Time constant of first order system

I am new to this site, so please bear with me. here is plot of the step response of first order system given enter image description here Now, the question is to find time constant as well as DC gain so I can find it's transfer function. here is my attempt, but I am not sure about finding the time constant. enter image description here I read somewhere that time constant for first order system is t=5*tau where t is the time when system reaches its steady state value, so tau=3/5 ? is this correct?

asked Mar 6, 2016 at 21:26 3 1 1 gold badge 1 1 silver badge 3 3 bronze badges \$\begingroup\$ time constant is simply tau. gain is OK. \$\endgroup\$ Commented Mar 6, 2016 at 21:32 \$\begingroup\$ how can I find Tau ? \$\endgroup\$ Commented Mar 6, 2016 at 21:34

\$\begingroup\$ I need to find Tau, so I can plug them into first order system equation \$\endgroup\$

Commented Mar 6, 2016 at 21:37

1 Answer 1

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Set \$ t = \tau \$ in your equation. This gives

where K is the DC gain, u(t) is the input signal, t is time, \$ \tau \$ is the time constant and y(t) is the output.

The time constant can be found where the curve is 63% of the way to the steady state output.

Easy-to-remember points are \$ \tau \$ @ 63%, \$ 3 \tau \$ @ 95\% and \$ 5 \tau \$ @ 99\%.

Your calculation for \$ \tau = \frac \$ appears to be based on the time the curve reaches \$ 5 \tau \$ but that is very difficult to pinpoint on the gentle slope of the curve. In the graphical solution below it looks more like 0.5 s rather than 0.6 s. (\$ 3 \cdot 63\% = 1.89 \$.)

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