Introduction

Transfer functions are a fundamental concept in control systems engineering. They provide a mathematical representation of the input-output relationship of a linear time-invariant (LTI) system. Transfer functions are used to analyze, design, and simulate control systems in various domains, including electrical, mechanical, and chemical systems.

In this tutorial, we will cover the following topics:

1. Definition of transfer functions
2. Derivation of transfer functions
3. Properties of transfer functions
4. Block diagrams and transfer function algebra
5. MATLAB examples

Definition of Transfer Functions

A transfer function, denoted as $G(s)$, is defined as the ratio of the Laplace transform of the output $Y(s)$ to the Laplace transform of the input $X(s)$, assuming zero initial conditions:

$$ G(s) = \frac{Y(s)}{X(s)} $$

where $s$ is the complex frequency variable in the Laplace domain.

Derivation of Transfer Functions

To derive the transfer function of an LTI system, we follow these steps:

1. Write the differential equation describing the system dynamics.
2. Apply the Laplace transform to both sides of the differential equation, assuming zero initial conditions.
3. Rearrange the equation to express the output $Y(s)$ in terms of the input $X(s)$.
4. The transfer function $G(s)$ is the ratio of $Y(s)$ to $X(s)$.

Example: Consider a simple RC circuit with input voltage $V_{in}(t)$ and output voltage $V_{out}(t)$ across the capacitor. The differential equation describing the system is:

$$ RC \frac{dV_{out}(t)}{dt} + V_{out}(t) = V_{in}(t) $$