Second Order Overdamped

This page demonstrates the steps in finding the step response of an overdamped second order system. Here the auxiliary equation has two real roots. It is also assumed that initial conditions are zero.

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

The steady state response, where $O$ is not changing, is when $cO = c$ that is $O = 1$.

The auxiliary equation is $ar^2 + br + c = 0$

If its roots are $r_{1}$ and $r_{2}$, the transient response is $k_{1}e^{r_{1}t} + k_{2}e^{r_{2}t}$

The complete response is $O = 1 + k_{1}e^{r_{1}t} + k_{2}e^{r_{2}t}$

The constants are determined by finding $\frac{dO}{dt}$, evaluating that and $O$ when $t = 0$, forming two simultaneous equations in $k_{1}$ and $k_{2}$ and solving them, as is shown here.

Note, in the general case $k_{1} = \frac{r_{2}}{r_{1}-r_{2}}$ and $k_{2} = \frac{r_{1}}{r_{2}-r_{1}}$.

ShowNextStep shows the next calculation. Restart does just that. If you change a, b and/or c, press ReCalculate and you can go through the steps with the new values. Note only overdamped systems are allowed.

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