page_title: Time constant
section_title: Relation of time constant to bandwidth
hierarchical_section_title: Time constant / Relation of time constant to bandwidth
caption_attribution_description: English: Frequency response of single pole system
context_page_description: In physics and engineering, the time constant, usually denoted by the Greek letter τ, is the parameter characterizing the response to a step input of a first-order, linear time-invariant system. The time constant is the main characteristic unit of a first-order LTI system.
  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. In the frequency domain 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 modeled 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.
context_section_description: 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) = V₀ 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 = 2πf):
  The notation f3dB 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.
caption: Frequency response of system vs. frequency in units of the bandwidth f3dB. The response is normalized to a zero frequency value of unity, and drops to 1/√2 at the bandwidth.
exif: {}