Second Order Frequency Response

How second order systems respond to sinusoidal/oscillating input. Similar to first order.

Gain and Phase for Second Order Systems

For a 2nd order system in standard input-output form:

The gain and phase of the frequency response are therefore:

Bode Plots, from Data Book

The plots show gain and phase shift for varying values of

Example

For the electrical system shown below with the values , , find:

• The undamped natural frequency
• The damping factor
• Sketch the magnitude of the frequency response
• At what frequency is this at it's maximum?
• Sketch a bode plot using matlab

The system equation is:

Undamped natural frequency:

Damping factor:

Using the graph from the data book

The graph peaks at approx , so:

Matlab plot:

R = 1000
C = 10e-7
L = 0.1
sys = tf([1],[L*C R*C 1]); figure; step(sys);
bode(sys);