Step Response

This pages demonstrates the algorithm and maths for finding the step response of a first/second order system assuming that initial conditions are zero.

We assume $a \frac{d^2O}{dt^2} + b \frac{dO}{dt} + c O = d I$ where $I$ is a unit step and $c$ and $d$ are positive.

If $a$ is zero, the system is first order. Otherwise the roots of the auxiliary equation $ar^2 + br + c = 0$ determine if the response is overdamped, critically damped, underdamped or an oscillator. The root(s) are given as $a$, $a$ and $b$, $a\pm ib$ or $\pm ib$.

The steady state response, where $O$ is not changing is $o_{ss} = \frac{d}{c}$. The transient response is determined by the auxiliary equation and its root(s). The full response is one of :

1. $O = o_{ss} - o_{ss} \ e^{at}$
2. $O = o_{ss} + k_{a} e^{at} + k_{b} e^{bt}$
3. $O = o_{ss} - o_{ss} e^{at} (cos(bt) - \frac{a}{b} sin(bt))$
4. $O = o_{ss} - o_{ss} \ e^{at} + o_{ss} \ a \ t \ e^{at}$
5. $O = o_{ss} - o_{ss} \ cos(bt)$

NextStep shows the next code step (at a high level) and calculation: with more details of the code below. PreviousStep goes back. Restart resets to the first step.

If you change a, b, c and/or d, press ReCalculate after which you can go through the steps with the new values.

At the end a plot is shown of the step response.

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