# Effect of zeroes in stability

I thought earlier that zeros have no direct effect on stability but when I read the below article of Wikipedia where it said under the heading "Effect of poles and zeros "

It says that adding zeros to transfer function pulls the root locus to left, making system more stable

Why this difference in concept about effect of zero on stability?

https://en.m.wikibooks.org/wiki/Control_Systems/Poles_and_Zeros

• Open loop zeroes do affect relative stability of the closed loop. This is what the root locus is all about. – Chu Nov 7 '19 at 12:10
• The difference in concept you ask about is the difference between you holding a false belief and being corrected. If you want to know why you held the false belief I don't think this site can help. – Andy aka Nov 7 '19 at 12:28
• It is important to realize there are 2 types of zeros, Left-Half Plane and Right-Half Plane. RHP zeros are gremlins when it comes to loop stability, burning phase and boosting gain. – sstobbe Nov 7 '19 at 16:46
• @Andyaka What do you mean?Which of my concept is false? – abtj Nov 7 '19 at 18:01

## 1 Answer

Consider a standard inverting amplifier built around an op amp with a standard two resistor feedback network. Now we place a small capacitor (pF) across the feedback resistor. If the capacitor value is carefully chosen stability can be improved. The added capacitor has added a zero in the loop transfer function. That is to say as frequency increases, the loop gain is increased above the zero frequency at a rate of 20dB/decade which actually reduces stability but a greater effect is caused by the added 90 degrees of phase lead, added to the loop phase, improving stability.

Note the frequency of this zero must be chosen carefully because the added capacitor also adds a loop pole at a higher frequency than the zero which can have a destabilising effect.

There can be a confusing element to the added capacitor adding a zero in the loop. That is that the added zero, when viewed from the point of view of the loop, can be viewed as a pole when viewed from the point of view of the closed loop response. That is to say it roles off the closed loop gain and therefore is actually a pole in the closed loop response.

It is the rate of conversion of the open loop response and the 1/B curves which determines stability. The zero reduces the rate of convergence between these two responses improving stability so long as the 1/B curve crosses the open loop curve before that higher frequency feedback pole comes in.