Phase and Gain relationships

[karankhosla]

In Michael R. Goodman's book "Study Notes in System Dynamics", a fair amount of attention is paid to understanding input/output gain and phase relationships of closed loop systems.

One interesting method to understanding these relationships is to:
1) initialize the system in steady state;
2) break any one link in the system such that the closed loop system becomes an open system;
3) introduce a sinusoidal pulse with an a determined period;
4) if necessary, rewrite any feedback structures as delays in the system;

Once these steps are completed one can go component by component relating the delay (TC or Time Constant) across those components to the original input period (P), ratio = (TC/P); depending on the type of delay (material, information, order of delay), this ratio yields corresponding gain and phase values from a chart.

Any suggestions as to where can I find out how they came up with the relationship of (TC/P) to phase and gain for various types of delays? Where can I find higher resolution charts (since the ones in the book are barely readable, P.232 "Study Notes in SD" ) or even the formula's themselves?

From my research, Bode Plots are as close to this as I can get but I don't see how I can modify them to match SD delay and feedback structures.

Are there more direct and less time consuming ways of determining phase and gain relationships across a closed loop system with delays?

Best

Karan

[karankhosla]

The appendix of Industrial Dynamics does in fact give the formula for calculating phase and gain from input period and delay period across system components; however, it doesn't explain the necessity for making the system open loop, nor does it explain why feedback structures have to represented as simple delays? Any thoughts?