Given the differential equation model of a physical system, determine the time constant of the system:

\(40 \frac{dx}{dt}+2x=f(t)\)

Concept:

Time constant \(\tau = \frac{{ - 1}}{{{\rm{real\ part\ of\ Dominant\ pole}}}}\)

Calculation:

\(40\frac{{dx}}{{dt}} + 2x = f\left( t \right)\)

Taking Laplace transform, we get

40 s X(s) + 2X(s) = 12(s)

\(\frac{{X\left( s \right)}}{{F\left( s \right)}} = \frac{1}{{40s + 2}}\)

\( = \frac{1}{{40\left( {s + \frac{1}{{20}}} \right)}}\)

Pole will be at -1/20. 

Time constant \( = \frac{1}{{pole}} = 20\)