# Relationship between damping ratio and quality factor

The damping ratio is a measure of describing how rapidly the oscillations decay from The key difference between critical damping and overdamping is that, The factors Q, damping ratio ζ, and exponential decay rate α are related such that . recognize the correlation between a damped sinusoid in the time The Q-factor or quality factor is defined as the ratio of energy stored to the. Damping is an influence within or upon an oscillatory system that has the effect of reducing, The key difference between critical damping and overdamping is that , in critical damping, the system returns to equilibrium in the minimum amount . The Q factor, damping ratio ζ, and exponential decay rate α are related such that.

This case is called critical damping. The key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time.

## Damping ratio

Definition The effect of varying damping ratio on a second-order system. It is particularly important in the study of control theory.

It is also important in the harmonic oscillator. The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient: Derivation Using the natural frequency of a harmonic oscillator and the definition of the damping ratio above, we can rewrite this as: This equation can be solved with the approach.

Using it in the ODE gives a condition on the frequency of the damped oscillations, Undamped: Is the case where corresponds to the undamped simple harmonic oscillator, and in that case the solution looks likeas expected. If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like. This case occurs forand is referred to as underdamped. If s is a real number, then the solution is simply a decaying exponential with no oscillation.

• Damping ratio (related to Quality factor)

This case occurs forand is referred to as overdamped. The case where is the border between the overdamped and underdamped cases, and is referred to as critically damped.

### Q factor - Wikipedia

This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required e. See Individual reactive components. Q factor and damping[ edit ] Main articles: For mathematical details about these systems and their behavior see harmonic oscillator and linear time invariant LTI system.

Such a system doesn't oscillate at all, but when displaced from its equilibrium steady-state output it returns to it by exponential decayapproaching the steady state value asymptotically.

It has an impulse response that is the sum of two decaying exponential functions with different rates of decay.

As the quality factor decreases the slower decay mode becomes stronger relative to the faster mode and dominates the system's response resulting in a slower system.

### Damping ratio (related to Quality factor) - calculator - fx​Solver

A second-order low-pass filter with a very low quality factor has a nearly first-order step response; the system's output responds to a step input by slowly rising toward an asymptote. Underdamped systems combine oscillation at a specific frequency with a decay of the amplitude of the signal.

As the quality factor increases, the relative amount of damping decreases. A high-quality bell rings with a single pure tone for a very long time after being struck.