Step response

The step response is the output signal of a linear, time-invariant system (LZI system) to which the step function is fed at the input . It is referred to as a transition function if the height of the input jump is 1 (unit jump function) or was divided by the height of the input jump.

Exemplary step response of a 2nd order LTI system

Mathematical description

The step response can also be calculated as a convolution of the step function with the impulse response : ${\displaystyle h(t)}$ ${\displaystyle \sigma (t)}$ ${\displaystyle g(t)}$

${\displaystyle h(t)=(g*\sigma )(t)=\int \limits _{-\infty }^{\infty }{g(\tau )\sigma (t-\tau )}\,\mathrm {d} \tau =\int \limits _{-\infty }^{t}{g(\tau )}\,\mathrm {d} \tau }$

The step response is thus the time integral of the impulse response.

In the discreet:

${\displaystyle h[n]=(g*\sigma )[n]=\sum _{k=-\infty }^{\infty }g[n-k]\sigma [k]=\sigma [n]\sum _{k=0}^{\infty }g[n-k]}$

Since the transfer function represents the Laplace transform of the impulse response , it can also be determined by Laplace transform of the time derivative of the step response:

${\displaystyle G(s)={\mathcal {L}}\left\{g(t)\right\}={\mathcal {L}}\left\{{\frac {dh(t)}{dt}}\right\}=s{\mathcal {L}}\left\{h(t)\right\}}$

Conversely, it follows: ${\displaystyle h(t)={\mathcal {L}}^{-1}\left\{{\frac {G(s)}{s}}\right\}}$

In practice, jump signals can be generated much more precisely than Dirac pulses (which are the input signal for the impulse response ). The above relationship also makes it easy to determine the transfer function of the system from the step response. Thus, the step response is an important parameter of the system behavior and of high relevance for the description of systems.

example

Step response of an RC system ( low pass )

The step function is suitable for a system as a test signal. If a step function with a level of 2 V is applied to the input of an electronic circuit  , then a change in voltage can also be determined at the output of the transmission element. The time course of this voltage is called the step response, it is the response of the system to the applied step function. The picture shows how the output signal slowly approaches the value at the input. If the step response looks like the picture, it can be concluded that it is a system with a memory. In this case, the memory is a capacitor . This is charged by the 2 V at the input via the resistor until the input voltage is reached. The system behaves like a PT1 element .

See also

• Overshoot

literature

• R. Parthier: Measurement technology - Basics and applications of electrical measurement technology for all technical fields and industrial engineers (4th edition - vieweg Verlag), ISBN 978-3-8348-0336-8