Comparison between theoretical and practical phase computation

Transfer functions which are a composite of delay terms cause problems for computer packages, such as MATLAB, in calculating the frequency response open loop transfer function, as the form of the open loop transfer function is irrational. So, the delay terms have to be approximated. There are several ways to evaluate a delay term. A calculator such as MATHEMATICA uses the exact representation of the delay called the Euler form. A computer package such as MATLAB uses a rational approximation for the delay. Using the MATLAB built-in function pade(n,tau), the open loop transfer function of a system with delay can be calculated and the Bode plot may be obtained using another MATLAB built-in function bode(transfer function). Unfortunately, it was noticed that the phase was shifted by an angle from the correct phase, due to the trigonometric properties of the arctan function. Consequently, methods to construct the correct Bode plot were studied and developed.