Library / Foundations: Maths / FMA-46
Foundations: Maths
First-order response to a constant input
Solution of the first-order equation dy/dt + ay = b with starting value y₀.
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- FMA-46
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- v1
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- Foundations: Maths
- Result
- y(t)
The equation dy/dt + ay = b has constant coefficients. Its solution is y(t) = b/a + (y0 − b/a)e−at. The steady value b/a is where the response ends up; the exponential term shows how the starting error dies away, with time constant 1/a. When a = 0 the response is a straight line, y = y0 + bt. A positive a gives decay towards the steady value; a negative a gives runaway growth.
Solution of the first-order equation dy/dt + ay = b with starting value y₀. For positive a, y decays or rises smoothly from y₀ towards the steady value b/a.
Calculator
- Steady value, b/a3
- Time constant, 1/a0.5 s
- Fraction of the way to the steady value reached86.47 %
- Time to reach 95 % of the change (about 3 time constants)1.498 s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With a = 2, b = 6 and a start from 0, the response heads for 3 and is at 2.594 after 1 s, 86.5 % of the way. This default, a = 2 and b = 6 starting from zero, gives y(1) = 2.594.
- After three time constants (1.5 s) the response has covered about 95 % of the change, so the steady value is effectively reached after four or five time constants.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Constant input b | −20 to 20 | −8.647 | 8.647 | 17.29 | +0.8 | +13.3 % |
| y0 Starting value y(0) | −20 to 20 | −0.1127 | 5.301 | 5.413 | +0.8 | +4.17 % |
| a Decay rate a (nonzero) | 0.1 to 10 1/s | 0.6 | 5.71 | 5.11 | +10% | −6.51 % |
| t Time | 0 to 10 s | 0 | 3 | 3 | +10% | +2.84 % |
Sweep of Time
| Time (s) | Response at time t |
|---|---|
| 0 | 0 |
| 1.25 | 2.754 |
| 2.5 | 2.98 |
| 3.75 | 2.998 |
| 5 | 3 |
| 6.25 | 3 |
| 7.5 | 3 |
| 8.75 | 3 |
| 10 | 3 |
Sweep one variable
Variables and how each one contributes
Response at time t
What the equation gives you. Everything else feeds it.
Decay rate a (nonzero)
The decay rate a in dy/dt + ay = b, in 1/s. Its reciprocal is the time constant. It must be nonzero; positive values give a stable response. Speeds up the approach to the steady value b/a, and also lowers that steady value in inverse proportion. Raising it by 0.5 (to 2.5) moves the result by −15.1 %, from 2.594 to 2.203.
Constant input b
The constant input b on the right-hand side, in the units of y per second. Raises the steady value b/a and the response at every time in direct proportion to b. Raising it by 1 (to 7) moves the result by +16.7 %, from 2.594 to 3.026.
Starting value y(0)
The starting value y(0) at time zero. Shifts the response by e−at of the change, so its effect fades as time passes. Raising it by 1 (to 1) moves the result by +5.22 %, from 2.594 to 2.729.
Time
The time at which the response is wanted, in seconds. Moves the response along its curve towards the steady value; early on it changes fast, later slowly. Raising it by 0.5 s (to 1.5 s) moves the result by +9.89 %, from 2.594 to 2.851.
Worked example with the values above
- Steady value: b/a = 6 / 2 = 3.
- Decay factor: e−at = e−2 = 0.13534.
- y = 3 + (0 − 3) × 0.13534 = 2.594.
Assumptions and limits
- Constant coefficients a and b and a first-order equation.
- A single starting value y(0) = y₀ and no further inputs, so the model is linear.
Common mistakes
- Using this formula for a = 0, where the response is linear in time.
- Taking b/a as the starting value instead of the final value.
Statistics and uncertainty
Measured response data usually give a and b together with scatter; the time constant 1/a is the quantity best read from the curve, because it is where the response has covered 63 % of the change. Fitting the exponential to many points is more reliable than reading one.
Where it is used
- Cooling or heating of a lumped body, charging of a capacitor, and any first-order filter.
- The unit step response of a first-order system with nonzero initial condition.
Maths you need
- Exponential function: e raised to a power
- Differential equation: a rule linking a quantity to its rate of change