# Time constant of second order system

I have a second order system with open-loop transfer function $$G(s)={a\over(s+b)(s+c)}$$, $$a,b,c$$ are already known. This system has unity feedback.

I am interested in the time consant of this system, but I need to know should I calculate based on the open-loop or closed-loop transfer function?

The second question is how to calculate the time consant of a second order system? On this webpage (Second Order Systems), it says a second order system may be the combination of two first order systems. If the time constant for the two first order system is $$\tau_1$$ and $$\tau_2$$, the time constant for the second order system is $$\tau^2=\tau_1 \tau_2$$. Can I compute according to this equation?

Time constant for a system should be calculated based on the closed loop transfer function.

For the open loop transfer function you are considering, the closed loop transfer function is given by: $$TF = \frac{G(s)}{1 + G(s)H(s)}$$

For a unity feedback: $$H(s) = 1$$ \begin{align} \therefore TF &= \frac{\frac{a}{(s+b)(s+c)}}{1 + \frac{a}{(s+b)(s+c)}}\\ &= \frac{a}{(s+b)(s+c) + a} \\ &= \frac{a}{s^2 +(b+c)s + (a + bc)} \end{align} Write the above equation in the form: $$TF = \frac{a}{s^2 + 2\zeta\omega_n s + \omega_n^2}$$ where: $$2\zeta\omega_n = (b+c)$$ and $$\omega_n^2 = (a + bc)$$.

The time constant is given by $$T = \frac{1}{\zeta\omega_n}$$. You would get this same value when you break the second-order system into two first order systems and then find their corresponding time constants. And finally, use the formula that you have stated.

The equation you give, $$\tau^2 = \tau_1\tau_2$$ is on the web page you linked to.

However you need to understand that $$\tau_1$$ and $$\tau_2$$ are first order time constants but $$\tau$$ is a second order time constant. The first and second order constants do not have exactly the same physical interpretation, and they are not even in the same physical units!

• What is the physical meaning of second order constant? and according to the equation $\tau^2=\tau_1\tau_2$, it should have the same physical unit with first order time constant. Is there anything I missed? Dec 2 '19 at 6:20