# Transfer Functions

• A transfer function is a representation of the system which maps from input to output
• Useful for system analysis
• Carried out in the Laplace Domain

## The Laplace Domain

• Problems can be easier to solve in the Laplace domain, so the equation is Laplace transformed to make it easier to work with
• Given a problem such as "what is the output given a differential equation in and the step input ?"
• Express step input in Laplace domain
• Express differential equation in Laplace domain and find transfer function
• Find output in Laplace domain
• Transfer back to time domain to get
FunctionTime domainLaplace domain
Input
Output
Transfer

The laplace domain is particularly useful in this case, as a differential equation in the time domain becomes an algebraic one in the Laplace domain.

## Transfer Function Definition

The transfer function is the ratio of output to input, given zero initial conditions.

For a general first order system of the form

The transfer function in the Laplace domain can be derived as:

## Step Input in the Laplace Domain

Step input has a constant value for

For a first order system, the output will therefore be:

## Example

Find the transfer function for the system shown:

The system has input-output equation (in standard form):

Taking the Laplace transform of both sides:

Rearranging to obtain the transfer function:

## Using Matlab

In matlab the tf function (Matlab docs) can be used to generate a system model using it's transfer function. For example, those code below generates a transfer function , and then plots it's response to a step input of amplitude 1.

G = tf([1],[2 3]);
step(G);


### Example

For the system shown below, where , , , plot the step response and obtain the undamped natural frequency and damping factor .

system = tf([1],[100 40 100]);
step(system, 15); % plot 15 seconds of the response

%function to obtain system parameters
[wn,z] = damp(system)


The script will output wn=1, and z = 0.2. The plotted step response will look like: