Second Ėorder systems:

An RLC circuit is shown below.



The response of this system can be solved for by using node equations.Assume a voltage, Ví between the resistor R and inductor L.Now writing the node equations at Ví and V0:


†††††††† (1)




Solving the other node equation, , for Ví and substituting into equation (1), yields



Second order systems are often characterized as follows:




where is the natural frequency and is the damping ratio.These terms result from the fact that if the damping ratio is zero, the system is an oscillator, and a unit step input would cause an output that is a unit sinusoidal varying about 1 with frequency radians/second.Shown below is the step response



For a damping ratio between 0 and 1, the output is a damped sinusoidal and the rate of damping increases as the damping ratio increases.This situation is referred to as an underdamped system.If the natural frequency is unchanged, but the damping ratio is increased to say 0.3, the step response looks like


If the damping ratio is increased to 0.7, the step response is:

If the damping ratio is larger than 1, the system is said to overdamped.Below is the result with a damping ratio of 1.5;

 The solution to the above equation for the underdamped case with a unit step driving function, is




The time to peak is given by:


This is found by differentiating y(t) and setting it to zero.


The peak overshoot is found by evaluating V0(t) at the time to peak and subtracting 1.

Maximum overshoot as percentage is



The damping ratio for a particular %OS is