page_title: Time constant
section_title: Step response with arbitrary initial conditions
hierarchical_section_title: Time constant / Step response with arbitrary initial conditions
caption_attribution_description: English: Response of single pole system to unit step with two initial conditions
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 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) = V₀ 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 modeled by a first-order LTI system.  Suppose that when started from rest, the motor takes ⅛ 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 ⅛ 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 ⅛ 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, ⅛ of a second later this particular motor will have gained an additional 0.63 × (100 − s) RPM.
caption: Step response of system for two different initial values V0, one above the final value and one at zero. Long-time response is a constant, V∞. Time axis in units of the time constant 
    
      
        
          τ
        
      
      {\displaystyle \tau }
    
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