Introduction to Transfer Functions

What is a Transfer Function?

A transfer function describes the relationship between the input and the output of a linear, time-invariant (LTI) system in the frequency domain. Instead of working with differential equations directly, we apply the Laplace transform and represent the system as a ratio of two polynomials in the complex variable s.

For a system with input u(t) and output y(t), the transfer function G(s) is defined as:

G(s) = Y(s) / U(s)

where Y(s) and U(s) are the Laplace transforms of the output and input, assuming zero initial conditions.

Why Use Transfer Functions?

Working in the s-domain turns differential equations into algebra. This makes it dramatically easier to:

• Analyze system stability
• Combine subsystems (in series, parallel, or feedback)
• Predict frequency response without simulating in time
• Design controllers using well-established graphical and algebraic tools

A First Example: The RC Circuit

Consider a simple RC low-pass filter. Its governing differential equation is:

RC * dy/dt + y(t) = u(t)

Taking the Laplace transform (zero initial conditions) and solving for Y(s)/U(s) gives:

G(s) = 1 / (RCs + 1)

This is a first-order transfer function with a single pole at s = -1/RC. We'll explore poles and zeros in detail in the next chapter.

General Form

Most transfer functions you'll encounter in this course take the general form:

G(s) = (b_m * s^m + ... + b_1 * s + b_0) / (a_n * s^n + ... + a_1 * s + a_0)

The numerator and denominator coefficients come directly from the physical parameters of the system (masses, resistances, spring constants, gains, time constants, and so on).

Key Takeaways

1. A transfer function is only defined for linear, time-invariant systems.
2. It is computed assuming zero initial conditions.
3. The order of the denominator polynomial is the order of the system.
4. Transfer functions let us analyze and design controllers algebraically instead of solving ODEs directly.

In the interactive notebook below, try changing the resistor and capacitor values and observe how the pole location and step response change.