Time-delay is frequently encountered in many control problems． The existence of time-delay induces difficulty in stability analysis
and may thereby lead to oscillations or degraded performance in closed-loop systems． This paper investigates the stability analysis
of linear time-invariant( LTI) systems with time delay depending upon the solution of the characteristic equation． Not only the absolute
stability but also the relative stabilities the transient performances are depended on the locations of the characteristic equation roots． The
existence of time delay transforms the closed-loop characteristic equation into a transcendental equation with an infinite number of roots．
By investigating the real part and the imaginary part of the transcendental equation，the characteristic equation can be transformed into a
rotation equation with two vectors related to the system parameters． The characteristic roots on the given region boundaries can be determined
exactly and concisely． Numerical examples are provided to illustrate the proposed algorithm．