The “ramp” variation in the measurand occurs when the measurand increases at a constant rate, as shown by the red line in the figure.
The sensor output in both cases, after a transient period, follows the slope of the measurand but with a delay. For the first-order case, the effective delay, also known as the dynamic lag, is equal to the time constant so this is often a useful way to determine that time constant. In the case of aircraft-borne temperature sensors, this response will result in a difference between aircraft ascent and descent profiles, and the time constant of the sensors can be determined from that difference.
For a second-order sensor, the delay is 2𝛄 / ω and, for the parameters shown, is larger than the first-order delay 𝜏 .