First order time constant calculator
A first-order system reaches 63.2% of a step in one time constant and 99.3% in five. Almost every sensor and thermal process behaves this way.
s
s
%
Value at that time
66.612 19
- Fraction of the step completed
- 0.776 870
- Still to go
- 13.387 81
- Time to reach your target
- 29.9573 s
- Time to 95%
- 30.000 s
- Time to 99.3%
- 50.000 s
- Rise time, 10% to 90%
- 21.9722 s
- Equivalent bandwidth
- 0.015 92 Hz
- Initial rate of change
- 6
Show the working
- Time constant = 10
- Initial value = 20
- Final value = 80
- Time elapsed = 15
- Target percentage of the step = 95
- step = 80 - 20 = 60
- Fraction of the step completed = 1 - exp(-15/10) = 0.776 87
- Value at that time = 20 + 60 * 0.776 87 = 66.6122
- Still to go = 80 - 66.6122 = 13.3878
- Time to reach your target = -10 * ln(1 - 95/100) = 29.9573
- Time to 95% = 10 * 3 = 30
- Time to 99.3% = 10 * 5 = 50
- Rise time, 10% to 90% = 10 * 2.197224577336219 = 21.9722
- Initial rate of change = 60 / 10 = 6
- Equivalent bandwidth = 1 / (2 * pi * 10) = 0.015 915 5
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- y(t) = initial + step x (1 - e^(-t/tau))
- time to a fraction f = -tau ln(1 - f)
- rise time from 10% to 90% is 2.197 tau
A sensor with a long time constant averages the process. That is sometimes a feature and often a control problem.