The characteristic equation is s² + 4s + k = 0.

Unity Feedback System and Open-Loop Transfer Function

In a unity feedback system, the output of the system is fed back and subtracted from the input to generate an error signal. This error signal is then used to control the system. The open-loop transfer function represents the relationship between the input and output of the system without considering the feedback.

The given open-loop transfer function is G(s) = k/s(s+4), where k is the gain and s represents the complex frequency variable.

Increasing Gain to Infinity

When the gain k is increased to infinity, it means that the system is highly amplified. In practical terms, this implies that the output of the system becomes very large compared to the input. Mathematically, it can be represented as k -> ∞.

Impact on Damping Ratio

The damping ratio (ζ) is a measure of how quickly a system returns to equilibrium after a disturbance. It is determined by the roots of the characteristic equation, which is obtained by setting the denominator of the transfer function equal to zero.

In this case, the characteristic equation is s^2 + 4s = 0. Solving this equation, we find two roots: s1 = 0 and s2 = -4.

The damping ratio (ζ) is given by the formula ζ = -s1 / (2 * sqrt(s1^2 + s2^2)), which simplifies to ζ = 0 / (2 * sqrt(0 + 16)) = 0.

Explanation

When the gain is increased to infinity, it means that the system amplifies the output signal to a very large extent. As a result, any disturbances or errors in the system will be significantly magnified. This leads to a highly responsive and unstable system.

In terms of the damping ratio, a value of 0 indicates that there is no damping present in the system. Damping is responsible for controlling the rate at which the system returns to equilibrium after a disturbance. Without damping, the system becomes highly oscillatory and does not return to equilibrium smoothly.

Therefore, when the gain is increased to infinity, the corresponding damping ratio is 0. This implies that the system becomes highly unstable and oscillatory, making it difficult to control and prone to instability.