The response of a stable system at large times is characterized by its amplitude and phase shift when the input is a sinusoidal wave. These two quantities can be obtained from the transfer function, of course, without inverse transform. The analysis of this frequency response can be based on a simple substitution (mapping) s = jω, and the information is given by the magnitude (modulus) and the phase angle (argument) of the transfer function. Because the analysis begins with a Laplace transform, we are still limited to linear or linearized models.

What Are We Up to?

• Theoretically, we are making the presumption that we can study and understand the dynamic behavior of a process or system by imposing a sinusoidal input and measuring the frequency response. With chemical systems that cannot be subject to frequency-response experiments easily, it is very difficult for a beginner to appreciate what we will go through. So until then, take frequency response as a math problem.

• Both the magnitude and the argument are functions of the frequency. The so-called Bode and Nyquist plots are nothing but graphical representations of this functional dependence.

• Frequency-response analysis allows us to derive a general relative stability criterion that can easily handle systems with time delay. This property is used in controller design.

Magnitude and Phase Lag

Our analysis is based on the mathematical property that, given a stable process (or system) and a sinusoidal input, the response will eventually become a purely sinusoidal function.