A second-order system is a system whose behavior is described by a second-order differential equation. Such systems contain two energy storage elements, for example:
• RLC circuits
• Mechanical mass–spring–damper systems
• Control systems with two poles
Second-order systems are very important in control systems and signal processing because they exhibit oscillatory behavior depending on system parameters.
Table of Contents
Standard Transfer Function
The standard transfer function of a second-order system is
\[
H(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}
\]
where
\(\omega_n\) = natural frequency
\(\zeta\) = damping ratio
\(s\) = Laplace variable
These two parameters completely determine the dynamic behavior of the system.
Characteristic Equation
The characteristic equation of the system is
\[
s^2 + 2\zeta\omega_n s + \omega_n^2 = 0
\]
The roots of this equation determine the system response.
Types of Second Order Systems
The behavior of the system depends on the value of the damping ratio (\(\zeta\)).
Underdamped System (\(0 < \zeta < 1\))
• Oscillatory response
• Output shows overshoot
• Common in many practical control systems
Critically Damped System (\(\zeta = 1\))
• Fastest response without oscillation
• No overshoot
Overdamped System (\(\zeta > 1\))
• No oscillations
• Slower response compared to critical damping
Undamped System (\(\zeta = 0\))
• Pure oscillations
• No energy loss
Step Response of an Underdamped System
For a unit step input, the response is
\[
y(t) = 1 – e^{-\zeta \omega_n t}
\left[
\cos(\omega_d t) +
\frac{\zeta}{\sqrt{1-\zeta^2}}
\sin(\omega_d t)
\right]
\]
where
\[
\omega_d = \omega_n\sqrt{1-\zeta^2}
\]
is the damped natural frequency.
Important Time-Domain Specifications
Second-order systems are characterized by several performance parameters.
Peak Time
Time required to reach the first maximum peak.
\[
t_p = \frac{\pi}{\omega_d}
\]
Maximum Overshoot
Amount by which the response exceeds the final value.
\[
M_p = e^{\left(\frac{-\zeta\pi}{\sqrt{1-\zeta^2}}\right)}
\]
Settling Time
Time required for the response to stay within 2% of final value.
\[
t_s \approx \frac{4}{\zeta\omega_n}
\]
Rise Time
Time taken to rise from 10% to 90% of final value.
For underdamped systems, it depends on \(\zeta\) and \(\omega_n\).
Poles of Second Order System
The poles are
\[
s = -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2}
\]
• Real part → determines speed of decay
• Imaginary part → determines oscillation frequency
Applications
Second-order systems are widely used in:
• RLC electrical circuits
• Control systems
• Mechanical vibration systems
• Communication filters
• Servo systems
Summary
| Parameter | Formula |
|---|---|
| Transfer function | \(\frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\) |
| Damped frequency | \(\omega_d = \omega_n\sqrt{1-\zeta^2}\) |
| Peak time | \(t_p = \frac{\pi}{\omega_d}\) |
| Overshoot | \(M_p = e^{-\frac{\zeta\pi}{\sqrt{1-\zeta^2}}}\) |
| Settling time (2%) | \(t_s \approx \frac{4}{\zeta\omega_n}\) |