Transfer function of a first-order system

After a unit step, a first-order system reaches about 63.2% of its final value after τ. After about five time constants, the remaining error is below one percent. These are ideal exponential approximations, not universal settling times with extra poles or dead time. The transfer function assumes a linear time-invariant first-order model and zero initial conditions. Application: Model simple sensor, thermal and actuator dynamics in the Laplace domain. The numerical example For K=2 and τ=0.5 s, G(s)=2/(0.5s+1); a unit step approaches a steady output of 2 anchors the idea in a concrete measurement case.

A transfer function is defined as output transform over input transform with zero initial conditions. Initial values add free-response terms and must be included separately for a specific simulation. Do not confuse G(s) with the complete output solution. The transfer function assumes a linear time-invariant first-order model and zero initial conditions. The corresponding units are K dimensionless for matching input/output units; τ in s; s in s⁻¹. For K=2 and τ=0.5 s, G(s)=2/(0.5s+1); a unit step approaches a steady output of 2. This checks that the reference quantities are compatible.

K is dimensionless only when input and output share units. A temperature sensor with voltage output and kelvin input, for example, has gain in V/K. τ remains in seconds and s in s⁻¹, making τs dimensionless. The transfer function assumes a linear time-invariant first-order model and zero initial conditions. Scope: This form follows from a linear constant-coefficient first-order differential equation with zero initial conditions. After a unit step, the output reaches about 63.2% of its final value after one τ. Dead time, additional poles, saturation and time-varying parameters are excluded; input and output units determine K. A practical application is Model simple sensor, thermal and actuator dynamics in the Laplace domain.

Write the quantity and unit from the variable list next to every value: K dimensionless for matching input/output units; τ in s; s in s⁻¹. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For K=2 and τ=0.5 s, G(s)=2/(0.5s+1); a unit step approaches a steady output of 2. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. The transfer function assumes a linear time-invariant first-order model and zero initial conditions.

The background states this starting condition: This form follows from a linear constant-coefficient first-order differential equation with zero initial conditions. After a unit step,. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. The transfer function assumes a linear time-invariant first-order model and zero initial conditions.