Final answer:
The angle of departure for the transfer function HG(s) can be found by identifying the complex poles of the denominator and using the root locus method to determine the angle by which the root locus leaves the pole as gain K increases.
Step-by-step explanation:
The question relates to the angle of departure in control systems, particularly within a unity feedback system. To find the angle of departure for the given open-loop transfer function HG(s) = K(s+1)(s+2)/(s²-4s+5), we first need to identify the poles of the transfer function. These are the roots of the denominator, which are complex in this case. Once the poles are identified, the angle of departure can be calculated using the angle criteria from the root locus method.
To find the poles, we solve the quadratic equation s²-4s+5=0. The roots of this equation are complex, and their angle with the positive real axis can be computed using the inverse tangent function. The angle of departure is the angle by which the root locus leaves the pole as the gain K increases to infinity. This angle is calculated based on the geometric properties of the root locus and the phase contributions from all poles and zeros of the transfer function.
Using tools from control system analysis, such as Nyquist or root locus plots, allows for a graphical solution to this problem. It's important to remember that the angle of departure is typically measured counterclockwise from the positive real axis.