# Shakshat Virtual Lab

# INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI

# Theory

**Systems:**
Systems are specified as zero-order, first-order, and second-order depending on the order (highest derivative term) of the differential equation. The following sections describe the system models, real life examples and equations of responses for the test inputs.

**Second-Order System:**The highest order term in the differential equation of the second-order systems is second derivative. The second-order systems follow the equation.

The transfer function of the second-order system is

An example of a second-order measurement system is a mass-spring-damper assembly shown in above Fig. 1

**1. Test Inputs**

The following test inputs are normally used for testing the dynamics of a system. For characterizing system in time domain, the test inputs used are impulse, step, and ramp. A swept frequency sine wave is used to characterize system in frequency domain.

### 1.1 Impulse Input

### 1.2 Step Input

The unit step function u(t) is defined as

The Laplace transform of unit impulse is given by

### 1.3 Ramp Input

The unit ramp function r(t) is defined as