
Applying directly the definition of
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...
Solving for ts the two following equations by using known properties of the
error functionIn mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics, materials science, 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 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...
input
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 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...
:
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
oscilloscopeAn oscilloscope is a type of electronic test instrument that allows signal voltages to be viewed, usually as a two-dimensional graph of one or more electrical potential differences plotted as a function of time or of some other voltage...
is 350
MHzThe hertz is a unit of frequency. It is defined as the number of complete cycles per second. It is the basic unit of frequency in the International System of Units , and is used worldwide in both general-purpose and scientific contexts...
, 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
overshootIn signal processing, control theory, electronics, and mathematics, overshoot is when a signal or function exceeds its steady-state value.It arises especially in the step response of bandlimited systems such as low-pass filters...
, whose input signal has a rise time : then its output signal has a rise time equal to
This result is a consequence of the
central limit theoremIn 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 also requires the random variables to...
, as reported in and proved by
Henry WallmanHenry Wallman was an American mathematician, known for his work in lattice theory, dimension theory, topology, and electronic circuit design.Wallman received his Ph.D...
in .
Factors affecting rise time
Rise time values in a resistive circuit are primarily due to stray
capacitanceIn electromagnetism and electronics, capacitance is the ability of a body to hold an electrical charge.Capacitance is also a measure of the amount of electric charge stored for a given electric potential. A common form of charge storage device is a parallel-plate capacitor...
and
inductanceInductance is the property in an electrical circuit where a change in the electric current through that circuit induces an electromotive force that opposes the change in current ....
in the circuit. Because every
circuitAn electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources, and switches....
has not only
resistanceThe electrical resistance of an object is a measure of its opposition to the passage of a steady electric current. An object of uniform cross section will have a resistance proportional to its length and inversely proportional to its cross-sectional area, and proportional to the resistivity of the...
, but also
capacitanceIn electromagnetism and electronics, capacitance is the ability of a body to hold an electrical charge.Capacitance is also a measure of the amount of electric charge stored for a given electric potential. A common form of charge storage device is a parallel-plate capacitor...
and
inductanceInductance is the property in an electrical circuit where a change in the electric current through that circuit induces an electromotive force that opposes the change in current ....
, a delay in voltage and/or current at the load is apparent until the
steady stateIn 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 circuitA 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. (Nise 2008)
The quadratic approximation for normalized
rise time for a 2
nd-order system,
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...
, no zeros is:
where ζ is the
damping ratioIn engineering, the damping ratio is a measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium. A mass suspended from a spring, for example, will, if pulled and released,...
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 2
nd-order system is:
where ζ is the damping ratio and ω
n is the natural frequency of the network.
See also
- Fall time
In electronics, fall time is the time required for the amplitude of a pulse to decrease from a specified value to another specified value...
- Frequency response
Frequency response is the measure of any system's output spectrum in response to an input signal. In the audible range it is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cables,...
- 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...
- 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...
- Transition time
The transition time 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 time
The 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...