Tham khảo tài liệu 'robotics process control book part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Frequency Analysis 137 The graph of G jw Gjw G j ej - G jw j G j in the complex plane is called the Nyquist diagram. The magnitude and phase angle can be expressed as follows G j G j tan 7 G jw 2 G jw 2 V G jw G -jw G jw G jw _ G jw arctan G jw Essentially the Nyquist diagram is a polar plot of G jw in which frequency w appears as an implicit parameter. The function A A w is called magnitude frequency response and the function w phase angle frequency response. Their plots are usually given with logarithmic axes for frequency and magnitude and are referred to as Bode plots. Let us investigate the logarithm of A w exp j w ln G jw ln A w j w The function ln A w 1 log w defines the magnitude logarithmic amplitude frequency response and is shown in the graphs as L w 20 log A w 20 ln A w . L is given in decibels dB which is the unit that comes from the acoustic theory and merely rescales the amplitude ratio portion of a Bode diagram. Logarithmic phase angle frequency response is defined as w 2 log w Example Nyquist and Bode diagrams for the heat exchanger as the first order system The process transfer function of the heat exchanger was given in . G jw is given as G jw Z1 Z1 T1jw - 1 T1jw 1 T1jw 2 1 Z1Z1T1w T1 w 2 1 j T1 w 2 1 Z Z1 e j arctan T1 7 T1w 2 1 The magnitude and phase angle are of the form A w Z1 77 2 1 w arctan T1w Nyquist and Bode diagrams of the heat exchanger for Z1 T1 min are shown in Figs. respectively. 138 Dynamical Behaviour of Processes Phase deg Gain dB Im Re Figure The Nyquist diagram for the heat exchanger. Frequency rad min Figure The Bode diagram for the heat exchanger. Frequency Analysis 139 Figure Asymptotes of the magnitude plot for a first order system. Frequency Characteristics of a First Order System In general the dependency G j on G j for a first order system described by can easily be found from
