Information about Time Constant
In physics and engineering, the time constant usually denoted by the Greek letter 
, (tau), characterizes the frequency response of a first-order, linear time-invariant (LTI) system. Examples include electrical RC circuits and RL circuits. It is also used to characterize the frequency response of various signal processing systems – magnetic tapes, radio transmitters and receivers, record cutting and replay equipment, and digital filters – which can be modelled or approximated by first-order LTI systems.
Other examples include time constant used in control systems for integral and derivative action controllers, which are often pneumatic, rather than electrical.
Physically, the time constant represents the time it takes the system's step response to reach approximately 63% of its final (asymptotic) value, ie about 37% below its final value.
where
represents the exponential decay constant and V is a function of time t
The time constant is related to the exponential decay constant by
where
is the initial value of V.
in the specific case where 
, otherwise referred to as a "decaying" exponential function:
Suppose
then
The term (tau) is referred to as the "time constant" and can be used (as in this case) to indicate how rapidly an exponential function decays.
Where:
After a period of one time constant the function reaches e-1 = approximately 37% of its initial value. In case 4, after five time constants the function reaches a value less than 1% of its original. In most cases this 1% threshold is considered sufficient to assume that the function has decayed to zero - Hence in control engineering a stable system is mostly assumed to have settled after five time constants as a rule of thumb. (in seconds) is
where R is the resistance (in ohms) and L is the inductance (in henries).
Similarly, in an RC circuit, the time constant (in seconds) is:
where R is the resistance (in ohms) and C is the capacitance (in farads). is
where rm is the resistance across the membrane and cm is the capacitance of the membrane.
The resistance across the membrane is a function of the number of open ion channels and the capacitance is a function of the properties of the lipid bilayer.
The time constant is used to describe the rise and fall of the action potential, where the rise is described by
and the fall is described by
Where voltage is in millivolts, time is in seconds, and is in seconds.
Vmax is defined as the maximum voltage attained in the action potential, where
where rm is the resistance across the membrane and I is the current flow.
Setting for t = for the rise sets V(t) equal to 0.63Vmax. This means that the time constant is the time elapsed after 63% of Vmax has been reached.
Setting for t = for the fall sets V(t) equal to 0.37Vmax, meaning that the time constant is the time elapsed after it has fallen to 37% of Vmax.
The larger a time constant is, the slower the rise or fall of the potential of neuron. A long time constant can result in temporal summation, or the algebraic summation of repeated potentials. by
, (tau), characterizes the frequency response of a first-order, linear time-invariant (LTI) system. Examples include electrical RC circuits and RL circuits. It is also used to characterize the frequency response of various signal processing systems – magnetic tapes, radio transmitters and receivers, record cutting and replay equipment, and digital filters – which can be modelled or approximated by first-order LTI systems.
Other examples include time constant used in control systems for integral and derivative action controllers, which are often pneumatic, rather than electrical.
Physically, the time constant represents the time it takes the system's step response to reach approximately 63% of its final (asymptotic) value, ie about 37% below its final value.
Differential equation
First order LTI systems are characterized by the differential equation
where
represents the exponential decay constant and V is a function of time t
The time constant is related to the exponential decay constant by
General Solution
The general solution to the differential equation is
where
is the initial value of V.
Control Engineering
The diagram below depicts the exponential function in the specific case where , otherwise referred to as a "decaying" exponential function:
)
Suppose
then
The term
(tau) is referred to as the "time constant" and can be used (as in this case) to indicate how rapidly an exponential function decays.
Where:
- t = time (generally always
in control engineering)
- A = initial value (see "specific cases" below).
Specific cases
- 1). Let
, hence 
, and so 
- 2). Let
, hence 
, ≈ 0.37A
- 3). Let
, and so 
- 4). Let
, hence 
, ≈ 0.0067A
After a period of one time constant the function reaches e-1 = approximately 37% of its initial value. In case 4, after five time constants the function reaches a value less than 1% of its original. In most cases this 1% threshold is considered sufficient to assume that the function has decayed to zero - Hence in control engineering a stable system is mostly assumed to have settled after five time constants as a rule of thumb.
Examples of time constants
Time constants in electrical circuits
In an RL circuit, the time constant (in seconds) is
where R is the resistance (in ohms) and L is the inductance (in henries).
Similarly, in an RC circuit, the time constant
(in seconds) is:
where R is the resistance (in ohms) and C is the capacitance (in farads).
Thermal time constant
See discussion page.Time constants in neurobiology
In an action potential (or even in a passive spread of signal) in a neuron, the time constant is
where rm is the resistance across the membrane and cm is the capacitance of the membrane.
The resistance across the membrane is a function of the number of open ion channels and the capacitance is a function of the properties of the lipid bilayer.
The time constant is used to describe the rise and fall of the action potential, where the rise is described by
and the fall is described by
Where voltage is in millivolts, time is in seconds, and
is in seconds.
Vmax is defined as the maximum voltage attained in the action potential, where
where rm is the resistance across the membrane and I is the current flow.
Setting for t =
for the rise sets V(t) equal to 0.63Vmax. This means that the time constant is the time elapsed after 63% of Vmax has been reached.
Setting for t =
for the fall sets V(t) equal to 0.37Vmax, meaning that the time constant is the time elapsed after it has fallen to 37% of Vmax.
The larger a time constant is, the slower the rise or fall of the potential of neuron. A long time constant can result in temporal summation, or the algebraic summation of repeated potentials.
Radioactive half-life
The half-life THL of a radioactive isotope is related to the exponential time constant by
See also
External links
- Conversion of time constant τ to cutoff frequency fc and vice versa
- All about circuits - Voltage and current calculations
Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
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Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. The American Engineers' Council for Professional Development, also known as ECPD,[1] (later ABET [2]
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Writing system: Greek alphabet
Official status
Official language of: Greece
Cyprus
European Union
recognised as minority language in parts of:
European Union
Italy
Turkey
Regulated by:
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step response of a dynamical system consists of the time (or more generally the evolution parameter) behavior of its outputs when its control inputs are Heaviside step functions, for a given initial state.
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A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and λ is a positive number called the decay constant.
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A resistor-inductor circuit (RL circuit), or RL filter or RL network, is one of the simplest analogue infinite impulse response electronic filters. It consists of a resistor and an inductor, either in series or in parallel, driven by a voltage source.
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second (SI symbol: s), sometimes abbreviated sec., is the name of a unit of time, and is the International System of Units (SI) base unit of time.
SI prefixes are frequently combined with the word second to denote subdivisions of the second, e.g.
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ohm (symbol: Ω) is the SI unit of electrical impedance or, in the direct current case, electrical resistance, named after Georg Ohm.
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Definition
An ohm is the electrical resistance offered by a current-carrying element that produces a voltage drop of one volt when a..... Read more.
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farad (symbol: F) is the SI unit of capacitance. It is named after the British physicist Michael Faraday.
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Definition
The farad is defined as the amount of capacitance for which a potential difference of one volt results in a static charge of one coulomb...... Read more.
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lipid bilayer or bilayer lipid membrane (BLM) is a membrane or zone of a membrane composed of lipid molecules (usually phospholipids). The lipid bilayer is a critical component of all biological membranes, including cell membranes, and so is absolutely essential for all
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