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In physics and engineering, the **time constant**, usually denoted by the Greek letter τ (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.^{ [1] }^{ [note 1] } The time constant is the main characteristic unit of a first-order LTI system.

- Differential equation
- Example solution
- Relation of time constant to bandwidth
- Step response with arbitrary initial conditions
- Examples
- Time constants in electrical circuits
- Thermal time constant
- Time constants in Biophysics
- Exponential decay
- Meteorological sensors
- See also
- Notes
- References
- External links

In the time domain, the usual choice to explore the time response is through the step response to a step input, or the impulse response to a Dirac delta function input.^{ [2] } In the frequency domain (for example, looking at the Fourier transform of the step response, or using an input that is a simple sinusoidal function of time) the time constant also determines the bandwidth of a first-order time-invariant system, that is, the frequency at which the output signal power drops to half the value it has at low frequencies.

The time constant 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.

Time constants are a feature of the lumped system analysis (lumped capacity analysis method) for thermal systems, used when objects cool or warm uniformly under the influence of convective cooling or warming.^{ [3] }

Physically, the time constant represents the elapsed time required for the system response to decay to zero if the system had continued to decay at the initial rate, because of the progressive change in the rate of decay the response will have actually decreased in value to 1 / *e* ≈ 36.8% in this time (say from a step decrease). In an increasing system, the time constant is the time for the system's step response to reach 1 − 1 / *e* ≈ 63.2% of its final (asymptotic) value (say from a step increase). In radioactive decay the time constant is related to the decay constant (**λ**), and it represents both the mean lifetime of a decaying system (such as an atom) before it decays, or the time it takes for all but 36.8% of the atoms to decay. For this reason, the time constant is longer than the half-life, which is the time for only 50% of the atoms to decay.

First order LTI systems are characterized by the differential equation

where τ represents the exponential decay constant and V is a function of time t

The right-hand side is the *forcing function**f*(*t*) describing an external driving function of time, which can be regarded as the system *input*, to which *V*(*t*) is the *response*, or system output. Classical examples for *f*(*t*) are:

The Heaviside step function, often denoted by *u*(*t*):

the impulse function, often denoted by *δ*(*t*), and also the sinusoidal input function:

or

where A is the amplitude of the forcing function, f is the frequency in Hertz, and *ω* = 2*π f* is the frequency in radians per second.

An example solution to the differential equation with initial value *V*_{0} and no forcing function is

where

is the initial value of V. Thus, the response is an exponential decay with time constant τ.

Suppose

This behavior is referred to as a "decaying" exponential function. The time τ (tau) is referred to as the "time constant" and can be used (as in this case) to indicate how rapidly an exponential function decays.

Here:

- t is time (generally
*t*> 0 in control engineering) *V*_{0}is the initial value (see "specific cases" below).

- Let ; then , and so
- Let ; then
- Let , and so
- Let ; then

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 – as a rule of thumb, in control engineering a stable system is one that exhibits such an overall damped behavior.

Suppose the forcing function is chosen as sinusoidal so:

(Response to a real cosine or sine wave input can be obtained by taking the real or imaginary part of the final result by virtue of Euler's formula.) The general solution to this equation for times *t* ≥ 0 s, assuming *V*(*t* = 0) = *V*_{0} is:

For long times the decaying exponentials become negligible and the steady-state solution or long-time solution is:

The magnitude of this response is:

By convention, the bandwidth of this system is the frequency where |*V*_{∞}|^{2} drops to half-value, or where *ωτ* = 1. This is the usual bandwidth convention, defined as the frequency range where power drops by less than half (at most −3 dB). Using the frequency in hertz, rather than radians/s (*ω* = 2*πf*):

The notation *f*_{3dB} stems from the expression of power in decibels and the observation that half-power corresponds to a drop in the value of |*V*_{∞}| by a factor of 1/2 or by 3 decibels.

Thus, the time constant determines the bandwidth of this system.

Suppose the forcing function is chosen as a step input so:

with *u*(*t*) the Heaviside step function. The general solution to this equation for times *t* ≥ 0 s, assuming *V*(*t* = 0) = *V*_{0} is:

(It may be observed that this response is the *ω* → 0 limit of the above response to a sinusoidal input.)

The long-time solution is time independent and independent of initial conditions:

The time constant remains the same for the same system regardless of the starting conditions. Simply stated, a system approaches its final, steady-state situation at a constant rate, regardless of how close it is to that value at any arbitrary starting point.

For example, consider an electric motor whose startup is well modelled by a first-order LTI system. Suppose that when started from rest, the motor takes 1/8 of a second to reach 63% of its nominal speed of 100 RPM, or 63 RPM—a shortfall of 37 RPM. Then it will be found that after the next 1/8 of a second, the motor has sped up an additional 23 RPM, which equals 63% of that 37 RPM difference. This brings it to 86 RPM—still 14 RPM low. After a third 1/8 of a second, the motor will have gained an additional 9 RPM (63% of that 14 RPM difference), putting it at 95 RPM.

In fact, given *any* initial speed *s* ≤ 100 RPM,1/8 of a second later this particular motor will have gained an additional 0.63 × (100 − *s*) RPM.

In an RL circuit composed of a single resistor and inductor, the time constant (in seconds) is

where *R* is the resistance (in ohms) and *L* is the inductance (in henrys).

Similarly, in an RC circuit composed of a single resistor and capacitor, the time constant (in seconds) is:

where *R* is the resistance (in ohms) and *C* is the capacitance (in farads).

Electrical circuits are often more complex than these examples, and may exhibit multiple time constants (See Step response and Pole splitting for some examples.) In the case where feedback is present, a system may exhibit unstable, increasing oscillations. In addition, physical electrical circuits are seldom truly linear systems except for very low amplitude excitations; however, the approximation of linearity is widely used.

In digital electronic circuits another measure, the FO4 is often used. This can be converted to time constant units via the equation .^{ [4] }

Time constants are a feature of the lumped system analysis (lumped capacity analysis method) for thermal systems, used when objects cool or warm uniformly under the influence of convective cooling or warming. In this case, the heat transfer from the body to the ambient at a given time is proportional to the temperature difference between the body and the ambient:^{ [5] }

where *h* is the heat transfer coefficient, and *A*_{s} is the surface area, *T* is the temperature function, i.e., *T*(*t*) is the body temperature at time *t*, and *T*_{a} is the constant ambient temperature. The positive sign indicates the convention that *F* is positive when heat is *leaving* the body because its temperature is higher than the ambient temperature (*F* is an outward flux). If heat is lost to the ambient, this heat transfer leads to a drop in temperature of the body given by:^{ [5] }

where *ρ* = density, *c*_{p} = specific heat and *V* is the body volume. The negative sign indicates the temperature drops when the heat transfer is *outward* from the body (that is, when *F* > 0). Equating these two expressions for the heat transfer,

Evidently, this is a first-order LTI system that can be cast in the form:

with

In other words, larger masses *ρV* with higher heat capacities *c*_{p} lead to slower changes in temperature (longer time constant *τ*), while larger surface areas *A*_{s} with higher heat transfer *h* lead to more rapid temperature change (shorter time constant *τ*).

Comparison with the introductory differential equation suggests the possible generalization to time-varying ambient temperatures *T*_{a}. However, retaining the simple constant ambient example, by substituting the variable Δ*T* ≡ (*T − T*_{a}), one finds:

Systems for which cooling satisfies the above exponential equation are said to satisfy Newton's law of cooling. The solution to this equation suggests that, in such systems, the difference between the temperature of the system and its surroundings Δ*T* as a function of time *t*, is given by:

where Δ*T*_{0} is the initial temperature difference, at time *t* = 0. In words, the body assumes the same temperature as the ambient at an exponentially slow rate determined by the time constant.

In an excitable cell such as a muscle or neuron, the time constant of the membrane potential is

where *r*_{m} is the resistance across the membrane and *c*_{m} 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 membrane voltage, 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.

*V*_{max} is defined as the maximum voltage change from the resting potential, where

where *r*_{m} is the resistance across the membrane and *I* is the membrane current.

Setting for *t* = for the rise sets *V*(*t*) equal to 0.63*V*_{max}. This means that the time constant is the time elapsed after 63% of *V*_{max} has been reached

Setting for *t* = for the fall sets *V*(*t*) equal to 0.37*V*_{max}, meaning that the time constant is the time elapsed after it has fallen to 37% of *V*_{max}.

The larger a time constant is, the slower the rise or fall of the potential of a neuron. A long time constant can result in temporal summation, or the algebraic summation of repeated potentials. A short time constant rather produces a coincidence detector through spatial summation.

In exponential decay, such as of a radioactive isotope, the time constant can be interpreted as the mean lifetime. The half-life *T*_{HL} or *T*_{1/2} is related to the exponential decay constant by

The reciprocal of the time constant is called the decay constant, and is denoted .

A **time constant** is the amount of time it takes for a meteorological sensor to respond to a rapid change in a measure, and until it is measuring values within the accuracy tolerance usually expected of the sensor.

This most often applies to measurements of temperature, dew-point temperature, humidity and air pressure. Radiosondes are especially affected due to their rapid increase in altitude.

- ↑ Concretely, a first-order LTI system is a system that can be modeled by a single first order differential equation in time. Examples include the simplest single-stage electrical RC circuits and RL circuits.

In classical mechanics, a **harmonic oscillator** is a system that, when displaced from its equilibrium position, experiences a restoring force *F* proportional to the displacement *x*:

In engineering, a **transfer function** of a system, sub-system, or component is a mathematical function that models the system's output for each possible input. It is widely used in electronic engineering tools like circuit simulators and control systems. In simple cases, this function can be represented as a two-dimensional graph of an independent scalar input versus the dependent scalar output. Transfer functions for components are used to design and analyze systems assembled from components, particularly using the block diagram technique, in electronics and control theory.

In electromagnetism, a **dielectric** is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor, because they have no loosely bound, or free, electrons that may drift through the material, but instead they shift, only slightly, from their average equilibrium positions, causing **dielectric polarisation**. Because of dielectric polarisation, positive charges are displaced in the direction of the field and negative charges shift in the direction opposite to the field. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarised, but also reorient so that their symmetry axes align to the field.

In signal processing, **group delay** and **phase delay** are two related ways of describing how a signal's frequency components are delayed in time when passing through a linear time-invariant (LTI) system. Phase delay describes the time shift of a *sinusoidal* component. Group delay describes the time shift of the *envelope* of a *wave packet*, a "pack" or "group" of oscillations centered around one frequency that travel together, formed for instance by multiplying a sine wave by an envelope.

A **low-pass filter** is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter design. The filter is sometimes called a **high-cut filter**, or **treble-cut filter** in audio applications. A low-pass filter is the complement of a high-pass filter.

In physics and engineering, the **quality factor** or ** Q factor** is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy lost in one radian of the cycle of oscillation. Q factor is alternatively defined as the ratio of a resonator's centre frequency to its bandwidth when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results. Higher

A **resistor–capacitor circuit**, or **RC filter** or **RC network**, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.

The **Drude model** of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials. Basically, Ohm's law was well established and stated that the current *J* and voltage *V* driving the current are related to the resistance *R* of the material. The inverse of the resistance is known as the conductance. When we consider a metal of unit length and unit cross sectional area, the conductance is known as the conductivity, which is the inverse of resistivity. The Drude model attempts to explain the resistivity of a conductor in terms of the scattering of electrons by the relatively immobile ions in the metal that act like obstructions to the flow of electrons.

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 zero to one in a very short time. The concept can be extended to the abstract mathematical notion of a dynamical system using an evolution parameter.

In system analysis, among other fields of study, a **linear time-invariant** (**LTI**) **system** is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined below. These properties apply (exactly or approximately) to many important physical systems, in which case the response *y*(*t*) of the system to an arbitrary input *x*(*t*) can be found directly using convolution: *y*(*t*) = (*x* ∗ *h*)(*t*) where *h*(*t*) is called the system's impulse response and ∗ represents convolution (not to be confused with multiplication). What's more, there are systematic methods for solving any such system (determining *h*(*t*)), whereas systems not meeting both properties are generally more difficult (or impossible) to solve analytically. A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers.

A **resistor–inductor circuit**, or **RL filter** or **RL network**, is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor, either in series driven by a voltage source or in parallel driven by a current source. It is one of the simplest analogue infinite impulse response electronic filters.

**Nondimensionalization** is the partial or full removal of physical dimensions from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. In some physical systems, the term **scaling** is used interchangeably with *nondimensionalization*, in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities intrinsic to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units.

A **Kelvin-Voigt material**, also called a **Voigt material**, is the most simple model viscoelastic material showing typical rubbery properties. It is purely elastic on long timescales, but shows additional resistance to fast deformation. It is named after the British physicist and engineer Lord Kelvin and German physicist Woldemar Voigt.

The **Cole–Cole equation** is a relaxation model that is often used to describe dielectric relaxation in polymers.

The **Mason–Weaver equation** describes the sedimentation and diffusion of solutes under a uniform force, usually a gravitational field. Assuming that the gravitational field is aligned in the *z* direction, the Mason–Weaver equation may be written

**Resonance fluorescence** is the process in which a two-level atom system interacts with the quantum electromagnetic field if the field is driven at a frequency near to the natural frequency of the atom.

An **RLC circuit** is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

Phasor approach refers to a method which is used for vectorial representation of sinusoidal waves like alternative currents and voltages or electromagnetic waves. The amplitude and the phase of the waveform is transformed into a vector where the phase is translated to the angle between the phasor vector and X axis and the amplitude is translated to vector length or magnitude. In this concept the representation and the analysis becomes very simple and the addition of two wave forms is realized by their vectorial summation.

In mathematics, the **exponential response formula** (ERF), also known as **exponential response and complex replacement**, is a method used to find a particular solution of a non-homogeneous linear ordinary differential equation of any order. The exponential response formula is applicable to non-homogeneous linear ordinary differential equations with constant coefficients if the function is polynomial, sinusoidal, exponential or the combination of the three. The general solution of a non-homogeneous linear ordinary differential equation is a superposition of the general solution of the associated homogeneous ODE and a particular solution to the non-homogeneous ODE. Alternative methods for solving ordinary differential equations of higher order are method of undetermined coefficients and method of variation of parameters.

**Anelasticity** is a property of materials that describes their behaviour when undergoing deformation. Its formal definition does not include the physical or atomistic mechanisms but still interprets the anelastic behaviour as a manifestation of internal relaxation processes. It is a behaviour differing from elastic behaviour.

- ↑ Béla G. Lipták (2003).
*Instrument Engineers' Handbook: Process control and optimization*(4 ed.). CRC Press. p. 100. ISBN 978-0-8493-1081-2. - ↑ Bong Wie (1998).
*Space vehicle dynamics and control*. American Institute of Aeronautics and Astronautics. p. 100. ISBN 978-1-56347-261-9. - ↑ GR North (1988). "Lessons from energy balance models". In Michael E. Schlesinger (ed.).
*Physically-based Modelling and Simulation of Climate and Climatic Change*(NATO Advanced Study Institute on Physical-Based Modelling ed.). Springer. NATO. p. 627. ISBN 978-90-277-2789-3. - ↑ Harris, D.; Sutherland, I. (2003). "Logical effort of carry propagate adders".
*The Thirty-Seventh Asilomar Conference on Signals, Systems & Computers, 2003*. pp. 873–878. doi:10.1109/ACSSC.2003.1292037. ISBN 0-7803-8104-1. S2CID 7880203. - 1 2 Roland Wynne Lewis; Perumal Nithiarasu; K. N. Seetharamu (2004).
*Fundamentals of the finite element method for heat and fluid flow*. Wiley. p. 151. ISBN 978-0-470-84789-3.

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