# 4: Frequency Response of First Order Systems, Transfer Functions, and General Method for Derivation of Frequency Response

- Page ID
- 7648

- 4.1: Definition of Frequency Response
- The steady-state response of a stable linear system to sinusoidal excitation also varies sinusoidally, as either sin(ωt+ϕ) or cos(ωt+ϕ) , where the frequency ω is the same as the excitation frequency, and ϕ is a phase angle. This steady-state sinusoidal response is generally called frequency response. Although the frequency of response is the same as that of excitation, the magnitude of response can vary greatly for different excitation frequencies.

- 4.2: Response of a First Order System to a Suddenly Applied Cosine
- we derive a complete solution in the conventional manner for the original standard 1st order ODE x˙−ax=bu(t) [Equation 1.2.1], with IC x(0)=x0 , and with the suddenly applied (at t = 0) cosine input u(t)=Ucosωt , t > 0, where U is a constant amplitude.

- 4.5: Derivation of the Complex Frequency-Response Function - Easy derivation of the complex frequency-response function for standard stable first order systems.
- This section is an example of a much easier method for deriving the frequency-response function of a system.

- 4.6: Transfer Function - General Definition
- The system transfer function is defined to be the ratio of the output transform to the input transform.