An increase in gain, in most systems, leads to:

Smaller damping ratio

The damping ratio is a measure of the relative stability of a system. It quantifies the rate at which the system's response decays after a disturbance. A higher damping ratio indicates a faster decay rate and therefore a more stable system.

When the gain of a system is increased, it means that the amplification of the input signal is also increased. This amplification can have a significant impact on the behavior of the system, including its damping ratio.

Positive feedback

Increasing the gain in a system often results in positive feedback. Positive feedback occurs when the output of a system is fed back into the input, reinforcing the original signal. This can lead to instability and oscillation in the system.

Effects on damping ratio

When positive feedback occurs due to increased gain, it can cause the system to become less stable and exhibit larger oscillations. This is because the positive feedback reinforces the input signal, leading to sustained oscillation. As a result, the damping ratio decreases.

Mathematical explanation

In a second-order system, the damping ratio (ζ) is defined as the ratio of the actual damping coefficient to the critical damping coefficient. The critical damping coefficient corresponds to the minimum value of damping required for a system to not oscillate after a disturbance.

The damping ratio can be expressed as ζ = C/Cc, where C is the actual damping coefficient and Cc is the critical damping coefficient. The actual damping coefficient is influenced by factors such as resistance, friction, and energy dissipation in a system.

When the gain is increased, the positive feedback amplifies the oscillations, making them more pronounced. This effectively reduces the damping coefficient and hence the damping ratio.

Conclusion

In most systems, an increase in gain leads to a smaller damping ratio. This is because the positive feedback resulting from increased gain amplifies the oscillations in the system, reducing the damping coefficient and making the system less stable.