OPA827: Question on Precision Labs 10.4 - Op Amps: Stability 4：how to get the relationship curve between Phase margin and overshooting %, and AC gain peaking.

Morning Yi, 

These come from fundamental control theory texts for 2nd order low pass response shapes, this article also reports those along with a new F-3dB/Fxover extension plot (Figure 4) for a 2nd order loop gain analysis. 

https://www.planetanalog.com/stability-issues-for-high-speed-amplifiers-introductory-background-and-improved-analysis-insight-5/

Hi Yi, 

Thank you for your interest in Precision Lab videos. 

As you can imagine, the mathematical derivations of the plots shown in the 4th video are beyond the scope of this video series.

In addition to the article Michael referenced above, I would highly recommend reading portion of chapter 9 of Gray and Meyer textbook:

• Gray, Paul R. Analysis and Design of Analog Integrated Circuits. 5th ed., Wiley, 2009. Sect. 9.1-9.4.

The textbook explores the topic of stability in great detail and shows full derivations for 2nd order system stability and phase margin equations. 

I hope you find this helpful, and happy reading!

Regards,
Vladimir

Hi Vladimir and Michael, Thanks for answer.

I check the control theory textbook ,there are detailed mathematical derivations about the damping-factor / pahse margin relation plots for 2nd order low pass system. 

However, there is an pre-assumption that our ampifier system is a second-order system. I doubt whether this assumption would be appliable or not, for all amplifiers. How do you think? 

The textbooks  are <Control System Engineering, 6th Nise Norman>,section 10.8 

or 

<Modern control system, 12th ed, Dorf > section 9.4

You are technically correct, the actual op amp systems are normally > 2nd order but the 2nd order analysis usually does a good job of estimating what you need without getting into the >10X more computational complexity of even 3rd order systems. 

I have gone down a 3rd order path on several designs, very quickly you get into equation complexity no one wants to deal with if they can avoid it - here is an example that did lead to something useful for my MFB tools 

https://www.planetanalog.com/include-the-op-amp-gain-bandwidth-product-in-the-rauch-low-pass-active-filter-performance-equations/