# Settling time

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Settling time is the time required for an output to reach and remain within a given error band following some input stimulus.

The settling time of an amplifier or other output device is the time elapsed from the application of an ideal instantaneous step input to the time at which the amplifier output has entered and remained within a specified error band, usually symmetrical about the final value. Settling time includes a very brief propagation delay, plus the time required for the output to slew to the vicinity of the final value, recover from the overload condition associated with slew, and finally settle to within the specified error.

Systems with energy storage cannot respond instantaneously and will exhibit transient responses when they are subjected to inputs or disturbances.[1]

## Definition

Tay, Mareels and Moore (1997) defined settling time as "the time required for the response curve to reach and stay within a range of certain percentage (usually 5% or 2%) of the final value."[2]

## Mathematical detail

Settling time depends on the system response and time constant.

The settling time for a 2nd order, underdamped system responding to a step response can be approximated if the damping ratio $\zeta \ll 1$ by

$T_s = -\frac{\ln (\text{tolerance fraction})}{\text{damping ratio} \times \text{natural freq}}$

Thus, settling time to within 2% = 0.02 is:

$T_s = -\frac{\ln(0.02)}{\zeta \omega_n}\approx\frac{3.9}{\zeta \omega_n}$

## References

1. ^ [Modern Control Engineering (5th Edition), Katsuhiko Ogata]
2. ^ Tay, Teng-Tiow; Iven Mareels; John B. Moore (1997). High performance control. Birkhäuser. p. 93. ISBN 0-8176-4004-5.