After reading this topic Delay time $({t_d})$ in Time response of a second-order control system for subjected to a unit step input underdamped case, you will understand the theory, expression, and plot.
A block diagram of the second order closed-loop control system with unity negative feedback is shown below in Figure 1,
The general expression for the time response of a second order control system or underdamped case is
\[c(t) = 1 – \frac{{{e^{ – \xi {\omega _n}t}}}}{{\sqrt {1 – {\xi ^2}} }}\sin \left[ {({\omega _n}\sqrt {1 – {\xi ^2}} )t + \theta } \right]…(1)\]
Also Equation 1, is plotted in Figure 2 as shown below
Delay time $(t_d)$
It is the time required by the response to reach 50% of output c(t) for the first time as shown in Figure 2. Its general expression is
\[{t_d} = \frac{{1 + 0.7\xi }}{{{\omega _n}}}\]
