# First Order Step Response

Modelling is about predicting the behaviour of a system. Often, need to know

• What is the output for a given input?
• Is the system stable?
• If the input changes quickly, how will the output change?

## First Order Systems

First order systems are those with only one energy store, and can be modelled by a first order differential equation.

TypeEquation
Electrical
Thermal
Mechanical
General

For the general form of the equation , the solution for a step input at time , with : T is the time constant of the system.

## Free and Forced Response

• Free response:
• The response of a system to its stored energy when there is no input
• Zero Input
• Non-zero initial Conditions
• Homogenous differential equation
• Forced response:
• The response of a system to an input when there is no energy initially in the system
• Non-zero input
• Zero initial Conditions
• Non-homogeneous differential equation
• Total system response is a linear combination of the two

## System Inputs

Different inputs can be used to determine characteristics of the system.

### Step Input

• A sudden increase of a constant amplitude input
• Can see how quickly the system responds
• Is there is any delay/oscillation?
• Is it stable?

### Sine Wave

• Can vary frequency and amplitude
• Shows frequency response of a system

### Impulse

• A spike of infinite magnitude at an infinitely small time step

### Ramp

• An input that starts increasing at a constant rate, starting at .

## Step Response

• The step response of the system is the output when given a step input
• System must have zero initial conditions
• Characteristics of a response:
• Final/resting value
• Rise time
• Delay
• Overshoot
• Oscillation (frequency & damping factor)
• Stability

For a system with time constant , the response looks something like this: The time constant of a system determines how long the system takes to respond to step input. After 1 time constant, the system is at about (63) % of its final value.

Time (s)% of final value
39.3%
63.2%
86.5%
95.0%
98.2%
99.3%