In physics and engineering, the time constant usually denoted by the Greek letter <math>\tau</math>, (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 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.
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Differential equation
First order LTI systems are characterized by the differential equation
- <math>
{dV \over dt} = - \alpha V, </math> where <math> \ \alpha </math> represents the exponential decay constant and V is a function of time t
- <math>
V \ = \ V(t) \, </math> The time constant is related to the exponential decay constant by
- <math>
\tau = \ { 1 \over \alpha } \, </math>
General Solution
The general solution to the differential equation is
- <math>
V(t) \ = \ V_o e^{-\alpha t} \ = \ V_o e^{-t / \tau} \, </math> where
- <math>
V_o \ = \ V(t=0) \, </math> is the initial value of V.
Control Engineering
The diagram below depicts the exponential function <math>y=Ae^{at}</math> in the specific case where <math>a<0</math>, otherwise referred to as a "decaying" exponential function: 
Suppose
- <math>y=Ae^{-at} \ = \ Ae^{-{t \over \tau}}</math>
then
- <math>\tau={ 1 \over a}</math>
The term <math>\tau</math> (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 <math>t>0</math> in control engineering)
- A = initial value (see "specific cases" below).
Specific cases
- 1). Let <math>t=0</math>, hence <math>y=Ae^0</math>, and so <math>y=A</math>
- 2). Let <math>t= \tau</math>, hence <math>y=Ae^{-1}</math>, ≈ <math> 0.37 A </math>
- 3). Let <math>y=f(t)=Ae^{-{t \over \tau}}</math>, and so <math> \lim_{t \to \infty}f(t) = 0 </math>
- 4). Let <math>t=5 \tau</math>, hence <math>y=Ae^{-5}</math>, ≈ <math> 0.0067 A </math>
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.
Examples of time constants
Time constants in electrical circuits
In an RL circuit, the time constant <math>\tau</math> (in seconds) is
- <math>
\tau \ = \ { L \over R } \, </math> where R is the resistance (in ohms) and L is the inductance (in henries). Similarly, in an RC circuit, the time constant <math>\tau</math> (in seconds) is:
- <math>
\tau \ = \ R C \, </math> 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 <math>\tau</math> is
- <math>
\tau \ = \ r_{m} c_{m} \, </math> 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
- <math>
V(t) \ = \ V_{max} (1 - e^{-t /\tau}) \, </math> and the fall is described by
- <math>
V(t) \ = \ V_{max} e^{-t /\tau} \, </math> Where voltage is in millivolts, time is in seconds, and <math>\tau</math> is in seconds. Vmax is defined as the maximum voltage attained in the action potential, where
- <math>
V_{max} \ = \ r_{m}I \, </math> where rm is the resistance across the membrane and I is the current flow. Setting for t = <math>\tau</math> 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 = <math>\tau</math> 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 <math>\tau</math> by
- <math>
T_{HL} = \tau \cdot \mathrm{ln2} \, </math>
Step Response with Non-Zero Initial Conditions
If the motor is initially spinning at a constant speed(expressed by voltage), the time constant <math>\tau</math> is 63% of Vinfinity minus Vo. Therefore,
- <math>
V(t) \ = \ V_o + (V_{infinity} - V_o) * (1 - e^{-t / \tau}) \, </math> can be used where the initial and final voltages, respectively, are:
- <math>
V_o \ = \ V(t=0) </math> and
- <math>
V_{infinity} \ = \ V(t=infinity) </math>
Step Response from Rest
From rest, the voltage equation is a simplification of the case with non-zero ICs. With an initial velocity of zero, V0 drops out and the resulting equation is:
- <math>
V(t) \ = \ V_{infinity} * (1 - e^{-t / \tau}) \, </math>
Coincidentally, the time constant will remain the same for the same system regardless of the starting conditions. For example, if an electric motor reaches 63% of its final speed from rest in ⅛ of a second, it will also take ⅛ of a second for the motor to reach 63% of its final speed when started with some non-zero initial speed. Simply stated, a system will require a certain amount of time to reach its final, steady-state situation regardless of how close it is to that value when started.
