The Bode diagram with the Ti-84 Plus (CE)
Explanation of the Bode diagram program : P-control
The Bode diagram is useful for determining the stability of a control system (phase margin and gain margin). It is an important subject in Electrical Engineering. With this Ti-84 Basic program, it is possible to plot the Bode diagram and to determine the proportional gain for a desired phase margin or gain margin. As an example, we treat a speed control system of which we have to determine the proportional gain for a phase margin of 60 degrees. Phase and gain margin are determined of an open loop system, See the figure below. The example is related to a real existing drive system at Avans University.
For Kp=1 the Bode diagram of Hopen is plotted, Kasyn=2.071Nm/A, J=0.018887kgm^2,
tauuni=10^-3 sec, tautacho=10^-3dec./A,
The Bode Program menu has 5 options.
1: New transfer function open loop= Hs, 2 The open loop function * Controller = H(s)*Hr
3: existing model is used for adapting the scaling of the axes 4 Closed loop =(Hs*Hr/(1+Hs*Hr)
Open loop transfer function. Green= amplitude, Red=phase/10 (to fit in amplitude screen). @ 0dB the phase=-13.48*10=-134.8 degrees. Which is a phase margin of 180-134.8=45.2 degrees
To achieve 0dB of total gain, the gain Kp should be -5.65 dB =10^(-5.65/20)=0.52. The corresponding frequency = 51.12 rad/sec.
If a P-control is added by inserting Hr(s)=0.52, and we run the program Bode again, then 0dB gives a phase margin of 180-120-= 60deg phase margin. See figures below. Remember that the real phase has to be multiplied by 10 (12*10=120deg). The corresponding frequency = 51.12 rad/sec, which = 10^1.71
Check with Simulink (Matlab) gives the same result, for Kp=0.52 we have a phase margin of 60.1 degrees at a corresponding radian frequency of 51 rad/sec