Dynamic characteristics: rise time, delay, response

Dynamic characteristics describe how a measurement system responds to inputs that change with time.

• Amplitude response (magnitude ratio): the ratio of output amplitude to input amplitude, $M(\omega) = |q_o|/(K|q_i|)$, for a sinusoidal input of frequency $\omega$. An ideal instrument has $M = 1$ over its working range.
• Frequency response: the variation of $M$ and phase with frequency. The range over which $M$ stays within a stated tolerance (e.g. $\pm 5\%$) is the bandwidth or useful frequency range.
• Phase response: the phase angle $\phi(\omega)$ between output and input. For distortion-free measurement of complex signals, $\phi$ should be proportional to frequency (a constant time delay).
• Delay time ($t_d$): the time for the output to reach 50% of its final value after a step input.
• Rise time ($t_r$): the time for the step response to rise from 10% to 90% (sometimes 0 to 100%) of the final value. For a first-order system $t_r = 2.2\tau$.
• Settling time ($t_s$): the time after which the output stays within a band (2% or 5%) of the final value.
• Dynamic error: the difference between the indicated value and the true value of a time-varying input, excluding static error. For a sinusoidal input, the amplitude error is $(1 - M)$ times the input amplitude, and the output also lags by $\phi$.

 y
 1.0 |           _.-~~~~~~~~~
 0.9 |--------.-'
 0.5 |------.'        <- delay time t_d
 0.1 |---.-'
     |  |<--- t_r --->|
     +--------------------> t