Steady state response refers to the response of the measurem

Steady state response refers to the response of the measurement system after a harmonic input has been applied for a period of time and the transients have decayed to zero. Determine the length of time that a first order system needs to achieve a steady response (to within 1% or better).

Solution

The unit step response of a first order system is given by

Response(t) = K * (1 - exp(-t/tau))

                       where K is the gain and tau is the time constant of the system.

for 1% steady response we have

0.99 K   = K * (1 - exp(-t/tau))

exp(-t/tau)   = 0.01

t/tau = -ln(0.01)

time   = 4.605 * tau    

Therefore length of time is equal to 4.605 times the time constant of the first order system.